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Racing to Zipf’s Law:
Race and Metro Population Size 1910-2020
Ricardo T. Fernholz∗Rory Kramer†
Claremont McKenna College Villanova University
January 24, 2023
Abstract
Urban systems follow a power law population distribution if the population has full
labor mobility. Comparing city population distributions for White and Black Ameri-
cans across U.S. metropolitan areas from 1910-2020 shows that the White distribution
conforms to both Zipf’s and Gibrat’s law, while Jim Crow restrictions on Black Ameri-
can mobility create deviations from both laws for the Black population in the first part
of the 20th century. A smaller but substantial deviation emerges in the 21st century
as a result of the reverse great migration. Comparisons using data on city population
distributions for different U.S. income groups suggest that this recent deviation is the
result of racially distinct mobility patterns rather than common mobility constraints
affecting all low-income households.
Keywords: Zipf’s law, power laws, race, mobility, random growth
∗Claremont McKenna College, 500 E. Ninth St., Claremont, CA 91711, rfernholz@cmc.edu
†Villanova University, 800 E. Lancaster Ave, Villanova, PA 19085, rory.kramer@villanova.edu
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1 Introduction
Scholarship on racial residential segregation has traditionally focused on urban populations,
in part because of data and methodological benefits and in part because of the argument that
the rise of racial segregation mirrored the migration of Black Americans from rural southern
areas into northern cities (Massey and Denton, 1993). Using metrics such as the dissimilarity
index, centralization index, and isolation index, this scholarship demonstrated that the U.S.
first saw “hypersegregation” in the second half of the 20th century and that segregation was
at its worst in northern rustbelt cities (Charles, 2003). Extreme racial segregation enacted
within individual urban areas, in other words, was a response to increased Black mobility
between metro areas. Yet scholarship on segregation has primarily focused on neighborhood
level segregation patterns, while scholarship on the geography of racism shows macro-level
impacts of racism at the state level (Baker, 2022).
Within metro areas, recent scholarship has expanded the study of racial segregation by
rethinking the scale of segregation, the history of segregation, and the location of interest.
First, Geographic Information Systems have allowed scholars to shift from using adminis-
tratively defined proxies of neighborhoods to egocentric or spatially adjusted neighborhood
concepts to better appreciate the scale of segregation (Wong, 1993; Lee et al., 2008). A
second line of scholarship uses recently released census microdata from the first half of the
20th century to argue that the prior segregation literature underestimated the extent of
segregation in the Jim Crow South because of its reliance on aggregated data and admin-
istratively defined neighborhood proxies (Logan and Parman, 2017; Logan and Martinez,
2018; Shertzer and Walsh, 2019). This literature also finds that segregation in the Jim Crow
South developed earlier than previously thought. While this research does not invalidate
the claim that northern industrial segregation was especially extreme and problematic, it
challenges the claim that such segregation was created whole cloth as a replacement to Jim
Crow style social control and instead suggests that it originated alongside Jim Crow before
expanding rapidly in the North.
Finally, a third line of research has followed the rise, fall, and revitalization of the urban
core by studying segregation outside of major, hypersegregated cities (Lichter, Parisi, and
Taquino, 2015; Parisi, Lichter, and Taquino, 2017). Urban segregation was accelerated by
the creation and subsidization of suburban development as a White space (Jackson, 1987;
Rothstein, 2017). Today, however, the suburbs are often sites of increasing diversification,
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by both international immigration and internal migration out of the city (Lacy, 2016). At
the same time, suburbs have experienced an increase in poverty and, as geographies created
for an idealized White middle class, are often ill-equipped to provide necessary social services
for poor residents (Kneebone and Berube, 2013; Allard, 2017).
Across all of these subfields, isolating specific urban or metro areas to measure internal
levels of segregation is taken as a methodological given, even when constructing national
measures of segregation patterns. But the isolation of individual metro areas from each
other does not engage with the process of sorting across metro areas, only within them. The
history of racial segregation and labor mobility demonstrates that racial segregation also
occurs at this more macro scale. The most egregious example was the sustained effort by
southern leaders to enforce vagrancy laws and “enticement” laws to restrict Black worker
mobility to the north in the post-bellum era (Anderson, 2016; Derenoncourt, 2022). Like
segregation, racialized restrictions on mobility similarly spread across the U.S. As Carpio
(2019) argues, “mobility and its management by state forces” was a key mechanism for the
racialization of Mexicans in 20th century California.
Some social network research has examined inter-metro mobility and segregation by fol-
lowing individual migration patterns (South and Crowder, 1998; Crowder, Pais, and South,
2012), but there remains no measure specific to cross-metro segregation. Prior research that
uses a national set of data instead of looking only at specific metro areas shows that in the
U.K., looking at national spatial and aspatial measures of local segregation shows complex
and varying threshold effects (Catney, 2018). In the U.S., using racial population distri-
butions across metro areas can demonstrate that some areas are disproportionately White
and/or Black, but historical patterns of racialized international migration or enslavement
likely explain many of those differences. In fact, scholarship on Black internal migration
indicates that the historical connection to southern spaces has led to a return migration for
some Black families out of the more heavily segregated North (Pendergrass, 2013; Robinson,
2014; Derenoncourt, 2022).
