New Evidence on Gibrat’s Law for Cities

Abstract

The aim of this work is to test empirically the validity of Gibrat’s law on the growth of cities, using data on the complete distribution of cities (without size restrictions) from three countries (the US, Spain and Italy) for the entire 20th century. In order to achieve this, different techniques are used. First, panel data unit root tests tend to confirm the validity of Gibrat’s law in the upper-tail distribution. Secondly, when the entire distribution is considered using non-parametric methods, it is found that Gibrat’s law does not hold exactly in the long term (in general, size affects the variance of the growth process but not its mean). Moreover, the log-normal distribution works well as a description of city size distributions across the whole century when no truncation point is considered.

1. Introduction

The relationship between the growth rate of a quantifiable phenomenon and its initial size is a question with a long history in statistics: do larger entities grow more quickly or more slowly? However, perhaps no relationship exists and the rate is independent of size. A fundamental contribution to this debate is that of Gibrat (1931), who observed that the distribution of size (measured by sales or the number of employees) of firms could be approximated well with a log-normal, and that the explanation lies in the tendency of the growth processes of firms to be multiplicative and independent of their sizes. This proposition became known as Gibrat’s law, and it prompted a deluge of work exploring the validity of this law in relation to the distribution of firms (see the survey by Santarelli et al., 2006). Gibrat’s law states that no regular behaviour of any kind can be deduced from the relationship between growth rate and initial size. The fulfilment of this empirical proposition also has consequences for the distribution of the variable—the law of proportionate effect implies that the logarithms of the variable will be distributed following the normal distribution (Gibrat, 1931).

In the field of urban economics, Gibrat’s law, especially since the 1990s, has given rise to numerous empirical studies testing its validity for city size distributions; these have arrived at a majority consensus, although not absolute at all, that it explains the growth of cities relatively well and tends to hold in the long term. This has yielded theoretical work explaining the fulfilment of Gibrat’s law in the context of external urban local effects and productive shocks, associating it directly with an equilibrium situation. These theoretical models include those of Gabaix (1999), Duranton (2007) and Córdoba (2008).

Returning to the empirical side, there is an apparent contradiction in these studies, because they usually accept the fulfilment of Gibrat’s law but at the same time claim that the distribution followed by city size (at least the upper-tail) is a Pareto distribution, which is very different from the log-normal. Eeckhout (2004) was able to reconcile both results by demonstrating that, as Parr and Suzuki (1973) claimed in a pioneering work, imposing size restrictions on cities by taking only the upper tail biases the analysis. Thus, if all cities are taken into account it can be found that the true distribution is log-normal and that the growth of these cities is independent of size. However, to date, the studies by Eeckhout (2004) and Giesen et al. (2010) are the only ones to consider the entire city size distribution. Nevertheless, these are short-term analyses,¹ and the phenomenon under study (Gibrat’s law) is a long-term result (Gabaix and Ioannides, 2004). In this paper, ‘long term’ means a lengthy period comprising several decades (a whole century), while ‘short term’ indicates cross-section data considered in an isolated way.

The aim of this work is to test empirically the validity of Gibrat’s law on the growth of cities, using data on the complete distribution of cities without size restrictions in three countries (the US, Spain and Italy) for the entire 20th century. Our results qualify this study as the most comprehensive empirical examination of Gibrat’s law to date, considering the number of cities (untruncated settlement size data from three countries), the long time-span (a whole century) and the different methodologies that we apply (parametric and non-parametric techniques).

Furthermore, the three countries selected give us information about two different urban behaviours. The US is an extremely interesting country in which to analyse the evolution of urban structure because it is a relatively young country whose inhabitants are characterised by high mobility. By contrast, the European countries have a much older urban structure and their inhabitants present greater resistance to movement; specifically, Cheshire and Magrini (2006) estimated mobility in the US to be 15 times higher than that in Europe. Spain and Italy have a consolidated urban structure and new cities rarely appear (the foundation of many European cities dates back to the Middle Ages), so urban growth is produced by population increases in existing cities. In the US, however, urban growth has a double dimension: as well as increases in city size, the number of cities also increases, with potentially different effects on city size distribution.

The following section offers a brief overview of the literature on Gibrat’s law for cities and the results obtained. Section 3 introduces the databases. From our results, we deduce that panel data unit root tests tend to confirm the validity of Gibrat’s law in the upper-tail distribution (section 4.1). In section 4.2, we consider the entire distribution and, using non-parametric methods, find that Gibrat’s law does not hold exactly in the long term as, in general, size affects the variance of the growth process but not its mean. The validity of the law in the short term (by decade) is even weaker. In section 5, we test whether the log-normal distribution is a good description of city size distributions over the entire century. The work ends with our conclusions.

