Reconsidering Zipf’s law for regional development: The case of settlements and cities in Croatia

Power law is an observed regularity that occurs when an event’s value or quantity is inversely proportional to the power of that event’s value (Newman 2005). Our attempts to understand natural occurrences form a rationale for the power law’s existence and are not founded in any valid theory (Krugman 1996a). Examples of mathematical regularities that are attributed to power laws include Pareto’s law and Zipf’s law.

The aim of our study is to check whether Zipf’s law holds for Croatian settlements, cities and agglomerations, and what implications Zipf’s law has for the country’s regional development. Our study looks at the differences between a city proper and an urban agglomeration by examining Zipf’s law as applied to Croatian cities and settlements. It explains the Republic of Croatia’s regional development differences by analysing cities and settlements, and through the comparison of the city proper and the urban agglomeration adherence to a rank-size rule.

There are several reasons for our study. First, the migration that occurred during and after the Croatian War of Independence caused differences in the sizes of cities and, consequently, differences in regional development (Dimou & Schaffar 2009). Second, the changing nature of cities, and their mutual competition and interaction, demand a continuous analysis that will contribute to conclusions about the country’s regional development (Jiang, Yin & Liu 2015). This notion is especially relevant within the context of the European Union’s regional development policies, where this study builds on Cieślik and Teresiński’s (2016) study to provide an empirical examination of Zipf’s law for new members of the European Union. Third, studies such as Šolak and Dobric’s (2010), and Dimou and Schaffar’s (2009), which examined the regularity of Zipf’s law in the Balkans, did not include cities and settlements in their analysis, nor did they compare the city proper with urban agglomeration. There exists a large number of unanalysed cities that could exhibit power law regularities and explain differences in the regional development. Therefore, this study is a necessary prerequisite to our understanding of Croatia’s regional economic development.

Zipf’s law is a statistical, mathematical regularity that shows that the probability or the frequency of an observed event is inversely proportional to its rank (Zipf 1949). There are numerous events in which cumulative distribution patterns follow Zipf’s law (Gan, Li & Song 2006): word frequency (Zipf 1949), the citation of scientific papers, web hits, copies of books sold, telephone calls, the magnitude of earthquakes, the diameter of moon craters, the intensity of solar flares, the intensity of wars, and the frequency of family names (Newman 2005). In economics, Zipf’s law has been observed in world income distributions (Sinclair 2001), the number of employees in firms (Knudsen 2001), and the gravity model of international trade (Jošić & Nikić 2013). Nonetheless, its most prominent use is in the urban economic studies of the rank-size distributions of cities (Auerbach 1913; Gabaix 1999b; Jiang & Jia 2011).

Zipf’s law states that the probability that a city has a size greater than S decreases with the size of the city, that is, 1/S (Gabaix 1999b). “The size S(i) of a city of rank i will follow a power law’s rank-size rule which states that the size of the city of rank i varies as 1/i, and the ratio of the second largest city to the largest city should be ½,” etc. (Gabaix 1999b, p 752–753). Zipf’s law aims to answer two essential questions: (1) Does the law hold for a particular country or a region? and (2) Why has this law emerged? (Jiang & Jia 2011).

F. Auerbach (1913) was the first to suggest the formula for the size distribution of cities based on the Pareto distribution:

R=CS−<!-- - -->α<!-- a -->$$\begin{array}{} \displaystyle \rm R = CS^{-\alpha} \end{array}$$(1)

where R is the rank of the city (cities are ranked from largest to smallest, rank=1 is the city with the highest population), C is a constant, S is the population of the city, and α is the Pareto exponent or the Zipf value. Another method of writing this equation is to use natural logarithms due to the right-skewed city size distribution, which provides a better fit for samples encompassing small cities (Gabaix 1999b; Jiang, Yin & Liu 2015):

ln(R)=ln(C)−<!-- - -->α<!-- a -->ln(S)$$\begin{array}{} \displaystyle \rm ln ({\it R})= {\it ln}\,({\it C})-\alpha ln\,({\it S}) \end{array}$$(2)

