Many empirical size distributions in economics and elsewhere follow Zipf?s law. Starting from the Gibrat assumption, it is essential to add a second as-sumption to explain this phenomenon.

Gerrit de Wit

English

Research Papers in Economics

  1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
SCALES-paper N200503 
Zipf’s Law in Economics 
 
 
Gerrit de Wit 
 
 
 
Zoetermeer, June, 2005 
 
 
 
 
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Zipf’s Law in Economics 
 
 
 
 
Gerrit de Wit* 
EIM Business & Policy Research 
 
 
 
 
Abstract 
Many empirical size distributions in economics and elsewhere follow Zipf’s 
law. Starting from the Gibrat assumption, it is essential to add a second as-
sumption to explain this phenomenon. 
 
 
 
 
 
 
Keywords: Zipf’s law, power law, Gibrat assumption, size distribution 
 
 
JEL classification: L11, D31, R12  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
* Tel.: +31-79-3430277. E-mail: gdw@eim.nl.  4
 
Many empirical size distributions in economics and elsewhere follow Zipf’s 
law, that is: they exhibit a Pareto distribution - also called Zipf distribution or 
power law - with an exponent near 1. In economics well-known examples 
are the distributions of firm sizes, incomes, and city sizes.
1 
2 
This intriguing phenomenon continues to fascinate researchers as the many 
alternative explanations offered in the literature show. Many of them start 
from assuming a proportional growth process (the Gibrat assumption) to-
gether with a second assumption. For example, in Simon (1955) the second 
assumption is the presence of a constant rate of small new cities, Levy and 
Solomon (1996) assume a minimum size below which cities cannot decline, 
Reed (2001) assumes that the time of observation is distributed exponen-
tially
3, while recently Fujiwara et al. (2004) introduce the law of detailed bal-
ance. See for an overview De Wit (2005).  
 
Till now, the often cited explanation of Gabaix (1999) seems by far the most 
elegant of all these explanations, because he claims that in his approach in 
effect only a proportional growth process suffices to arrive at Zipf’s law. Un-
fortunately, this claim – and three other related claims - of Gabaix appear to 
be incorrect, as I will show in this letter. Hence, the conclusion must be that 
it is still in debate what the most satisfactory explanation of Zipf’s law is.  
 
Gabaix considers a fixed number of cities exhibiting proportional growth. 
Furthermore, there is a minimum size (smin) below which cities cannot de-
cline. Gabaix (pp. 749-750) proofs that these assumptions lead the city size 
distribution to converge to a Pareto distribution with an exponent ζ equal to: 
 
 
c −
=
1
1
ζ                   ( 1 )  
 
where c is defined by smin/sav and sav denotes average city size. Hence, he 
concludes that “as the minimum city size becomes lower (smin tends to 0, 
hence becomes almost invisible), the Pareto exponent converges to 1. So, in 
essence, all we need is an infinitesimally small minimum city size, to ensure 
that the steady state distribution will be Pareto with an exponent very close 
to 1.” Because the required minimum city size may be infinitesimally small, 
for all practical purposes it is negligible so that the main message of Gabaix 
(pp. 739, 741, 760) becomes that “some proportional growth process among 
cities automatically leads their distribution to converge to Zipf’s law”. 
 
 
1
 For presentational reasons I will refer to city sizes in the remainder of this letter. 
2
 Zipf’s law as an established empirical regulary is not wholly undisputed. See e.g. De Wit (2005) 
for firm sizes and Eeckhout (2004) for city sizes.  
3
 Reed is not aware of the fact that Steindl (1965, p. 53) already assumed this, albeit in a 
slightly less general context.   5
Unfortunately, Gabaix’s proof contains a flaw.
4 Removing this flaw leads to 
the following expression defining the Pareto exponent ζ implicitly:  
 
  ⎥
⎦
⎤
⎢
⎣
⎡
−
− −
=
) / ( ) / (
1 ) / ( 1
N c N c
N c
N ζ
ζ
ζ
ζ
    ( 2 )  
 
where N denotes the total number of cities. This equation is not new: see 
Malcai et al. (1999). The fact that Gabaix found the incorrect equation (1) in-
stead of (2), has far reaching consequences for the rest of his paper. In fact, 
it led Gabaix to make four claims that now appear unjustified. 
 
First, what is the true condition for the minimum city size to arrive at a Pareto 
exponent near 1? As is clear from equation (2), this condition will be de-
pendent on the number of cities N in the sample. If we take, as an example, 
the sample of the 135 American metropolitan areas that Gabaix uses in his 
paper, one finds from equation (2) that in order to get the Pareto exponent 
near 1 – say in the interval [0.95, 1.05] - smin/sav should be in the interval 
[0.13, 0.17].
5 
This gives a completely other picture of the required minimum city size: a 
negligible minimum city size appears not to suffice to ensure a Pareto distri-
bution with an exponent near 1. On the contrary, if the minimum city size be-
comes too low, the Pareto exponent will become substantially smaller than 
1. In the limit it even converges to 0 instead of 1! Hence, the conclusion that 
a negligible small minimum city size – not worth mentioning - suffices to get 
Zipf’s law should be revised. Instead, it should read that there is a second – 
rather specific - condition necessary for the minimum city size (loosely 
speaking: it should be small but not too small and certainly not negligible) to 
explain Zipf’s law. 
This also explains the puzzling result of Gabaix (p. 749) that only the intro-
duction of an infinitesimal small minimum city size is sufficient to change the 
steady state distribution from a lognormal distribution with ever increasing 
variance (the standard result of Gibrat, 1931) to a Pareto distribution with an 
exponent of 1. This result is not true! With the correct equation (2) for the 
Pareto exponent, we now find that the introduction of an infinitesimal small 
minimum city size makes the steady state distribution to converge to a 
Pareto distribution with exponent 0 (instead of 1, which Gabaix finds from 
equation (1)). A steady-state distribution converging to a Pareto distribution 
with an exponent of 0 indeed corresponds with a steady-state distribution 
converging to a lognormal distribution with ever increasing variance, as one 
would expect. See, e.g., De Wit (2005). 
 
