DOES THE SIZE MATTER? Zipf's Law for cities Revisited

Several authors (Berry 1970, Krugman 1996 or Eaton and Eckstein 1997, among many others) have experienced amazement about the accurate functioning of the law of "least effort" established by Zipf (1949) in most places. Cities, ranked by population, seem to follow almost exactly a log/log function, in which the logarithm of the 'mass' (population, density, etc.) correlates almost perfectly with the logarithm of the order of that mass. This log/log function, advanced by Pareto in the nineteenth century, has seduced quite a number of researchers, for its presence, hypothetically, both in natural phenomena (earthquakes, meteorites, living species) and in the ones which derive from society (language, or cities), which has led to investigate its theoretical basis (Simon 1955, Brakmar et al. 1999, Gabaix 1999). While some authors (Rosen and Resnick 1980, Fan and Casetti 1994) have discussed the linear validity of Zipf's Law, introducing nonlinear models, technical literature has focused on the 'upper tail' of the urban hierarchy, large metropolitan areas, tend to silence the fact that the log / log function does not appear to be a general model. This paper attempts to show that when taking into account all the cases (ie, all populated localities in a particular territory), the log/log model seems to be only a special case of 'the big' ones. In fact it shows that a log/lin model tends to be more efficient, even with 'folded tails'. This has led to the hypothesis which was tested in this study, that the logarithm of the urban mass tends to have a 'normal distribution', leading its cumulative distribution (and ordered by rank) to be spread in a logistical structure, in 'S'. In this sense, the repeated observation of fulfillment of the Law of Zipf in the size of the cities would be just "the tip of the iceberg", in which small and medium cities also take their part, and where a "law" of a higher level appears. The presented research questions if this "normal" appearance of the logarithm of the mass could be shaped in a simple and elegant form, and makes some experiments in this regard.