According to Zipf (1949), the Pareto exponent, α, is employed as a measure of population concentration among cities of different sizes. The rank-size rule is validated when α=1 because the values are then centred around an average value (Jiang & Jia 2011). Zipf’s law is a specific example of rank size distribution. Smaller values of the Pareto coefficient imply that the urban system is highly concentrated. On the other hand, higher values of the Pareto exponent imply more equality between cities and less hierarchy. The city rank–size relationship appears linear in the logarithms, yielding a very high R2 (close to 1) (Gan, Li & Song 2006). Gabaix and Ibragimov (2011) showed that the previous standard approach for estimating the Pareto exponent provided biased estimations for small samples. The authors derived a simple but effective solution to correct this bias by using the variable R –1/2 instead of R (Deliktas, Önder & Karadag 2013). This correction is optimal since it minimizes the small-sample bias in the OLS estimator. The corrected formula is presented in equation (3):

ln(R−<!-- - -->1/2)=ln(C)−<!-- - -->α<!-- a -->′<!-- ' -->ln(S)$$\begin{array}{} \displaystyle \rm ln ({\it R-{\rm 1/2}})= {\it ln}\,({\it C})-\alpha {^\prime} ln\,({\it S}) \end{array}$$(3)

In order to test Zipf’s law on settlements and cities in Croatia, two hypothesis were tested:

H1

The acceptance of Zipf’s law depends on the settlement size in Croatia.

For the purpose of testing the aforementioned hypothesis, the sizes of settlements and cities were measured based on the administrative definition of the city, which encompassed urban, historical, natural, economic, and social definitions. Data were taken and aggregated from the Croatian Bureau of Statistics Census of Population, Households and Dwellings 2011. Croatian Law on local and regional self-government, Article 5, states that “The town is a unit of local self-government where the seat of the county is located, as well as any other place with more than 10,000 inhabitants.” The population of the city proper for Croatian cities is defined as cities containing a local self-government unit, while the urban agglomeration population adds the population of the surrounding settlements to the population of the city proper. Both the city proper and the urban agglomerations were ranked according to their size, that is, by population (Jiang and Jia 2011).

The hypotheses were tested using methods of inferential statistics. Zipf’s law was tested using a simple regression model based on Equation 2. The sample size comprised of 6,756 settlements with 127 towns and the city of Zagreb. There was no issue with biased estimation. Rank is a sequential variable, and the size of the city in terms of population is a quantitative variable. The dependent variable in the analysis is the variable RANK, which takes the form of a natural logarithm. Regressions were made with respect to the different sizes of the independent variable SIZE and took the form of a natural logarithm.

Statistics that are available after 1991 show that the number of inhabitants in Croatia has been in constant decline (Croatian Bureau of Statistics: Census of Population, Households and Dwellings 2011). This decline has been caused by the Croatian War of Independence, the low rate of both natural and economic growth, emigration, an aging population, etc. (Croatian Bureau of Statistics: Census of Population, Households and Dwellings 2011).

According to the 2011 census there were 6,756 settlements with 127 cities, as well as the capital city of Zagreb, which holds special urban status in the Republic of Croatia. Table 1 presents the structure and population size of Croatia’s settlements. Settlements are divided up according to population size. There is a total of 6,756 settlements in Croatia. Of these 150 settlements have no inhabitants; hence, there are 6,606 settlements with at least one inhabitant in Croatia. The hypotheses were tested on these 6,606 settlements with at least one inhabitant. The majority of the settlements were populated by up to 1,000 inhabitants (92.17%); however, the proportion of the population in the settlements with up to 1,000 inhabitants out of the total population was only 28.71%. In comparison, the capital city of Zagreb comprises 16.06% of the total Croatian population. Additionally, population distribution of Croatia’s largest cities does not follow the usual Zipf’s law, which states that the second largest city in a country will be approximately ½ the population of the first city, which is the rationale for testing Zipf’s law on the sample of Croatian settlements.

