Methodological Advances in Gibrat’s and Zipf’s Laws: A Comparative Empirical Study on the Evolution of Urban Systems

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.

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 interurban 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 specialisation 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 interurban 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 (Gibrat 1931), and Zipf’s law stating the proportionality of a given city size to its rank (Zipf 1949).¹

More in details, Gibrat’s law states 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 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):

$$ \log P(t)= \log P(0)+\varepsilon (1)+\varepsilon (2)+\cdots +\varepsilon (t) $$

(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.

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 Auerbach² in 1913). Zipf’s law states that the size of the cities in a country is proportional to their rank. This means that, for example, in Botswana, the size of the largest city, Gaborone, is roughly twice the size of Francistown, the second largest city, three times the third largest city, Molepolole, and so on. Formally, this can be written as:

$$ {P}_i={KR}_i^{-q} $$

(2)

Equation 2 is known as the rank-size rule and is usually expressed in logarithmic form, as follows:

$$ \log {(P)}_i= \log (K)-q \log \left({R}_i\right) $$

(3)

where P _i is the population of city i, R _i is the rank of the ith city and K is a constant. Zipf’s law holds precisely, when the coefficient q is equal to 1.

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 (Brakman et al. 2001; Reggiani and Nijkamp 2015; Singer 1930). In fact, the q-coefficient 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 is 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 1990s, to an increasing number of empirical studies, testing cities and economic growth at 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 2007). 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 analysed together, at least from the theoretical point of view, given their possible complementarity. 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.

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 1 (in Eq. 3). More specifically, the larger the q-coefficient, the more unequal the distribution; this finding 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 implies that smaller cities face a greater volatility in 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. Hence, larger cities face a greater volatility in growth than smaller cities.

Given these premises, there are very well-known problems associated with empirical verification of Gibrat’s law (and Zipf’s law as well) that are often ignored or that are not jointly considered. First, it should be noted that empirical investigations of the relationship ‘Gibrat’s law vs. Zipf’s law’ in the Cordoba view are still rare. This means that, typically, scholars investigate about the validity or non-validity of the laws singularly, i.e. without any empirical comparison between the two. Namely, if Zipf’s law does not hold the corresponding ‘behaviour’ of Girat’s law is unknown (or unexplored). At the same time, if Gibrat’s law deviates we do not know the implications on Zipf’s law, at least empirically.

Second, the results critically depend on the choice of the geographic unit, i.e. municipalities vs. urban areas (González-Val et al. 2013). Third, even though the geographic unit is the same, i.e. municipality, this does not necessary mean that two municipalities in two different countries are comparable. This is because the ‘legal’ or even ‘intrinsic’ definition of municipality (or urban area) might change by countries (e.g. a Japan municipality is quite different from a German one). Fourth, scholars adopt truncated samples in the analysis. However, the way in which this should be done it has been addressed only recently (see for instance, Bee et al. 2013; Ioannides and Skouras 2013; Fazio and Modica 2015). Finally, it is not well clear whether the two laws apply over a long time period or span or, on the contrary, they are in operation even in the short run.

For all these reasons, there are very few analyses that run international comparisons of the two laws (to our knowledge the only two papers are Rosen and Resnick (1980) and Soo (2005)). This chapter, then, aims at building the basis for a rigorous analysis on the empirical link between Zipf’s law and Gibrat’s law. To do that, we will present five case studies that explore the empirical link between the two laws but also underline the common shortcomings of the empirical analysis and the difficulties in the comparison of the results between countries. Notice that we focus only on five countries because of the illustrative purpose of this work. Indeed, this contribution wants to be just a baseline for further research in this field. At the same time, this chapter highlights caveats in empirical research. Finally, it does not pretend to be a complete analysis and further works are needed for a better comprehension of the issue.

The chapter is then organised as follows. Section 2 describes the main characteristics of the data, while subsequent sections illustrate the results of the empirical analysis devoted to testing Gibrat’s law (Sect. 3), as well as the link between Gibrat’s law and Zipf’s law (Sect. 4). The chapter concludes with some methodological considerations and directions for future research (Sect. 5).

