Zipf's Law states the population of a town/city is inversely proportional to its rank. So the second biggest city should be half that of the biggest, the third biggest one third and so on. It often works pretty well & has also been used to describe the frequencies of word use.

Ed Glaeser has a nice article in the NYT with the background & evidence for the US (h/t Greg Mankiw). The Law implies that the log of the rank is equal to -1 times the log of population. Xavier Gabaix showed formally that Zipf’s Law will result if the population growth rate of an area is independent of that area’s initial population (which it usually is & that is known as Gibrat's Law). For the big US cities (NY,LA,Chicago) it doesn't work well, over-prediciting their size, but otherwise its pretty good.

http://economix.blogs.nytimes.com/2010/04/20/a-tale-of-many-cities/

So here is the evidence for Ireland:

As with the US, it doesn't work too well for the big cities (Dublin, Belfast) the difference being that the law under-predicts population rather than over-predicts in the US case.

## 2 comments:

That was an interesting NYT article. Enda raised a good point in relation to it; how can econometric models adapt to power law distributions?

http://en.wikipedia.org/wiki/Power_law

I made a silly error. The model cannot predict the size of the largest city: its everywhere else relative to it.

If you do the regression, the slope is -1.14 with a 95% CI= [-1.178, -1.115] so one can reject the model which implies a slope of -1.

Taking the top cities out doesn't change that.

Not sure about power law distributions: doesn't it lead to log-linearity or something?

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