Human capital divergence and the size distribution of cities: Is Gibrat’s law obsolete?

Abstract

Chinese

This article studies how the changing geographic distribution of skilled workers in the US affects theoretical models that use Gibrat’s law to explain the size distribution of cities. In the empirical literature, a divergence hypothesis holds that college share increases faster in cities where college share is larger, and a growth hypothesis maintains that the rate of city population growth is also directly related to initial college share. Examining the divergence hypothesis, the classic test for Gibrat’s law is shown to be a test for $β$ -convergence. Testing shows that there has been absolute, not relative, divergence in human capital since the 1970s. However, the combination of even absolute divergence and the growth hypothesis is shown to violate the condition that a city’s population growth is independent of its size. Additional testing finds that the relation between college share and city growth is concave rather than monotonic. These results imply that stochastic growth models can survive the challenge posed by divergence in the distribution of human capital.

Keywords

divergence Gibrat’s law human capital skill ratio Zipf’s law

Introduction

Applying the standard that human capital is either based on, or proxied by, years of schooling, much research has been devoted to changes in the geographic distribution of workers with different levels of educational attainment. In the United States, as attainment has increased nationally, the distribution of human capital across cities has become more unequal. Skill intensity ratio (SIR), defined as the ratio of adults with at least a bachelor’s degree to those with less education, is a common measure of an area’s human capital level. Table 1 shows that the mean and variance of SIR have risen steadily for metro areas in the US, with both increasing threefold from 1970 to 2010.

Table 1.

Testing for $σ$ -divergence.

	SIR		Log SIR
	Mean	SD	Mean	SD
1970	0.13	0.06	−2.13	0.38
1980	0.20	0.09	−1.68	0.37
1990	0.25	0.11	−1.45	0.39
2000	0.32	0.14	−1.22	0.40
2010	0.39	0.17	−1.03	0.41
Homogeneity of variance
	$F$	$p$	Result
SIR	376.97	0.01	Reject
Log SIR	0.81	0.52	Fail to reject

Notes: Cross-section observations = 316 metro areas. Skill Intensity Ratio (SIR) is the ratio of adults (aged 25 years or older) with at least a bachelor’s degree to those with less education. Levene’s test is used to determine if decennial samples have statistically constant variance.

The trend of human capital divergence across US cities has been noted in a number of papers, including Moretti (2004, 2012, 2013), Berry and Glaeser (2005), Wheeler (2006), Hammond and Thompson (2010), Glaeser (2012) and Diamond (2016). In addition to increasing dispersion, these authors show that the level of human capital has been rising more, in absolute terms, in areas where it was initially more abundant. A second trend documented in the literature is that the population growth rates of US cities over the same period have also been an increasing function of their initial human capital levels (Glaeser and Saiz, 2003; Glaeser and Shapiro, 2003; Glaeser et al., 1995; Shapiro, 2006; Simon, 1998, 2004; Simon and Nardinelli, 2002).

The interaction of these two empirical trends from the literature implies that US cities have been diverging in skill and size, with low-skill workers concentrating in slow-growing cities where economic opportunities may be limited. In addition to concerns over inequality, there is another reason why the trends merit special attention. This article makes a new theoretical observation that the potential for diverging human capital levels to generate divergence in urban growth rates has important long-run implications for the distribution of city sizes.

An established literature on the size distribution of cities suggests a pattern that is remarkable for its simplicity and stability over time and across countries. Zipf’s law is an empirical regularity in which log of city population rank regressed on log of city population yields an estimated slope coefficient that is approximately equal to minus one. Indeed, the city size distribution is so regular that conformity with Zipf’s law may well represent a necessary condition of acceptance for any model that seeks to explain urban growth and urbanisation.

Beginning with Gabaix (1999), research has identified one possible explanation for the emergence of Zipf’s law in a system of cities. If the growth rate of cities is random with constant variance, then city sizes should follow a lognormal distribution. The type of local population growth that generates the regular city size distribution is referred to as a Gibrat process. Consequently, Gibrat’s law for cities has been advanced under the hypothesis that the growth rate of a city is independent of its size. Gibrat’s law is not necessary for Zipf’s law to hold, but it provides a dynamic rationale for the static distribution of city sizes associated with the law.

The size distribution of cities must be maintained by a pattern of productivity growth such that relative wages adjust to compensate for differences in cost of living associated with city size. As noted by Combes et al. (2008), these differences cannot be generated by skill differences associated with city size, but must represent a true compensating differential. The measurement of wage gains associated with movement is complicated by the finding in De la Roca and Puga (2017) that workers may experience unequal and permanent productivity gains from working in big cities. Thus, it seems that measurement of dynamic changes in the urban wage premium is a difficult matter and not a promising venue for testing Gibrat’s law.

Instead, empirical tests of Gibrat’s law have been based on either the size distribution of cities conforming to the lognormal or city growth being independent of city population. Black and Henderson (2003), González-Val et al. (2014) and Michaels et al. (2012) find instances in which one or the other condition fails. Similarly, research by Glaeser et al. (2014) and Desmet and Rappaport (2017) shows that over sufficiently long periods the relation between city size and growth rate varies from slightly above to slightly below zero. However, a review of these tests by González-Val et al. (2014) concludes that the predominant result has tended to support Gibrat’s law. As well, a measure of theoretical support has been provided in studies by Eeckhout (2004), Duranton (2006, 2007), Rossi-Hansberg and Wright (2007), Córdoba (2008) and Lee and Li (2013).

Recent testing of Zipf’s law has revived debate over appropriate geographic units subject to analysis. González-Val (2015) uses incorporated places in the US and finds that population growth rates are independent of size over the 1990–2000 period. Arribas-Bel et al. (2015) analyse 844 employment centres located within 357 metro statistical areas (MSAs) and conclude that Zipf’s law holds. Dobis et al. (2020) consider the effect of proximity to larger cities on growth rates and find that it is consequential for smaller US cities. Moreno-Monroy et al. (2020) estimate the rank-size rule exponent and find values of −0.93 for the US, −1.07 for China, −1.02 in Brazil and −1.11 in India. For these countries, sample sizes are large and results are roughly consistent with Zipf’s law.¹

This article examines the effects of changes in the distribution of skilled workers in the US on the relation between population level and population growth rate, that is, on a fundamental rationale for stochastic growth models that can generate the empirical distribution of city sizes. The classic test for Gibrat’s law is first shown to be a test for $β$ -convergence. Then, testing finds that while there has been absolute divergence in the SIR of US cities since 1970, there has been no relative divergence. These results shed new light on how the geographic distribution of human capital has been changing.

