A Comparative Analysis of Gibrat's and Zipf's Law on Urban Population

Modica, Marco; Reggiani, Aura; Nijkamp, Peter Working Paper A Comparative Analysis of Gibrat's and Zipf's Law on Urban Population Quaderni - Working Paper DSE, No. 1008 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Modica, Marco; Reggiani, Aura; Nijkamp, Peter (2015) : A Comparative Analysis of Gibrat's and Zipf's Law on Urban Population, Quaderni - Working Paper DSE, No. 1008, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4270 This Version is available at: https://hdl.handle.net/10419/159846 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ ISSN 2282-6483 A Comparative Analysis of Gibrat’s and Zipf’s Law on Urban Population Marco Modica Aura Reggiani Peter Nijkamp Quaderni - Working Paper DSE N°1008 1 A Comparative Analysis of Gibrat’s and Zipf’s Law on Urban Population Marco Modica* Aura Reggiani** Peter Nijkamp*** *CNR – CERIS Institute for economic research on firms and growth. Via Bassini 15, 20133 Milano, Italy; email: [email protected] **Department of Economics, University of Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy; email: [email protected] ***Department of Spatial Economics, VU University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands; email: p.nijk[email protected] Abstract The regional economics and geography literature on urban population size has in recent years shown interesting conceptual and methodological contributions on the validity of Gibrat’s Law and Zipf’s Law. Despite distinct modeling features, they express similar fundamental characteristics in an equilibrium situation. Zipf’s law is formalized in a static form, while its associated dynamic process is articulated by Gibrat’s Law. Thus, it is likely that both Zipf’s Law and Gibrat’s Law share a common root. Unfortunately, empirical investigations on the direct relationship between Gibrat’s Law and Zipf’s Law are rather rare and not conclusive. The present paper aims to answer the question whether (a generalisation of) Gibrat’s Law allows us to infer Zipf’s Law, and vice versa? In our conceptual and applied framework, particular attention will be paid to the role of the mean and the variance of city population as key indicators for assessing the (non-) validity of the generalised Gibrat’s Law. Our empirical experiments are based on a comparative analysis between the dynamics of the urban population of four countries with entirely mutually contrasting spatial-economic and geographic characteristics: Botswana, Germany, Hungary and Luxembourg. We arrive at 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 one, then the variance is dependent on city size. Key-words: rank-size rule, Zipf’s law, (generalised) Gibrat’s Law, hierarchical structure, spatial interaction, city growth . JEL Classification: C46, D30, O40, R11 2 1.Gibrat’s Law vs Zipf’s Law: Preliminary Considerations Cities all over the world offer an amazing variety in terms of size and growth rates. Despite these differences, systems of cities do not exhibit a random pattern, but a strict regularity in terms of urban hierarchies and inter-urban connectivity. The genesis of such hierarchical perspectives on city size and urban systems can already be found in the seminal contributions of Christaller (1933) and Lösch (1940). The validity of these frameworks has extensively been tested in subsequent statistical experiments in many countries around the world. The conceptual foundation for the existence of central place hierarchies rests on various pillars: agglomeration advantages in cities (depending on city size), smart specialization of industries (depending on scale advantages in different size classes of cities), and transportation and logistics costs (depending on distance frictions between cities or between cities and their hinterlands). Urban hierarchies and inter-urban connectivity are therefore two sides of the same coin (see Paelinck and Nijkamp 1976). Clearly, it ought to be added that the spatial range of interurban linkages has extended drastically over recent decades. Whereas a century ago, most cities were at best part of an interlinked regional or national system, nowadays cities are often part of a globally connected network. Surprisingly, despite the complex evolution of current socio-economic spatial networks, two robust empirical regularities seem to hold: Gibrat’s law affirming that city growth does not depend on size, and Zipf’s law stating the proportionality of a given city size to its rank. 1 More in details, in 1931, Gibrat observed that the growth rate of a city’s population