Abstract
The regional economics and geography literature has in recent years shown interesting conceptual and methodological contributions on the validity of Gibrat’s law and Zipf’s law. Despite distinct modelling features, they express similar fundamental characteristics in an equilibrium situation. Zipf’s law is formalised in a static form, while its associated dynamic process is articulated by Gibrat’s law. Thus, it seems that both Zipf’s law and Gibrat’s law share a common root. Unfortunately, although several studies analyse both the laws looking at the validity of these regularities, very few empirical investigations assess the implication of one law from the other one (i.e. deviations from Zipf’s law result in a deviation from Gibrat’s law). Moreover, due to heterogeneity in data sources, comparative analyses between countries are difficult to perform. The present chapter aims at building the basis for further innovative research in this field, while it also aims to provide some caveats in empirical research. Specifically, we pay particular attention to the role of the mean and the variance of city population as key indicators for assessing the (non-)validity of the so-called generalised Gibrat’s law. Our empirical experiments are based on illustrative case studies on the dynamics of the urban population of five countries with entirely mutually contrasting spatial-economic characteristics: Botswana, Germany, Hungary, Japan and Luxembourg. We provide evidence on the following results: if (i) the mean is independent of city size (first necessary condition of Gibrat’s law) and (ii) the coefficient of the rank-size rule/Zipf’s law is different from 1, then the variance is dependent on city size.
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Notes
- 1.
Another way to refer to Zipf’s law is a Pareto distribution, with a shape parameter equal to 1. It is investigated using the so-called rank-size rule. We note here that the slope coefficient of the rank-size rule represents the inverse form of the parameter of the conventional Pareto distribution. For more details, we refer inter alia to Adamic (2000) and Parr (1985). In this paper, we refer to Zipf’s law (Zipf’s distribution), when the rank-size coefficient is exactly equal to 1. In all the other cases, we refer to the rank-size rule (rank-size distribution).
- 2.
The population of a city is inversely proportional to the number indicating its rank among the cities of a given country (Auerbach 1915, p. 384).
- 3.
The volatility is a measure of fluctuation of a process. We will use the variance as an indicator of the volatility of an underlying proportionate growth process.
- 4.
For all countries, we have data over all cities from the biggest to the smallest one.
- 5.
The authors wish to thank Uwe Blien and Anette Haas (IAB, Germany), for kindly providing the data used in our study on German cities (sections ‘Testing Gibrat’s’ Law: Method and Results’ and ‘Gibrat’s Law and Zipf’s Law: A Comparative Study’).
- 6.
According to the IMF classification
- 7.
Another example, not included in this analysis, is France with more than 30,000 municipalities and quite same surface area of Germany.
- 8.
We report here only the condition on the estimated β. However, it should be noted that another condition is necessary to affirm that Gibrat’s law is in operation; indeed, the error terms have to be serially uncorrelated. We, then, add one more lag in Eq. 4 to this very additional condition. In most of the cases, the error terms result serially uncorrelated. Moreover, notice that when β is lower than 1, this means that size converges towards its mean, namely, the larger a city, the smaller the expected growth. On the contrary, when β is greater than 1, the larger a city, the larger the expected growth.
- 9.
where *** indicates a significance level at 1 %
- 10.
It suggests a change in the variance over the time, and then this might also imply changes in the dependence of the variance over the size.
- 11.
Notice that in the years in which the q-coefficient is equal to that of previous year (i.e. 1999 and 2000), Gibrat’s law holds. The stability of the process should not lead to any changes in the hierarchical structure of cities.
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Modica, M., Reggiani, A., Nijkamp, P. (2017). Methodological Advances in Gibrat’s and Zipf’s Laws: A Comparative Empirical Study on the Evolution of Urban Systems. In: Shibusawa, H., Sakurai, K., Mizunoya, T., Uchida, S. (eds) Socioeconomic Environmental Policies and Evaluations in Regional Science. New Frontiers in Regional Science: Asian Perspectives, vol 24. Springer, Singapore. https://doi.org/10.1007/978-981-10-0099-7_3
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