Table 1

Croatian settlement structure and population

Figure 1 is a graphical representation of Zipf’s law for settlements in the Republic of Croatia based on a sample of 6,606 settlements with at least one inhabitant. Settlements without populations were not included in the analysis.

Figure 1

Figure 1 shows the concave shape of the curve, especially with respect to the smaller sized settlement. Table 2 depicts the results of OLS regression testing on Zipf’s law for settlements in Croatia.

Table 2

OLS for testing Zipf’s law as applied to settlements in Croatia

Table 2 depicts R squared (R2=0.7887), which indicates that a good proportion of the variance in the dependent variable is explained by the independent variable. The Pareto coefficient is -0.5561 and is statistically significant at a 1 percent significance level. The value of the Pareto coefficient indicates a dispersion of population size relative to the settlements’ size. Therefore, Zipf’s law does not hold true for all settlements in the Republic of Croatia.

As the analysis showed that Zipf’s law did not hold true for settlements of all sizes in the Republic of Croatia, we decided to test this by consecutively omitting groups of settlements with small population sizes from the analysis. Figure 2 shows the results of the regression analyses in which settlements with less than 100, 500, 1000, and 2000 inhabitants were omitted consecutively.

Figure 2

Figure 2 displays Zipf’s law for settlements in the Republic of Croatia that contain populations larger than (a) 100, (b) 500, (c) 1000, and (d) 2000 inhabitants based on samples of (a) 3,953, (b) 1,187, (c) 529, and (d) 221 observations, respectively. If the settlement’s size was greater than 100 inhabitants, Zipf’s law holds perfectly. The Pareto coefficient is -0.9941 and is statistically significant at a level of 1 percent, indicating a perfect match with Zipf’s law, or rank-size distribution. R squared is 0.9843, which indicates that the independent variable model explains the model well.

In cases in which settlements have populations larger than (b) 500, (c) 1000, or (d) 2000 inhabitants (Ln (Size >500), Ln (Size >1000), Ln (Size >2000)), Zipf’s law also holds true. The values of the Pareto coefficients are (b) –1.1401, (c) -1.1013 and (d) -1.0263 respectively. The independent variable is significant for all cases at a significance level of 1 percent, while the value of R squared is very high (above 0.99). The values of the Pareto coefficient are slightly higher than 1, which indicates that the concentration of settlements is high and uniformly distributed.

Hypothesis 1 was tested by calculating the Pareto coefficient in relation to different sizes of settlement based on 20 truncation points (Figure 3). Figure 3 shows that the values of the Pareto coefficient equal 1 (absolute term) for settlements whose size is greater than 100 inhabitants and 2000 inhabitants. For the area in between these two sizes the value of the Pareto coefficient is higher than 1 with a maximum value for settlement size above 500 inhabitants. Therefore, Hypothesis 1 has been confirmed.

Figure 3

Regarding settlements in Croatia, Zipf’s law holds true for the majority of settlement sizes in Croatia. Rank-size distribution does not hold true for the lower and upper-tail of the settlement size distribution.

The second goal of the analysis was to test Zipf’s law for the city proper and urban agglomerations in Croatia – presented in Table 3 and Figure 4.

Table 3

OLS for testing Zipf’s law as applied to the city proper and urban agglomeration in Croatia

Figure 4

OLS regression for the definition of the city proper in the Republic of Croatia showed that Zipf’s law cannot be confirmed. The Pareto coefficient value is -0.7397, which depicts a highly concentrated urban system or divergent city sizes. Out of the 127 Croatian cities and urban agglomerations in Croatia there are six cities (city proper populations) with less than 1000 inhabitants (Obrovac, Vrlika, Skradin, Klanjec, Rab and Cabar); this points to the previously described exceptions in the definition of city size that affected the hypothesis’ confirmation. On the other hand, Zipf’s law for urban agglomeration population in Croatia is valid. The value of the Pareto coefficient was -0.9501 and is statistically significant. It can be concluded that Zipf’s law is valid for urban agglomerations in Croatia but cannot be confirmed for the city proper sample.