To complement the above scholarship, we build on economic history and urban studies
findings about city size patterns and geographic mobility and introduce a measure of national
level patterns of segregation that is not, at its core, a measure of neighborhood level or intra-
metro area segregation patterns. Following Gabaix (1999) and Dittmar (2011), we exploit
the fact that an urban system in which there is free mobility of labor and other factors of
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production will be coherent (defined in the next section) and follow Gibrat’s law, with city
population growth rates and variances that are independent of size. According to Fernholz
and Fernholz (2020), if a subpopulation within a larger urban system is coherent, then the
subpopulation city size distribution will follow a power law. Thus, a subpopulation city
size distribution that deviates from Zipf’s law, or a power law, implies a deviation from
Gibrat’s law and, potentially, restrictions on that subpopulation’s ability to migrate within
the larger urban system. Because our theoretical approach is general and can be applied to
any urban system, the idea of measuring mobility restrictions via deviations from Zipf’s law
and coherence could help scholars quantify the extent to which different racial, ethnic, or
other groups are able to freely migrate internally within different countries or across regions
of the world.
We apply our measure of mobility to the U.S. White and Black populations from 1910-
2020, and uncover a large and dramatic deviation from Zipf’s law for the U.S. Black popu-
lation in the early 20th century. This deviation is driven by slower Black population growth
rates for the U.S. cities with the largest Black populations, in violation of coherence and
Gibrat’s law but consistent with Jim Crow era laws used to restrict Black mobility. While
the U.S. White population city size distribution approximately follows Zipf’s law from 1910-
2020, the Black population only converges to a power law distribution in the middle part of
the 20th century, and then shows some movement away from coherence in the 21st century.
This latter change is quantitative evidence of the impact of the “reverse Great Migration”
of Black Americans back to the South on city population growth patterns. To investigate
whether this finding is the outcome of the expanding employment spatial mismatch identi-
fied by prior scholars, we examine the city population distribution for different U.S. income
groups in recent decades and find no evidence that the growing Black divergence is an arti-
fact of growing economic immobility. Instead, this divergence appears to be unique to the
disproportionate migration of Black people as part of the reverse Great Migration.
2 Power Laws, Zipf ’s Law, and City Size Distributions
A power law, or Pareto distribution, for the city size distribution refers to a log-linear
relationship between the population size of each city and its rank. Thus, if a city with
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population sihas population size rank rigiven by
log ri=β0+β1log si,(2.1)
where β0and β1<0 are constants, then the city size distribution follows a power law. If
β1=−1, then the city size distribution satisfies Zipf’s law. Note that (2.1) implies that the
slope of the log-log plot of city size versus rank will be a straight line with slope β1<0.
Following much of the literature, we use the terms power law and Zipf’s law interchangeably
since we are not concerned with the particular slope of the log-log plot of size versus rank,
as long as that slope is not far from minus one.
Soo (2005) documents that city size distributions in many different countries follow power
laws of the form (2.1) with a constant β1approximately equal to minus one. Dittmar
(2011) shows that historical metro area populations in Europe follow a power law after 1500,
once labor and capital mobility had risen and locational fundamentals had become less
important for city growth. Furthermore, scholars note that Zipf’s law holds most strongly
when an urban system (e.g., a province or nation) is coherent and co-evolving (Arshad,
Hu, and Ashraf, 2018). Research in this field has demonstrated that power laws hold in
contemporary U.S. census “places” (Eeckhout, 2004), “natural cities” (Jiang and Jia, 2011),
and metropolitan areas (Krugman, 1996; Gabaix, 1999).
We define a coherent urban system as one in which there is free mobility of labor and
other factors of production between cities or regions in the system. If city populations
follow power laws, this is evidence that those cities constitute a coherent urban system,
as Dittmar (2011) demonstrates for Europe after 1500. Conceptually, Zipf’s law should
apply for subpopulations of cities in a coherent system if those subpopulations are randomly
selected (Fernholz and Fernholz, 2020). For example, if a power law holds for the whole
metro population, it should hold for the male population and female population separately,
as the two do not segregate residentially. Similarly, if a power law holds for American metro
areas and racial groups have equal mobility among those metro areas, then power laws should
hold for each individual racial group as well. Using the insight that a power law city size
distribution is evidence of coherence, we argue that if subpopulations have similar or distinct
city size distributions, then this can determine whether those subpopulations have similar
or different levels of urbanization and coherence.
Importantly, the specific rank order of cities is irrelevant to this application of Zipf’s
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law to subpopulations. That a given city may rank high for one subpopulation and low
for another will not affect the urban system’s coherence, as long as the distributions of
subpopulations across metro areas follow power laws. For example, Memphis is only the
43rd largest metro area overall, but it has the eleventh largest Black population in the U.S.,
thanks in part to the recent “return migration” of Black Americans from the Northeast and
Midwest (Pendergrass, 2013). On the other hand, while Boston has the tenth largest U.S.
metro area in total population, it only ranks 21st in terms of Black population. For our
purposes, it does not matter whether Boston or Memphis’ rank order is consistent across
racial groups, just that the overall distribution for an individual racial group reflects a power
law.
We argue that we can identify restrictions on Black mobility within the U.S. not only
by examining local histories of inclusion or exclusion, but also by seeing whether or not the
Black population adheres to Zipf’s law across metro areas to the same extent as the White
population. Below, we illustrate how Zipf’s and Gibrat’s laws enable a bird’s eye view of
racial restrictions on geographic mobility across an urban system.
