Rank-Size Rule

The rank-size rule is a mathematical description of the sizes of cities within a given urban hierarchy. It states that the rank of a city with size S relative to the largest city in that hierarchy is proportional to some negative power. It was first proposed by George Kingsley Zipf in 1949. The negative power should be close to 1 in absolute value, which implies that the second largest city is half the size of the largest, the third largest city is one-third the size of the largest, and so on. In cases where this power is greater than 1, it suggests that the second largest city is more than half the size of the largest city, the third largest city is more than one-third the size of the largest city, and so on.

The rank-size rule is commonly expressed as follows:

where Ri is the rank of the ith largest city, Si is the city's size, and β is the exponent coefficient. A log transformation can be used to estimate β as follows:

The rank-size rule has two striking features. One is its excellent empirical fit. Numerous studies have shown that a linear regression of log-rank on log size (as shown in Equation 2) generates an excellent correlation (high R2 value). For example, Rosen and Resnick (1980) used data from 44 countries and found that R2 values were above 0.95 for 36 countries, with only Thailand having an R2 value lower than 0.9. This astonishing regularity led Paul Krugman (1995) to say that the rank size is “a major embarrassment for economic theory: one of the strongest statistical phenomenon we know, lacking any clear basis in theory” (p. 44).

The other striking observation is about the power exponent. For the 44 countries studied by Rosen and Resnick (1980), the estimated coefficient ranges from 0.809 for Morocco to 1.963 for Australia.

Several studies have noted that estimating Equation 2 of the rank-size rule yields an ordinary least squares (OLS) bias through standard errors. The following version corrects for the bias, giving the standard errors of the exponent the form (2/ni) β, where ni is the corresponding sub-sample size. Instead of Equation 2, they offered the following:

This corrected version is known as the “rank minus half” rule. The estimated coefficients (βs) are slightly higher for the OLS-bias-corrected model than for the original uncorrected version.

The rank-size rule is remarkably consistent and frequently referenced. Its causes are highly debated, with some suggesting that it is purely a statistical phenomenon. The applications of the rank-size rule are growing and leading to interesting questions.