2. Gibrat’s Law for Cities: An Overview of the Literature

In the 1990s, numerous studies began to appear that empirically tested the validity of Gibrat’s law. Table 1 shows the classification of all the studies on urban economics that we know of to date. The countries considered, statistical and econometric techniques used and sample sizes are heterogeneous, although Table 1 shows that most of the recent empirical works have chosen between one of two techniques to test for Gibrat’s law: panel unit root tests to consider a small sample size, and non-parametric kernel regressions when the sample size is large; we apply these two methodologies in Section 4. While the results are rather mixed, the acceptance of the law is the predominant outcome, albeit by a slight margin.

Table 1.

Empirical studies on Gibrat’s law: a survey

Study	Country	Period	Truncation point	Sample size	GL	EcIss
Eaton and Eckstein (1997)	France and Japan	1876–1990 (F) 1925–1985 (J)	Cities >50 000 inhabitants (F) Cities > 250 000 inhabitants (J)	39 (F), 40 (J)	A	par (gr reg); non-par (tr mat, lz)
Davis and Weinstein (2002)	Japan	1925–1965	Cities >30 000 inhabitants	303	A	par (purt)
Bosker et al. (2008)	West Germany	1925–1999	Cities >50 000 inhabitants	62	M	par (purt); non-par (ker)
Giesen and Südekum (2011)	West Germany	1975–1997	Cities >100 000 inhabitants	71	A	non-par (ker)
Brakman et al. (2004)	Germany	1946–1963	Cities >50 000 inhabitants	103	A	par (purt)
Clark and Stabler (1991)	Canada	1975–1984	Seven most populous cities	7	A	par (purt)
Resende (2004)	Brazil	1980–2000	Cities >1 000 inhabitants	497	A	par (purt)
Eeckhout (2004)	US	1990–2000	All cities	19 361	A	par (gr reg); non-par (ker)
Ioannides and Overman (2003)	US	1900–1990	All MSAs	112 (1900) to 334 (1990)	A	non-par (ker)
Gabaix and Ioannides (2004)	US	1900–1990	All MSAs	112 (1900) to 334 (1990)	A	non-par (ker)
Black and Henderson (2003)	US	1900–1990	All MSAs	194 (1900) to 282 (1990)	R	par (purt)
González-Val (2010)	US	1900–2000	All cities	10 596 to 19 296	A	non-par (ker)
Michaels et al. (2012)	US	1880–2000	Minor Civil Divisions and counties	10 864	M	non-par (dsf)
Guérin-Pace (1995)	France	1836–1990	Cities >2 000 inhabitants	675 (1836) to 1782 (1990)	R	par (corr)
Soo (2007)	Malaysia	1957–2000	Urban areas >10 000 inhabitants	44 (1957) to 171 (2000)	R	par (purt)
Petrakos et al. (2000)	Greece	1981–1991	Urban centres >5 000 inhabitants	150	R	par (gr reg)
Henderson and Wang (2007)	World	1960–2000	Metro areas >100 000 inhabitants	1220 (1960) to 1644 (2000)	R	par (purt)
Anderson and Ge (2005)	China	1961–1999	Cities > 100 000 inhabitants	149	M	par (rank reg); non-par (tr mat)

Key: Gibrat’s Law: GL	EcIss: Econometric Issues		gr reg: growth regressions	corr: coefficient of correlation (Pearson)
A: Accepted	par: parametric methods		ker: kernel regressions	lz: Lorenz curves
R: Rejected	non par: non-parametric methods		rank reg: rank regressions	dsf: discrete-step function
M: Mixed Results	purt: panel unit root tests		tr mat: transition matrices

Both Eaton and Eckstein (1997) and Davis and Weinstein (2002) accept its fulfilment for Japanese cities, although they use different sample sections (40 and 303 cities respectively) and time horizons. Brakman et al. (2004) come to the same conclusion when analysing the impact of bombardment on Germany during the Second World War, concluding that, for the sample of 103 cities examined, bombing had a significant but temporary impact on post-war city growth. Nevertheless, nearly the same authors in Bosker et al. (2008) obtain a mixed result with a sample of 62 cities in West Germany: correcting for the impact of the Second World War, Gibrat’s law is found to hold only for about 25 per cent of the sample. Giesen and Südekum’s (2011) recent work cannot formally reject Gibrat’s law for the 71 largest German cities in 1997.

Moreover, both Clark and Stabler (1991) and Resende (2004) also accept the hypothesis of proportionate urban growth for Canada and Brazil respectively. The sample size used by Clark and Stabler (1991) is tiny (the seven most populous Canadian cities), although the main contribution of their work is to propose the use of data panel methodology and unit root tests in the analysis of urban growth. Resende (2004) applies the same methodology to his sample of 497 Brazilian cities. However, Henderson and Wang (2007) strongly reject Gibrat’s law and a unit root process in their world-wide dataset on all metro areas over 100 000 inhabitants from 1960 to 2000.

