Talk:Pareto index

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1 empirical values for wealth distribution
2 wealth or income?
3 Sources
- 3.1 small thought-experiment

[edit] empirical values for wealth distribution

Since Vilfredo Pareto came up with the Pareto distribution while measuring economic inequality I would like to know which values have been empirically found to best fit reality (maybe for selected countries). I am also not clear what is meant by "embodies": Which index leads to the 80/20 rule? A table with the index and the corresponding percentages would be nice. Common Man 02:42, 19 October 2005 (UTC)

The 80-20 rule holds if α = log₄(5) (about 1.16). Michael Hardy 16:59, 6 July 2006 (UTC)

This distribution is more valid for the very wealthy than for others apparently. Based on Cap Gemini's World Wealth Report I found values rangin from 1.10 (in Latin America) to 1.46 (Asia-Pacific), with a world-wide value 1.36. This is pretty soft data but it is quite consistent with another study from the Boston Consulting Group giving an average around 1.43. Lachambre 15:12, 6 July 2006 (UTC)

[edit] wealth or income?

I've seen respectable sources saying that what Pareto wrote about was the distribution, not of wealth, but of income. Does anyone know? (Maybe I'll go to the library and try to dig it out from his books.) Michael Hardy 17:03, 6 July 2006 (UTC)

He did start writing about income (much more data is available on income than on wealth) but his distribution also empirically fits well to the repartition of wealth (especially for the rich).

[edit] Sources

Please provide source support for this article or it risks deletion. The article does not adequately explain the topic and without sources there's no way to fix that problem or even veiry the Pareto index. Please add sources or I will recommend deletion. --Cplot 17:39, 7 November 2006 (UTC)

You are certainly quite wrong to say it does not adequately explain what the Pareto index is. Here's what the article says:

One of the simplest characterizations of the Pareto distribution, when used to model the distribution of wealth, says that the proportion of the population whose wealth exceeds any positive number x > x_m is

$\left(\frac{x_\mathrm{m}}{x}\right)^\alpha$

where x_m is the wealth of the poorest people (the subscript m stands for minimum). The Pareto index is the parameter α.

That is crystal-clear. Michael Hardy 20:27, 7 November 2006 (UTC)

I can't tell if you're being sarcastic or not, but just in case you're not, I'll respond. As it stands now, the article looks like some sort of hoax. I know Pareto was a real academic. And I know he worked in the field of economics. However, the article does not refer to anything I'm familiar with in Pareto"s work and the mathematics you quote is just nonsense as it stands. How is the wealth of the poorest people determined? The poorest one? The poorest quintile? The poorest 99%?. The variable α could be any number because any complex number can serve as the exponent of a quotient. So is the Pareto index any exponent of a quotient?

So the prose don’t expalin the concept. The mathematics are meaningless as they stand. So I hope you were being sarcastic. --Cplot 20:59, 7 November 2006 (UTC)

The article is not a hoax.
I was not being sarcastic.
The mathematics is not nonsense.
The mathematical part of this article is absolutely clearly written.

What in the world could you mean by suggesting that the mathematics is meaningless? Be specific. Michael Hardy 21:29, 7 November 2006 (UTC)

The reason I said the math was meaningless is because the expression is neither a function, an equation, a forumla, nor anything else that would provide context to someone who doesn't already know what this index is. It still doesn't do that with the latest changes. The prose could be made to stand on their own, but they are no more clear than the mathematical expression. The references might provide answers (I haven't yet followed them), but the article really provides too little information for a general reader (or anyone not familiar with the topic) to understand the topic. --Cplot 05:52, 8 November 2006 (UTC)

It does do that. It does EXACTLY that. Clearly and efficiently. Here it is:

the proportion of the population whose income exceeds any positive number x > x_m is

$\left(\frac{x_\mathrm{m}}{x}\right)^\alpha$

where x_m is a positive number

What could be clearer? The proportion of the population whose income exceeds x_m itself is (according to the above)