2 Data: Descriptive Statistics

We have selected in our empirical study five distinct countries characterised by different socio-economic typologies, but also different ‘data characteristics’ (i.e. in terms of time series and time span and different ‘intrinsic’ definitions of municipality). These countries are Botswana, Germany, Hungary Japan and Luxembourg. The selection of these five countries, although mainly illustratively, may be representative of countries with different characteristics. However, our work is to be considered only as a demonstrative analysis with any attempt to be a rigorous (in statistical terms) study from which we can draw meaningful results.

Table 1 is an overview of the countries analysed in our empirical research. They range significantly in size, economic development and geographic position. This selection was done deliberately, in order to test whether likely common findings in Zipf’s law and Gibrat’s law would be robust in the case of contrasting countries.

Table 1 Rationale for country selection

In Table 2 we report, for each country, some economic indicators (e.g. 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 five of these countries.⁴ In particular, we collected data from the Central Statistics Office of Botswana, the Institute for Employment Research⁵ (IAB) in Germany, the Hungarian Central Statistical Office, the Statistics Bureau of Japan and the STATEC (Institute National de la Statistique et des Etudes Economiques) of Luxembourg.

Table 2 Spatial-economic characteristics of the five countries under analysis

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-advanced⁶ 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. Japan is an Asian country, and it is the most populated country among the selected ones; however, its population is spread in very few municipalities. Luxembourg, like Germany, was a founding member of the European Community in 1957; it is a small country, but geographically very central in Europe and a high income per capita. Moreover, its subdivision in municipalities is a purely administrative fact rather than a subdivision de facto. Clearly, other choices could have been made, but the present set of countries aims to represent a sufficiently interesting collection of cases for preliminary investigation addressing all the shortcomings listed in the previous section.

Indeed, it should be noted that an extensive debate, concerning the type of spatial unit under analysis, is very much alive; however, several studies have been carried out using metropolitan areas, i.e. by considering the entire population in a given city, as well as all population of suburban areas. Nevertheless, our object is to carry out comparative analysis between the five countries, by also including all the cities in the countries under analysis. We then do not constrain the analysis only to the larger cities. For these reasons, we consider in this work all 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 fulfil our scopes exactly. Indeed, the ‘intrinsic’ definition of municipalities is quite different in the selected countries. To underline the different ‘intrinsic’ definitions of municipality, we report in Tables 2 and 3 descriptive statistics of the five countries. We can see how Japan shows a number of municipalities of 1719 against a total population of about 128 million in an area of 378 km². At the same time, Germany shows quite the same area (i.e. 357 km²) with a slightly lower population (about 82 million), but it shows a huge number of municipalities in comparison to Japan (12,262).⁷ It is evident that in the case of Japan, the municipality covers an area much more larger than that in the case of Germany. Then, the concept of municipality in Japan might be much more close to that of urban areas with respect to the concept of German municipality, and, thus, the two objects may be difficult to compare.

Table 3 Descriptive statistics of the five countries under analysis

Another shortcoming that should be addressed is that the selected countries show temporal horizons consistently differ. Indeed, for Botswana, we have only two census observations (2000 and 2010). In Hungary we cover a period of 30 years (1980–2011), but with only four census observations. For Germany, however, although the time span is 15 years, we have annual data (1993–2007). On the contrary Japan data cover the period 1995–2010 and four census observations. Finally, Luxembourg has a long time series considering all census data from 1821 until 2011 for Luxembourg; however, the time span between two subsequent observations is not constant even in the same country and especially before World War II.

Following on from the above observations, in Sect. 3, we focus our attention on the validity of Gibrat’s law, in order to design an analytical framework that is useful for the analysis on 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). In particular, the adopted model is the following OLS model:

$$ {g}_t^i={\beta}_t{g}_{t-1}^i+{\varepsilon}_t^i $$

(4)