In the case of the US, there are also several studies that statistically accept the fulfilment of Gibrat’s law, whether at the level of cities—Eeckhout (2004) is the first to use the entire sample without size restrictions and González-Val (2010) generalises this analysis for the entire 20th century—or with metropolitan statistical areas (MSAs) (Ioannides and Overman, 2003, whose results reproduce Gabaix and Ioannides, 2004). Also for the US, however, Black and Henderson (2003) reject Gibrat’s law for any sample section, although their database of MSAs differs from that used by Ioannides and Overman (2003). Michaels et al. (2012) use data from Minor Civil Divisions and counties to track the evolution of populations across both rural and urban areas in the US from 1880 to 2000, finding that Gibrat’s law is a reasonable approximation for population growth only for the largest units.

Other works exist that reject the fulfilment of Gibrat’s law. Thus, Guérin-Pace (1995) finds that in France, using a wide sample of cities with over 2000 inhabitants during the period 1836–1990, there seems to be a fairly strong correlation between city size and growth rate, a correlation that is accentuated when the logarithm of the population is considered. This result opposes that obtained by Eaton and Eckstein (1997) when considering only the 39 most populated French cities. Petrakos et al. (2000) and Soo (2007) also reject the fulfilment of Gibrat’s law in Greece and Malaysia respectively. In the case of China, Anderson and Ge (2005) obtain a mixed result with a sample of 149 cities of more than 100 000 inhabitants: Gibrat’s law seems to describe well the situation prior to the Economic Reform and ‘One Child’ policy period, but later Kalecki’s reformulation seems to be more appropriate.

What we wish to emphasise is that, with the exception of Eeckhout (2004), Giesen et al. (2010) and González-Val (2010), none of these studies considers the entire distribution of cities, because all of them impose a truncation point, whether explicitly by taking cities above a minimum population threshold or implicitly by working with MSAs.² This is usually for practical reasons concerning data availability. Consequently, most studies focus on analysing the most populous cities, the upper-tail distribution. However, any analysis using this kind of sample will have a local character because the behaviour of the entire distribution cannot be extrapolated from that of large cities (for an analysis of the effect of sample size on Gilbrat’s law, see Gonzzález-Val et al., 2013).

3. Databases

We use un-truncated settlement size data from three countries: the US, Spain and Italy.³ Our database includes decennial census data for each decade of the 20th century.⁴ Table 2 shows the number of cities for each decade and the descriptive statistics.

Table 2.

Number of cities and descriptive statistics by year and country

Year	Cities	Mean	Standard deviation	Minimum	Maximum	Country population (CP)	Percentage of CP in our sample
US
1900	10 596	3 376.04	42 323.90	7	3 437 202	76 212 168	46.9
1910	14 135	3 560.92	49 351.24	4	4 766 883	92 228 496	54.6
1920	15 481	4 014.81	56 781.65	3	5 620 048	106 021 537	58.6
1930	16 475	4 642.02	67 853.65	1	6 930 446	123 202 624	62.1
1940	16 729	4 975.67	71 299.37	1	7 454 995	132 164 569	63.0
1950	17 113	5 613.42	76 064.40	1	7 891 957	151 325 798	63.5
1960	18 051	6 408.75	74 737.62	1	7 781 984	179 323 175	64.5
1970	18 488	7 094.29	75 319.59	3	7 894 862	203 302 031	64.5
1980	18 923	7 395.64	69 167.91	2	7 071 639	226 542 199	61.8
1990	19 120	7 977.63	71 873.91	2	7 322 564	248 709 873	61.3
2000	19 296	8 968.44	78 014.75	1	8 008 278	281 421 906	61.5
Spain
1900	7 800	2 282.40	10 177.75	78	539 835	18 616 630	95.6
1910	7 806	2 452.01	11 217.02	92	599 807	19 990 669	95.7
1920	7 812	2 621.92	13 501.02	82	750 896	21 388 551	95.8
1930	7 875	2 892.18	17 513.90	79	1 005 565	23 677 095	96.2
1940	7 896	3 180.65	20 099.96	11	1 088 647	26 014 278	96.5
1950	7 901	3 479.86	26 033.29	64	1 618 435	28 117 873	97.8
1960	7 910	3 801.71	33 652.11	51	2 259 931	30 582 936	98.3
1970	7 956	4 240.98	43 971.93	10	3 146 071	33 956 047	99.4
1981	8 034	4 701.40	45 995.35	5	3 188 297	37 742 561	100.0
1991	8 077	4 882.27	45 219.85	2	3 084 673	39 433 942	100.0
2001	8 077	5 039.37	43 079.46	7	2 938 723	40 847 371	99.6
Italy
1901	7 711	4 274.84	14 424.61	56	621 213	32 963 000	100.0
1911	7 711	4 648.11	17 392.98	58	751 211	35 842 000	100.0
1921	8 100	4 863.80	20 031.61	58	859 629	39 397 000	100.0
1931	8 100	5 067.10	22 559.85	93	960 660	41 043 000	100.0
1936	8 100	5 234.38	25 274.48	116	1 150 338	42 398 000	100.0
1951	8 100	5 866.12	31 137.52	74	1 651 393	47 516 000	100.0
1961	8 100	6 249.82	39 130.55	90	2 187 682	50 624 000	100.0
1971	8 100	6 683.52	45 581.66	51	2 781 385	54 137 000	100.0
1981	8 100	6 982.33	45 329.33	32	2 839 638	56 557 000	100.0
1991	8 100	7 009.63	42 450.26	31	2 775 250	56 778 000	100.0
2001	8 100	7 021.20	39 325.47	33	2 546 804	56 996 000	99.8