$\left(\frac{x_\mathrm{m}}{x_\mathrm{m}}\right)^\alpha$

and this is obviously 1. That means everyone's income exceeds x_m. The proportion of the population whose income exceeds (for example) x_m + 2 is (according to the above)

$\left(\frac{x_\mathrm{m}}{x + 2}\right)^\alpha$

and that can be computed in an instant with elementary-school arithmetic, provided one knows the two quantities x_m and α. Anyone not familiar with the topic would see all of that instantly by reading the quoted part above (except the sort of people who don't like math anyway). Are you just trying to be difficult? Michael Hardy 22:26, 8 November 2006 (UTC)

I should add that I did not know what the Pareto index is until I read a characterization amounting to essentially what it says above, and that's all it took. I understood it then. It is perfectly comprehensible to those who don't already know anything about this topic. Michael Hardy 22:29, 8 November 2006 (UTC)

...and I should further add that this particular case is both extremely elementary and extremely simple; I'd expect all 18-year-olds to understand it at once (except perhaps those who will never work professionally with anything slightly mathematical). It does look as if you have a serious reading-comprehension problem. Why don't you try being specific about what parts of this formula you would expect someone to have trouble understanding? Michael Hardy 22:32, 8 November 2006 (UTC)

[edit] small thought-experiment

OK, a thought-experiment. I have many years experience teaching mathematics to a wide range of different sorts of students. I think questions on tests should be shorter and simpler than those on homework assignments. Say we have a class of 16-year-olds and we want to give them a question that's really simple and intended to test whether they're conscious and their body temperature is near 98.6° Fahrenheit. Here's what we give them:

[edit] quiz question

The proportion of persons in a certain population whose income exceeds any positive number x ≥ x_m is

$\left(\frac{x_\mathrm{m}}{x}\right)^\alpha.$

For this population, the two quantities x_m and α are:

$x_\mathrm{m} = \$ 33750\mbox{ per year}\,$

$\alpha = 4/3 \approx 1.3333\dots\,$

What proportion of the population are those having incomes exceeding $80,000 per year?

[edit] answer

The student is expected to write:

The proportion of people whose income exceeds $x per year is

$\left({33750 \over x}\right)^{4/3}.$

Therefore, the proportion of people whose income exceeds $80000 per year is

$\left({33750 \over 80000}\right)^{4/3} \approx 0.3164\dots$

i.e. nearly 31 and two-thirds percent of the population make more than $80000 per year.

[edit] (end of quiz question section)

I submit that the above is far less challenging than many problems that secondary-school students are required to do, and it's shorter and simpler than most. And it is reasonable to expect 16-year-olds to understand the question when it is phrased just as above---very nearly identical to what you see in this article. (Not ALL 16-year-olds---not those who will major in French poetry, perhaps---but all who will go into any even slightly mathematical field.)

"Cplot", do you have a problem with that assertion? Michael Hardy 23:51, 8 November 2006 (UTC)

[edit] responses

The arithmatic is fine. I can certainly now see how one could calculate the approximate proportion of a population with an income above a certain arbitrary number, given the Pareto index. That arithmatic demonstrates how someone might use the Pareto index. What I think the article should make clear is what the Pareto index is. What social significance does it have? How does one derive it? Is it simply a variable for statistical estimation? What are the upper (none?) and lower bound (1?)? I get no ansswers to these questions from the article or your recent arithmatic.

Also as the forumla stands now it needs to define what the forumla cacluates. Something like:

$\left(\frac{x_\mathrm{m}}{x}\right)^\alpha \approx$ Y_x (where Y is the proportion of the population (sample?) with an income above x)

Again, this forumla then shows how to use the index, but not how the index itself is defined or derived.