Second, starting point of Gabaix’s analysis is a situation in which no new cit-
ies are created. An interesting robustness conclusion of the paper is that re-
 
4
 Gabaix overlooks the fact that in his framework city sizes are normalized, so that the maxi-
mum city size is 1 instead of infinity. Hence, when using the normalization condition at the 
end of the proof, Gabaix should have set the upper bound of city sizes equal to 1 instead of 
infinity. 
5 To get an idea of the influence of the sample size: for a much larger sample - say 10.000 cities 
- one finds that smin/sav should be in the interval [0.06, 0.11] in order to get the Pareto expo-
nent in the interval [0.95, 1.05].   6
sults are not affected if the appearance of new cities is introduced, as long 
as the appearance rate is not too large. This would be important news: the 
arrived results would be valid in a far greater domain. Besides, it would dis-
prove the result of Steindl (1965) that the appearance rate of new cities has 
in all circumstances a direct influence on the exponent of the Zipf distribu-
tion. 
Gabaix’s robustness result is based on the following reasoning (pp. 762-
763). Gabaix shows that - as long as the appearance rate of new cities is 
smaller than or equal to the growth rate of existing cities – older age cohorts 
will eventually always dominate newer age cohorts completely. Since older 
age cohorts have the time to converge to Zipf’s law the total population will 
also converge to Zipf’s law. 
However, this reasoning is incorrect. For, by assumption, the average 
growth rate of each age cohort is positive.
6 Hence, parameter c – by defini-
tion equal to smin/sav – decreases to zero as time goes on. It follows from the 
correct equation (2) that in that case the Pareto exponent itself also de-
creases to zero. In particular, older age cohorts will never converge to Zipf’s 
law as Gabaix finds from the incorrect equation (1).
7 Thus, Steindl’s (1965) 
result is incorrectly challenged by Gabaix. 
 
Third, Gabaix stresses the lack of convergence of the rivalling Simon model 
when the appearance rate of new cities becomes small. Simultaneously, he 
claims that there are no convergence problems in his own approach (p. 
745). However, also in his approach convergence to the steady-state distri-
bution becomes increasingly troublesome the smaller the minimum city size 
is chosen.
8 In the limit with an infinitesimally small minimum city size, there 
would be no convergence altogether! This potential convergence problem in 
the Gabaix approach can be solved when an extra condition on the exis-
tence of a sizable minimum city size is introduced. 
 
Fourth, when comparing his approach to those of others, Gabaix (p. 741 and 
pp. 754-755) stresses that other approaches leading to a Pareto distribution 
“stopped short of explaining why the Pareto exponent should be 1”, while his 
approach only needs the proportional growth assumption to arrive at a 
Pareto distribution with an exponent of 1. This alleged advantage of the 
Gabaix approach no longer holds. We now see that also in the Gabaix ap-
proach a rather specific condition on the minimum city size is needed to ar-
rive at a Pareto distribution with an exponent near 1. 
 
In summary, there are many different explanations for the appearance of 
Zipf’s law, all starting from a proportional growth process. Including that of 
Gabaix, they all need a second assumption to do the trick. It remains a mat-
 
6
 Otherwise it cannot be larger than the appearance rate of new cities. 
 
7
 It is not even possible to “repair” Gabaix’s result by introducing a sizable minimum city size, 
because - for any given minimum size - parameter c will eventually diminish to zero as time 
goes on so that the Pareto exponent will eventually always diminish to zero too. 
8
 Gabaix did not recognize that because in his Monte-Carlo simulations he does not investigate 
how the convergence behaviour depends on the minimum city size.   7
ter of empirics (or of taste if empirics are not conclusive) which of these ex-
planations should be preferred in a specific situation. 
 
 
 
 
Acknowledgements 
 
This letter is written under the SCALES-initiative (Scientific AnaLysis of En-
trepreneurship and SMEs), which is financed by the Netherlands Ministry of 
Economic Affairs. The helpful comments of Martin Carree are acknowl-
edged. 
 
 
 
References 
 
De Wit, Gerrit, 2005. Firm Size Distributions, An overview of steady-state 
distributions resulting from firm dynamics models. International Journal of 
Industrial Organisation, forthcoming. 
Eeckhout, Jan, 2004. Gibrat’s Law for (All) Cities. American Economic Re-
view 94, 1429-1451. 
Fujiwara, Yoshi, C. Di Guilmi, H. Aoyama, M. Galegati, and W. Souma, 
2004. Do Pareto-Zipf and Gibrat Laws Hold True? An Analysis with Euro-
pean Firms. Physica A, 335, 197-216.. 
Gabaix, Xavier, 1999. Zipf’s Law for Cities: an Explanation. The Quarterly 
Journal of Economics 114, 739-767. 
Gibrat, R., 1931. Les Inegalites Economiques, Paris: Sirey. 
Malcai Ofer, Ofer Biham, and Sorin Solomon, 1999. Power-law Distributions 
and Lévy-stable Intermittent Fluctuations in Stochastic Systems of Many 
Autocatalytic Elements, Physical Review E 60, 1299-1303. 
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Letters 74, 15-19. 
Simon, Herbert A., 1955. On a Class of Skew Distribution Functions. Bio-
metrika 52, 425-440. 
Steindl, Josef, 1965. Random Processes and the Growth of Firms. A Study 
of the Pareto Law. Griffin & Company. 

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