does not depend on the size of the city. In other words, although cities can grow at different rates, no systematic behaviour exists between their growth and their size, so that, according to Gibrat (1931), we cannot affirm that larger cities grow faster than smaller ones or vice versa. Analytically, we can write the following logarithmic expression, as in Steindl (1968): log ( ) log (0) (1) (2) ( ) P t P t ε ε ε = + + + + K (1) where P(t) is the size of a certain city at time t, P(0) is the initial population, and ε (t) is a random variable (indicating random shocks), i.i.d random variable with mean μ and variance σ 2 . Equation (1) identifies the logarithm of the size of a given city as the sum of the initial size and past growth rates. 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). 3 This law can now be interpreted as follows: “A variate subject to a process of change is said to obey the law of proportionate effect if the change in the variate at any step of the process is a random proportion of the previous value of the variate” (Chesher, 1979, p. 403). The implication of Gibrat’s law is that the growth processes of cities have “a common mean (equal to the mean city growth rate) and a common variance” (Gabaix, 1999, p. 741), that is, both the mean and variance have to be independent from the size of the cities. The second well-known spatial regularity is given by the so-called Zipf’s law (on the basis of a first study by Auerbach 2 in 1913). In 1949, Zipf observed and established that the sizes of the cities in a country are proportional to their rank. This means that in Botswana, for example, the size of largest city, Gaborone, is roughly twice the size of Francistown, the second largest city, three times the third largest city, Molopolole, and so on. Formally, this can be written as: q i i P KR − = (2) Equation (2) is known as the rank-size rule and is usually expressed in logarithmic form, as follows: log( ) log( ) log( ) i i P K q R = − (3) where P i is the population of city i , R i is the rank of the i th-city and K is a constant. Zipf’s law holds precisely, when the coefficient q is equal to one. Several interpretations of the Zipf coefficient, q , have been proposed in the literature. In principle, the q -coefficient can be seen as an indicator of the hierarchical degree of a system of cities (Singer, 1930). In fact, the qcoefficient measures how unequal the city distribution is: the higher the q -coefficient, the more unequally distributed is the city system. On the contrary, the smaller the value of q , the more even is the system of cities (in the extreme, when q= 0, we have a very even system of cities all of the same size; when q = ∞ , instead, we have only one city hosting the entire population). In summary, Gibrat’s law expresses the growth process of a certain variable (firm, city, income, wealth, etc.), independent of its size, while Zipf’s law presents the static relationship of the size of this variable with its rank. In the field of spatial economics, these two regularities have given rise, especially since the late ’90s, to an increasing number of empirical studies, testing cities and economic growth at 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). 4 various spatial levels (national, regional, local), by means of Gibrat’s law and Zipf’s law. It seems that in the majority of urban studies, Zipf’s law and Gibrat’s law are generally confirmed by empirical data (Eeckhout, 2004; Gonzalez-Val, 2010; Ioannides and Overman, 2003; Gabaix and Ioannides, 2004; Giesen and Suedekum, 2011). However, other studies seem to reject these two empirical regularities (Black and Henderson, 2003; Cuberes, 2011; Gonzalez-Val et al., 2012; Henderson and Wang, 2005). These contrasting results have prompted a continuous debate in the literature, on the (non)validity of Gibrat’s law and/or Zipf’s law. These two laws are often theoretically treated together, given their possible complementarily. Indeed, Champernowne (1953) and Simon (1955) have shown that rank-size distributions arise naturally, if Gibrat’s law is satisfied. Gabaix (1999) has demonstrated that Gibrat’s law leads to a Zipf distribution, while Cordoba (2003) argues that a weak version of Gibrat’s law leads to more general rank-size distributions, where weak means that only the mean of the city growth is independent from city size, while its variance can change according to size (Cordoba, 2003). In this setting, Cordoba, for the first time, shows an unknown relationship between the two laws; indeed, he shows that Zipf’s law might imply Gibrat’s