The aim of this paper was to test whether the Zipf’s law holds true for Croatian settlements, cities and agglomerations, and what implications this has for the country’s regional development. It tested Zipf’s law on data for settlements and cities in Croatia using the 2011 Census of Population Survey. The results of the analysis have shown that Zipf’s law when applied to settlements in Croatia holds true for the majority of settlement sizes. Although universal, it is limited to the scope of observation (Jiang, Yin & Liu 2015).

The rank-size distribution does not hold true for extremely small or extremely large sizes of settlement, which is in accordance with previous studies (Gabaix 1999b; Newman 2005). Although Eeckhout (2004) argues that the entire distribution of city sizes is lognormal, studies by authors that affirm that the distribution of city sizes have a Pareto tail and a lognormal body (Bee, Riccaboni & Schiavo 2011, 2013; Clauset, Shalizi & Newman 2009; loannides and Skouras, 2013; Luckstead and Devadoss, 2017; Malavergne, Pisarenko & Sornette 2011) are consistent with our results. Our study follows the study of Fazio and Modica (2015), which examined alternative city size distributions depending on differing truncation points and proved the validity of Eeckhout’s (2009) study, stating that the choice of Pareto or lognormal distribution depends on the truncation point, wherein the upper tail is longer than assumed.

Moreover, our study is consistent with Veneri’s (2016) findings in which the actual size of the city is more accurately identified through its economic functions, which in our study coincide with the definition of urban agglomeration. Urban agglomeration comprises the population of the city proper and includes economic functional areas that are comprised of the commuting flows of urban populations. Our analysis shows that the Croatian system of cities, especially cities based on economic functions, comprise a hierarchical structure. This finding is consistent with Modica’s (2017) observation of the hierarchical structure of systems of cities in the European Union’s Member States who are experiencing a transition from socialism to capitalism.

Krugman (1996b) found that the size and growth of the few first ranked cities can be slowed down through the opening up of trade and reduced government intervention. However, international trade is not to be assumed to be the key determinant of the regional convergence process in Croatia due to work by Modica (2017) who found the differences in the concentration and dispersion processes of cities in the European Union Member States; namely, the effect of the Schengen treaty induced a concentration of population in larger cities, while the introduction of a single currency has enabled the dispersion processes.

Regional growth in Croatia is primarily determined using the quality of human capital, investments in fixed assets, and structural features of individual Croatian counties (Mikulić & Nagyszombaty 2015). Hence, recommendations concerning regional convergence and regional development include policies that concern education (Deliktas, Önder & Karadag 2013), as demographic changes have not been found to influence rank-size distribution in the long run (Rosen & Resnick 1980; Dimou & Schaffar 2009). The stress should be put on smaller cities that have a tendency to grow faster; therefore, there is room for policy implementation in both very large and very small cities.

Cieślik and Teresiński’s (2016) study of rank-size distribution of Polish cities used a smaller sample (306 and 908 observations) than our study and did not find evidence of a rank-size distribution for samples of different sizes. Our study uses a larger sample (6,606 settlements), that is, all cities and settlements in the Republic of Croatia, and tests illustrate the validity of Zipf’s law based on samples of settlements and cities of different sizes (6,606; 953; 1,187; 529; and 221 observations). Moreover, by examining the differences between the city proper and urban agglomerations, our study found that when the city proper and the urban agglomerations for 127 Croatian cities were examined, the Zipf’s law holds true only in cases of urban agglomerations. Policywise, this implies that populations have started to move out of cities towards secondary urban centres (Dimou & Schaffar 2009). Possible consequences for regional development based on our study include a lack of institutional and demographic policy effects in accordance with the Yule distribution (Rosen & Resnick 1980; Dimou & Schaffar 2009; Newman 2005). he rich-get-richer mechanism could lead to limitations in urban agglomeration development and policies should target secondary urban centres to achieve a more even regional development. As mentioned above, policies aimed at raising regional productivities of secondary urban centres through education, structural features of each county, and investment in fixed assets, should be the primary focus for a more even urban development.