Long tails

Mark Penn wrote an article about the lucrative opportunities supposedly available to bloggers. The article has been widely criticized (see for example Kaus or Lemieux or Yglesias ). The article makes a number of dubious claims but the one I will focus on is:

... It takes about 100,000 unique visitors a month to generate an income of $75,000 a year. ...

which Penn later defends as follows:

... We say it takes "about 100,000 unique visitors a month to generate an income of $75,000 a year" and Technorati states those who had 100,000 or more unique visitors the average income is $75,000.

It should be clear that the first statement does not follow from the second since the average income of bloggers with 100000 or more visitors can be expected to be more (perhaps much more) than the average income of bloggers with just 100000 visitors. And in fact the Technorati article Penn is citing says:

... Among active bloggers that we surveyed, the average income was $75,000 for those who had 100,000 or more unique visitors per month (some of whom had more than one million visitors each month). The median annual income for this group is significantly lower — $22,000.

So 100000 visitors can in fact be expected to generate at most $22000 in income. The actual number will be less since bloggers at the lower end of the range 100000 to infinity visitors can be expected to earn less than the median.

This suggests the following mathematical puzzle. Suppose we assume that annual income is strictly proportional to unique visitors per month. And suppose the distribution of bloggers with 100000 or more unique visitors a month obeys a power law. This means the density of bloggers with x visitors is proportional to 1/x**a for some value a. Then given average income of $75000 and median income of $22000 find the annual income for bloggers at the lower bound with 100000 unique visitors per month.

You can give your answer in the comments, I will give mine in a later post.