where g _i (t) is the deviation of the logarithm of the population of city i from the mean of the logarithms of the city populations at time t and ε is the error term. β is the parameter to be estimated and provides an estimate of the divergence/convergence of the size distribution towards its mean (Bottazzi et al. 2001, p. 1184). Gibrat’s law holds if β is equal to 1.⁸ As an indicator of the volatility of the growth process, we use two different measures: (i) the variance of growth, σ ² _t , and (ii) the variance ratio, θ _t , between the variance of g at time t, σ(g _t ) and the variance of g at time t-1, σ(g _t-1 ). The reason why we report both the indicators is because Gibrat’s law is a dynamic concept, while Zipf’s law is static. Including both the measures allows us to be more confident in taking into consideration both the aspects of dynamicity and staticity. Moreover, the variance ratio is also adopted for testing mean reversion and stationarity of the series so it is a further control for Gibrat’s law. Formally, it is given by:

$$ {\theta}_t=\frac{\sigma^2\left({g}_t^i\right)}{\sigma^2\left({g}_{t-1}^i\right)} $$

(5)

Gibrat’s law holds if θ _t in Eq. 5 is equal to 1. Tables 4a and 4b shows the results; the column p-value shows the p-value of a t-test with null hypothesis β = 1; accordingly, the significance has to be interpreted in the relation of our H ₀. Two main conclusions arise from here. Firstly, looking at the parameter β, Gibrat’s law does not always hold (over time). Germany and Luxembourg are clear examples of this intermittency: we can see periods where Gibrat’s law holds and others where it does not. Secondly, it seems that the effect of the (non-)validity of the law is lengthy: namely, Gibrat’s law in general holds (or does not hold) continuously for two or three periods. Consequently, the first result of our analysis is that testing Gibrat’s law requires a data set of considerable length or as many possible observations as we can. It is also interesting to note that in general, when Gibrat’s law does not apply, the variance at time t is higher than the variance at the previous times; this is denoted by the parameter θ in Eq. 5 greater than 1. This is, of course, consistent with the idea of Gabaix (1999) which, according to Gibrat’s law, both the mean and variance of the growth rate have to be independent with respect to the size.

Table 4a Model A estimates (Countries: Botswana, Germany, Hungary, Japan and Luxembourg; different years)

In more detail, by observing β in Tables 4a and 4b, we can see that in Botswana and Luxembourg, Gibrat’s law holds quite often (β = 1). In Germany it does not apply more than half of the time and in Hungary and Japan Gibrat’s law never holds.

Table 4b Model A estimates (Country: Luxembourg, different years)

It should be noted, however, that, although this analysis is suitable for an ‘internal’ validity of the law, this does not allow for an international comparison between countries, due to the availability of the data worldwide. Indeed, even in the five case studies shown before, our data present different characteristics that may affect the results: for Germany we have only annual data, and this is not enough for a robust analysis, since a longer time window seems to be more appropriate. Second, Botswana shows very few observations (we have only one test). Finally, the concept of municipality in Japan seems to differ to the other. Indeed, a sample of only 1716 municipalities might cover different economic characteristics, the one for Germany (12,280) resulting in a higher level of aggregation for the former.

4 Gibrat’s Law and Zipf’s Law: A Comparative Study

4.1 Role of the Adopted Parameters

In the previous sections, we have shown that Botswana and Luxembourg seem to obey Gibrat’s law, while this seems not to be the case for Germany, Japan and Hungary. The final step in our analysis is then the investigation of the relationship between Gibrat’s law and the rank-size/Zipf’s law, by means of the rules (a), (b) and (c) (outlined in Sect. 1).

From the operational viewpoint, we investigate the relationship between the q-coefficient in Eq. 3 and the estimated parameters β and θ from Eqs. 4 to 5 and σ ² _t , on the basis of Cordoba’s propositions (a), (b) and (c). In particular, we estimate the q-coefficients in the rank-size rule (3) by means of a modification proposed by Gabaix and Ibragimov (2011), according the following:

$$ \log \left({P}_i\right)= \log (K)-q \log \left({R}_i-0.5\right) $$

(6)

where P _i , K, q and R _i are the same as in Eq. 3.

In Tables 5a and 5b, we report the estimated q-coefficients and the parameters β, θ and σ ² _t , according to Eqs. 4, 5 and 6, respectively, for each of the five countries. Overall, by means of these five parameters, we can experiment with the propositions (a), (b) and (c) in Sect. 1.