3 Theory
Suppose that there are i= 1, . . . , N cities, and that the population size of each city sievolves
according to
log si(t+ 1) −log si(t) = µ+σBi(t),(3.1)
where σ > 0 and each Biis independent and normally distributed with mean zero and
variance one. In the setup (3.1), both the average and variance of city population growth
rates are equal to each other and do not depend on size or any other distinguishing factor.
We refer to an urban system with city population growth rates and variances that are
independent of size as in (3.1) as a coherent system. Note that a coherent urban system
follows Gibrat’s law (Gibrat, 1931).
A coherent urban system is one in which there is free mobility of labor and other factors of
production and city population growth is subject to random fluctuations caused by changes in
city-specific amenities that do not vary on average across city characteristics. Such an urban
system is unified and coheres, with free movement of people across different cities within
the system. Gabaix (1999), Rossi-Hansberg and Wright (2007), and Dittmar (2011) present
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models of coherent urban systems with no labor mobility restrictions and no distortions
and show that city size dynamics like (3.1) endogenously emerge in the equilibrium of their
models. These models establish the equivalence of free mobility of factors of production and
Gibrat’s law (population growth rates and variances that are independent of size), both of
which are, therefore, equivalent definitions of coherence.
Let s(k)(t) denote the population size of the k-th largest city at time t, so that
s(1)(t)≥s(2)(t)≥ · · · ≥ s(N)(t),(3.2)
for all t. We use the parameters k= 1, . . . , N to denote the rank of cities. Let s∗
(k)denote
the size of the k-th largest city in the stationary city size distribution. Fernholz and Fernholz
(2020) show that the model (3.1), when representing the top Nranks of a larger system,
yields a stationary city size distribution that follows
Elog s∗
(k)−log s∗
(k+1)≈ −α(log(k)−log(k+ 1)) ,(3.3)
where α > 0 is a constant and the approximation becomes more precise as kincreases. Recall
that a linear relationship between log city size and log rank represents a power law. Thus,
the result (3.3) implies that a coherent urban system will have a stationary power law city
size distribution.
Fernholz and Fernholz (2020) extend this result in two directions. First, they show that
an urban system that follows a power law and has equal population growth rates must also
be fully coherent and follow Gibrat’s law with city size dynamics as in (3.1). Thus, if city
population growth rates do not depend on size, then the presence of a power law distribution
indicates coherence. Second, they explain that (3.3) also describes the stationary city size
distribution for any subpopulation with city growth dynamics as in (3.1). Any subgroup of
the full population with similar behavior should also follow the same city size distribution
as the full population. In a coherent urban system, then, any subpopulation without unique
mobility restrictions should follow a power law roughly as well as the total population.
A natural extension of (3.1) posits that city population growth rates and variances may
vary depending on the size of a city. This is equivalent to a model in which the size of each
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city sievolves according to
log si(t+ 1) −log si(t) = µri(t)+σri(t)Bi(t),(3.4)
where ri(t) = kif city iis the k-th largest city at time t. Note that this definition implies
that ri(t) denotes the size-rank of city iat time t. The rank-based variation in growth rates
and volatilities in (3.4) captures the real-world possibility that labor and capital mobility
are imperfect, and that individuals may not be able to costlessly move themselves or their
capital to the most desirable cities. In urban systems like this that are not coherent, we
might expect that the growth rates of the largest cities are on average below those of smaller
cities, in violation of Gibrat’s law. Systematically, then, this argues that constraints on labor
mobility would disproportionately affect the growth rate of larger cities below their expected
size given a power law distribution. The same would be true of subpopulations.
Fernholz (2017) shows that if the rank-based growth rate parameters µ1, . . . , µNare such
that higher-ranked, larger cities grow more slowly than lower-ranked, smaller cities, then the
model (3.1) yields a stationary city size distribution that is no longer a power law with a
linear log-log plot of city size versus rank.
Proposition 3.1. Suppose that the rank-based growth rates µ1, . . . , µNsatisfy
µ1< µ2<· · · < µN,(3.5)
and the rank-based variance parameters satisfy σk=σ > 0, for all k= 1, . . . , N . Then, the
stationary city size distribution of the model (3.4) satisfies
Elog s∗
(k)−log s∗
(k+1)≈ −α(k) (log(k)−log(k+ 1)) ,(3.6)
where α(k)>0is a strictly increasing function of rank k.
We refer the reader to Fernholz (2017) for the proof of Proposition 3.1, as well as a formal
characterization of the approximations in (3.3) and (3.6). This proposition implies that a
version of the model (3.4) in which lower-ranked, smaller cities have higher average growth
rates will have a stationary city size distribution in which the slope of the log-log plot of
city size versus rank is steeper at lower ranks. Thus, if some factor restricts the growth of
larger cities relative to smaller cities, in violation of coherence and Gibrat’s law, then the
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city size distribution will deviate from a power law in such a way that large, higher-ranked
cities will be smaller than a power law predicts. As before, this is true for the full population
city size distribution as well as for subpopulation city size distributions. In sum, if mobility
for a (sub)population is constrained across metro areas, then the pattern of deviation from
a power law distribution should be that the largest cities have a smaller population size
than expected. Thus, with regard to racial population distributions, we hypothesize that
the restricted mobility of Black Americans in the early 20th century would reduce the size
of the largest Black metro areas.