law. Moreover, this setting is related to Garmestani et al. (2007), that have proposed a model that lead to different growth processes for clustering of cities that are aggregated according to similarity of their sizes. More in details, Cordoba (2008, p. 1463) proposes a: “ generalisation of Gibrat’s law that allows size to affect the variance of the growth process but not its mean ”. In particular, one of the implications of Cordoba’s generalised model is that non-proportionality of the variance is required to take into account a q -coefficient different from one (in Eq. (3)). More specifically, the larger the q -coefficient, the more unequal is the distribution, and this makes a growth process more volatile. 3 On the basis of Cordoba’s results, we can outline the following relationships between Zipf’s law and Gibrat’s law: (a) If q =1, Zipf’s law holds. In order that Gibrat’s law applies, neither the mean nor the variance of growth can depend on size. (b) If q >1, the distribution is more unequal. In order that Gibrat’s law applies, it is necessary that the mean is independent of the city size, but not the variance; indeed, the associated growth process requires that smaller cities face a greater volatility of growth than larger cities. (c) If q <1, the distribution is more evenly distributed. Again, in order that Gibrat’s law applies, it is necessary that the mean is independent of the city size, but not the variance. Here, the associated growth process requires that larger cities face a greater volatility of growth than 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. 5 smaller cities. It seems, therefore, plausible that Zipf’s law and Gibrat’s law show features of close kinship, in the sense that they show a bidirectional predictive frame that is able to reveal distinct modeling outcomes, even though in equilibrium it expresses identical fundamental characteristics. A test of this proposition calls for evidence-based research. Starting from these considerations, the present paper aims to answer the following research question: can (a generalisation of) Gibrat’s law allow us to infer Zipf’s law and vice versa, by empirically analysing the link between these two laws, in the context of urban growth, and, in particular, the dynamics of city size distributions? In this framework, particular attention will be paid to the role of the mean and variance of the city population as a key indicator for assessing the validity (or non-validity) of the generalised Gibrat’s law. Starting from these considerations, the main challenge and aim of this paper is to empirically explore the above mentioned relationship “Gibrat’s law vs Zipf’s law”, according to the three statements (a)-(c) above. It should be noted that empirical investigations of the relationship “Gibrat’s law vs Zipf’s law” are still rare in the sense that typically scholars investigate about the validity or non-validity of the laws singularly without any empirical comparison between the estimated coefficients of the two. Here, instead, we want to compare directly the implications of the validity and non-validity of one law to the other in a bidirectional way. Consistently with Eeckhout (2004), we focus our empirical investigation on the entire city size distributions of four countries (Botswana, Germany, Hungary and Luxembourg) and not only on the upper tail, 4 as other studies have done (see among others, Giesen and Suedekum, 2011; Guerin-Pace, 1995; Rosen and Resnick, 1980 and Soo, 2005, 2007). Notice that we focus on those particular countries because of their heterogeneity in terms of socio-economic and spatial characteristics. Our results find evidence of the existence of Gibrat’s law for two out of the four preselected countries; we then test the empirical relationship between the (generalised) Gibrat’s law and Zipf’s law, by considering the dynamics of the hierarchical structure of the various city systems, on the basis of the mean and variance indicators. The paper is then organised as follows. Section 2 describes the rationale underlying the selection of the four countries under analysis, by focusing on their different spatial economic characteristics and related statistics, while subsequent sections illustrate the results of the empirical analysis devoted to testing 4 Eeckhout (2004) shows that if the city growth does not depend on city size “then the estimated OLS coefficient of the socalled rank-size rule varies depending on the truncation city size, i.e. the inclusion of smaller (larger) cities in the sample, leads to a smaller (larger) coefficient” (Fazio and Modica, 2012, p. 3). 