The data for the US are the same as those used by González-Val (2010). Our base, created from the original documents of the annual census published by the US Census Bureau,⁵ consists of the available data of all incorporated places without any size restriction for each decade of the 20th century. The US Census Bureau uses the generic term ‘incorporated place’ to refer to a governmental unit incorporated under state law as a city, town, borough or village, which has legally established limits, powers and functions. Incorporated places in Alaska, Hawaii and Puerto Rico are excluded because of data limitations.

The alternative would be to use data from metropolitan areas. Both units have advantages. As Glaeser and Shapiro (2003) indicate, metro areas represent urban agglomerations, covering huge areas that are meant to capture labour markets. Metropolitan areas are attractive because they are more natural economic units. Legal cities are political units that usually lie within metropolitan areas and their boundaries make no economic sense, although some factors, such as human capital spillovers, are thought to operate at a very local level. Furthermore, Eeckhout (2004) argues that there are statistical reasons that justify the use of incorporated places (untruncated data) rather than metro areas.

The percentage of the total US population that our sample of incorporated places represents can appear low compared with other studies using MSAs. The population of incorporated places increases from representing less than half of the total population of the US in 1900 (46.9 per cent) to 61.5 per cent in 2000, while the number of cities increases by 82.11 per cent from 10 596 in 1900 to 19 296 in 2000. The population excluded from the sample is what the US Census Bureau calls ‘population not in place’. Incorporated places do not cover the whole territory of the US and some territories are excluded from any recognised place. For example, more than 74 million people (26.64 per cent of the total US population) lived in a territory that, at least officially, was not in a place in 2000.⁶ Most of these people (61.58 per cent in 2000) are rural population.

Although the people living outside incorporated places are excluded from our sample, they are included in some MSAs because these are multicounty units and this population is counted as inhabitants of the counties. MSAs cover huge geographical areas and include a large proportion of the population living in rural areas. This explains why the percentage of the total population represented by MSAs is higher than our sample of incorporated places. However, although the sample of incorporated places covers a lower percentage of the total US population, the population of incorporated places is almost entirely urban (94.18 per cent in 2000) compared with 88.35 per cent of the urban population in the MSAs.

For Spain and Italy, the geographical unit of reference is the municipality.⁷ The data come from the official statistical information services. In Italy, this is the Istituto Nazionale di Statistica and for Spain our sources are the censuses by the Instituto Nacional de Estadística, (INE).⁸

Municipalities are the smallest spatial units (local governments), thus they are the administratively defined ‘legal’ cities. The main difference between these municipalities and the incorporated places is that municipalities are the lowest spatial sub-division in Spain and Italy, so they represent the whole territory of the country. Municipalities comprise the total land area and therefore all the population too (see Table 2). However, in the US, a large amount of land area and population is not included in any place, as noted previously.⁹

4. Testing for Gibrat’s Law

4.1. Parametric Analysis: Panel Unit Root Testing

Clark and Stabler (1991) suggest that testing for Gibrat’s law is equivalent to testing for the presence of a unit root. This idea is also emphasised by Gabaix and Ioannides (2004). Therefore, most recent studies now apply unit root tests (see Table 1).

Some authors (Black and Henderson, 2003; Henderson and Wang, 2007; Soo, 2007) test the presence of a unit root by proposing a growth equation, which they estimate using panel data. Nevertheless, as pointed out by Gabaix and Ioannides (2004) and Bosker et al. (2008), this methodology presents some drawbacks. First, the periodicity of our data is by decades and we have only 11 temporal observations (decade-by-decade city sizes over a total period of 100 years), when the ideal would be to have at least annual data. Secondly, the presence of cross-sectional dependence across the cities in the panel can give rise to estimations that are not very robust. The literature has established that panel unit root and stationarity tests that do not explicitly allow for this feature among individuals present size distortions (Banerjee et al., 2005).

For this, we use one of the tests especially created to deal with this question: Pesaran’s (2007) test for unit roots in heterogeneous panels with cross-section dependence is calculated based on the CADF statistic (cross-sectional ADF statistic, see later). To eliminate cross-dependence, the standard Dickey–Fuller (or augmented Dickey–Fuller, ADF) regressions are augmented with the cross-section averages of lagged levels and first differences of the individual series, such that the influence of the unobservable common factor is asymptotically filtered.

