Dittmar J. (2008) The emergence of Zipf’s Law

Dittmar, Jeremiah (2008) “Cities, Institutions, and Growth: The Emergence of Zipf’s Law”, Job Market Paper.


This paper is available on line.


Zipf’s Law is a simple power law holding that the number of cities with population greater than N is proportionate to 1/N, this results in a log-linear relation between city population and city size rank (p.2). However, as shown by pre-modern European urban history, this law is not universal nor a-temporal (p.3). This outlying is very significant since economists have recognized that “there is an optimal level of urban concentration and that both over- and under- concentration can be very costly in terms of productivity growth” (p.4). When respecting the Zipf’s Law, city growth is considered random, so what prevented “normal” urban expansion and what later on made it possible?

Why cities matter?

Cities were true capitalist islands in a “feudal sea” (p.7). Towns are important indicators of prosperity, there are also commonly hubs of technical innovation (p.8). The author identifies three homogeneous integrated urban systems: Western, Eastern and Ottoman Europe. In Ottoman Europe, the central had deprived cities from most of their autonomy and a social and religious divide arose between urban dwellers (mostly civil servants and commonly Muslims) and their rural counterparts who remained Christian (p.13). The lands East of the Elbe River were initially fairly similar to Western Europe, but after 1500 under the regime known as Second Serfdom, peasants were tied to their lords’ estates and cities lost their freedom to power-hungry rural gentry and absolutist rulers. In Western Europe rural-urban mobility and urban liberties were common (p.14).

The emergence of Zipf

Using the same data as Bosker et al., the author determines that the Zipf’s Law was not relevant either in Western nor in Eastern Europe in 1400 and 1500, but is present in Western Europe after 1600 but does not appear East before 1800 (p.16). Large cities consistently failed to keep up with lesser one’s growth rate, “for instance [between 1500 and 1700] the ten largest Eastern cities were on average ½ the size of the counter-factual Zipf-consistent population in 1700” (p.18). On the other hand, as could be expected from “dictatorial” regimes, in 1800, the four largest Eastern cities (Vienna, Berlin, Moscow and St. Petersburg) were substantially larger than predicted by Zipf’s Law (p.19).

Geographical determinism (location near to rivers and seas) is often assumed to be at the origin of urban hierarchy and of the Zipf’s Law (and the distortions thereof). But the landscape did change much in 500 years, so mere location cannot explain much. Similarly industrial specialization cannot account for city size distribution since Zipf’s Law appeared in Western Europe around 1600, long before the Industrial Revolution (p.20).

The author identifies the main constraint on a size-independent urban growth rate (i.e. small cities grew faster than large ones) as transport costs which forced cities to rely on local foodstuff production. As grain price would double every 200 km when carried overland, inland cities had to rely on their “contado” creating a acute land constraint. Unsurprisingly food prices would increase with the city size, but as this correlation declined with the emergence of efficient grain trade and greater agricultural productivity the land constraint that had had checked larger cities growth gradually disappeared (p.22). Correspondingly, “the economies where city growth was concentrated also experienced relatively high rates of productivity growth in agriculture”.

The impact of the Second Serfdom

The limitations on rural-urban mobility in Eastern Europe proved particularly detrimental on city growth because it was “installed at a time when cities were relatively unhealthy places and required in-flows of migrants from the rural sector in order to grow. […] Cities in this ear were characterized by endemic excess mortality and hence city growth was closely tied to immigration” (p.26). “Broadly, between 1400 and 1750, the correlations between size and growth were smaller (more negative) in the East”. In other words, Eastern Europe (unlike Western Europe) departed from randomness of city growth stipulated by Zipf’s Law especially between 1500 and 1600 and 1700 and 1750 (p.32).

The growth rate of Eastern cities should have outstripped the one of Western cities because of their location in emerging economies which should have allowed them to benefit from a catch-up advantage (observable before 1500). But this advantage was cancelled by the impact of Second Serfdom, which mean that they grew at the same rate as their Western counter parts thus losing ground in absolute terms (p.37). The impact of the limitations imposed on labour mobility was very significant and may have cut city growth in Eastern Europe by ⅖.

Symptomatically, the presence of institutions associated with the Second Serfdom are correlated with significant delay in the emergence of the Zipf’s Law. In the same way exposure to these institutions made city growth non-random (i.e. coefficient of variation in the evolution much lower than in the West, indicating an element affecting city growth across the region, in this case impeding the growth of the larger ones; p.38).


More generally, the author associates a number of factors with urban growth: administrative centres, availability of navigable rivers, vicinity of the Atlantic Ocean, presence of a university, location on a Roman-ear settlement (indicating a propitious location) have all been recognized as important, the presence of Second Serfdom institution is also relevant. The impact of the Second Serfdom on the development of cities clearly indicates its importance on the “Great European Divergence” between East and West after 1500 (p.42). But historians differ on their interpretation of the causal link, indeed it not clear whether its the original weakness of the Eastern European urban centres which allowed the rise of the Second Serfdom or if the reverse is true.






One Response to Dittmar J. (2008) The emergence of Zipf’s Law

  1. Dave Parker says:

    The very fact of widely disparate urban primacy ratios should tell us that Zipf’s law doesn’t apply at the top. It also seems to depart from direct oubservation at the bottom, perhaps as smaller places with some urban characteristics become subsumed in their rural surroundings or aren’t selected / don’t develop sufficient urban critical mass for legal or administrative promotion or a fuller commercial role. Nor is the slope quite what it should be, at least in less urbanised societies – there are usually more than twice as many places of 5-10,000 as of 10-20,000, etc – logically enough since at some point up the size pyramid there aren’t any. But with those caveats it seems reliable enough for predicting at lower levels of the size hierarchy for which we may lack direct evidence. Perhaps that’s all we’re entitled to ask of it.

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