6 Gibrat’s law (Section 3), as well as the link between Gibrat’s law and Zipf’s law (Section 4). The paper concludes with some methodological considerations and directions for future research (Section 5). 2. Choice of Case Studies: Descriptive Analysis and Statistics We have selected in our empirical study four distinct countries characterised by different socioeconomic typologies: Botswana, Germany, Hungary and Luxembourg. The selection of these four countries, although mainly illustratively, may be representative of countries with different characteristics; the selection has been made according to size, population density and GDP per capita, according to the representation in Table 1. Moreover, in Table 2 we report, for each country, some economic indicators (such a GDP per capita, growth rate and percentage of investment over GDP), as well as some other important indicators for the mobility and transportation system (such as the length of railways and roadways, and the number of cars per thousand people). We collected data from the National Institute of Statistics for all four of these countries. 5 In particular, we collected data from the Central Statistics Office of Botswana, the Institute for Employment Research 6 (IAB) in Germany, the Hungarian Central Statistical Office and the STATEC-Institute National de la Statistique et des Etudes Economique of Luxembourg Some points are worth noting here. Botswana is the only non-OECD country, while all the others are OECD countries. Botswana shows the features of a non-advanced 7 economy; however, it exhibits a trend towards an increase in population and economic growth. Germany was a founding member of the European Community in 1957 (which became the European Union (EU) in 1993); it is central in Europe and is a large country in terms of surface area and population, with an advanced economy. Hungary joined the EU in 2004; it is located in central Europe, but shows a non-advanced economy and a decreasing population. Luxembourg, like Germany, was a founding member of the European Community in 1957; it is a small country, but very central in Europe with a high income per capita. Clearly, other choices could have been made, but the present set of countries aims to represent a sufficiently interesting collection of cases for in-depth investigation. 5 For all countries we have data over all cities from the biggest to the smallest one. 6 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 3 and 4). 7 According to the IMF classification. 7 < Table 1. About Here > < Table 2. About Here > It should be noted that an extensive debate concerns the type of spatial unit under analysis: several studies have been carried out using metropolitan areas, i.e. by considering the entire population in a given city, as well as all populations of suburban areas. Nevertheless, our object is to carry out comparative analysis between the four countries, by also including all the cities in a given country. For these reasons, we consider in our analysis the entities legally defined as cities or villages in their countries, although we are aware that the administrative definition given by legal borders might not fulfill our scopes exactly. In order to have a comparable unit, in all countries we have selected those localities which are similar to a municipality. 8 Another concern is due to the fact that we have different temporal horizons, which, sometimes, are short. This is the case for Botswana, where we have only two census observations (2000 and 2010), as well as for Hungary, where, although the time span is 30 years (1980-2011), we have only four census observations. For Germany, however, although the time span is 15 years, we have annual data (1993- 2007), so we can conduct a more precise analysis. Finally, Luxembourg have a long time series considering all census data from 1821 until 2011. Following on from the above observations, in Section 3 we focus our attention on the validity of Gibrat’s law, in order to design an analytical framework that is useful for meeting the ultimate goal of our analysis: a comparison of Gibrat’s’ law and Zipf’s law. 3. Testing Gibrat’s’ Law: Method and Results In this section we will use an OLS regression model and report the results from the parametric analysis. We check dynamic deviations from the proportionality of mean growth and variance to size, by using a method firstly proposed by Kalecki (1945) and subsequently utilised, among others, by Bottazzi et al. (2001). In particular, the adopted model is the following OLS model: 1 i i i t t t t g g β ε − = + (4) 8 We encountered some difficultly in making the right choice for Botswana where we also had data for small localities but we chose to collect all localities with ID code 100, namely villages and cities. 