Cities and Countries

By Rose, Andrew K. | Journal of Money, Credit & Banking, December 2006 | Go to article overview

Cities and Countries


Rose, Andrew K., Journal of Money, Credit & Banking


Cities are a standard unit of observation in urban economics, just as countries are a norm in international economics. The distribution of city sizes has been extensively studied. A couple of striking empirical regularities characterize the distribution of cities within a country. The rank (by size) of a city is almost perfectly inversely related to its size (at least for the largest cities), a stylized fact known as "Zipf's Law." It is also well known that growth in cities seems to be approximately proportionate, independent of city size; this is known as "Gibrat's Law." In this short paper, I consider both of these well-known characteristics of city size distributions, and show that they work about as well when one considers countries instead of cities.

1. EMPIRICAL CHARACTERISTICS OF CITY SIZE DISTRIBUTION

The focus of this paper is a pair of well-known empirical regularities that characterize the distribution of population size across cities. Zipf's Law states that when cities are ranked by the size of their populations, city size is inversely correlated with rank. Gibrat's Law states that the size of a city is uncorrelated with its growth rate. Both stylized facts are long established, well known, and essentially undisputed to the best of my knowledge. Accordingly, I now briefly provide results that use recent data and are representative of the larger literature.

1.1 City Size and City Rank: Zipf's Law

"Zipf's law for cities" states that the number of cities with population greater than S is approximately proportional to 1/S. (1) The relationship fits well, and Zipf's Law characterizes the cities of different countries at different points of time. (2) A vast literature documents Zipf's Law, while a smaller literature attempts to explain it. Among the more recent references are Eeckhout (2004), Gabaix (1999), Krugman (1996), and Rossi-Hansberg and Wright (2004); Gabaix and Ioannides (2004) provide a recent survey and Nitsch (2005) a recent meta-analysis.

Two methods have been used in the literature to document Zipf's Law: graphs and regressions. Both begin by ranking cities by the size of their population (New York is currently #1 in the United States, Los Angeles #2, and so forth). One then compares the natural logarithm of city rank to the natural logarithm of city population, using either (1) graphical or (2) regression techniques. Table A1 lists the populations of the largest American cities in 2000 (the most recent census).

Figure 1 presents a typical set of graphs. The top-left graph is a scatter-plot of the rank of the fifty largest America combined statistical areas (CSAs) in 2000 (on the ordinate or y-axis) against their sizes (on the abscissa or x-axis). Collectively the fifty CSAs covered almost 152 million people at the time of the 2000 census, around 56% of the population of the United States. (3) A line with slope of -1 is provided to facilitate comparison. Clearly Zipf's Law works well. These results do not depend much on the exact year; the top-fight graph is the analog for 1990 census data. The exact definition of "city" does not matter much either; analogs for the 200 largest metropolitan and micropolitan statistical areas (MSAs) in 2000 and 1990 are provided in the bottom pair of graphs of Figure 1 (the 200 MSAs included almost 212 million people in 2000, over three-quarters of the population of the United States). (4)

[FIGURE 1 OMITTED]

Analogous regression results are tabulated in Table 1; these corroborate the graphical results. Each row in the table reports a regression of the log of city rank on the log of city size (and an unreported intercept). (5) Consider the results for the 50 largest CSAs in 2000, which are presented in the top row. The slope coefficient is -1.03, close to the Zipf value of - 1. (6) I follow Gabaix and Ioannides (2004) and approximate the standard error by [beta][(2/N) (5) where [beta] is the slope coefficient and N is the sample size; this delivers a standard error of 0. …

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