4 Data
Data for this project come from U.S. decennial censuses and the American Community
Survey (ACS). The U.S. Census Bureau first identified metropolitan areas for the 1950
census. As part of their reports on demographic trends in metropolitan areas from the 1960
census, the bureau released reports using the 1960 Census demarcations of metropolitan
areas for 1910-1960. We use those reports for data through 1960. For each census after 1960,
we use the contemporary U.S. Census demarcations of metropolitan areas and data from
the Census Bureau’s decennial Summary File 1. Any error from including overly rural areas
from 1910-1950 in metro areas as defined by 1960 should bias our estimates toward Zipf’s
law for the Black population, as Black Americans disproportionately lived in rural areas in
the first half of the century.
As a supplementary analysis of our findings from the more recent Census Bureau data,
we also analyze ACS data on population distributions for different income levels. These data
use the 2008-2012 and 2016-2020 five-year ACS estimates for metro areas. The 2008-2012
ACS estimates best correspond to our results from the 2010 Census, while the 2016-2020
ACS estimates are the closest available estimates at the time of writing for the 2020 Census.
In order for a coherent system that follows Gibrat’s law such as (3.1) to yield a sta-
tionary city size distribution, it is necessary to introduce a minimum value for city size or
a city birth/death process (Gabaix, 2009), or to model only the top Nranks of a larger
urban system (Fernholz and Fernholz, 2020). Thus, in all figures and tables we present, we
restrict our analysis to those metros with populations equal to at least 1% of the largest city
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population for each group.1In each case, we are looking at the top N— with Nvarying
across decades — ranks of a larger urban system.
5 Empirical Evidence
Figure 1 plots the log population of U.S. metro areas (as a fraction of the total) versus their
log population-rank in 1910, 1940, 1980 and 2020 (we limited all figures to only 4 plots for
readability). Recall that a linear relationship between log city size and log rank represents
a power law, or Zipf’s law. According to Figure 1, the U.S. metro size distribution has
approximately followed Zipf’s law throughout the last century, although the distribution did
become slightly more curved in 2020.
Figure 2 presents a similar log-log plot of metro population versus rank for 1910 (be-
fore the Great Migration) and 1950 (near the beginning of the second wave of the Great
Migration), but considers the White and Black U.S. populations separately. According to
the figure, the distribution of Black metro populations deviated substantially from Zipf’s
law in 1910, with only the very largest, top 10-20 ranked metros following a straight line in
the plot. By 1950, however, the distribution of Black metro populations follows the straight
line pattern of a power law much more closely, with almost all of the top 100 ranked met-
ros generating a relatively linear log-log plot. In contrast, the distribution of White metro
populations is close to a straight line at all ranks in both 1910 and 1950.
Figures 3 and 4 more closely examine this transition towards Zipf’s law for U.S. Black
metro populations through the 20th century. According to Figure 3, the log-log plot of Black
metro populations versus rank consistently grew more linear from 1910-1960, with the 1960
distribution following the straight line pattern of Zipf’s law for nearly 100 ranked metro areas.
There is much less change in Figure 4, which shows the Black metro population distribution
from 1970-2020. The different log-log plots in this figure are much closer together than those
in Figure 3, which suggests that the transition of the Black metro size distribution that
occurred in the early part of the 20th century was largely finished by the later part of the
century.
In Figure 5, we plot the average growth rates of different population-ranked U.S. cities for
Black and White populations from 1910-1930 and 1930-1950. These multi-decade average
1In the appendix, we instead restrict results to metro areas with 0.1% of the largest city population.
10
Rank
Population Share
12510 20 50 100 200
0.001 0.01 0.1
1910
1940
1980
2020
Figure 1: U.S. city size distribution, 1910-2020.
Rank
Population Share
12510 20 50 100 200
0.001 0.01 0.1
1910 White
1910 Black
1950 White
1950 Black
Figure 2: U.S. White and Black city size distributions, 1910-1950.
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Rank
Population Share
12510 20 50 100 200
0.001 0.01 0.1
1910
1930
1940
1960
Figure 3: U.S. Black city size distribution, 1910-1960.
Rank
Population Share
12510 20 50 100 200
0.001 0.01 0.1
1970
1990
2000
2020
Figure 4: U.S. Black city size distribution, 1970-2020.
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growth rates are constructed by ranking each city by population size at the start of each
decade, calculating the (natural) log growth rate of each ranked city over the ensuing decade,
removing any aggregate population growth so that the rank growth rates add up to zero,
and smoothing the resulting estimated rank growth rates using a Guassian kernel smoother
with a range of half of the total ranks for the corresponding decade (Fernholz, 2017). Finally,
these single-decade estimates of city population growth rates are then averaged across two
decades to construct the multi-decade estimates shown in the figure.2
Figure 5 shows a clear deviation from coherence and Gibrat’s law for Black city population
growth rates from 1910-1930, with the largest Black cities growing more than 10% slower on
average than the smallest Black cities over these two decades. This substantial variation in
growth rates across different ranks for the Black subpopulation diminishes substantially and,
in fact, larger cities grew faster than expected between 1930-1950 at the height of the Great
Migration. In contrast, the figure shows that White city population growth rates for both
1910-1930 and 1930-1950 conform fairly closely to Gibrat’s law, with much less variation in
average log growth rates across different ranks.