The test of the unit root hypothesis is based on the t-ratio of the OLS estimate of b_i in the following cross-sectional augmented DF (CADF) regression

Δ y_{i t} = a_{i} + b_{i} y_{i, t - 1} + c_{i} {\bar{y}}_{t - 1} + d_{i} Δ {\bar{y}}_{t} + e_{i t}

(1)

where, a_i is the individual city-specific average growth rate; and ${\bar{y}}_{t}$ is the cross-section mean of y_it, ${\bar{y}}_{t} = N^{- 1} \sum_{j = 1}^{N} y_{j t}$ .

We will test for the presence of a unit root in the natural logarithm of city relative size $(y_{i t} = \ln s_{i t})$ . City relative size (s_it) is defined as

s_{i t} = \frac{S_{i t}}{{\bar{S}}_{t}} = \frac{S_{i t}}{\frac{1}{N} \sum_{i = 1}^{N} S_{i t}}

From a long-term temporal perspective of steady-state distributions, it is necessary to use a relative measure of size (Gabaix and Ioannides, 2004). The null hypothesis assumes that all series are non-stationary, and Pesaran’s CADF is consistent under the alternative that only a fraction of the series is stationary.

However, the problem with Pesaran’s test is that it is not designed to deal with such large panels, especially when so few temporal observations are available $(N \to \infty, T = 11)$ . For this reason, we must limit our analysis to the largest cities (although the next section offers a non-parametric analysis of the entire sample).¹⁰

As noted previously in the literature review in section 2, traditionally most studies focus only on the upper-tail distribution, because of the data availability. Recent papers (Eeckhout, 2009; Levy, 2009) discuss that the behaviour of the upper-tail distribution (log-normal or Pareto) can differ from that of the rest of the sample. Moreover, as Levy (2009) argues, while the upper tail of the US city size distribution in 2000 includes only 6 per cent of the cities, it accounts for almost 23 per cent of the total US population. Therefore, a separate analysis for the largest cities is important.

Table 3 shows the results of Pesaran’s test, both the value of the test statistic and the corresponding p-value, applied to the upper-tail distribution until the 500 largest cities in the initial period have been considered for all the decades. All the statistics are based on univariate AR(1) specifications including constant and trend. The null hypothesis of a unit root is not rejected in the US or Italy for any of the sample sizes considered, which provides evidence in favour of the long-term validity of Gibrat’s law. Spain’s case is different since, when the sample size is greater than the 200 largest cities, the unit root is rejected. The analysis in the next section reveals that the reason for this rejection is a positive relationship between the relative size and the growth rate for the largest cities.

Table 3.

Panel unit root tests, Pesaran’s CADF statistic

Cities (N)	US	Spain	Italy
50	–0.488 (0.313)	–0.915 (0.180)	4.995 (0.999)
100	0.753 (0.774)	0.050 (0.520)	5.983 (0.999)
200	1.618 (0.947)	–2.866 (0.002)	–1.097 (0.136)
500	1.034 (0.849)	–12.132 (0.000)	5.832 (0.999)

Notes: test-statistic (p-value). Pesaran’s CADF test: standardised Z tbar statistic, $Z [\bar{t}]$ . Variable: relative size (in natural logarithms), sample size: (N, 11).

4.2. Non-parametric Analysis: Kernel Regression Conditional on City Size

At this point, we perform an analysis of the entire distribution, not just the upper tail. As a first approximation to city growth, Figure 1 shows the scatter plots of growth against relative city size for three representative decades (the behaviour for the remaining adjacent periods is similar) in the US, Spain and Italy. These graphs seem to support that growth is independent of size, although they also point to a great variance across observations, especially in the case of the US. In this section, we analyse the relationship between growth and initial size using two different non-parametric tools.

Figure 1.

Scatter plots of city growth against city size.

We perform a non-parametric analysis using kernel regressions (Ioannides and Overman, 2003). It consists of taking the following specification

g_{i} = m (s_{i}) + ε_{i}

(2)

where, g_i is the growth rate $(\ln s_{i t} - \ln s_{i t - 1})$ normalised (subtracting the contemporary mean and dividing by the standard deviation in the relevant decade); and s_i is the logarithm of the ith city’s relative size. Instead of making assumptions about the functional relationship m, $\hat{m} (s)$ is estimated as a local mean around the points and is smoothed using a kernel, which is a symmetrical, weighted and continuous function in s.