Table 5a The Zipf’s and Gibrat’s parameters (Countries: Botswana, Germany, Hungary, Japan and Luxembourg; different years)

Table 5b The Zipf’s and Gibrat’s parameters (Countries: Botswana, Germany, Hungary, Japan and Luxembourg; different years)

Concerning the coefficient q (Eq. 6), it should be noted that we interpret the q-coefficient as a measure of hierarchy of city size distribution. In this sense a positive change in the estimated q-coefficient denotes a situation where larger cities have grown more than smaller ones (in relative terms); thus, an increasing q-coefficient (see Eq. 6) reflects the tendency towards agglomeration economies in the country at hand (see Sect. 1).

For example, an increasing/decreasing q-coefficient – indicating changes in the growth rate between large and small cities – should lead to a generalised Gibrat’s law. It appears that the q-coefficients, together with the β-, θ- and σ ² _t -parameters, offer insights into different aspects of the same growth process: the q-coefficient captures the output of the growth process, while the β-, θ- and σ ² _t -parameters take into account the mean and variance of the growth process.

In the latter context (regarding the role of the mean and variance), it seems worthwhile to test the different dynamics of the large cities vs. the small cities, in order to explore in more detail where a greater volatility shows up. We can then split, for each country, our sample into two halves by defining two subsamples: one for the large cities and the other one for the smaller cities. This rule is as arbitrary as the other rules typically proposed in the literature (this issue has been addressed by Bee et al. 2013; Ioannides and Skouras 2013; Fazio and Modica 2015; the reader might consult these papers for more details). We then estimate the parameters β and θ according to Eqs. 4–5 and σ ² _t for these two subsamples. In this way we can analyse, firstly, whether Gibrat’s law holds separately for large and small cities and, secondly, whether the growth process is more volatile. In this way we will also show how the truncation rule may affect the results on Gibrat’s and Zipf’s law.

In the next subsections, the role of the various parameters q, β, θ and σ ² _t in capturing the relationship ‘rank-size rule vs. Gibrat’s law’ will be illustrated with reference to the empirical analyses in each of the five countries.

4.2 Botswana

Starting with Botswana, we can see that the estimated q-coefficient is greater than 1 for both the years 2001 and 2011, indicating a predominance of larger cities. In particular, in 2011 the estimated q-coefficient is slightly greater than that 1 in 2001, thus showing a tendency – in the last decade – towards a higher economic development. By considering the relationship with Gibrat’s law, we then investigate condition (b) of Sect. 1. Considering the entire sample, we have already shown that Gibrat’s law holds in 2011 with an estimated β-parameter not significantly different from 1 (β = 0.995). However, considering the two subsamples, we find evidence of Gibrat’s law for large cities (β _BIG = 0.979) but not for small ones (β _small = 0.761***).⁹ This indicates that larger cities of the subsample of small cities (i.e. medium-size cities) have an expected growth lower than smaller ones. This is still consistent with proposition (b) of Sect. 1 predicting that the associated growth process requires that smaller cities face a greater volatility of growth than larger cities. For this reason we now analyse the behaviour of the variance.

The variance ratio for the entire sample in Botswana is greater than 1 (θ = 1.078), indicating a greater volatility of the process in 2011. This latter condition is not enough to investigate our statements (b) of Sect. 1, because it only refers to the ‘temporal’ non-stability of the variance, without considering the ‘spatial aspect’, namely, the (non)independence of the variance with respect to the size of the cities.¹⁰ For this reason, we analyse the two subsamples separately, as previously anticipated, and we look both at the variance ratio, θ, and variance of growth, σ ² _t .

The variance ratio, θ, for large cities shows a striking stability (θ _BIG = 1), while, for the small cities, it is slightly greater than 1 (θ _small = 1.011), implying an increasing change in the underlying volatility of the growth process for small cities. Moreover, the variance of growth σ ² _BIG is less than σ ² _small.

Given these facts, we can affirm that at time t (2011), the variance is unchanged for large cities but increases for the small ones, indicating a dependence of variance with respect to size; in particular, smaller cities face a greater volatility than large cities.