The pattern of slower population growth for the largest Black cities from 1910-1930 shown
in Figure 5 aligns with the hypotheses of Proposition 3.1. In addition, the deviation from
a power law for Black metro populations in 1910 as well as, to a lesser extent, in 1930 and
1940 (Figures 2-3) closely matches with the predictions of Proposition 3.1. According to the
proposition, if some factor — here, racist policies designed to restrict Black mobility out of
the Jim Crow South — constrains the growth of the largest cities within an urban system,
then the resulting city size distribution will feature a log-log plot of city size versus rank in
which the slope is less steep for top-ranked, larger metros. Therefore, when interpreted via
Proposition 3.1, Figures 2, 3, and 5 demonstrate that the U.S. Black city size distribution in
the early 20th century deviated from a power law because population growth in the largest
Black cities was far lower than expected given the growth in the smallest Black cities during
this period. This models the substantial constraints on the ability of the Black population to
freely move from one U.S. metro area to another in the first decades of the 20th century. Such
mobility constraints dampened the growth rate of the largest, most desirable Black metro
areas during this period, and as Proposition 3.1 reveals, led to a non-coherent and curved size
distribution for the Black population. The higher than expected growth between 1930 and
2We show multi-decade growth rates for ease of presentation and interpretability. Figures for single
decade growth rates are in the appendix, and the results are substantively similar.
13
050 100 150
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1910-1930 Black
1910-1930 White
1930-1950 Black
1930-1950 White
Figure 5: Growth rates of different size-ranked U.S. White and Black cities, 1910-1950.
1950 in the largest Black metros, then, could be interpreted as the normal, expected growth
during those decades combined with the delayed in-migration due to constraints slowing
Black mobility in the prior decades.
Figures 6 and 7 plot smoothed average growth rates of different ranked Black and White
U.S. city populations from 1950-1990 and 1990-2020. In contrast to Figure 5, Figures 6 and
7 show that both the Black and White population growth rates follow Gibrat’s law more
closely from 1950-2020. Even though there were slightly lower population growth rates for
the very largest Black cities from 1970-1990, these figures suggest that the constraints on
mobility within the U.S. faced by Black Americans in the early 20th century were mostly
gone by the second half of the century.
Dittmar (2011) introduces a method for quantifying the changes in size distributions that
we observe in Figures 1-4. Let sidenote the population of city iand ridenote the population
rank of city i. Following Dittmar (2011), we estimate the regression
log ri=β0+β1log si+i,(5.1)
14
050 100 150
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1950-1970 Black
1950-1970 White
1970-1990 Black
1970-1990 White
Figure 6: Growth rates of different size-ranked U.S. White and Black cities, 1950-1990.
050 100 150 200 250 300
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1990-2010 Black
1990-2010 White
2010-2020 Black
2010-2020 White
Figure 7: Growth rates of different size-ranked U.S. White and Black cities, 1990-2020.
15
using the non-parametric estimator based on the median developed by Theil (1950). For
this estimator, the slope coefficient β1is calculated as the median of all slopes generated by
pairs of data points. Correspondingly, the constant β0is calculated as the median over all
data points iof
log ri−ˆ
β1log si,(5.2)
where ˆ
β1is the median slope estimator of β1.
The median regression estimator of Theil (1950) is designed specifically for applications in
which outliers are potentially significant, and provides less biased estimates in such scenarios.
It also performs well in small samples. Furthermore, the median estimator is competitive
with OLS estimates of β0and β1in environments in which there are no major outliers
(Dittmar, 2011). Given these properties and the substantial deviations from a power law
that we observe in Figures 2 and 3, the median estimator of Theil (1950) is particularly
well-suited for our application.
The first three columns of Table 1 report the mean squared errors (MSE) in percent
between the actual metro size distributions and the estimated power law distributions (5.1)
using the median estimator for U.S. White and Black populations for each decade from 1910-
2020. In this table, the actual and estimated metro size distributions are constructed using
only those metros with populations equal to at least 1% of the largest city population for each
group. The last three columns of Table 1 report the number of cities with populations above
this 1% threshold from 1910-2020. This table shows a dramatic decline in the deviation
between the actual distribution and the estimated power law distribution for the Black
population from 1910-1960. The mean squared errors for Black metro populations are similar
to those for White metro populations after 1970, except for 2010 and 2020 which we discuss
below.
Adherence to Gibrat’s law and Zipf’s law as measured by Figures 1-7 and Table 1 demon-
strates the coherence of an urban system for a given population. Confirming prior research,
our analyses show the U.S. as a whole is a coherent urban system that follows Gibrat’s law
with approximately equal population growth rates across different ranked cities. Coherence
also holds for the White population, which has historically had relatively free domestic geo-
graphic mobility across all the years in our data. On the other hand, coherence only became
true for the Black population after the Great Migration, and did not reach parity with White
coherence until 1970.
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All White Black All White Black
1910 1.1 1.8 95.9 154 131 148
1920 1.2 1.3 88.9 154 141 156
1930 0.9 1.0 31.5 153 142 132
1940 0.9 1.1 25.9 156 147 119
1950 1.3 1.5 14.1 172 169 120
1960 2.5 2.7 9.2 184 184 120
1970 3.9 4.1 3.4 205 209 118
1980 2.8 2.9 2.6 181 225 117
1990 3.1 3.4 3.8 197 261 129
2000 3.2 4.2 5.1 203 276 133
2010 3.9 5.0 8.6 221 299 153
2020 4.6 5.9 10.8 223 313 161
Table 1: First three columns: Mean squared errors (%) relative to estimated power law
distributions for U.S. White and Black populations, 1910-2020. Last three columns: Number
of cities with populations greater than or equal to 1% of largest city population for U.S. White
and Black populations, 1910-2020.