Our theoretical discussion demonstrates that, even under the assumption of no relative divergence, absolute divergence in human capital levels along with the association between human capital and city growth documented in literature are sufficient to violate the conditions for Gibrat’s law to hold. This theoretical challenge to stochastic growth models has not been noted in the literature.

The next contribution of the research is empirical. The finding of a positive relation between initial human capital and the rate of subsequent population growth is reexamined in light of consequences for Gibrat’s law for cities. The empirical literature has assumed a linear relation between lagged SIR and city growth. In contrast, testing performed below indicates that the relation may be concave. If the effect of human capital on urban growth weakens as educational attainment increases over time, then the effect is less likely to produce a long-run relation between population level and growth rate that would violate Gibrat’s law. Hence, our conclusion is that stochastic growth models can survive the challenge posed by divergence in the location of skilled workers.

Measurement

There are several measures of variation that could be used to analyse changes over time in the distribution of city characteristics. In the literature on human capital divergence, insufficient attention has been paid to the conceptual problems that measuring changes in inequality entails. This section draws on two bodies of literature with highly-developed efforts for grappling with these measurement issues – works on regional economic growth and works on individual economic inequality.

$σ$ -divergence versus $β$ -divergence

Beginning with Barro and Sala-i-Martin (1990), the literature on economic growth has distinguished two types of convergence: $σ$ -convergence occurs when measures of relative dispersion in per capita income fall over time, and $β$ -convergence is when the growth rate in per capita income is negatively correlated with its initial level. Subsequently, Sala-i-Martin (1996) demonstrated that the two measures convey different but separately valuable information, and that $β$ -convergence is a necessary, but not sufficient, condition for $σ$ -convergence.

In contrast, papers in the literature on human capital divergence have not formally distinguished between $σ$ - and $β$ -type differences, even though they convey different information on changes in the distribution. Whereas $σ$ -divergence indicates that dispersion in human capital across cities is increasing, $β$ -divergence means initially more educated cities are becoming still more educated.

In the literature on regional growth, the connection between Gibrat’s law and the concept of $β$ -convergence in population measures is strong. Gibrat’s law implies that there is neither $β$ -convergence nor $β$ -divergence of city population size. In fact, the standard test for Gibrat’s law, following Gabaix (1999), is identical to testing for $β$ -convergence, although this connection has not been noted in the urban literature.

Previous testing for convergence in US regional income has generally found a significant pattern of $β$ -convergence. However, for the 1970–1998 period, Young et al. (2008) show that $β$ -convergence for US counties was not accompanied by $σ$ -convergence. In fact, they find statistically significant $σ$ -divergence for counties in some states. Similarly, a recent paper by Ganong and Shoag (2017) has argued that regional convergence in the US has slowed and one explanation may be the relative preference of high-skilled workers for larger cities with more amenities and higher living costs. An implication of these papers is that there could be a connection between the finding of slower income convergence and the failure of human capital levels to converge. This suggests the need for the separate testing for $σ$ - versus $β$ -convergence of human capital performed in this article.

Absolute versus relative divergence

The theoretical literature on measuring individual income inequality has emphasised that when distributions of a variable are compared over time in a growing economy, it is important to distinguish between relative and absolute measures of change. Foster (1998) provides a seminal discussion of absolute versus relative index numbers. For example, absolute measures of income dispersion, such as variance and standard deviation, naturally increase over time when average income is growing. Therefore, it is common to measure $σ$ -convergence using changes in the Gini coefficient, coefficient of variation or variance of logarithms, because these relative measures have the property of scale invariance, that is, doubling each untransformed component of the distribution has no effect on the measure of dispersion. Similarly, for $β$ -convergence, estimating an equation relating absolute differences in income and initial income levels can indicate divergence, when regressing the rate of income growth, a measure of relative change, on initial levels shows no relation or even convergence.

In addition to failing to distinguish between $σ$ - and $β$ -divergence, the literature on human capital divergence in the US has insufficiently distinguished relative and absolute measures, mostly reporting increasing absolute differences in variation. The findings on US cities contrast with studies that do not find divergence across cities in Germany (Suedekum, 2008) and Norway (Rattsø and Stokke, 2014). Rattsø and Stokke argue that the contradictory findings for European and US samples may result from the use of absolute changes in college share in US studies, whereas the European papers analyse relative changes. Similarly, Broxterman and Yezer (2020) have pointed out that absolute divergence in the US has been the result of growth in human capital per worker that has proceeded without relative divergence. At issue is whether the US differences are simply a statistical artefact of growth in the share of college-educated workers over time, or alternatively, if there has been divergence beyond what would be expected if average educational attainment were growing at the same rate in every city.

Alternative measures in the data

Following convention in the literature on human capital divergence, metropolitan areas rather than incorporated cities are the unit of analysis. Demographic data on a panel of 316 MSAs in the contiguous United States are obtained for the decadal years 1970–2010 by aggregating county-level tabulations to metro definitions promulgated by the US Office of Management and Budget for the 1990 census. The general concept of a metropolitan statistical area is that it represents a local labour market. Using the official delineations, the sample excludes small urbanised areas, that is, the smallest MSA in the panel in 1970 has 27,559 inhabitants.

The underlying data come from the 1970, 1980, 1990 and 2000 US censuses and the 2008–2012 multi-year American Community Survey (ACS), referred to as ‘2010’ in reference to the year on which the five-year period is centred. For the census data, MSA-level values are accessed from the US Department of Housing and Urban Development’s State of the Cities Data Systems. The authors perform their own aggregations from the ‘summary tape file datasets’ for the ACS.

Testing for $σ$ -divergence in human capital is accomplished by computing changes in a scale-invariant index of spread. Table 1 compares the variance across MSAs of log SIR, which has the property of scale invariance, with the variance of SIR, which lacks it. Values for the five census years from 1970 to 2010 are reported. In contrast to the large increase in the standard deviation of SIR, from 0.06 to 0.17, the analogous range in values for log SIR, 0.37–0.41, is quite narrow. More formally, Levene’s test for homogeneity of variance rejects the null hypothesis of equal variance for SIR, but fails to reject for log SIR. Thus, while there has been absolute divergence in human capital across cities, there has been neither $σ$ -divergence nor $σ$ -convergence based on the homoscedasticity of log SIR.

