Talk:Pareto index

definition of x_m unclear and highly important
After reading the article and also the discussions here, I would like to point out that there is no mention of the correct choice of x_m, neither here nor in the Pareto Distribution article. Let's say I do a worldwide study and get x_m = $0.2 for some super-poor farmer and I assume 80-20 for the worldwide wealth distribution, so a=1.16. Then I would get a proportion of 3e-6 for having >$10k in wealth. Now that super-poor farmer dies and the next poorest person has $3 in wealth. Suddenly, my proportion for having >$10k jumps to 8e-5.

The point I am trying to make here is that the choice of x_m is crucially important, especially for wealth distributions. And this article should definitely mention that and possibly also provide useful instructions for determining x_m. Because contrary to what is written in the Pareto Distribution article, x_m is not the minimum of all observations, or at the very least that approach doesn't work for Pareto-based wealth estimates. — Preceding unsigned comment added by Fxtentacle (talk • contribs) 10:32, 15 July 2019 (UTC)

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 &alpha; = 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)


 * Wataru Souma brings a selection of indices for different countries and centuries. roughly: historical data over the past centuries is about 1.13 to 1.89. Data about not-too-rich people in Japan during the past decade shows a range of 1.3-2.6, and older data about the US between the world wars ranges between 1.3 and 1.9.Ladypine (talk) 11:48, 7 August 2011 (UTC)

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).

The problem for wealth, it seems to me, is that even if you can identify it, which you often can't, it's very difficult to make comparisons. Prices only have meaning for verified, public, detailed, and honest transactions. There are few of these of any importance.

The value added taxes, of which Canada's Goods and Services Tax is an interesting example, represent a new and major innovation in human history. They are the first mechanism that I know of to produce relatively sound information about the very important numbers, sales of goods, that result from the concatenation of many many unimportant events, individual retail sales. Yes, those clay tablets of Nineveh should be filed as fiction.

This changes a great deal of both economics and governmental administration from here on out. Canada's application of the tax to a broad swath of services is the first accommodation, since the invention of public schooling, of government policy to the information-based economy.

It is very little remarked, but I think it is also important, that consumption taxes are the only way you can tax Milovan Djillas's "New Class," roughly the 80th to 95th percentiles in gross incomes. Before Canada's tax reforms of 1977 this slice of the population paid the lowest percentage of their income in total taxes of any of the population. This should not surprise us. They are the accountants, the lawyers, and the writers of tax regulations. Capitalism pari passu, that's what Djilas was talking about. (Parenthetically, this punches a rather large hole in the economists' dogma that "sales taxes" are always regressive. Taxing lawyers and accountants will be, whenever it occurs, a major step toward progressivity of taxation.)

Book values of capital assets do not even measure the value of capital assets, let alone the wealth that they represent to their owners. "Assessed" "values" of real estate are a joke to everyone but the assessors, to whom they are merely one, albeit often a delicious one, indicator of the fees that might be available for making the assessment. ("Two and two make what?" "What do you want them to make?"  Accountancy students' humour.)

Even income is a pretty flaky concept. The Canadian Income Tax Act has two sections, Section 1, which runs about two pages and creates an income tax, and Section 2, a thousand pages plus in my thirty year-old copy, which tried to define income. Major accounting firms have entire rooms of documents making this effort, and Congressman Emanuel Celler once showed me his personal working copy of the US income tax act. It ran to five 10-foot steel shelves. So. Wealth. Income. Two interesting notions. Always interesting to read and think about. Two great reminders of the value of the pinch of salt. David Lloyd-Jones (talk) 15:45, 23 August 2015 (UTC)

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&deg; Fahrenheit.  Here's what we give them:

quiz question

 * The proportion of persons in a certain population whose income exceeds any positive number x &ge; xm is


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


 * For this population, the two quantities xm and &alpha; 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?

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.

(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)

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:


 * $$\left(\frac{x_\mathrm{m}}{x}\right)^\alpha \approx$$ Yx (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: xM and &alpha;. In other words (if I understand correctly) the xM parameter serves as the intersection of the function and &alpha; 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 xM 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 xM 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 &alpha; 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 &alpha;. 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 xm 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)