In summary, statement (b) (Sect. 1), 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 associated growth process requires that smaller cities face a greater volatility of growth than larger cities’, is satisfied for the whole sample.

4.3 Germany

Germany shows a U-shaped q-coefficient: it decreases until 1999 and then it increases. In fact Germany shows a lower degree of agglomeration between 1993 and 1999, namely, larger cities become less ‘heavy’ in the city system. After 1999, Germany shows again a process of concentration indicated by the increasing q-coefficient. By considering the relationship with Gibrat’s law, we then investigate condition (b) of Sect. 1.

Considering the entire sample, we have 2 years in which Gibrat’s law holds. In particular, in 1999–2000, the estimated β-parameters are not significantly different from 1.

Now if we focus on the period 1993–1999, where a decreasing q-coefficient applies, we can note that β _BIG is significantly lower than 1, whereas β _small is not significantly different from 1 in most cases, indicating a situation in which the larger the city, the lower the expected growth. On the contrary, in the period 2000–2007,¹¹ where an increasing q-coefficient applies, we can notice that β _BIG are often not significantly different from 1, whereas β _small are (most of the time) significantly greater than 1, indicating a situation in which the larger the city, the larger the expected growth (in the small subsample, i.e. medium-size cities). In this situation we can figure out the following growth processes: when q-coefficient is decreasing, we have modifications on the growth process of large cities; in particular, the larger the city, the lower the expected growth. On the other hand, when q is increasing, small cities present a different growth process, namely, the larger the city, the larger the expected growth. By considering the entire sample, the variance ratio, θ, is often close to unity, but slightly lower than 1 until 1999 when it becomes stable (and equal to 1). Again, this latter condition says few things about the independence of the variance from the size.

We then analyse the two subsamples separately, and in particular we analyse those years where Gibrat’s law holds (according to proposition (b) of Sect. 1), i.e. 1999 and 2000.

In these years, it should be noted, firstly, that when Gibrat’s law holds, the variance ratios for large cities, θ _BIG, are less than 1, that is, the variance at time t is lower than that at time t–1. Instead, the variance ratios for small cities, θ _small, are greater than 1. At (any) time t, large cities face a lower volatility, while small cities face a greater (or almost stable) volatility. Second, the variance of growth σ ² _BIG is always less than σ ² _small. Then, both these facts are again consistent with proposition (b) of Sect. 1. Again we can affirm that for those years in which Gibrat’s law holds (β = 1), statement (b) (Sect. 1) is satisfied for the whole sample.

4.4 Hungary

In Sect. 3, we have shown that Gibrat’s law does not hold in Hungary. We have already mentioned the role of the capital (Budapest) in this country, which attracts most of the total population of the country. In percentage terms, more than half of the total population of Hungary lives in the Budapest urban area. Moreover, it faces a migration flow from its rural area to the centre of the city. The evolution of the q-coefficient in this country reflects the tendency to agglomeration in the large cities: indeed the estimated q-coefficient is increasing and greater than 1. In this situation we can check for statement (b) of Sect. 1. Unfortunately Gibrat’s law never holds, since the β-coefficients are always significantly greater than 1. This means that size diverges towards its means, namely, the larger a city, the larger the expected growth. Then, it is straightforward that the variance ratio, θ, increases for both large and small cities. However, it should be noted that the variance of small cities, θ, is always greater than the variance of large cities and that the variance of growth σ ² _BIG is always less than σ ² _small; thus, although we cannot formally show evidence of generalised Gibrat’s law (in particular regarding statement (b)), we again show a greater volatility for small cities when q > 1.

4.5 Japan

In Sect. 3, we have shown that Gibrat’s law does not hold (Table 4a). So the evolution of the q-coefficient in this country reflects the tendency to agglomeration in the large cities and again Gibrat’s law never holds, since the β-coefficients are always significantly greater than 1. This means that size diverges towards its means, namely, the larger a city, the larger the expected growth. However, as we have repeatedly said throughout this chapter, Japan shows a higher level of aggregation on the definition of its municipalities, so that the comparison between countries is difficult. Nevertheless, we suppose that Japan show characteristics of aggregation more pronounced of that for Germany, as represented by the q indicator.

