# Mathematical inequality measures for the social sciences

In summary, the conversation discusses the shortcomings of using the Gini coefficient for measuring inequality and explores alternative methods, such as majorization, for studying economic inequality. The example of a merger between two regions with different levels of wealth distribution is used to demonstrate how the Gini coefficient can be misleading and how majorization can provide a more accurate representation. The conversation also touches on the relationship between income and wealth inequality and the importance of pro forma calculations in mergers.

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This doesn't really fit in any other place. I'm interested in alternatives to the Gini coefficient for studying inequality, such as were discussed in this thread and this one.

What do I see as the shortcomings of Gini? By example: in Lower Slobovia, everyone has nothing. It's a perfectly equal society, so Gini = 0. In Upper Slobovia, on the other hand, everyone is wealthy - each citizen has exactly one million quatloos. Again, Gini = 0. Now, Upper and Lower Slobovia merge to form Greater Slobovia, and during this merger, each former Upper Slobovian gives 1/3 of his wealth to a corresponding Lower Slobovian. The result? Inequality actually goes up. Gini is now, if I did the math right, 0.17. But inequality in each former region is still zero.

The actual practical problem is when comparing a part of a polity with the whole: a city or state with the nation, or a single nation with "Europe". People do this comparison all the time, but as the example above shows it's not entirely valid. There's a sort of scale dependency.

It would also be helpful for an alternative also to have a clear and simple relationship between income inequality and wealth inequality - just as income and wealth have a clear relationship. (One being the derivative of the other)

Are there less flawed alternatives out there?

This doesn't really fit in any other place. I'm interested in alternatives to the Gini coefficient for studying inequality, such as were discussed in this thread and this one.

What do I see as the shortcomings of Gini? By example: in Lower Slobovia, everyone has nothing. It's a perfectly equal society, so Gini = 0. In Upper Slobovia, on the other hand, everyone is wealthy - each citizen has exactly one million quatloos. Again, Gini = 0. Now, Upper and Lower Slobovia merge to form Greater Slobovia, and during this merger, each former Upper Slobovian gives 1/3 of his wealth to a corresponding Lower Slobovian. The result? Inequality actually goes up. Gini is now, if I did the math right, 0.17. But inequality in each former region is still zero.

The realm of mathematical inequalities dealing with marjoization was actually uncovered by economists looking into economic inequality. The case you give is a simple robinhood transfer which implies ##\mathbf d_1 \succeq \mathbf d_2## where ##\mathbf d_1## is your pre-transfer but 'pro forma' for merger distribution (i.e. bivariate, highly dispersed) and ##\mathbf d_2## is after the re-distribution scheme. I doubt it this will be a one sized fits all approach but if you are interested in applying mathematical inequalities, it's a rich area to look into.

Note: despite what wikipedia says, the Gini Coefficient is actually strictly Schur-convex (and simple calculation tells you it cannot be schur concave). For a reference, see, e.g. page 563 of Olkin's "Inequalities", second edition.

If you accept that ##\mathbf d_1## does majorize ##\mathbf d_2## then this indicates your problem is not with the Gini Coefficient per se but elsewhere. My read: you're not pro forma'ing the numbers properly for the merger (or possibly are making an arithmetic error with Gini calculations). Alternatively your post is not about metrics like Gini, but the appropriate way to interpret and pro forma numbers for a merger.

The point is if you merge entities A and B -- you need a look at their numbers, combined pro forma for the merger, before any new-initiatives are implemented (e.g. re-distribution) and then after the inititiative. That's part of the process in M&A though mostly for corporates as I'm not so familiar with country mergers

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So I am not an expert on majorization. Didn't even know the word. Can you fill me in?

So I am not an expert on majorization. Didn't even know the word. Can you fill me in?

so for our purposes here: you have two vectors ##\mathbf d_1## and ##\mathbf d_2##. Let's cut to the chase and assume they each have real non-negative entries, are already sorted from biggest entry to smallest and there are ##n## components in total.

