Harvard economist Edward L. Glaeser has an interesting post today on a law named after the linguist George Kinglsley Zipf. Zipf found that the frequency of any word in a text is inversely proportional to its rank in the frequency table. The same relationship has been observed in rankings unrelated to language, including the size distribution (population) of cities.
University of Minnesota economist Thomas Holmes and UBC economist Sanghoon Lee found, however, that Zipf’s Law does not hold for fixed geographical boundaries. In other words, Zipf’s Law for cities appears to result from urban sprawl rather than population density. Their work was published as “Cities as Six-by-Six-Mile Squares: Zipf’s Law?”, in the book Agglomeration Economics edited by Edward Glaeser (University of Chicago Press, 2010)
“Zipf’s Law” is one of the great curiosities of urban research. The law claims that the number of people in a city is inversely proportional to the city’s rank among all cities. In other words, the biggest city is about twice the size of the second biggest city, three times the size of the third biggest city, and so forth. ….
But [researchers have shown that] Zipf’s Law seems to be mainly a product of city or metropolitan area boundaries, not the natural distribution of population.
Professors Holmes and Lee ignored political boundaries and split America up using a six-by-six-mile grid. Their cities are squares crafted without any attention to actual boundaries. Using Census Block level data, they calculate the population of each square in the grid. It turns out that Zipf’s Law doesn’t work for these fixed geographic areas.
Professors Holmes and Lee find that “for squares above 1,000 in population, a Zipf’s plot has a piecewise linear shape, with a kink at around a population of 50,000,” and “below the kink the slope is 0.75; above the kink, it is around 2.”
In other words, in dense areas population drops far more quickly with rank than Zipf’s Law would suggest, and in less dense areas, population drops off far too slowly to be compatible with Zipf’s Law. Zipf’s Law is a bust at describing the population levels of areas within fixed boundaries.
Edward L. Glaeser, “A Tale of Many Cities”, Economix, 20 April 2010.
Would the findings of Holmes and Lee apply also to countries of Asia and Europe, where cities have a denser core, with less urban sprawl? This topic should be added to someone’s research agenda.
A concern that I have with population statistics is that definitions of “city” are varied and arbitrary. Some cities include all surrounding suburbs – even rural farmland! – whereas others are restricted to an urban core, surrounded by suburbs. The approach of Holmes and Lee has the advantage that boundaries are fixed. The disadvantage is that it fails to distinguish a densely-populated square surrounded by densely-populated squares from one surrounded by empty squares.