Tests for $β$ -convergence in human capital intensity across cities take the form commonly associated with tests of Gibrat’s law. This entails testing the relation between an area’s initial human capital level and its subsequent rate of growth. For illustration purposes, three different measures of human capital growth are used as the dependent variable:

y_{it} = β_{0} + β_{1} (\frac{S_{i, t - 10}}{U_{i, t - 10}}) + \sum_{j = 1980}^{2000} δ_{j} d_{j} + ε_{it},

(1)

where $y_{it} = {Δ (\frac{S_{it}}{U_{it}}), Δ (\frac{S_{it}}{U_{it}}) / (\frac{S_{i, t - 10}}{U_{i, t - 10}}), \frac{Δ S_{it}}{S_{i, t - 10}} - \frac{Δ U_{it}}{U_{i, t - 10}}}$ for city $i$ in census year t, $S$ is the number of college graduates and $U$ is the number non-college graduates. The sample includes four observations per city, one for each change period, 1970–1980, 1980–1990, 1990–2000 and 2000–2010. The $d_{j}$ variables are indicators for the respective change periods excluding the 2000–2010 reference group. Identification of equation (1) is based on the fact that lagged skill intensity is being related to future city change.

The first test is performed for a relation between initial SIR and absolute change in SIR. This specification is not scale invariant and therefore not a true test for $β$ -divergence. However, because this test has been common in the literature on human capital divergence, it is included here. Estimates of equation (1) with $y_{it} = Δ (\frac{S_{it}}{U_{it}})$ reported in Model (1) of Table 2 confirm the standard finding in the literature that ${\hat{β}}_{1} > 0$ . The result is positive and highly statistically significant. Measured in absolute terms, human capital appears to have been concentrating in cities with the highest initial rates of human capital during the period from 1970 to 2010.

Table 2.

Testing for $β$ -divergence.

	$Δ$ ${SIR}_{t}$	$% Δ$ ${SIR}_{t}$	$% Δ$ $S_{t}$ – $% Δ$ $U_{t}$
	(1)	(2)	(3)
${SIR}_{t - 10}$	0.244^*** (0.015)	−0.035 (0.036)	−0.012 (0.039)
$R^{2}$	0.511	0.629	0.661

Notes: OLS estimates of equation (1). $N = 1264$ . Cross-section observations = 316 metro areas. Time series = 4: 1980, 1990, 2000, 2010. The dependent variable in Model (1) is the absolute change in the ratio of adults (aged 25 years or older) with at least a bachelor’s degree ( $S$ ) to those with less education ( $U$ ) in city $i$ : $Δ (SI R_{it}) = Δ (S_{it} / U_{it}) = S_{it} / U_{it} - S_{i, t - 10} / U_{i, t - 10}$ . For Model (2), the dependent variable is the growth rate of SIR: $% Δ (S I R_{i t}) = Δ (S I R_{i t}) / S I R_{i, t - 10}$ . In Model (3), the dependent variable is the difference in growth rates of skilled and unskilled adults: $% Δ S_{i t} - % Δ U_{i t} = Δ S_{i t} / S_{i, t - 10} - Δ U_{i t} / U_{i, t - 10}$ .

***

$p < 0.01$ . Heteroskedasticity consistent standard errors clustered at the metro level in parentheses.

The next step requires testing for possible relations between skill intensity level and either its rate of growth or the difference in growth rates between skilled and unskilled workers. Estimates of equation (1) with $y_{it} = Δ (\frac{S_{it}}{U_{it}}) / (\frac{S_{i, t - 10}}{U_{i, t - 10}})$ or $y_{it} = \frac{Δ S_{it}}{S_{i, t - 10}} - \frac{Δ U_{it}}{U_{i, t - 10}}$ are reported in Models (2) and (3) of Table 2. The estimate of ${\hat{β}}_{1}$ is not statistically different from zero in either model. Thus, there is no evidence of $β$ -divergence in this test for relative difference in differences. Put another way, SIR appears to follow Gibrat’s law in that past SIR is not related to its rate of growth in the next time interval.

The literature on income inequality has shown that it is reasonable to find absolute divergence when there is relative stability, if income per capita is increasing. Similarly, the results in Tables 1 and 2 indicate that, as expected in an economy with rising educational attainment, measures of the absolute difference in differences are positive and statistically significant, even though there is no relative divergence as indicated by either $σ$ - or $β$ -divergence measures. However, the next section shows that even a persistent absolute divergence in SIR combined with a relation between SIR and city growth would violate the Gibrat process that can generate the empirical distribution of city sizes.

Theoretical implications of human capital level divergence for Gibrat’s law

The simplest model of the evolution of cities that comports with stylised fact is stochastic growth based on Gibrat’s law. Gabaix (1999) first demonstrated that if city growth is characterised by Brownian motion with drift, and the mean and variance of the growth process are similar across cities, then a distribution of cities consistent with Zipf’s law with an exponent of unity will be generated. Thus, Gibrat’s law is sufficient, but not necessary, for Zipf’s law to hold.

Suppose the evolution of population ( $N$ ) in city $i$ is given by

\frac{d N_{i}}{N_{i}} = μ (N_{i}) d t + σ (N_{i}) d B_{t},

(2)

where $μ$ is the growth rate of cities with a population of $N$ , $σ$ is the variance in growth rate of cities of this size and $d B_{t}$ is white noise. If the drift and variance are constant over a range of city sizes, then the Zipf’s law exponent over that size range will be unity.² Therefore, the stylised fact that estimates of the Zipf’s law exponent are not significantly different from unity over a broad range of city sizes, countries and time periods suggests that $μ (N_{i})$ and $σ (N_{i})$ do not vary significantly with city size. Furthermore, direct testing of equation (2) has established that relative growth rates do not vary with city size over substantial city size ranges and/or time periods. Note that this is precisely the test performed in the literature on $β$ -convergence of regional incomes.