To estimate $\hat{m} (s)$ , the Nadaraya–Watson method is used, as it appears in Härdle (1990, ch. 3), based on the following expression

\hat{m} (s) = \frac{n^{- 1} \sum_{i = 1}^{n} K_{h} (s - s_{i}) g_{i}}{n^{- 1} \sum_{i = 1}^{n} K_{h} (s - s_{i})}

(3)

where, K_h denotes the dependence of the kernel K (in this case an Epanechnikov) on the bandwidth h. We use the same bandwidth (0.5) in all the estimations to permit comparisons between countries.

Starting from this calculated mean $\hat{m} (s)$ , the variance of the growth rate g_i is also estimated, again by applying the Nadaraya–Watson estimator

{\hat{σ}}^{2} (s) = \frac{n^{- 1} \sum_{i = 1}^{n} K_{h} (s - s_{i}) {(g_{i} - \hat{m} (s))}^{2}}{n^{- 1} \sum_{i = 1}^{n} K_{h} (s - s_{i})}

(4)

The estimator is very sensitive, both in mean and in variance, to atypical values. Eeckhout (2004) finds that some outliers have a huge impact on the variance of growth rates. For this reason, in the same way as Eeckhout (2004), we eliminate some atypical observations from the sample: the 5 per cent of smallest cities because they usually have much higher growth rates in mean and variance. This is logical; these are cities of under 200 inhabitants, where the smallest increase in population is very large in percentage terms.

According to Gabaix and Ioannides

Gibrat’s law states that the growth rate of an economic entity (firm, mutual fund, city etc.) of size S has a distribution function with mean and variance that are independent of S (Gabaix and Ioannides, 2004, p. 2346).

Thus, they distinguish between Gibrat’s law for means and Gibrat’s law for variances. As the growth rates are normalised, if Gibrat’s law in mean is strictly fulfilled, the non-parametric estimate will be a straight line on the zero value. Values different from zero involve deviations from the mean. Moreover, the estimated variance of the growth rate will also be a straight line on the value one, which would mean that the variance does not depend on the size of the city. To be able to test these hypotheses, we construct bootstrapped 95 per cent confidence bands (calculated from 500 random samples with replacement).

We offer a first approach to the behaviour of city growth from a short-term perspective—namely, considering each decade individually. Figures 2, 3 and 4 show the non-parametric estimates for the US, Spain and Italy respectively, corresponding to the same three representative decades shown in Figure 1.

Figure 2.

Non-parametric estimates (bandwidth 0.5) of the growth rate and its variance for the US, by decade.

Figure 3.

Non-parametric estimates (bandwidth 0.5) of the growth rate and its variance for Spain, by decade.

Figure 4.

Non-parametric estimates (bandwidth 0.5) of the growth rate and its variance for Italy, by decade.

Two different behaviours can be observed: in the US the estimate of growth is very close to the zero value (this value falls within the confidence bands for most of the distributions, supporting Gibrat’s law even in the short term), while in Spain and Italy a different pattern of growth can be seen. From the beginning of the century until the mid-century, the city growth exhibits clearly divergent behaviour in both European countries, although Gibrat’s law can only be rejected for some values at the upper-tail distribution. However, in the second half of the century, the growth changes gradually to an inverted U-shaped pattern. These results confirm that, as Gabaix and Ioannides (2004, p. 2353) indicate, “the casual impression of the authors is that in some decades, large cities grow faster than small cities, but in other decades, small cities grow faster”.

There is a negative relationship between the estimated variance of growth and city size in the three countries for most of the decades (this is especially true for the US, where Gibrat’s law can be rejected at the upper tail), although in Spain and Italy the behaviour of the variance is irregular, particularly in the first decades of the century.

Moreover, to analyse the entire 20th century, we build a pool with all the growth rates between two consecutive periods. This enables us to carry out long-term analysis. Figure 5 shows the non-parametric estimates of the growth rate of a pool for the entire 20th century for the US (1900–2000; 152 475 observations), Spain (1900–2001; 74 100 observations) and Italy (1901–2001; 73 260 observations). For the US, the value zero is always in the confidence bands, so that the growth rates being significantly different for any city size cannot be rejected. For Spain and Italy, the estimated mean grows with the sample size, although it is significantly different from zero only for the largest cities. One possible explanation is historical: both Spain and Italy have suffered wars on their territories during the 20th century, so that for several decades the largest cities attracted most of the population (the ‘safe harbour effect’; see Glaeser and Shapiro, 2002). However, the estimations by decade indicate that this tendency would have reversed in the second half of the century. Therefore, we find evidence in favour of Gibrat’s law for means for the US throughout the 20th century. Support is weaker in Spain and Italy because the largest cities show some divergent behaviour.

Figure 5.

Non-parametric estimates (bandwidth 0.5) of the growth rate and its variance: all the twentieth century.

Figure 5 also shows the non-parametric estimates of the variance of growth rates of a pool for the entire 20th century for the US, Spain and Italy. As expected, while for most of the distribution the value one falls within the confidence bands, indicating that there are no significant differences in variance, the tails of the distribution show differentiated behaviours. In the US, the variance clearly decreases with the size of the city, while in Spain and Italy the behaviour is more erratic and the biggest cities also have high variances.