The analysis carried out over these three countries provides an important first conclusion. On the basis of statement (b), we have shown that when q > 1 and Gibrat’s law holds (β = 1), the variance (θ- and σ ²-parameters) is actually greater for small cities. Moreover, when the q-coefficient is greater than 1 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 the q-coefficient is greater than 1, but increasing, small cities present the opposite growth process: namely, the larger the city, the larger the expected growth. Consequently, we find evidence of the generalised Gibrat’s law – as in statement (b) of Sect. 1 – in the countries displaying q > 1, where the independence of the mean with respect to the size is in operation, while the same is not true for the variance.

A reasonable criticism, at this point in the analysis, could be that, generally, small entities (cities, firms and so on) present a greater volatility than larger ones. We can anticipate that we will find the opposite evidence in the case of q < 1.

We now move to the situation where q < 1. By considering the relationship with Gibrat’s law, we investigate condition (c) of Sect. 1 which predicts that the associate growth process requires that smaller cities face a lower volatility of growth than larger cities.

4.6 Luxembourg

Luxembourg shows an estimated q-coefficient lower than 1. It increases until 1930, and after that it is not significantly different from 1. Between 1821 and 1922, we are in condition (c) of Sect. 1. Considering the entire sample, we have already shown that Gibrat’s law holds in the years 1871 and 1880 and in 1922 with estimated β-parameters not significantly different from 1. Moreover, considering the two subsamples, most of the time we find evidence of Gibrat’s law for both large and small cities. At a first glance, it seems that large and small cities face the same underlying growth processes. However, to test statements (c), we need to take into account the behaviour of the variance.

The variance ratio for the entire sample in Luxembourg in the period 1821–1922 is always greater than 1, indicating a greater volatility of the process as time goes by. Indeed, when we split the sample in two halves, the variance ratios for the large cities show values always greater than 1, while, for the small cities, they are always below 1. 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 ² _BIG is always greater than σ ² _small. 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 associated growth process requires that smaller cities face a lower volatility of growth than larger cities’, is satisfied for the whole sample in 1821–1922.

In the period 1930–2011, the q-coefficient is not statistically different from 1. By considering the relationship with Gibrat’s law, we then investigate condition (a) of Sect. 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 β-parameters not significantly different from 1). Considering the two subsamples, 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 behaviours (i.e. θ _BIG = .963 and θ _small = 1.162; σ ² _BIG = 1.93 and σ ² _small = 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 ² _BIG = 0.55 and σ ² _small = 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.

4.7 Synthesis

A synthesis of the above results – summarising the possible empirical link between Gibrat’s and Zipf’s law – is presented in Table 6.

Table 6 Relationship between Zipf/rank-size and generalised Gibrat

The analysis carried out in this section prompts several interesting conclusions. We gave some empirical clues about the presence of a ‘generalised Gibrat’s law’, as theoretically predicted by Cordoba (2003). In particular, we have controlled for statements (a), (b) and (c) of Sect. 1. In more detail, we have evidence that when q > 1 (statement (b)) and Gibrat’s law holds (β = 1), the variance ratio (θ-parameter) and 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 (θ-parameter) and the variance of growth, σ ², are 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 1 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 1.

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 but also to underline some common shortcomings of the empirical analysis on Gibrat’s and Zipf’s law.

For these purposes we empirically explored the link between the rank-size exponent, q, and the necessary conditions for Gibrat’s law (i.e. 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 (Sect. 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 1; in particular, we empirically show what Cordoba (2003) calls a ‘generalised Gibrat’s law’ for different countries with different spatial-economic characteristics: Botswana, Germany, Hungary, Japan and Luxembourg. We found clues 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 seem to support 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 identical, but offer new perspectives on the same multifaceted 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.

However, we also present some common shortcomings that may affect international comparisons between results or even the correctness of the results.

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, e.g., on Mandelbrot’s principles) or the persistent existence of spatial population or socio-economic disparities (based, e.g., 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 and (c) the relationship between recent strong evolutionary trends in the digital world and the development of cities (and systems of cities).