Talk:Hoover index

Hoover index is not Robin Hood index

See this discussion and http://www.bmj.com/cgi/content/full/312/7037/1004 --DL5MDA (talk) 06:37, 1 November 2008 (UTC)

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All of the Reliable-Sources say that "Hoover Index" is another name for "Robin Hood Index".

Robin Hood Index, Booty Index, Hoover Index, Pietra Index, etc. all refer to the same inequality-index.

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— Preceding unsigned comment added by 96.39.179.76 (talk) 09:12, 8 October 2021 (UTC)

I moved a remark on the Robin Hood index from here to the discussion of that Index. --DL5MDA (talk) 07:06, 1 November 2008 (UTC)

The redistribution is not between the two halves of the population.

The redistribution that the Robin-Hood (Hoover) Index refers to is from the people with income above their equal-share, to people with income below their equal-share.

I.e. from people with above-mean income to people with below-mean income.

To correct that sentence in the article it was only necessary to remove the parenthesized words that referred to halves of the population. That correction has been done.

The Robin-Hood (Hoover) Index is the fraction of the whole combined income that would have to be redistributed to make all incomes equal.

That's what the article now says.

The arithmetical-mean income is usually very close to the 65th %ile.

...which can be explained as a consequence of the Palma Proposition.

Mean Income & Robin-Hood Index, from Lorenz Curve

Very plausibly, there could be a situation in which the only information immediately-available about an income-distribution is its Lorenz curve. In such an instance, one would have reason to get the national mean income, &/or the Robin-Hood index, from the Lorenz curve.

Of course it goes without saying that, for this psurpose, one would want a Lorenz curve that is a smooth curve, rather than an approximation by a small number of straight lines.

Yes, the Robin Hood is represented on the graph as the greatest vertical distance between the Lorenz curve & the 45-degree equality-line.

But it’s not easy to judge where that maximum vertical distance is. Because the Lorenz curve bows away from the equality-line & back, but is, on the average, parallel to it, it’s much easier to find the place on the Lorenz curve that’s farthest from the equality-line. …to find the point on the Lorenz curve that is farther from the closest point on the equality-line (rather than the point directly above it).

That’s the same point that also has the greatest vertical distance to the equality-line. …but it”s much easier to find.

Why is that point that’s farthest from the equality-line, also the point that’s _vertically_ farthest from it? …and why does that point represent the mean income?:

On the Lorenz graph, income is represented as the slope of the Lorenz curve. The equality line, the Lorenz-curve for equality, is 45 degrees because, with equality, any cumulative percent of the population should have the same cumulative percent of the national income. …and of course uniform income means uniform slope—a straight line.

The average income is also the income that everyone would have with equality. Therefore, the point on the Lorenz curve that’s at the mean income will be the point where the curve is parallel to the equality-line.

At that point, the Lorenz curve is going in the same direction as the equality-line, and therefore its distance from the equality-line isn’t changing. That distance has stopped changing, because it’s become as big as it’s going to get. After that point where the Lorenz curve is parallel to the equality-line, the Lorenz curve begins to curve back towards the equality-line.

So the point on the Lorenz curve that’s parallel to the equality-line is the point that’s farthest from the equality-line. And that point where the Lorenz-curve is parallel to the equality line is the point where the slope is the same as that of the equality-line, and is therefore the point of average income.

The Robin-Hood is equal to the amount by which the combined income of the people with income less than their equal-share (the people with income less than the mean) is less than what it would be with equality. With equality their combined share of the income would be the same as their combined share of the people. …and it would be represented by the height of the equality-line directly above that point on the Lorenz curve. So the vertical distance from that mean-income point on the Lorenze curve, up to the equality-curve is the difference between the cumulative income of the below-mean incomes, and what that cumulative value would have been with equality.

But why is that point on the Lorenz curve also the one that has the greatest _vertical_ distance to the equality-line.

Well, because the Lorenz curve is parallel to the equality line at that mean-income point, then anywhere before that point its slope will be less than that of the equality-line, and so its vertical distance from the equality-line will be increasing. And, after that point where it’s parallel, it will have a slope greater than that of the equality-line, and so then the vertical distance will be decreasing. So, the mean-income point must have the greatest vertical distance from the equality-line.

…in addition to having the greatest _direct_ distance, to the point nearest to it on the equality-line.

Yes, because the vertical distance has to be measured anyway, it’s briefer to speak only of the vertical distance. But the greatest _direct_ distance, to the nearest point on the equality-line, is easier to find.