Our results, obtained with a sample of all incorporated places without any size restriction, are similar to those obtained by Ioannides and Overman (2003) with their database of MSAs. To sum up, the non-parametric estimates (Figure 5) show that the mean of growth (Gibrat’s law for means) seems to be independent of size in the three countries in the long term (although in Spain and Italy the largest cities present some divergent behaviour). However, the variance of growth (Gibrat’s law for variances) depends negatively on size: the smallest cities present clearly higher variances in all three countries (although in Spain and Italy the behaviour is more erratic). In the short term (Figures 2, 3 and 4), the evidence supporting Gibrat’s law is weaker, as it corresponds to a law that is thought to hold mainly in the long term (Gabaix and Ioannides, 2004). This points to Gibrat’s law holding weakly (growth is proportional in means but not in variance).¹¹

Finally, as González-Val (2012), we perform a non-parametric estimation of growth using a resistant smoothing approach. Kernel estimation of regression functions has been receiving a great deal of attention in the recent literature examining Gibrat’s law (Ioannides and Overman, 2003; Eeckhout, 2004; González-Val, 2010; Giesen and Südekum, 2011) and the most widely used estimator is the Nadaraya–Watson estimator. Thus, the previous results can be compared with those of other studies. However, as argued before, the Nadaraya–Watson estimator is known to be highly sensitive to the presence of outliers in the data, so we have to exclude some observations.

Next, we try to reduce this sensitivity by using a resistant smoothing technique, the LOcally WEighted Scatter plot Smoothing (LOWESS) algorithm. It is based on local polynomial fits; see Härdle (1990, ch. 6). The advantages of LOWESS are that it is a free-functional form method and that it is robust to atypical values. Therefore, it allows us to obtain robust non-parametric estimates of growth and variance, using the entire sample, including the smallest 5 per cent of the distribution observations, which we previously excluded. Figure 6 shows the results for the 20th-century pool of observations for the US, Spain and Italy.¹² (Figure 6 is analogous to Figure 5, but includes the smallest 5 per cent of observations.)¹³ Their inclusion produces an increase in the estimates of both growth and variance at the lower tail of the distribution; this increase is much greater in the case of variance, as the dispersion of these observations is very high. Thus, small cities exhibit higher growth (except for the case of Italy) and variance than the rest of the cities, indicating again that Gibrat’s law does not hold exactly.

Figure 6.

Non-parametric estimates of the growth rate and its variance: all the 20th century, LOWESS (100 per cent of the sample).

For the rest of the sample, the results estimated by LOWESS are very similar to those estimated by the Nadaraya–Watson estimator, both in growth and in variance (Figure 5). This is logical because the Nadaraya–Watson estimator estimates a local mean around each point of the grid, so the estimates for the medium-sized cities or the upper-tail distribution do not depend on the values of the smallest observations, indicating that our previous results excluding the smallest 5 per cent of observations are robust for most of the distribution. Their inclusion only increases growth and variance at the lower tail of the distribution.

5. What About City Size Distribution?

Proportionate growth implies a log-normal distribution and this is a statistical relationship (Gibrat, 1931; Kalecki, 1945). However, if there is a lower bound to the distribution (which can be very low), the resulting distribution is Pareto (Gabaix, 1999) so, as Eeckhout (2004) shows, city size distribution follows a log-normal distribution only when we consider all cities without any size restriction.¹⁴ Our results show that the growth process leads to a log-normal distribution with standard deviation that increases in time t in the three countries if all cities are considered. This result is theoretically predicted: under a Brownian motion (Gabaix, 1999; Ioannides and Overman, 2003), the sample standard deviation should increase with the passing of time as a function of t (Anderson and Ge, 2005). Furthermore, it can be shown that by introducing a small change to one of the assumptions in Kalecki’s classic model,¹⁵ the same standard framework can be obtained, combining log-normality with a variance that increases over time.

We carried out Wilcoxon’s log-normality test (rank–sum test), which is a non-parametric test for assessing whether two samples of observations come from the same distribution. The null hypothesis is that the two samples are drawn from a single population and, therefore, that their probability distributions are equal, in our case, the log-normal distribution. Wilcoxon’s test has the advantage of being appropriate for any sample size. The results of the test¹⁶ indicate that the null hypothesis of log-normality cannot be rejected at 5 per cent for all the periods of the 20th century in Spain and Italy. In the US, a temporal evolution can be deduced: in the first decades, log-normality is rejected and the p-value decreases over time, but from 1930 the p-value begins to grow until log-normal distribution is not rejected at 5 per cent from 1960 onwards. In fact, if instead of 5 per cent we take a significance level of 1 per cent, the null hypothesis would only be rejected in 1920 and 1930.

