Zipf's Law for Cities and Beyond: The Case of Denmark

American Journal of Economics and Sociology, The,  Jan, 2001  by Thornbjorn Knudsen

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THORNBJORN KNUDSEN [*]

ABSTRACT. Zipf's law for cities is one of the most conspicuous and robust empirical facts in the social sciences. It says that for most countries, the size distribution of cities must fit the power law: the number of cities with populations greater than S is inversely proportional to S. The present paper answers three questions related to Zipf's law: (1) does the Danish case refute Zipf's law for cities?, (2) what are the implications of Zipf's law for models of local growth?, and (3) do we have a Zipf's law for firms? Based on empirical data on the 61 largest Danish cities for year 2000, the answer to (1) is NO--the Danish case is not the exception which refutes Zipf's law. The consideration of (2) then leads to an empirical test of (3). The question of the existence of Zipf's law for firms is tested on a sample of 14,541 Danish production companies (the total population for 1997 with 10 employees or more). Based on the empirical evidence, the answer to (3) is YES in the sense that the growth pattern of Dani sh production companies follows a clean rank-size distribution consistent with Zipf's law.

PREDICTION IN ECONOMICS, and in the social sciences generally, is a rather scarce commodity (Reder 1999) and perhaps an unattainable ideal (Aumann 2000). According to Aumann, the value of a good theory lies in its usefulness in structuring reasoning and, therefore, one empirical fact to be cited in favour of a theory is its diffusion in some population of scientists. In other words, the more use of a particular theory, the better. As Reder notes, economists tend to place higher value on technique than content; clever theoretical ideas are valued over the assiduous gathering and careful presentation of data. And since mainstream economics, in any case, has a sufficiently flexible theoretical basis to rationalize contrary empirical facts also, data do not play the prominent role they do in the natural sciences. As important reasons for the gap between theory and applied work, Reder points to the rather low status of empirical facts and the tendency to use the term "prediction" when "retrodiction" would be more suitable (pp. 27-29). In contrast to this generally disappointing state of affairs (see, e.g., Reder 1999), there exists one exceptional case, a notable empirical success story in which theory must bow to facts. This case is zipf's law for cities, which has important implications for the admissibility of theoretical growth models. In economics one very rarely finds empirical relationships which deserve to be called laws. Zipf's law for cities, however, is one of the most conspicuous empirical facts in economics and in the social sciences in general (Brakman et al. 1999). It is surely an outstanding empirical regularity deserving the status of an experimental law (Gabaix 1999).

According to Zipf's law, the growth pattern of cities almost everywhere follows the power law--the number of cities with populations greater than S is proportional to 1/S. Put differently, if we rank a sample of cities according to population size, and then place the log of the size on the X-axis and the corresponding log of the rank on the Y-axis, there should appear a straight line with slope -1. Should the numerical value of the slope exceed 1, cities are more dispersed than predicted whereas a slope less than one indicates that cities are more even sized than the prediction. Surprisingly, we actually see a slope of about 1 when data on American metropolitan areas are used. Both Gabaix (1999) and Krugman (1996) obtained a slope of -1.005 (std.dev. 0.010) and an [R.sup.2] of .986 for the 135 American metropolitan areas listed in the Statistical Abstract of the United States for 1991. Similar results have been reported for most countries in contemporary times (Rosen and Resnick 1980). The support of Zipf's law for previous periods has included samples of cities in India (Zipf 1949), China (Rozman 1990), the Netherlands (Brakman et al. 1999) and the United States (Krugman 1996; Zipf 1949).

Although most evidence corroborates Zip's law, some evidence has been reported which seems to refute the prediction of a slope of -1. Thus, Brakman et al. (1999) compare data from the Netherlands in 1600, 1900, and 1990 and, despite a very good fit for all three regression models ([R.sup.2] = 0.96 or better), obtain estimates that deviate from the slope of -1 predicted by Zipf's law. In their study, only data for 1900 fit this prediction. Both the estimates for 1600 and 1990 obtain a lower value, indicating that cities are more even-sized than predicted. Inspired by this deviation, we have sampled data to test the Danish case for the year 2000. Since Denmark and the Netherlands are both small countries and to some extent comparable in development, it is interesting to ask if the Danish case will show yet another deviation from Zipf's law for cities.

At this point it should be noted that Zipf's law is a special case of what is known as the rank-size distribution, which states an inverse linear relationship between the logarithmic size of a city and its logarithmic rank without any constraints on the slope. In the study of size distributions of firms, the term "Pareto distribution" is often used synonymously with the term rank-size distribution (see, e.g., Ijiri and Simon 1977). Furthermore, Zipf's law, or rather the rank-size distribution, applies to a much wider number of phenomena than size distributions of cities (Zipf 1949). For example, rank-size distributions have been shown to fit empirical observations of relative income (Zipf 1949), the relative size of business firms (Ijiri and Simon 1977), the number of biological species per genus (Zipf 1949) and the relative frequency of a word (Estoup 1916; Zipf 1935, 1949; Irmay 1997). Moreover, Irmay (1997) showed that Zipf's law is approximately equal to Benford's (1938) logarithmic distribution of first significant digits in a table of numbers. That Zipf's law fits size distributions of cities and business firms, relative frequencies of large texts of words (in several languages) and can account for first digits in various tables of numbers begs a second question: What is the explanation for Zipf's law and why do we see it in the case of cities?