The standard stochastic growth rationale for Gibrat’s law holding in the data requires that $μ$ be relatively constant and hence that the expectation of $d N_{i} / N_{i}$ not vary with $N_{i}$ . Careful empirical study by Ioannides and Overman (2003) demonstrates that the mean and variance in city population growth rates vary only slightly across city sizes in US data.

Turning to the specific case of the relation between divergence in SIR and stochastic growth models, population growth is easily expressed as a function of the growth rates of skilled and unskilled workers and the concentration of human capital in a city,

\frac{Δ N_{i}}{N_{i}} = \frac{Δ S_{i}}{S_{i}} \frac{S_{i}}{N_{i}} + \frac{Δ U_{i}}{U_{i}} \frac{U_{i}}{N_{i}} .

(3)

Consider the case in which there is absolute divergence in SIR but no relative divergence. Assume that two cities differ in initial skill intensity so that $S_{1} / N_{1} = θ (S_{2} / N_{2})$ , where $θ > 1$ , and unskilled workers are growing at rate $λ$ while skilled workers are increasing nationally at rate $ϕ λ$ , where $ϕ > 1$ . It follows from (3) that

\begin{matrix} \frac{Δ N_{i}}{N_{i}} = ϕ λ (\frac{S_{i}}{N_{i}}) + λ (\frac{U_{i}}{N_{i}}) \\ = ϕ λ (\frac{S_{i}}{N_{i}}) + λ (1 - \frac{S_{i}}{N_{i}}) \\ = λ + λ (ϕ - 1) (\frac{S_{i}}{N_{i}}) . \end{matrix}

(4)

Using (4), the difference in population growth rates between the two locations can be written as a function of initial skill level,

\frac{Δ N_{1}}{N_{1}} - \frac{Δ N_{2}}{N_{2}} = λ (θ - 1) (ϕ - 1) (\frac{S_{2}}{U_{2}}) .

(5)

Equality of growth rates requires that either there is no change in national skill intensity ratio ( $ϕ = 1$ ) or that the skill intensity ratio be identical everywhere ( $θ = 1$ ). If neither condition holds, as appears to be the case for US cities, there will be a systematic difference in the population growth rates of cities based on their initial skill ratios. Absolute divergence is sufficient to eventually produce an association between city size and population growth rate. Such differences in growth rates, if they persist, would violate Gibrat’s law as cities with higher skill intensity ratios would become the largest and fastest-growing cities. Relative divergence would make the conditions for Gibrat’s law to hold in the long run more problematic. Even if overall SIR were not growing, variance in SIR would still rise over time, as SIR growth would be positive in high SIR cities.

There is an abundance of empirical evidence, discussed at length in Glaeser et al. (2014), that since 1970 lagged SIR has been positively related to population growth rates in cities. As discussed above, if this association between SIR and the rate of city growth were found to be temporary, then it would not represent a violation of Gibrat’s law. Instead it would be considered part of the pattern of idiosyncratic growth documented in the literature. Desmet and Rappaport (2017), for example, examine two centuries of change and report extended periods in which population growth was either directly or inversely related to city size.

Indeed, Glaeser et al. (2014) identify a number of variables which have been associated for temporary periods with faster rates of city growth. For example, variation in the industrial mix of cities has been shown to produce periods of unequal city growth. But again, these effects should be temporary and periods of unusual industrial expansion should generally be followed by periods of sluggish growth, thus reversing the sign of the relation between industrial composition and growth.

In their treatment of models of city size distributions, Gabaix and Ioannides (2004) devote an entire section to discussing possible deviations from Gibrat’s law that do not change the city size distribution. Writing (2) in discrete time yields

\ln N_{it} - \ln N_{it - 1} = μ (x_{it}, t) + ε_{it},

(6)

where $x_{it}$ could be one or more time-varying characteristics that vary across cities rather than the constant traditionally associated with Gibrat’s law. Less consequential for the discussion here is the fact that $ε_{it}$ need not be identically and independently distributed. It is only necessary that if $ε_{it}$ contains a mean reverting component, it must also contain a non-zero unit root component. Most important is the requirement that either $x_{it}$ is mean reverting or, if not, that $d μ / d x_{it}$ vary in sign across time.

Consider climate as an example. Climatic variables may be mean reverting, although periods of drought or unusual temperature may show significant persistence in some areas. Alternatively, the effects of climate on city growth may be temporary. The effect of air conditioning on the relation between hot summer temperatures and economic growth is one example of an effect that is likely temporary over a sufficiently long time horizon. If either of these propositions holds, then the finding of a temporary association between climate and population growth rate does not imply a permanent departure from Gibrat’s law. However, Rappaport (2007) argues that there is a permanent component to differences in climate and that the income elasticity of demand for mild winters and summers may be greater than unity, giving areas with superior climate a permanent advantage over rival cities. Such permanent effects could be a challenge for Gibrat’s law.

The specific $x_{it}$ variable of interest here is SIR. Given that the range of $S / U$ is $[0, \infty]$ , the variable may not be mean reverting, or at least may take a substantial time to reach an upper limit. Furthermore, empirical evidence shows that, at least over the past four decades, $d μ / d (S_{it} / U_{it}) > 0$ . If this pattern were to persist, then even if SIR is only characterised by absolute but not relative divergence, future city growth would become inconsistent with Gibrat’s law. The future expectation for Gibrat’s law holding becomes even more problematic if there is relative divergence in the SIR of cities.

The strong evidence for a relation between absolute changes in SIR and its initial level found above along with the documented empirical relation between absolute differences in SIR and subsequent city growth, when combined with the theory in this section, have implications Gibrat’s law. In particular, this suggests the need for a further test regarding the second derivative of population growth with respect to SIR. Gabaix (1999) shows that, if $d μ / d (S_{it} / U_{it}) > 0$ , but $d^{2} μ / d (S_{it} / U_{it})^{2} < 0$ , Gibrat’s law can still hold in the long run. Of course, if the relation between SIR and population growth is convex rather than concave, the case for Gibrat’s law holding in the future becomes even more problematic.

