ZIPF’S LAW FOR CITIES: AN EXPLANATION*
XAVIER GABAIX
Zipf ’s law is a very tight constraint on the class of admissible models of local
growth. It says that for most countries the size distribution of cities strikingly fits a
power law: the number of cities with populations greater than S is proportional to
1/S. Suppose that, at least in the upper tail, all cities follow some proportional
growth process (this appears to be verified empirically). This automatically leads
their distribution to converge to Zipf ’s law.
I. INTRODUCTION
Zipf ’s law for cities is one of the most conspicuous empirical
facts in economics, or in the social sciences generally. The
importance of this law is that, given very strong empirical
support, it constitutes a minimum criterion of admissibility for
any model of local growth, or any model of cities. Since George
Zipf ’s1 original explanation [1949], many explanations have been
proposed, but all pose considerable difficulties. The present paper
proposes a simple and robust account for the regularity.
To visualize Zipf ’s law, we take a country (for instance, the
United States), and order the cities2 by population: No. 1 is New
York, No. 2 is Los Angeles, etc. We then draw a graph; on the y-axis
we place the log of the rank (N.Y. has log rank ln 1, L.A. log rank ln
2), and on the x-axis the log of the population of the corresponding
city (which will be called the ‘‘size’’ of the city). We take, like
Krugman [1996a, p. 40], the 135 American metropolitan areas
listed in the Statistical Abstract of the United States for 1991.
We see a straight line, which is rather surprising (there is no
tautology causing the data to generate automatically a straight
line). Furthermore, we find its slope is 21. We can run the
regression,
* I thank Edward Glaeser and Paul Krugman for their guidance, and Richard
Arnott, Robert Barro, Olivier Blanchard, John Campbell, David Cutler, Donald
Davis, Aaron Edlin, Murray Gell-Mann, Elhanan Helpman, Yannis Ioannides,
Richard Johnson, Jed Kolko, David Laibson, Eric Maskin, Daniele Paserman, Jose´
Scheinkman, Andrei Shleifer, Didier Sornette, Olivier Vigneron, Romain Waczi-
arg, William Wheaton, two anonymous referees, and seminar participants at
several universities for their helpful comments. I also thank Jonathan Eaton and
Zvi Eckstein for their data.
1. The original discoverer of Zipf ’s law, however, seems to be Auerbach [1913].
2. The term ‘‘city’’ is, strictly speaking, a misnomer; ‘‘agglomeration’’ would be
a better term. So, the ‘‘city’’ of Boston should include Cambridge. Indeed Rosen and
Resnick [1980] show that Zipf ’s law holds better the more carefully agglomerations
are constructed.
r
1999 by the President and Fellows of Harvard College and the Massachusetts Institute of
Technology.
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(1) ln Rank 5 10.53 2 1.005 ln Size,
(.010)
where the standard deviation is in parentheses, and the R2 is .986.
The slope of the curve is very close to 21. This is an expression of
Zipf ’s law: when we draw log-rank against log-size, we get a
straight line, with a slope, which we shall call z, that is very close3
to 1. In terms of the distribution, this means that the probability
that the size of a city is greater than some S is proportional to 1/S:
P(Size . S) 5 a/Sz, with z . 1. This is the statement of Zipf ’s law.4
3. In fact, the regression above is not quite appropriate. Indeed, Monte-Carlo
simulations show that it understates the true z by .05 on average, and understates
the standard deviation on the estimate, which is around .1. But even given those
minor corrections, the estimates of z all remain around 1. See Dokkins and
Ioannides [1998a] for state-of-the-art measurement of z.
4. There are slight variations on the expression of Zipf ’s law. The most
common one is the ‘‘rank-size rule,’’ which subsection III.4 discusses. Its expression
is less convenient than the above probabilistic representation. Also, Gell-Mann
[1994, p. 95] proposes the modification P(Size . S) 5 a/(S 1 c)z, where c is some
constant. This paper sticks to the traditional representation (with c 5 0) of Zipf ’s
law, for two reasons. First, there is an immense empirical literature that studies
this representation. Second, theory turns out to say that the representation with
the constant c 5 0 is the one we should expect to hold.
FIGURE I
Log Size versus Log Rank of the 135 largest U. S. Metropolitan Areas in 1991
Source: Statistical Abstract of the United States [1993].
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