Here, the extent of racist labor mobility restrictions during Jim Crow is evident. While
the mean squared error for the total population and the White population are quite close
to zero throughout the 20th century, the same is not true of the Black MSE. The deviation
starts at nearly 100 in 1910, remains nearly that high in 1920 before a dramatic fall between
1920 and 1930 from 88.9 to 31.5 and then a steady decline through 1970 at which point it is
slightly lower than the White MSE (3.4 compared to 4.1). Not only was the Great Migration
a move north, it was also an urbanization of large portions of the Black population. For
example, Detroit boomed during that period, growing from less than 500,000 residents (of
whom roughly 6,000 were Black) in 1910 to 1.6 million in 1930, of whom over 120,000 were
Black, increasing the Black population of Detroit from roughly 1% of the total population to
nearly 8%. While Black residents remained a minority population in these cities — primarily
because of the continued White migration to these same urban economies from both Europe
and the South — they became part of a coherent urban economic system, even in the face
of southern efforts to restrict their migration north.
As mentioned earlier, we restricted the sample for each analysis to metro areas with at
17
least 1% of the population of the largest metro area. For example, for our analysis of 2020
MSAs, this means that we only include MSAs that had a population of at least 201,405
(1% of the population of New York, the largest MSA in 2020). In our analysis of 2020
White populations, each MSA had to have at least 87,122 White residents (again, 1% of the
population of the New York MSA). The last three columns of Table 1 report the number of
metros included in the mean squared error calculations from the first three columns of this
table. Thus, this table shows how many metro populations are equal to at least 1% of the
largest city population for U.S. White and Black populations from 1910-2020. This provides
a measure of the size of the coherent system under study.
Given the differences in Black and White total population and the historical difference in
mobility constraints, it is not surprising that the White urban system typically has a larger
footprint, save 1910 and 1920. This may be an artifact of the inaccuracies of the Census
Bureau’s backdating of metro areas, but it also may indicate that Jim Crow White America
may have constituted multiple, overlapping urban systems that did not merge into one until
after the Great Depression or World War II. On the other hand, the Black urban system saw
itself shrink in extent before growing in the late 20th century. This is likely caused by the
transition out of the post-Civil War, predominantly agrarian and highly constrained system
into the post-World War II urban system. We also note here that the White urban system,
starting in 1970, is far larger than the total population system. Here, we theorize that this
captures the fact that the U.S. population in total is a mix of racialized urban systems with
different loci but all share New York as the site of their largest population.3
Table 1 also shows that, beginning in 1990, the mean squared error for a power law started
rising for both Black and White Americans. This rise is larger for the Black population, with
the MSE growing from 5.1% in 2000 to 8.6% in 2010 to 10.8% in 2020. This rise could indicate
decreased coherence, however, it also could be a limitation of using cross-sectional data to
model an ongoing process of spatial realignment within a coherent urban system.4
Turning back to the figures, there is a small but noticeable decline in the 2020 population
shares of the top two metros for Black residents. New York had the largest Black population
3In the appendix, we instead restrict results to metro areas with 0.1% of the largest population. In this
analysis, the smallest cities give a clear curve to the log-log plot of size versus rank, indicating that the
populations do not fit a power law distribution when including metro areas with small populations of a given
racial group. See Figures A-1 and A-2 in the appendix.
4For a similar point about neighborhood diversity and segregation processes see Bader and Warkentien
(2016).
18
of all metros from 1990-2020, but its growth stalled between 1990 and 2000 and it lost roughly
100,000 Black residents between 2000 and 2010. New York’s stalled growth and decline in
comparison to the growth of the total Black population led to its Black population being well
below expectations for its status as the top rank in a power law distribution. The second
largest metro area in 1990 and 2000 was Chicago and it saw the same pattern — slow growth
between 1990 and 2000 and then a slight decline between 2000 and 2010. In fact, Chicago’s
decline was large enough that it was supplanted in the rankings by the growing Atlanta
MSA in 2010. Similar patterns exist for other mid-Atlantic and midwestern metro areas,
and Atlanta is joined by Miami, Houston, and Dallas with fast growing Black populations.
We hypothesize that the growing deviation from the median estimator from 1990-2020 is
not driven by a loss of spatial mobility for Black residents. Instead, this change is a realign-
ment of the Black population from northern postindustrial cities toward growing southern
economies that reflects negative social experiences in northern cities and that is not yet
complete and not purely economic in origin (Pendergrass, 2013; Robinson, 2014; Derenon-
court, 2022). Similar patterns of stagnation are also apparent among the White population
in major northern and midwestern MSAs, but that stagnation is not matched by significant
growth in southern MSA White populations and thus it did not lead to a similarly large
increase in deviation from the median estimator for the White distribution.
However, the above argument does not explain why both White and Black populations
saw increased deviations from a power law since 1990. That both populations saw increased
deviations indicates that, along with the realignment for Black populations, a common con-
straint on mobility for all residents may be reducing the population growth rate of the largest
cities relative to smaller cities.
One potential constraint on mobility in recent decades is the growing crisis of afford-
ability for working class people in metros like New York and Los Angeles. This constraint
affects both White and Black populations, since it impedes all lower-income Americans’
ability to freely move to different regions of the country. However, because class is a racial-
ized category, the affordability crisis has racially unequal impact on mobility opportunities.