Testing for concavity in the relation between human capital level and urban growth

The most common approach to testing Gibrat’s law is the parametric city growth equation. In addition to checking for the classic Gibrat relation, the primary focus of the testing here is determining whether and how rates of urban growth are related to human capital levels in the data. Testing begins by estimating the following panel equation:

\begin{matrix} \frac{Δ N_{it}}{N_{i, t - 10}} = β_{0} + β_{1} (\frac{S_{i, t - 10}}{U_{i, t - 10}}) + β_{2} {(\frac{S_{i, t - 10}}{U_{i, t - 10}})}^{2} \\ + β_{3} {(\frac{S_{i, t - 10}}{U_{i, t - 10}})}^{3} + α N_{i, t - 10} + \sum_{j = 1980}^{2000} δ_{j} d_{j} \\ + \sum_{j = 1980}^{2000} γ_{j} (d_{j} \times N_{i, j - 10}) + ε_{it} . \end{matrix}

(7)

Equation (7) follows directly from Glaeser et al. (2014), with identification based on using lagged SIR to explain subsequent population growth. The form of the relation between human capital and urban growth is reflected in estimates of $β_{1}$ , $β_{2}$ and $β_{3}$ . Gibrat’s law implies $α = 0$ , and estimates of the $γ_{j}$ coefficients test the hypothesis that the relation between population level and its subsequent rate of growth varies by decade.

Estimates of equation (7) are reported in Table 3. Skill intensity ratio enters the equation sequentially in first-, second- and third-degree polynomial functions. Odd-numbered models are estimated using ordinary least squares (OLS) and even-numbered models with iterated re-weighted least squares (IRLS) for robust regression. Across the six models in Panel A, estimates of $α$ , the coefficient on initial population, are either negative or not significant. As expected based on the urban growth literature, estimated values of $β_{1}$ , the coefficient on linear SIR, are greater than zero and statistically significant. When quadratic and cubic terms are included in Models (3)–(6), findings are that ${\hat{β}}_{2} < 0$ and ${\hat{β}}_{3} > 0$ significantly, with magnitudes indicating a single hump-shaped relation between population growth and human capital for the range of values in the data.³ The finding of concavity does not appear to be an artefact of outliers, as there are only small differences in the $\hat{β}$ s using IRLS and OLS. The final result to note is that population and human capital (plus year fixed effects) only explain about 10% of the variation in city growth rates. The lack of power across these equations is a finding typical in the literature.

Table 3.

Urban growth and human capital.

A. Baseline specification	Population growth ${rate}_{t, t - 10}$
	(1)	(2)	(3)	(4)	(5)	(6)
${Population}_{t - 10}$	−0.055 (0.041)	−0.072 (0.053)	−0.078* (0.041)	−0.097* (0.052)	−0.066 (0.042)	−0.087* (0.052)
${SIR}_{t - 10}$	0.218*** (0.053)	0.258*** (0.031)	0.706*** (0.159)	0.785*** (0.086)	1.180*** (0.339)	1.235*** (0.186)
${SIR}_{t - 10}^{2}$			−0.655*** (0.165)	−0.702*** (0.108)	−1.926*** (0.715)	−1.870*** (0.455)
${SIR}_{t - 10}^{3}$					0.921** (0.450)	0.824*** (0.321)
$R^{2}$	0.092	0.072	0.108	0.095	0.112	0.099
B. Robustness: Additional controls
${Population}_{t - 10}$	−0.161*** (0.039)	0.125*** (0.044)	−0.187*** (0.040)	−0.144*** (0.043)	−0.185*** (0.040)	−0.143*** (0.043)
${SIR}_{t - 10}$	0.290*** (0.059)	0.341*** (0.033)	0.746*** (0.140)	0.806*** (0.083)	0.866*** (0.280)	0.888*** (0.166)
${SIR}_{t - 10}^{2}$			−0.556*** (0.137)	−0.614*** (0.094)	−0.866 (0.567)	−0.825** (0.386)
${SIR}_{t - 10}^{3}$					0.221 (0.345)	0.147 (0.267)
Manufact ${share}_{t - 10}$	−0.564*** (0.122)	−0.283*** (0.047)	−0.510*** (0.125)	−0.229*** (0.047)	−0.506*** (0.125)	−0.228*** (0.047)
Prof Svcs ${share}_{t - 10}$	−0.622 (0.186)	−0.386*** (0.077)	−0.704*** (0.183)	−0.429*** (0.077)	−0.707*** (0.184)	−0.432*** (0.077)
Trade ${share}_{t - 10}$	0.092 (0.310)	0.405*** (0.142)	0.145 (0.312)	0.411*** (0.140)	0.138 (0.313)	0.405*** (0.140)
Coastal indicator	0.012 (0.014)	−0.005 (0.007)	0.009 (0.014)	−0.008 (0.006)	0.009 (0.014)	−0.008 (0.007)
Heating degree days	−0.241*** (0.028)	−0.231*** (0.014)	−0.251*** (0.028)	−0.239*** (0.014)	−0.251*** (0.028)	−0.239*** (0.014)
Precipitation	−0.157*** (0.047)	−0.192*** (0.020)	−0.156*** (0.048)	−0.189*** (0.020)	−0.155*** (0.048)	−0.188*** (0.020)
$R^{2}$	0.431	0.312	0.441	0.323	0.441	0.324
Robust regression	N	Y	N	Y	N	Y

Notes: Estimates of equation (7). Models include year fixed effects and year-by-population interactions, not reported. $N = 1264$ . Cross-section observations = 316 metro areas. Time series = 4: 1980, 1990, 2000, 2010. The dependent variable is the rate of change in total population of city $i$ : $Δ N_{it} / N_{i, t - 10} = (N_{it} - N_{i, t - 10}) / N_{i, t - 10}$ . Population (in 10 millions), heating degree days (10,000 per year) and precipitation (100 inches per year) are scaled for ease of interpretation of regression coefficients. Skill Intensity Ratio is the ratio of adults (aged 25 years or older) with at least a bachelor’s degree to those with less education. Robust regression is by iterated re-weighted least squares using bisquare weights.

$p < 0.10$ , $* * p < 0.05$ , $* * * p < 0.01$ . Standard errors in parentheses. For odd-numbered models, standard errors are heteroskedasticity consistent and clustered at the metro level.