##\mathbf d_1 \succeq \mathbf d_2## is equivalent to saying

##\sum_{k=1}^r d_{k, 1} \geq \sum_{k=1}^r d_{k,2}##

for ## r = \{1,2, ..., n-1\}##and in total the vectors each have the same sum i.e. ##\sum_{k=1}^n d_{k, 1} = \sum_{k=1}^n d_{k,2}##. So with respect to your example the pro forma for merger, but pre wealth redistribution, the total sum is equal to the pro forma for merger post transfer sum. That's the nature of a zero sum wealth transfer -- it moves stuff around but doesn't change size of the pie. (Second order effects for years down the line, incentives etc. are not part of this topic.)

An underlying principle: we are able to transform ##\mathbf d_1## into ##\mathbf d_2## by a finite number of robin-hood transfers (in this case 1 transfer only is needed).

Majorization in and of itself it gives you a look at distributions and which ones are more unequal than others. When you combine these things with Schur convex functions (like gini coeffiicents), well, the more equal distribution, the smaller the value the function maps to, and for more unequal distributions, the function maps to a bigger result.

You can consider this a refinement of Jensen's Inequality.
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##\mathbf d_1 \succeq \mathbf d_2##

can be interpreted as saying that the cumulative % of income held by people at the top in country one, dominates that of country 2. If you want to put things in buckets, it would, roughly, read the top 1% 's share of income in (1) exceeds that in (2), the top 2%'s share of income in (1) beats that of (2), the the top 3%'s share of income in (1) beats that in (2), the top 4% ... all the way down, until they 'tie' with the trivial point that the top 100% has 100% of income.

Main idea is it's a nice, heavily refined way of looking at inequality. Not a one size fits all, but it does fit the simple example in OP quite well.

I have lingering concerns that the main point of the post has to do with a "change in accounting". Income inequality necessarily is inequality with respect to some reference class -- if you change entity being used as a ruler (e.g. via a merger), then the inequality calculations necessarily change as well.

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This doesn't really fit in any other place. I'm interested in alternatives to the Gini coefficient for studying inequality, such as were discussed in this thread and this one...

Are there less flawed alternatives out there?
To be honest, I see a lot opportunity for abuse and not much of a point to inequality measures. Or to say it another way, I think people read into it or try to say something bad(when high) when in reality it doesn't say much useful at all - or worse, says something its proponents would rather not hear. So my question is: what do you want this alternative to tell us about?

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Are there less flawed alternatives out there?
Alternative flaws, perhaps: http://www.researchgate.net/publication/235282975_Empirical_Hyperbolic_Distributions_Bradford-Zipf-Mandelbrot_for_Bibliometric_Description_and_Predictionwww.researchgate.net/publication/235282975_Empirical_Hyperbolic_Distributions_Bradford-Zipf-Mandelbrot_for_Bibliometric_Description_and_Prediction ; name/pick your social hypothesis and there's a "defense/argument" designed/tailored to it. You looking for something in particular, or just browsing esoteric statistical methods?

Bystander said:
You looking for something in particular, or just browsing esoteric statistical methods?

The sorts of statements I am trying to make sense of are northern/southern/eastern/western/red/blue states/cities have higher/lower inequalities than others. One quickly realizes that this is not as unambiguous as it sounds. Calculating the Gini coefficient of the combined population of the N states under consideration gives a different number from calculating the Gini coefficient using the N states average income as data points, which in turn gives a different number than calculating the N different Ginis and averaging them (which seems to be the most common way to report this, and most probably the goofiest). The first question to ask seems to be "If Gini is not the right thing to use, what is?"

Bystander said:
Alternative flaws, perhaps

The sorts of statements I am trying to make sense of are northern/southern/eastern/western/red/blue states/cities have higher/lower inequalities than others.
So your question is specifically about gini as related to geography? Or geography as a proxy for something else?

For the example in the OP, inequality went up, but income went up for one and down for the other.

I haven't read it all yet, but how about a real reunification example: Germany:
https://www.diw.de/documents/publikationen/73/diw_01.c.43930.de/dp540.pdf

One quick takeaway is that just after reunification, East German Gini was lower than West German. After unification, the East German Gini gradually rose above the West German, and East Germany's economy grew faster than West Germany's.

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Checked it with "PREVIEW," and it worked; try "Bradford-Zipf," that's my entry point for the search. ... , and, you are correct, "page not found" is what comes up this a.m..

russ_watters said:
o your question is specifically about gini as related to geography? Or geography as a proxy for something else?

I would say it is about subsamples. Most people seem to be writing about geography, but you can separate in ways different from place: time, educational attainment, maybe even other sociological factors.