For example, the subprime mortgage market disproportionately targets Black borrowers for
higher rates. These higher rates imply higher costs and less equity, compounding the impact
of the housing bubble on Black families compared to White ones (Rugh and Massey, 2010;
Ghent, Hernandez-Murillo, and Owyang, 2014; Loya and Flippen, 2021). Thus, if a trigger
19
for moving was an underwater loan or inability to afford housing, that would lead to a greater
migration of Black residents compared to White residents, especially in highly segregated
and expensive metros like New York and Chicago (Rugh, Albright, and Massey, 2015). This
unequal impact is consistent with the larger increase in MSE for the Black population in
recent decades as shown in Table 1.
In Figures 8 and 9, we show the log-log plot of metro population versus rank, for different
income groups, constructed using ACS data for 2008-2012 and 2016-2020. If affordability is
the primary cause of the decline in adherence to Zipf’s law, low-income households should
show more deviation from a power law. Instead, the results show a clear difference in the
slope of the power law for low-income versus high-income populations in both figures. Low-
income households (those earning under $50,000 per year) are less concentrated in the largest
metros than high-income households (those earning over $200,000) and are dispersed among
a much larger set of cities. In other words, low-affordability and high-wealth cities in the
21st century have fewer low-income households than we would expect if they had the same
distribution as high-income households.
To further investigate the extent of deviation from a power law for different income
populations, following the procedure used to generate Table 1, we report the MSE between
the actual metro size distributions and the estimated power law distributions (5.1) using the
median estimator for different U.S. income groups in 2008-2012 and 2016-2020 in Table 2.
Unlike our measures of racial population distribution, Table 2 shows that no income group has
an MSE substantially different from that of the total population. That is, at low, medium,
and high income levels, labor mobility appears to be coherent, even though the power law
distributions for low- and high-income households are distinct. This may indicate that labor
mobility is fairly universal across U.S. income groups, but that the specific urban systems of
low- and high-income households are different, possibly due to housing affordability concerns.
Fernholz and Fernholz (2020) show that one potential cause of a flatter power law dis-
tribution curve, such as those shown in Figures 8 and 9 for low-income groups, is a lower
common volatility of city population growth. This corresponds to a lower value for the
parameter σfrom (3.1). A lower population growth volatility σfor low-income households
relative to high-income households implies less movement across cities over time for low-
income households, and hence a less dynamic, but still coherent, urban system. This type of
stagnant urban system is different from the incoherent system we observe for the Black pop-
20
Rank
Population Share
12510 20 50 100 200
0.001 0.01 0.1
Under $25,000
$25,000-$50,000
$100,000-$200,000
Over $200,000
Figure 8: U.S. city size distribution for different income groups, 2008-2012.
Rank
Population Share
12510 20 50 100 200
0.001 0.01 0.1
Under $25,000
$25,000-$50,000
$100,000-$200,000
Over $200,000
Figure 9: U.S. city size distribution for different income groups, 2016-2020.
21
2008-2012 2016-2020 2008-2012 2016-2020
All 4.3 5.0 235 239
Under $25,000 2.7 3.4 303 298
$25,000-$50,000 3.3 4.0 323 345
$50,000-$100,000 4.4 5.2 265 287
$100,000-$200,000 4.3 4.7 171 190
Over $200,000 3.7 4.7 104 113
Table 2: First two columns: Mean squared errors (%) relative to estimated power law
distributions for different U.S. income groups, 2008-2012 and 2016-2020. Last two columns:
Number of cities with populations greater than or equal to 1% of largest city population for
different U.S. income groups, 2008-2012 and 2016-2020.
ulation in the first part of the 20th century, since the latter implies constraints on population
movements to larger, more desirable cities. This stagnant urban system is also distinct from
the growing deviation we observe for the Black population in the 21st century, where the
most populous metropolitan areas are losing population to fast-growing southern cities, as
the theorized lower volatility would not lead to an increased MSE.
In sum, while different U.S. income groups may have different population growth volatil-
ities, thus explaining the different sloped population distribution curves for these groups
(Figures 8 and 9), all U.S. income groups appear to share a similar level of coherence. Black
and White Americans, on the other hand, do not share a similar level of coherence; com-
pared to a power law distribution, the Black population is now underrepresented in the
largest metro areas but overrepresented in the moderately large metro areas. It is unclear if
this is due to a temporary transition to a different population-rank order of cities.
We note, however, that it would take a shift of over one million for New York to be
replaced as the largest metro area via migration to Atlanta (equivalent to a Black population
increase of nearly 50%) or migration out of New York (a 35% decline in its Black population)
for that transition to occur. At the current rates of change, it would take at least 30 years
for the two cities to switch ranks. Thus, we expect the observed deviation to persist for
many years to come. Alternatively, the current Black population distribution may represent
a stable deviation from a power law that reflects Black Americans’ greater likelihood of
making residential mobility decisions in response to contemporary racism and the legacy
of prior generations’ restricted labor mobility rather than in response to purely economic
22
factors. Future research should continue to explore the causes and consequences of this
growing deviation from the expected Black population distribution.