To illustrate the economic significance of these results, the effect on population growth of increasing SIR by one standard deviation (0.14) can be compared for two cities at different points in the distribution. The Sheboygan, WI metro area has SIR at the first quartile value of 0.22 for the year 2000, and Wilmington-Newark, DE has the third quartile value of 0.38. As a result of the hypothetical SIR shock, quadratic OLS Model (3) and cubic OLS Model (5) predict that Sheboygan’s rate of population growth would have been 4.7% or 4.3% points higher, respectively, over the 2000–2010 decade. Because of the estimated concavity, the growth rate for Wilmington-Newark would have changed much less, rising by just 1.6 or 0.1 points. Suppose the mayors of these two cities had policy goals of increasing their growth rates. The results in Table 3 suggest that the efficacy of attracting more college-educated workers as a policy lever, were that even achievable, would be subject to a great deal of randomness and would also depend critically on how well-educated on average a city was already.

As a robustness exercise, natural amenity variables and measures reflecting the concentration of local employment in industries which have experienced divergent growth trends at the national level are added to the specification. The industry mix measures are aggregated from the same source data as the demographic variables (i.e. decennial US censuses and the ACS) and the amenity variables are taken directly from the County and City Data Book: 2000 published by the US Census Bureau. These variables have been reported to be significant in growth equations published elsewhere.⁴ Thus, their omission could be biasing estimates in Panel A if they are conditionally correlated with included regressors.

As expected based on previous research, coefficients on amenity and industry mix variables reported in Panel B of Table 3 are significant and have expected signs. For example, cold and wet areas with large manufacturing sectors are found to grow more slowly during the study period. The city size effects remain negative and estimates are now statistically significant. The new estimates of $β_{1}$ increase and $β_{2}$ decrease in magnitude, but remain significant, while estimates of $β_{3}$ are no longer significant. Although the conditional curves in Panel B are somewhat less concave than baseline estimates in Panel A, the results of the robustness testing reinforce the finding that the relation between population growth and human capital is concave.

To visualise general forms of the relations estimated thus far, Figure 1 fits by decade quadratic and cubic trend curves to population growth rate and initial skill intensity ratio in standardised (z-score) form. It is clear that the quadratic and cubic models fit quite similar concave relations to these data. Further inspection of the plots raises several issues. First, holding the distribution of SIR constant, it appears that the curves are becoming less steep and curved over time. Second, assuming the polynomial functional forms are reasonable approximations of the true underlying relationship, the plots show that a number of cities are likely located in a region where the derivative of population growth with respect to SIR is negative. In other words, there are metros on the downward-sloping portion of the estimated relation. Third, although the non-linear nature of the relation is apparent, a particular functional form that remains constant throughout the range of SIR values is not clear. The remainder of the testing examines these issues.

Figure 1.

Non-linear fits by decade.

In regard to the first two points above, population growth rate is next regressed by decade on initial population and a quadratic in skill intensity ratio,⁵

\begin{matrix} \frac{Δ N_{it}}{N_{i, t - 10}} = β_{0} + β_{1} (\frac{S_{i, t - 10}}{U_{i, t - 10}}) \\ + β_{2} {(\frac{S_{i, t - 10}}{U_{i, t - 10}})}^{2} + α N_{i, t - 10} + ε_{it} . \end{matrix}

(8)

In estimates of equation (8) reported in Panel A of Table 4, ${\hat{β}}_{1}$ and ${\hat{β}}_{2}$ have the expected signs and are significant at the 1% level in all but the 2000 cross-section. For both OLS and IRLS estimates, values for the $β$ coefficients of interest are falling over time, but at a diminishing rate. These results suggest that the slope of the relation between population growth and human capital is becoming less steep at the origin, and that the rate at which the slope falls as SIR increases is also falling. However, the relations appear to be stabilising.

Table 4.

Critical points in the relation between urban growth and human capital.

A. MSA sample ( $N = 316$ )	Population growth ${rate}_{t, t - 10}$
	1980		1990		2000		2010
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
${Population}_{t - 10}$	−0.054*** (0.089)	−0.422*** (0.086)	−0.079 (0.061)	−0.068 (0.067)	−0.057 (0.045)	−0.072 (0.053)	−0.052 (0.043)	−0.074 (0.074)
${SIR}_{t - 10}$	2.84*** (0.59)	2.54*** (0.50)	1.07*** (0.32)	1.41*** (0.29)	0.56** (0.23)	0.73*** (0.19)	0.41*** (0.13)	0.51*** (0.13)
${SIR}_{t - 10}^{2}$	−5.74*** (1.23)	−4.84*** (1.24)	−1.31*** (0.49)	−1.64*** (0.48)	−0.47 (0.29)	−0.64** (0.27)	−0.34*** (0.12)	−0.42*** (0.13)
$R^{2}$	0.12	0.12	0.05	0.11	0.05	0.08	0.04	0.05
Robust regression	N	Y	N	Y	N	Y	N	Y
Critical Point	0.25	0.26	0.41	0.43	0.60	0.57	0.60	0.61
Freq in excess	14	14	12	11	4	6	16	16
Mean ( ${SIR}_{t - 10}$ )	0.13		0.20		0.26		0.32
Max ( ${SIR}_{t - 10}$ )	0.45		0.63		0.79		1.14
B. Robustness: CBSA sample ( $N = 929$ )
${Population}_{t - 10}$	−0.310*** (0.090)	−0.189*** (0.056)	0.009 (0.048)	0.020 (0.041)	−0.036 (0.030)	−0.006 (0.040)
${SIR}_{t - 10}$	1.39*** (0.36)	0.81*** (0.20)	0.92*** (0.14)	0.95*** (0.10)	0.52*** (0.10)	0.47*** (0.08)
${SIR}_{t - 10}^{2}$	−2.45*** (0.74)	−1.24** (0.50)	−1.03*** (0.24)	−1.01*** (0.17)	−0.45*** (0.12)	−0.43*** (0.11)
$R^{2}$	0.03	0.02	0.05	0.09	0.04	0.03
Robust regression	N	Y	N	Y	N	Y
Critical Point	0.25	0.31	0.38	0.43	0.62	0.57
Freq in excess	22	12	20	15	9	13
Mean ${SIR}_{t - 10}$	0.10		0.16		0.19
Max ${SIR}_{t - 10}$	0.63		0.92		1.15

Notes: The rate of change in total population of city $i$ , $Δ N_{it} / N_{i, t - 10} = (N_{it} - N_{i, t - 10}) / N_{i, t - 10}$ , is regressed by decade on initial population level and a quadratic of initial SIR, as in Models (3) and (4), Panel A of Table 3. Critical Point is SIR value at which $d (\frac{Δ N_{it}}{N_{i, t - 10}}) / d SIR = 0$ . Frequency in Excess is the number of locations with ${SIR}_{t - 10}$ greater than the Critical Point value for that model. Values for 2010 are not reported for the core-based statistical area (CBSA) sample because of additional challenges in matching geographic definitions between census and ACS data at the CBSA level. Robust regression is by iterated re-weighted least squares (IRLS) using bisquare weights.