Obviously the article doesn't need to walk the reader through all of the probability and statistics topic, but I think it could be much clearer about it's relationship to those topics. Why not aim for the French poetry major and if you fall short, at least you tried. I can see how the article was even worse before you arrived, but it's got some ways to go. Perhaps a graph of the distribution depicting your numerical example would be helpful. --Cplot 01:47, 9 November 2006 (UTC)

I will answer at greater length later. But I am puzzled as to what you mean by suggesting that "what the formula calculates" is insufficiently specified. The words explicitly say "the proportion of the population whose income exceeds...x." Why is that not enough? Michael Hardy 16:02, 9 November 2006 (UTC)

I think the article does make clear what the Pareto index is. It also makes clear that there is no upper bound and that there is a lower bound. Certainly this could be explained at greater length, but it's hardly fair to say that those points are unclear in the article as now written. I'll try to add something for the French poetry major, however. Michael Hardy 05:02, 12 November 2006 (UTC)

In discussing this with you, researching it elsewhere and reflecting on it, I'm beginning to gain a greater understanind of the Pare–to index. Periodically I look back at the article to see if I could have learned those things from the article on its own and invariably, the answer is no. There are small bits of information needed to understand the index, but they are inadequately explained, typically in misordeed and sometimes misstated. For example, the article ends with "Economists therefore sometimes state that the Pareto law as stated here applies only to the upper tail of the distribution." This makes it sound like "sometimes" the law (meaning here the distribution) is thought about as one-tailed. This is always a one-tailed distribution. This fact is inherently a definition of the pareto index. Somethling like that should be stated up front (this is also where I think a simple graph depicting a one-tialed distribution would help).

The article could make it clearer what purpose these two parameters serve: x_M and α. In other words (if I understand correctly) the x_M parameter serves as the intersection of the function and α defines the slope. That these parameters define the shape of the continuous function estimating discrete income distribution (again a graph would help convey this).

I also think the article failes to make clear that the Pareto index (along with x_M are parameters estimated through statistical sampling. Too much is made of Pareto’s mistaken view that the distribution is automatically 80/20. This is interesting information to add later on in the article, but it's not part of the Pareto index. So right now I think the article is only written to an audience whom already know what the Pareto index is. That's the only audience it now addresses. The goal of wikipedia is to reach a wider audience than that. I would jump in and try to improve the article, but I still don't think I understand the topic enough to fix these problems. --Cplot 17:39, 12 November 2006 (UTC)

I'm surprised to see that "Pareto's mistaken view that the distribution is automatically 80/20" is mentioned at all, since I don't think Pareto had any such view, and I don't remember it being attributed to him in this article. I don't think that it's anywhere near being the sort of article that only explains the topic to those who already know it. I think it is crystal clear to anyone who's never heard of it before, except perhaps non-mathematicians. More later.... Michael Hardy 18:02, 13 November 2006 (UTC)

You wrote:

(if I understand correctly) the x_M parameter serves as the intersection of the function

I have no idea at all what that means. I know various meanings of the word "intersection" in mathematics and none of them fits.

and α defines the slope.

Wrong. Maybe what you have in mind is that if you take the logarithm of the expression given in the article as a function of x, that's a straight line whose slope is α. But tell me this: you complain bitterly, and wrongly, that the article could have been understood only by those who already know something about the Pareto index, and preach in holier-than-thou tones about clarity in writing. The fact is, you are the one who needs to learn about clarity in writing and about supplying "bits of information needed to understand the index, but they are inadequately explained". How is it that you're talking about a "slope" before you've said anything about a line whose slope you have in mind? And what do you mean by "intersection"? All I can do with that is hazard a wild guess that you mean "intercept", but that's only a guess, and at any rate that is not what x_m is. What is actually is is specified in the article in such terms that any high-school student who's good in math would understand it right away. On the other hand, what you mean by "intersection" cannot reasonably be expected to be understood by anyone.

The distribution is indeed always one-tailed. The statement about the upper tail does not mean that it's not one-tailed.

You are right that some things in the article could be made clearer to those who have always found mathematics confusing. If you would limit your message to that and stop making these false and unjust assertions, I would listen to you more sympathetically.

Also, could you use standard spelling and grammar, such as "arithmetic", and "persons who already know"? Michael Hardy 19:10, 13 November 2006 (UTC)

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