Calculating the Gini coefficient of the combined population of the N states under consideration gives a different number from calculating the Gini coefficient using the N states average income as data points, which in turn gives a different number than calculating the N different Ginis and averaging them

These points all scream convexity. (And in particular schur convexity here.)

Equivalently, while ignoring a special case: if you want something like this to not happen you'll need to choose a function that isn't convex -- or even monotonic.

But it may be worse than that -- convexity just provides predictable inequalities for what will happen-- in general the only functions you can actually interchange with expectations are linear and affine combinations.
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Main idea is you can't interchange a function and an expectation -- if people want the Gini coefficient for the combined population of a country, then compute the Gini coefficient on the combined population of a country. That's really it.
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Most reporters and pundits aren't numerate enough to appreciate the difference. Come to think of it, there may be a 'More or Less' podcast episode on this.

I would say it is about subsamples. Most people seem to be writing about geography, but you can separate in ways different from place: time, educational attainment, maybe even other sociological factors.
Got it. I'm thinking along similar lines. I have an idea, but before I present it, I want to point out that I don't think Gini is necessarily a useful thing to corellate with these other factors. E.G., I don't need Gini to tell me that a person with an MD is almost certainly going to have a vastly higher income than a person with a GED. Or that the person with the GED is vastly more likely to be from a single-mother household. These statistics are already tracked and are much more focused and therefore useful for gauging the impact of certain societal factors.

Editorially, I think Gini is popular today specifically because it allows one to avoid looking at direct and specific evidence of why the disparity exists, enabling one to speculate without the burden of quality evidence. However, if my arm is being twisted into looking at and making sense of Gini (and it is, but not by you), then such corellations become important.

The main obstacle I see to a focused Gini analysis is that change in Gini is intrinsically linked to growth. Advocates of Gini work hard to try to show that increasing Gini reduces growth:
http://www.oecd.org/economy/growth-and-inequality-close-relationship.htm
[summary of a not-yet-released paper]

...but the reality is that the best controlled data we have, the year-to-year change of a static sample of people, growth is corellated with increasing inequality and contraction is corellated with decreasing inequality. This is true in basically all market or partial-market economies, including the US, EU states and even China. And looking at the world as a whole, Gini has risen steadily over tha past 200 years, as incomes have risen and poverty decreased.

I don't think discussions can move forward without establishment and agreement that the corellation between growth and Gini is fundamental. I believe you understand and accept this issue (just wanted it on the table as a fundamental principle)...so I'll move on:

Because increasing Gini is a fundamental property of economic growth, there is no way to say "this" policy will yield x Gini while "that" policy will yield "y" Gini. What they might yield is a Gini growth rate...under a stable economic growth rate. But if economic growth changes, Gini growth does too. So that's what I would like to explore. The corellation may be as simple as:

Gini growth rate = k * GDP Growth Rate

The coefficient "k" would represent the structural differences between societies that affect the corellation.

Tonight, I'll see if I can find some data to check the corellation and/or see if anyone is already doing it.

Bystander

## 1. What are mathematical inequality measures for the social sciences?

Mathematical inequality measures are tools used to quantify the level of inequality within a given population or society. They use statistical calculations to measure the distribution of resources, opportunities, or outcomes among individuals in a group.

## 2. Why are mathematical inequality measures important in the social sciences?

These measures provide a way to objectively analyze and compare the level of inequality within different societies or populations. They can also help identify areas of social and economic disparity that may require attention or policy changes.

## 3. What are some examples of mathematical inequality measures?

Some commonly used measures include the Gini coefficient, the Lorenz curve, and the Theil index. Each of these measures uses different mathematical formulas to calculate inequality, but they all aim to provide an overall measure of inequality within a population.

## 4. How do mathematical inequality measures differ from other measures of inequality?

Unlike qualitative or descriptive measures, mathematical inequality measures provide a quantitative value that can be compared across different populations or time periods. They also take into account the entire distribution of resources, rather than just the extremes or averages.

## 5. What limitations should be considered when using mathematical inequality measures?

It's important to remember that these measures are based on statistical calculations and may not always accurately reflect the lived experiences of individuals within a population. Additionally, they may not capture other forms of inequality, such as social or political power dynamics.