$p < 0.05$ , $* * * p < 0.01$ . Standard errors in parentheses. For odd-numbered models, standard errors are heteroskedasticity consistent.

The critical points in Table 4 indicate the level of SIR above which population growth rate begins to fall with increasing human capital. The critical point values increase over time, which is not surprising given the overall rightward shift in the distribution of SIR. The table also reports mean and maximum SIR values by decade. All of the estimated critical points are greater than mean SIR, but lower than maximum, for their respective time periods. In other words, there are high-skill MSAs with population growth rates that are expected to fall with further increases in average human capital values in all four time periods. The number of such MSAs is referred to in the table as ‘frequency in excess’. Relative to the 1970s, these counts fall during the 1980s and 1990s, but rebound in the first decade of the 2000s. According to Models (7) and (8), for example, the population growth rate from 2000 to 2010 is inversely related to human capital level for 16 MSAs with SIR levels in 2000 greater than 0.60.

An important concern with parametric growth regressions is that the number of cities included in the sample can be consequential for parameter estimates (Eeckhout, 2004; González-Val et al., 2013). In particular, inclusion of small cities tends to produce refutations of Gibrat’s law. Because the sample of MSAs used here is truncated to exclude small cities, based on the findings in Levy (2009), the distribution of sizes is expected to be Pareto consistent with the law. Nonetheless, as an additional robustness exercise, equation (8) is re-estimated using a sample of 929 core-based statistical areas (CBSAs) delineated for the 2000 census. In addition to MSAs, CBSAs include ‘micropolitan’ statistical areas based on counties with urbanised areas of at least 10,000 inhabitants in 1990. Applying the definitions retroactively, the smallest CBSA in 1970 has a population of just 2665.

In Panel B of Table 4, the model fits are generally worse with the non-truncated sample. However, the estimated critical points and number of locations on the downward sloping portion of the urban growth–human capital relation using CBSAs are quite similar to the previous results using the sample of MSAs. While inclusion of small ‘cities’ may result in refutation of the Gibrat relation, the robustness exercise shows that it does not have the same impact on the finding of a concave human capital relation.

A well-known drawback of the use of polynomial terms to estimate a potential non-linear relation is that the functional form chosen may not accurately represent the underlying data-generating process. Researchers in the related inequality literature have used semi-parametric methods to flexibly estimate the Kuznets (bell-shaped) curve relation between GDP per capita and income inequality, e.g. Barrios and Strobl (2009) and Lessmann (2014). The final tests here draw inspiration from this literature by similarly estimating a semi-parametric functional form that places fewer restrictions on the underlying relationship than did the estimates above which use higher-order SIR terms. The approach involves estimating a piecewise functional form for the relation between population growth rates and SIR. This, of course, allows the effect of initial population level to remain linear.

The piecewise linear function relates growth rates to initial SIR and population as follows:

\frac{Δ N_{it}}{N_{i, t - 10}} = β_{0} + \sum_{j = 1}^{J} β_{j} S_{ij} + α N_{i, t - 10} + ε_{it},

(9)

where the $S_{ij}$ values provide a linear representation of SIR in city $i$ and the $β_{j}$ s are point estimates of the relation between SIR and population growth rate at a particular level $j$ of SIR. For purposes here, the range of observed SIR values in each decade is divided into four or five even-sized intervals. The SIR value for each observation in the data can then be expressed as a linear combination of the two endpoints of the interval containing the observed value. The weights from creating these linear combinations comprise the $S_{ij}$ values in equation (9) and take values between zero and unity. The coefficients on the weight variables are estimates of the relation between population change and SIR at the endpoints. These estimates can trace, with a flexible functional form, the relation between population change and SIR in a given year.

Table 5 shows results from estimating equation (9) by decade using OLS and IRLS. For the 1980 and 1990 cross-sections, six $S_{ij}$ variables define five intervals of even length. The $S_{ij}$ variable for ${SIR}_{t - 10} = 0$ is the reference group. As expected, the estimated coefficients values trace out a concave pattern, initially rising and then falling. However, a concave-down relation cannot be concluded decisively based on these results. Although the coefficient estimates are statistically significant, differences between them are not.⁶ For 2000 and 2010, four intervals are used in order to have sufficient observations in the top interval for statistical power. The estimated coefficients again exhibit a concave pattern. However, the estimates are generally not significant in the OLS models. Overall, these results suggest a concave relation between urban growth and human capital, consistent with the findings from estimating polynomial functional forms, that has perhaps become less curved over time.

Table 5.

Piecewise linear functional form.