However, the shape of the distribution in the US for the period 1900–50 is not far from log-normality either. Graphs located above in Figure 7 show the empirical density functions estimated by adaptive Gaussian kernels for 1900, 1950 (the last year in which log-normality is rejected at 5 per cent) and 2000. The reason for this systematic rejection seems to be an excessive concentration of density in the central values that is higher than would correspond to the theoretical log-normal distribution (dotted line). Starting in 1900 with a very leptokurtic distribution with a great deal of density concentrated in the mean value, from 1930 (not shown), when the growth of the urban population slows, the distribution loses kurtosis and the concentration decreases, not rejecting log-normality statistically at 5 per cent from 1960.

Figure 7.

Empirical city size distributions Above: Estimated density function (ln scale) and the theoretical lognormal (dotted line) for the US in 1900, 1950 and 2000. Below: Rank–size plots (ln scale) for the US (2000), Spain (2001) and Italy (2001).

Finally, there is one important issue related to the upper-tail distribution concerning the different behaviour of the largest cities discussed by some papers (Eeckhout, 2009; Levy, 2009). Following Levy (2009), the graphs situated below in Figure 7 show the empirical observations, the best log-normal fit and the Pareto fit for the upper tail (the largest 150 cities) and for the whole sample, for the three countries in the last year of our samples (the behaviour in the rest of the periods is similar). The Pareto exponent is estimated using Gabaix and Ibragimov’s Rank-½ estimator. A non-linear and clearly concave behaviour can be observed and the log-normal distribution provides a quite good fit for most of the distribution, especially for the small and medium city sizes. However, the largest cities’ behaviour is different and the rank–size relationship remains almost linear, confirming that the Pareto distribution is a better description of the behaviour of the upper-tail distribution.

6. Conclusions

The aim of this study is simple: to provide additional information on whether Gibrat’s law, a well-known empirical regularity in the literature on urban economics, holds. Briefly, this law states that the population growth rate of cities is a process deriving from independent multiplicative shocks, which implies two statistical conclusions. First, the empirical city size distribution can be well fitted by a log-normal, although if there is a lower bound the resulting distribution is Pareto (Gabaix, 1999); second, the growth rate is on average independent of the initial size of the urban centres and its evolution is fundamentally stochastic without any fixed pattern of behaviour.

This article contributes in several ways. On the one hand, it uses a database covering untruncated settlement size data from three countries (the US, Spain and Italy) with different urban histories over a long time-span (the entire 20th century). As far as we know, this is the widest-ranging attempt to test the geographical and temporal validity of this law, focusing on robust results. On the other hand, it employs different methods (parametric and non-parametric).

There are two basic conclusions. First, the panel data unit root tests carried out confirm that, in the long term, Gibrat’s law always holds for the upper tail of the distribution for the US and Italy, and only for the 200 largest cities of Spain. However, from the use of non-parametric techniques considering all the cities, also over the long term, such as kernel regressions conditional on city size, we deduce that Gibrat’s law does not hold exactly for the whole distribution. Gibrat’s law for means seems to hold for the US and, to a lesser extent, for Spain and Italy. In these two European countries, there is a positive relationship between city size and growth, although this divergent behaviour is only significant for the largest cities. Nevertheless, we also find that, in general, the variances depend negatively on size, which points to a weak version of Gibrat’s law where growth is proportional in means but not in variance. Moreover, small cities clearly exhibit higher growth (except in the case of Italy) and variance than the rest of the cities, even when we estimate using a non-parametric resistant smoothing approach. In the short term, as could be anticipated, the evidence regarding the validity of the law is more mixed.

Secondly, the log-normal distribution works well as a description of empirical city size distributions across the entire century when no truncation point is considered (the largest cities’ behaviour is Pareto rather than log-normal). Wilcoxon’s rank–sum test indicates that, except for the US in the first half of the century, the log-normal distribution can never be rejected.

Footnotes

Acknowledgements

Earlier versions of this paper were presented at: the 55th North American Meeting of the Regional Science Association International (New York, 2008); the XXXIII Symposium of Economic Analysis (Zaragoza, 2008); the 67th International Atlantic Economic Conference (Rome, 2009); the 24th Annual Congress of the European Economic Association (Barcelona, 2009); the XXXV Reunión de Estudios Regionales (Valencia, 2009); the 9th Annual Conference of the European Economics and Finance Society (Athens, 2010); and at the 50th Congress of the European Regional Science Association (Jönköping, 2010). All the comments made by participants at these events were highly appreciated. Suggestions and observations received from two anonymous referees have also improved the version originally submitted. All remaining errors are ours.

Funding

The authors acknowledge financial support from the Spanish Ministerio de Educación y Ciencia (ECO2009-09332 and ECO2010-16934 projects), the DGA (ADETRE research group), FEDER and the Generalitat (2009SGR102).

Notes

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