Income distributions


It would seem that the income distribution can be fairly acurately modeled by a lognormal distribution.  If we fit the U.S. income distribution to a lognormal distribution, we can then use that to do some calculations of the number of people "in the tail" (high wealth individuals).  Then the next step is to calculate tax revenues from the income distribution.

The standard lognormal distribution is given by:

{tex}\frac{1}{x\sigma\sqrt{2\pi}} e^{\frac{-\ln^2x}{2\sigma^2}}{/tex}

For our purposes here, it is useful to add a normalization factor:

{tex}\frac{1}{x/k_n\sigma\sqrt{2\pi}} \exp\left(\frac{-\ln^2(x/k_n)}{2\sigma^2}}\right){/tex}

It is easier to fit (for me at least) to the CDF, which is given by:

{tex}\frac{1}{2}  \mathop{\mathrm{erfc} \left(-\frac{\ln(x/k_n)}{\sigma \sqrt{2}}\right){/tex}

Following is a plot of the raw data from the census and the fit.


Now that we have a model, we can check some of the claims we have heard in the press.  The first repeated in the Wikipedia article states that 450,000 household make greater than $1 million.  The total number of households from the census is 118,682,000.  If we take that and multiply by (1- cdf(>$1 million), we get ~57,000.  Hmmmm, this is quite a ways off from the 450,000 number.  Where have we gone wrong?  Is the log-normal distribution just not a good model or is the cited 450,000 number questionable.  Let's set that question aside for now and see what kind of calculations we can do with the model.

The annual U.S. deficit is approximatley $1.3 trillion for 2011.  Just for fun, let's see what it type of tax increases would be balance that deficit. 

  • Increase taxes to 40% for everyone.  That only eliminates 35% of the deficit!
  • Increasing taxes to 63% eliminates all the deficit.  Perhaps a political no starter?
  • It looks like a tax of about 420% on the rich should balance the budget!

I created a little web app that shows the tax structure on top of the income distribution.  You can play with changing the tax rate for different income levels and see how big of a dent it puts in the deficit.  Check it out at:

Kind of interesting, maybe we'll return to the question of how well this model reflects reality....