Population growth ${rate}_{t, t - 10}$
1980	(1)	(2)	1990	(3)	(4)
${Population}_{t - 10}$	−0.545*** (0.092)	−0.427*** (0.086)	${Population}_{t - 10}$	−0.070 (0.063)	−0.063 (0.068)
${SIR}_{t - 10} = 0.09$	0.296*** (0.093)	0.209* (0.108)	${SIR}_{t - 10} = 0.13$	0.365** (0.146)	0.265** (0.121)
${SIR}_{t - 10} = 0.18$	0.417*** (0.077)	0.336*** (0.141)	${SIR}_{t - 10} = 0.26$	0.407*** (0.134)	0.349*** (0.113)
${SIR}_{t - 10} = 0.27$	0.392*** (0.101)	0.311*** (0.084)	${SIR}_{t - 10} = 0.39$	0.463*** (0.141)	0.408*** (0.121)
${SIR}_{t - 10} = 0.36$	0.336*** (0.099)	0.289** (0.129)	${SIR}_{t - 10} = 0.52$	0.414*** (0.143)	0.365*** (0.134)
${SIR}_{t - 10} = 0.45$	0.287*** (0.096)	0.243 (0.162)	${SIR}_{t - 10} = 0.65$	0.406*** (0.147)	0.357** (0.164)
$R^{2}$	0.127	0.121	$R^{2}$	0.054	0.106
Robust regression	N	Y		N	Y
2000	(5)	(6)	2010	(7)	(8)
${Population}_{t - 10}$	−0.064 (0.047)	−0.078 (0.053)	${Population}_{t - 10}$	−0.058 (0.044)	−0.081* (0.046)
${SIR}_{t - 10} = 0.20$	0.037 (0.076)	0.126** (0.054)	${SIR}_{t - 10} = 0.29$	0.064 (0.045)	0.095*** (0.037)
${SIR}_{t - 10} = 0.40$	0.113 (0.133)	0.194*** (0.048)	${SIR}_{t - 10} = 0.58$	0.110*** (0.040)	0.140*** (0.034)
${SIR}_{t - 10} = 0.60$	0.107 (0.077)	0.254** (0.099)	${SIR}_{t - 10} = 0.87$	0.085 (0.054)	0.126** (0.062)
${SIR}_{t - 10} = 0.80$	0.107 (0.090)	0.206*** (0.063)	${SIR}_{t - 10} = 1.16$	−0.037 (0.040)	−0.001 (0.103)
$R^{2}$	0.053	0.083	$R^{2}$	0.041	0.051
Robust regression	N	Y		N	Y

Notes: Standard errors in parentheses. Standard errors for odd-numbered models are heteroskedasticity consistent. Cross-section observations = 316 metro areas. The dependent variable is the percent change in total population of city $i$ : $Δ N_{it} / N_{i, t - 10} = (N_{it} - N_{i, t - 10}) / N_{i, t - 10}$ . Robust regression is by iterated re-weighted least squares (IRLS) using bisquare weights. The ${SIR}_{t - 10} = X . XX$ variables designate the endpoints of intervals that are used in linear combination to describe SIR values observed in the data. The endpoints, call them $S_{j}^{*}$ s, are set at intervals along the distribution of SIR for any given year (the endpoint values vary because the distribution of SIR is changing). The values of the $S_{ij}$ s for any observation in a particular year are based on the SIR value for that city, ${SIR}_{i}$ . There are then two possibilities. First, and least likely, is the knife-edge case in which $S_{j}^{*}$ exactly equals one of the endpoint $S_{j}$ values. In this case, $S_{ij} = 1$ for that particular $S_{j}$ and 0 for the other $S_{j}$ s. More commonly, ${SIR}_{i}$ lies on the interval between two $S_{j}^{*}$ s, call them $S_{k}^{*}$ and $S_{k + 1}^{*}$ . In this case, the values of the $S_{ij}$ s for city $i$ are set equal to 0 for all cases except $S_{k}^{*}$ and $S_{k + 1}^{*}$ , and the values of $S_{k}$ and $S_{k + 1}$ are set so that SIR is a linear combination of these two endpoint values, or so that ${SIR}_{i} = S_{i, k} (S_{k}^{*}) + S_{i, k + 1} (S_{k + 1}^{*})$ , where $S_{i, k} + S_{i, k + 1} = 1$ .

$p < 0.10$ , $* * p < 0.05$ , $* * * p < 0.01$ .

Given that SIR is diverging in absolute value naturally due to the rise in average SIR in the economy, the potential for SIR effects to influence Gibrat’s law is evident. However, a positive relation between SIR and population growth may not have implications for Gibrat’s law if it is concave. The results in this section, subject to substantial stressing, show that the positive relation reported in previous studies is not linear: it appears concave and not monotonic increasing. These empirical findings provide assurance that the divergence in absolute SIR, which is likely to continue as average SIR rises, may not have permanent implications for the validity of stochastic growth models that generate the empirical size distribution of cities.

Conclusions

This article makes a new observation that findings in the literature regarding divergence of human capital and a positive relation between initial human capital level and subsequent population growth could challenge stochastic models of city growth based on Gibrat’s law. However, analysis of the evidence supports a conclusion that Gibrat’s law and its implications for the size distribution of cities can survive the apparent challenge from these stylised facts.

First, measures of divergence in the skill intensity of cities reported in the literature lack the property of scale invariance. Given the national rise over time in average educational attainment, absolute divergence across locations could be a statistical artefact. Examination of scale-invariant measures of divergence in the measurement section of the article indicates that the dispersion of human capital has not increased relatively and there is evidence that human capital is growing at a rate independent of its initial value. Indeed, testing for both $σ$ - and $β$ -divergence of the human capital level of cities shows that there was neither divergence nor convergence over the 1970–2010 period.

However, there is rising absolute divergence as a natural consequence of the general rise in education within the labour force. The theoretical section demonstrates that, even when relative divergence is lacking, absolute divergence is sufficient to cause a positive relation between city size and city growth rates that could violate Gibrat’s law. Further analysis of the empirical relation between initial human capital and city growth in the testing section demonstrates that it is diminishing over time and concave. This limits the potential for absolute divergence to create a systematic relation between population levels and growth rates that would violate Gibrat’s law.

This article provides important clarification of previous empirical findings from the literature on urban growth and the distribution of human capital. Overall, the news for stochastic growth models in the results presented here is favourable. Indeed, it appears that the growth rate of human capital in cities is actually unrelated to past levels. This does not mean that absolute divergence will not be found as national education levels rise, but it does mean that there is no relative divergence using either $σ$ - or $β$ -divergence measures. Finally, the observed rise in absolute divergence does not portend a violation of the stochastic growth process provided that the relation between the human capital level of cities and their future growth is, consistent with the new empirical evidence presented here, not monotonic. On the other hand, the findings presented here do not demonstrate that Gibrat’s law is correct, only that it is not likely to be violated by the rise in absolute skill intensity divergence between cities.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Daniel Broxterman

Notes

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Human capital divergence and the size distribution of cities: Is Gibrat’s law obsolete?

Abstract

Keywords

Introduction

Measurement

σ -divergence versus β -divergence

Absolute versus relative divergence

Alternative measures in the data

Theoretical implications of human capital level divergence for Gibrat’s law

Testing for concavity in the relation between human capital level and urban growth

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Notes

References

$σ$ -divergence versus $β$ -divergence