Saturday, December 16, 2006

Them that's got shall get, them that's not shall lose, so the Bible said

I have been thinking about the Lorenz curve. It seems to me that the second derivative of the Lorenz curve is important if people take cues from their socioeconomic peers.

A person who is located on a part of the Lorenz curve with high second derivative can see that those people who are just above him in the economic pecking order have significantly greater income than he does, while those who are just below him have significantly less. He may therefore be highly incentivized to seek promotion, move to Wall Street, or do whatever else it takes to improve or maintain his position, since the prospect of greatly increased or reduced income is very real to him.

On the other hand, a person who is located on a part of the Lorenz curve with low second derivative sees that his immediate economic superior does not actually make much more money than he does. It may be that everyone he comes into any social contact with has about the same income, and substantial changes in income seem unrealistic, like winning the lottery or a peasant becoming king. Although there may be significant inequality elsewhere in the society, his own socioeconomic neighborhood is in effect an incentive-sucking egalitarian mini-society.

For economists who prize inequality as the means and result of properly incentivizing productive behavior: one problem with too much inequality is that you cannot make the Lorenz curve too steeply rising in one place without flattening it elsewhere. If the second derivative is exceptionally high at the upper end of the income spectrum, that may be creating too much relative equality elsewhere.

Based on this simple theoretical argument, one might propose that the second derivative of the Lorenz curve should ideally be constant across social strata, in which case the ideal Gini coefficient is 0.33. This is apparently about the same as the Gini coefficient of Canada or France. A quadratic Lorenz curve would also mean that the top one percent of earners receive two percent of all income, not sixteen percent as is apparently the case in the United States.

But on the other hand, a Lorenz curve with constant second derivative may not be desirable after all, because at the top end of the income distribution the (constant) reward associated with advancement becomes small compared to the individual's existing income.

If instead we say that the second derivative of the Lorenz curve should everywhere be proportional to the individual's income (i.e. proportional to the first derivative of the curve), then the Lorenz curve should assume the form of an exponential:

L(X) = exp(k X) / (exp(k) - 1) - 1/(exp(k) - 1)

for some choice of growth constant k.

The resulting Gini coefficient is
G = (1 - 2 * (exp(k) - k - 1)/k/(exp(k) - 1))
and can thus be anything depending on the choice of k.


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