14 two halves, the variance ratios for the large cities show values always greater than one, while, for the small cities, they are always below one. This implies an (increasing) change in the underlying volatility of the growth process for large cities, in contrast to a (decreasing) change in the underlying volatility of the growth process for small cities. Moreover, in those years, the variance of growth σ 2BIG is always greater than σ 2small. Given these facts, we can affirm that at time t , the variance is increased for the large cities but decreased for the small cities, indicating a dependence of variance with respect to size; in particular, smaller cities face a lower volatility than large cities. In summary, statement c), which affirms: “if q< 1, In order that Gibrat’s law occurs, it is necessary that the mean is independent from the city size but not the variance, indeed the associate growth process requires that smaller cities face a lower volatility of growth than larger cities”, is satisfied for the whole sample in 1821-1930. In the period 1935-2011, the q -coefficient is not statistically different from one. By considering the relationship with Gibrat’s law, we then investigate condition a) of Section 1 which predicts that the associated growth process requires that smaller cities face the same growth as larger cities. Considering the entire sample, we have already shown that Gibrat’s law holds most of the time (estimated βparameters not significantly different from one). Considering the two sub-samples, we find similar evidence for both large and small cities, even with same exceptions. However it is interesting to note that in those years where Gibrat’s law does not hold, the estimated parameters β , θ and σ 2 show very different behaviour (i.e. θ BIG =.965 and θ small =1.162; σ 2BIG =1.93 and σ 2small = 1.31 in 1970), but, in general, in those years where Gibrat’s law holds, the differences between the estimators are not so large (i.e. θ BIG =1.03 and θ small =1.01 and σ 2BIG =0.55 and σ 2small = 0.53 in 1947). In summary, statement a) which affirms: “if q= 1, then in order that Gibrat’s law occurs neither the mean nor the variance of growth can depend on size” is satisfied for the whole sample. 4.6 Synthesis A synthesis of the above results – confirming the hypotheses by Cordoba (2003) – is presented in Table 6. < Table 6. About Here > The analysis carried out in this section prompts several interesting conclusions. We have been able to empirically verify the presence of a “generalised Gibrat law”, as theoretically predicted by Cordoba (2003). In particular, we have verified statements a), b) and c) of Section 1. In more detail, we have shown that when q >1 (statement b)) and Gibrat’s law holds ( β =1), the variance-ratio ( θparameter ) and 15 the variance of growth, σ 2, are actually higher for the small cities, in comparison to that for large cities, indicating a larger volatility for small cities. On the contrary, when q< 1 (statement c)) and Gibrat’s law holds ( β = 1), the variance-ratio ( θparameter ) and the variance of growth, σ 2 , is actually lower for the small cities, in comparison to that for large cities, indicating a larger volatility for the smaller ones. When q= 1 (statement a)) and Gibrat’s law holds, our findings agree with previous research, as both the mean and variance appear to be independent from the size. Moreover, when the q -coefficient is greater than one but decreasing, we have modifications on the growth process of large cities, but not on those of small cities; in particular, the larger the city, the lower the expected growth. On the other hand, when q >1 but increasing, small cities present the opposite growth process, namely the larger the city, the larger the expected growth. We have, of course, an opposite behaviour when the q -coefficient is less than one. 5. Conclusion The aim of our research work was to explore specific conditions leading to a generalisation of Gibrat’s law in connection with the different typologies of rank-size distribution. For this purpose we empirically explored the link between the rank-size exponent, q , with the necessary conditions for Gibrat’s law (that is mean and variance of the growth have to be independent from the size). We started our analysis based on the conclusion of Cordoba (2003, p. 3): “ Pareto distributions with larger exponents (more unequal