Econintersect:
Zipf’s Law has been explained in a procedural manner by Dr. Richard S. Wallace:
Take all the words in a body of text, for example today’s issue of the New York Times, and count the number of times each word appears. If the resulting histogram is sorted by rank, with the most frequently appearing word first, and so on (“a”, “the”, “for”, “by”, “and”…), then the shape of the curve is “Zipf curve” for that text. If the Zipf curve is plotted on a log-log scale, it appears as a straight line with a slope of -1.
The Zipf curve is a characteristic of human languages, and many other natural and human phenomena as well. Zipf noticed that the populations of cities followed a similar distribution. There are a few very large cities, a larger number of medium-sized ones, and a large number of small cities. If the cities, or the words of natural language, were randomly distributed, then the Zipf curve would be a flat horizontal line.
Economist Edward L. Glaeser has presented what might be called an corrollary to Zipf’s Law:
…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.
However Glaeser observed that the six largest U.S. cities fell below the straight line for a perfect Zipf’s Law fit.
Click on graph for larger image.
Glaeser did not remark on the fact that the linear regression line drawn is “steeper” than the Zipf’s Law specified slope (~ -1.1 instead of the specified -1). Econintersect has drawn a second trend line which is anchored at the low population end (lower right corner) and forced to the position where slope = -1.
Click on graph for larger image.
Now, the decision to keep the trend line starting point anchored to the same low population point used by Glaeser was entirely arbitrary. Other lines with slope = -1.0 could have been used. One of them is shown in the next graph, which has been selected to approximate trend line for the middle of the population (excluding the six largest cities and the smallest).
Click on graph for larger image.
But there is no reason that two populations might not be considered divided roughly in the middle of the log population scale. If that is done, two well-fitted lines can be drawn with different slopes.
Click on graph for larger image.
This indicates that the large cities grow less rapidly with respect to population rank change than do small cities.
The above data was aggregated by metropolitan designation and not by land area. When the same kind of analysis is carried out with unit land area populations an entirely different conclusion results. In those cases the land area units with the most dense populations grow much faster with increasing density rank (slope = -2 for high density and -0.75 for lower densities), with the demarcation occurring at about 50,000 per six square miles. (See a chapter by Thomas Holmes and Sanghoon Lee in the book “Agglomeration Economics”.)
Sources:
Zipf’s Law (Dr, Richard S. Wallace, A.L.I.C.E. AI Foundation)
A Tale of Many Cities (Edward L. Glaeser, Economix, New York Times, 20 April 2010)