distributions) require more volatile growth processes ”. As far as we know, the conventional methodologies (Section 3) used to test Gibrat’s law do not address this issue. In particular, a greater (lower) volatility of the variance is usually not empirically envisaged. We showed, instead, that, according to Cordoba (2003), the variance can be dependent on size if the rank-size coefficient is different from one; in particular, we verified what Cordoba (2003) calls a “generalised Gibrat’s law” for different countries with different spatial-economic characteristics: Botswana, Germany, Hungary and Luxembourg. We found strong evidence of this generalised Gibrat’s law for Botswana and Luxembourg. We found weak evidence of Gibrat’s law for Germany and no evidence for Hungary. Our results confirm the propositions provided by Cordoba (2003). In particular, when q =1, neither the mean nor the variance of growth depend on size; when q >1, the mean is independent of the city size, but not the variance, and small cities face a greater volatility in growth than larger cities; alternatively, when q <1, the mean is independent from the city size, but not the variance, and large cities face a greater volatility in growth than smaller ones. Gibrat and Zipf have offered complementary perspectives on city size and systems of cites in a given country. Their contributions are not necessarily 16 identical, but offer new perspectives on the same multi-faceted prism of the space-economy. These results might be useful to ’relax’ Gibrat’s law in its strict interpretation, by reinforcing the hypothesis that small entities face a greater volatility in the growth process. Our analysis prompts various intriguing research questions in the future. While Gibrat’s law and Zipf’s law mirror important organised structures in the topology of systems of cities, other relevant structural patterns may be investigated as well, such as the existence of fractal structures in urban systems (based, for example, on Mandelbrot’s principles) or the persistent existence of spatial population or socioeconomic disparities (based, for example, on Herfindahl’s index). Clearly, the dynamics of such processes deserve due attention. In addition, the above applied investigation also calls for more fundamental research into the functional or behavioural backgrounds of such regularities. Three research directions are important here; (a) the interdependence between population indicators and broader socio-economic indicators for a system of cities; (b) the degree of various cities in the same national system; (c) the relationship between recent strong evolutionary trends in the digital world and the development of cities (and systems of cities). 17 References Adamic, L. A., 2000. Zipf, power-laws, and Pareto – a ranking tutorial, retrieved from www.hpl.hp.com Auerbach, F., 1913. Das Gesetz Der Bevolkerungs Konzentration, Petermanns Geographische Mitteilungen. 59, 73-76. Auerbach, F., 1915. The law of population concentration. Scientific American Supplement. 76, 384. Black, D., Henderson V., 2003. Urban evolution in the USA. Journal of Economic Geography. 3(4), 343- 372. Bottazzi, G., Dosi, G., Lippi, M., Pammolli, F., Riccaboni, M., 2001. Innovation and corporate growth in the evolution of the drug industry. International Journal of Industrial Organization. 19(7), 1161-1187. Champernowne, D. G., 1953. A model of income distribution. The Economic Journal. 63(250), 318-351. Chesher, A., 1979. Testing the law of proportionate effect. Journal of Industrial Economics. 27, 403-411. Christaller, W., 1933. Die Zentrale Orte in Süddeutschland. Eine Ökonomisch-Geografische Untersuchung Über Die Gesatzmässigkeit Der Verbreitung Und Entwicklung Der Siedlungen Mit Städtischen Funktionen, Gustav Fischer, Jena. Cordoba, J., 2003. On the distribution of city sizes. Urban/Regional. 0302002, EconWPA. Cordoba, J., 2008. A generalized Gibrat’s law. International Economic Review. 49(4), 1463-1468. Cuberes, D., 2011. Sequential city growth: empirical evidence. Joournal of Urban Economics. 69(2), 229- 239. Eeckhout, J., 2004. Gibrat’s law for (all) cities. American Economic Review. 94(5), 1429-1451. Fazio, G., Modica, M., 2012. Pareto or log-normal? A recursive truncation approach to the distribution of (all) cities. Working Papers 2012_10, Business School, Economics, University of Glasgow, Glasgow. Gabaix, X., 1999. Zipf’s law for cities: an explanation. The Quarterly Journal of Economics. 114(3), 739- 767. Gabaix, X., Ibragimov, R., 2011. Rank-1/2: a simple way to improve the OLS estimation of tail exponents. Journal of Business & Economic Statistics. 29(1), 24-39. Gabaix, X., Ioannides, Y. M., 2004. The evolution of city size distributions, in J.V. Henderson and J.-F. Thisse (eds), Handbook of Regional and Urban Economics, Vol. 4. Elsevier, Amsterdam, 2341–2378. Garmestani, A.J., Allen, C.R., Gallagher, C.M., Mittelstaedt, J.D., 2007. Departures from Gibrat’s law, discontinuities and city size distributions. Urban Studies. 44(10), 1997-2007. Gibrat, R., 1931. Les Inégalités Economiques, Libraire du Recueil Siray, Paris. Giesen, K., Suedekum, J., 2011. Zipf’s law for cities in the regions and the country. Journal of Economic Geography. 11(4), 667-686. 18 González-Val, R., 2010. The evolution of U.S. city size distribution from a long-term perspective (1900- 2000). Journal of Regional Science. 50(5), 952-972. González-Val, R., Lanaspa, L., Sanz, F., 2012. New evidence on Gibrat’s law for cities. Working Papers 2012/18. Institut de Economia de Barcelona (IBE), Barcelona. Guerin-Pace F., 1995. Rank-size distribution and the process of urban growth. Urban Studies. 32(3), 551- 562. Henderson, J.V., Wang, H.G., 2007. Urbanization and city growth: the role of institutions. Regional Science and Urban Economics. 37, 283-313. Ioannides, Y. M., Overman, H. G., 2003. Zipf’s law for cities: an empirical examination. Regional Science and Urban Economics. 33(2), 127-137. Kalecki, M., 1945. On the Gibrat distribution. Econometrica. 13(2), 161-170. Lösch, A., 1940. The Economics of location. Yale University Press, New Haven (translated by W. Woglam and W. Stolper in 1954). Paelinck, J.H.P., Nijkamp, P., 1976. Operational Theory and Method in Regional Economics, Saxon House, Farnborough. Parr, J. B., 1985. A note on the size distribution of cities over time. Journal of Urban Economics. 18, 199- 212. Rosen, K. T., Resnick, M., 1980. The size distribution of cities: An examination of the Pareto law and primacy. Journal of Urban Economics. 8(2), 165 -186. Simon, H., 1955. On a class of skew distribution functions. Biometrika. 42(3/4), 425-440. Singer, H. W., 1930. The “Courbe Des Populations”. A parallel to Pareto’s law. The Economic Journal. 48(182), 254-263. Soo, K. T., 2005. Zipf’s law for cities: a cross-country investigation. Regional Science and Urban Economics. 35(3), 239-263. Soo, K. T., 2007. Zipf's Law and Urban Growth in Malaysia. Urban Studies. 44(1), 1-14 Steindl, J., 1968. Size distributions in Economics, in International Encyclopedia of the Social Sciences, Vol. 14, Silks, ed. (New York, Macmillan and the Free Press). Zipf, G., 1949. Human behavior and the principle of least effort: an introduction to human ecology, Addison-Wesley, Cambridge. 19 Table 1. Rationale for countries selection Population Density GDP p.c High Low High Low Size High Germany Botswana Germany Botswana Low Luxembourg Hungary Luxembourg Hungary Table 2. Spatial Economic Characteristics of the Five Countries under Analysis Country Year Km 2 (thousands) Pop. (milion) Density % Urban pop. % pop. growth Railway *1000 km Roadway *1000 km GDP p. c. Growth (%) Botswana 2011 581 1.85 3.19 62.00% 1.47 0.90 25.80 9,481 5.1% Germany 2007 357 81.78 229 74.00% -0.20 41.90 644.50 40,403 2.7% Hungary 2011 93 9.99 107.4 69.00% -0.18 8.10 197.50 13,045 1.70% Luxembourg 2011 2,5 0.51 205.6 85.00% 1.13 0.27 5.20 106,958 1.00% Table 3. Descriptive Statistics of the Five Countries under Analysis Country Year N. cities ln(Mean) ln (Variance) ln (Median) Skewness Kurtosis Botswana 2011 461 7.15 1.22 6.97 0.78 2.23 Germany 2007 12,262 7.42 1.50 7.30 0.34 0.17 Hungary 2011 3,154 6.75 1.34 6.70 0.40 0.95 Luxemburg 2011 116 7.81 0.92 7.61 0.86 1.45 20 Table 4a. Model A Estimates (Countries: Botswana, Germany, Hungary and Luxembourg; Different Years) Country Year β Robust s.e. θ R 2 N. obs. Botswana 2011 .995** .0149 1.0782 .92 460 Germany 1994 .999 .0003 1.0009 .99 12,280 1995 .999* .0002 .9983 .99 12,291 1996 .999* .0003 .9987 .99 12,291 1997 .998 .0002 .9978 .99 12,291 1998 .998 .0002 .9984 .99 12,291 1999 1.00 *** .0003 1.0011 .99 12,293 2000 .999*** .0003 .9998 .99 12,294 2001 1.001 .0001 1.0024 .99 12,294 2002 1.001 .0001 1.0017 .99 12,294 2003 1.00* .0001 1.0009 .99 12,293 2004 1.001 .0001 1.0027 .99 12,292 2005 1.001 .0001 1.0031 .99 12,293 2006 1.001 .0001 1.0027 .99 12,993 2007 1.001 .0002 1.0039 .99 12,259 Hungary 1990 1.055 .0019 1.1215 .99 3,121 2001 1.034 .0022 1.0806 .99 3,121 2011 1.028 .0025 1.0674 .99 3,121 Luxembourg 1851 .941* .0264 .9557 .93 116 1871 .993** .0276 1.0779 .92 116 1880 1.005 ** .0306 1.0667 .95 116 1890 1.048 .0183 1.0855 .93 116 1900 1.109 .0324 1.2833 .96 116 1910 1.079 .0187 1.1803 .99 116 1922 1.01** .0123 1.0473 .97 116 1930 1.103 .019 1.2406 .98 116 1935 1.001*** .007 1.0059 .99 116 1947 1.014* .0061 1.0355 .99 116 1960 1.069 .0132 1.1725 .98 116 1970 1.044 .0156 1.1198 .97 116 1981 1.016** .0144 1.0576 .98 116 1991 .993 ** .0104 .9964 .99 116 2001 0.955 .0076 .9199 .99 116 2002 0.994 .0015 .989 .99 116 2003 .995** .0034 .9926 .99 116 * Significant at 1% ** significant at 5% *** significant at 10% 21 Table 4b. Model A Estimates (Country: Luxembourg; Different Years) Country Year β Robust s.e. θ R 2 N. obs. Luxembourg 2004 .997** .0018 .9947 .99 116 2005 .996** .0024 .9926 .99 116 2006 .997** .0015 .9961 .99 116 2007 0.992 .0029 .9851 .99 116 2008 0.992 .0023 .985 .99 116 2009 .996** .0019 .994 .99 116 2010 .996* .0016 .9924 .99 116 2011 .999** .0015 1.00 .99 116 *** Significant at 1% ** significant at 5% * significant at 10% 22 Table 5a. The Zipf’s and Gibrat’s Parameters (Countries: Botswana, Germany, Hungary and Luxembourg; Different Years) N. obs. 464 461 12,280 12,291 12,291 12,291 12,291 12,293 12,294 12,294 12,294 12,294 12,293 12,293 12,293 12,293 12262 3,121 3,121 3,121 3,154 * Significant at 1% ** significant at 5% *** significant at 10% σ Small - 5.065 - 6.030 4.347 4.235 4.033 3.863 3.582 4.644 2.872 3.114 2.904 2.889 2.775 2.728 2.847 - 1.005 1.429 1.410 θ Small - 1.011 - 1.009 1.004 1.006 1.002 1.000 1.012 .998 1.006 1.002 .998 1.004 1.003 1.000 1.005 - 1.194 1.151 1.070 β Small - 0.761 - 1.001** 1.00** 1.001** .999** .997** 1.004 .998** 1.002 1.00** .998 1.001 1.001** .999** 1.001 - 1.076 1.054 1.010** σ BIG - 3.632 - 2.998 2.583 2.101 2.099 1.906 1.716 1.536 1.419 1.278 1.188 1.270 1.162 1.074 1.438 - .8129 1.404 1.219 θ BIG - 1.000 - .994 .994 .995 .994 .994 .996 .998 .999 .999 1.001 1.001 1.002 1.001 1 - 1.046 .999 1.020 β BIG - .979** - .996 .996 .997 .997 .997 .998 .998 .999 .999** .999** 1.000** 1.001 1.001 1.002 - 1.018 0.993 1.005 θ - 1.0078 - 1.0009 .9983 .9987 .9978 .9984 1.0011 .9998 1.0024 1.0017 1.0009 1.0027 1.0031 1.0027 1.0039 - 1.12 1.08 1.06 β - .995** - .999 .999* .999* .998 .998 1.00 *** .999*** 1.001 1.001 1.00* 1.001 1.001 1.001 1.001 - 1.055 1.034 1.028 Robust s.e. .0746 .0746 .0178 .0178 .0178 .0177 .0177 .0177 .0177 .0177 .0177 .0177 .0177 .0177 .0178 .0178 .0178 .0285 .0300 .0309 .0316 q-coefficient 1.137 1.173 1.399 1.397 1.396 1.394 1.392 1.390 1.390 1.390 1.391 1.392 1.392 1.393 1.396 1.398 1.401 1.129 1.186 1.223 1.258 Year 2001 2011 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 1980 1990 2001 2011 Country Botswana Germany Hungary 23 Table 5b. The Zipf’s and Gibrat’s Parameters (Countries: Botswana, Germany, Hungary and Luxembourg; Different Years) N. obs. 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 * Significant at 1% ** sign ificant at 5% *** significant at 10% σ Small - .7822 .6157 1.904 .6451 .6763 6178 .5105 .5981 .7215 5250 .7906 1.308 1.268 .881 1.337 2.177 3.356 2.465 2.812 2.254 3.498 2.501 2.219 1.787 2.137 θ Small - .764 .981 .871 .965 .969 1.035 1.140 1.048 1.081 1.009 1.109 1.162 1.150 1.145 .871 .983 1.00 1.020 .989 .991 .955 .955 .981 .974 .996 β Small - .730 .887** .811** .944** .947** .986** 1.042** 1.01** 1.033* .982** 1.006** 1.016** 1.022** 1.053** .910 .990** .997** 1.00** .992** .994** .974** .976 .989** .986* .997** σ BIG - .6141 1.296 1.693 3.327 2.651 1.332 2.095 2.574 1.045 .5543 1.395 1.930 1.815 1.395 .874 1.291 1.366 1.209 1.529 1.109 1.412 1.291 1.672 1.247 1.073 θ BIG - .978 1.070 1.060 1.160 1.340 1.210 1.025 1.240 .976 1.034 1.059 .963 .922 .920 .907 .997 1.00 1.00 .992 1.00 .988 .998 1.00 1.00 1.00 β BIG - .949** .962** 1.003** .995** 1.115 1.091 0.998** 1.090 .986** 1.013** 1.011** 0.965 0.942 0.949 0.949 .998** 1.00** 1.00** .995** 1.001 0.994 .999** 1.00** 1.00** 1.00** θ - .9557 1.0779 1.0667 1.0855 1.2833 1.1803 1.0473 1.2406 1.0059 1.0355 1.1725 1.1198 1.0576 .9964 .9199 .989 .9926 .9947 .9926 .9961 .9851 .985 .994 .9924 1.00 β - .941* .993** 1.005 ** 1.048 1.109 1.079 1.01** 1.103 1.001*** 1.014* 1.069 1.044 1.016** .993 ** 0.955 0.994 .995** .997** .996** .997** 0.992 0.992 .996** .996* .999** Robust s.e. .0660 .0652 .0676 .0702 .0772 .0885 .0965 .0984 .1100 .1101 .1121 .1214 .1278 .1302 .1287 .1235 .1229 .1225 .1221 .1217 .1215 .1207 .1200 .1197 .1194 .1194 q - coefficient .5031 .4965 .5154 .5350 .5881 .6744 .7350 .7500 .8377** .8391** .8543** .9252** .9735** .9923** .9803** .9409** .9365* .9330** .9302** .9270** .9253** .9195** .9140** .9120** .9094** .9094** Year 1821 1851 1871 1880 1890 1900 1910 1922 1930 1935 1947 1960 1970 1980 1991 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Country Luxembourg