6 Discussion
Scholars studying racial segregation have long appreciated the importance of scale to un-
derstanding residential segregation. Primarily, that appreciation has focused on problems
of identifying the local neighborhood and of incorporating spatial dependence into measures
of residential segregation (Barros and Feitosa, 2018; Dean et al., 2019; Kramer, 2017, 2018;
Lee et al., 2008; Wong, 1993). We advance an alternative methodological appreciation for
the scale of residential segregation — we focus on segregation across metro areas instead
of segregation between neighborhoods within a metro area. Expanding on scholarship that
shows that metro areas in a coherent urban system follow power law population distribu-
tions, we argue that the same should be true of subpopulations of those metro areas, if those
subpopulations have the same mobility as other area residents. Next, we apply that insight
to racial populations in the U.S. and explore both the level of coherence and the size of the
U.S. urban system for White and Black Americans. Here, we discuss the implications of
our findings that both groups in the U.S. generally do demonstrate a coherent urban system
after 1970, though the sizes and scopes of those systems vary, and focus on other potential
applications of using Zipf’s law and Gibrat’s law as a test of urban system coherence.
Generally, our analysis shows that the U.S. urban system moved toward greater and
shared racial coherence during the 20th century as expected, and that racial differences re-
emerged as the Black population distribution shifted away from a power law distribution in
the first part of the 21st century. We argue that this reflects the stagnation of the Black
population in large northern and midwestern cities and the reverse migration to growing
southern metro areas. Similar stagnation did not show the same impact on White population
distributions because it is a less intense reshuffling of White American residential patterns.
While we demonstrate that adherence to Zipf’s law can identify the extent and coherence
of an urban system for subpopulations, we do note some limitations. First, deviations
from Zipf’s law are not inherently evidence of an incoherent urban system. The previous
paragraph hypothesizes that the growing deviation in the 21st century was not because the
urban system of Black Americans was growing less coherent, but rather that a snapshot of
23
a gradual transition from one system to a new one will appear to be incoherent. This is no
different from concerns for other measures of segregation in which single time measures of
areas undergoing gentrification or racial turnover can, depending on how far into the process
it is, appear to be evidence of integration (Bader and Warkentien, 2016). Second, it is not
clear what threshold of mean squared error from a predicted power law should be used to infer
coherence. Nor is it clear what minimum city size threshold should be used for inclusion in
such analyses of city population distributions and coherence. As the appendix demonstrates,
the lower end of the city size distribution can radically alter a distribution that otherwise
fits a power law well. Instead of picking specific thresholds as arbitrary cutoffs, we chose to
explore multiple thresholds and focus on comparisons across time and subpopulations rather
than comparisons to some precise cutoff between coherent and incoherent systems.
According to theory, a frictionless urban system should have no deviation from a power
law (Gabaix, 1999; Rossi-Hansberg and Wright, 2007). Externally imposed constraints like
vagrancy laws in the 20th century restricted mobility and caused Black Americans to take
decades longer to enter a coherent urban system than the rest of the country’s population.
However, constraints do not have to be explicitly focused on limiting labor mobility. Future
research could use the deviations of individual metro areas from their predicted population
size to explore the impact of non-explicit constraints on labor mobility, such as the extent
of zoning regulation (Trounstine, 2018), segregation by housing type or cost (Owens, 2019),
social network effects (Krysan and Crowder, 2017), or legacies of local historical racial regimes
(Baker, 2022).
In this paper, we focused on internal migration in the U.S. and racial differences in
adherence to Zipf’s law because there were documented histories of legal and extra-legal
efforts to restrict Black mobility in the 20th century, making this a good case study to
demonstrate that city size distributions can capture subpopulation mobility differences. As
with other measures of residential racial segregation, this method of analyzing city size
distributions can be used to study any type of subpopulation, not only racial or ethnic
groups. For example, scholars interested in following the possible rise of the “creative class” in
cities could use this method to determine whether or not creative class jobs are in a coherent
urban system, or if they are overly represented in a few “winner” cities. If distributions show
divergence from a power law distribution, that would identify not only topics of interest for
understanding what is and is not part of a coherent urban system, but also where a particular
24
population is over- or under-represented compared to expectation. Nor is this U.S.-specific;
an analysis of distributions of refugee populations either across or within countries could
reveal what context and other factors create more or less coherent urban systems of migration
for that unique political status. Alternatively, exploring the urban system coherence of other
marginalized populations, such as Roma in Europe or Uyghurs in China, could demonstrate
whether or not the antebellum South’s efforts to restrict Black out-migration were unique. In
short, any subpopulation should have equally coherent city distribution patterns. Differences
in coherence, we argue, are evidence of differences in mobility.
Before the Civil Rights Movement, Whites limited Black mobility through legal and
extralegal methods (Anderson, 2016). While legal efforts to enforce segregation and limit
Black mobility were reversed in the 1960s, scholarship on segregation within metro areas
provides overwhelming evidence that such efforts did not ameliorate segregation and, if
anything, segregation grew after the Civil Rights Movement (Charles, 2003). In this paper,
we shift further outward, to study segregation across metro areas instead of within them.
We find that the Great Migration was not just a movement north by Black Americans, but
a movement into a coherent urban system, largely catching up with the White American
urban systems in coherence, if not in scope. However, in the 21st century, there is evidence
that the reverse great migration is shifting those coherent urban systems. By recognizing
that a co-evolving urban system should also co-evolve for subpopulations if all residents are
equally able to migrate, scholars can not only demonstrate that city size distributions adhere
to Zipf’s law, but use that to test for whom and to what extent an urban system is coherent
and co-evolving.
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Appendix
In this appendix, we recreate Figures 1 and 2 using those cities with populations equal to
at least 0.1% of the largest city population for U.S. White and Black populations. Figure
A-1 shows a steep decline in population shares for the smallest cities in all years, which is
clearly inconsistent with the linear relationship between log size and log rank of a power
law distribution. The contrast between the approximately straight lines in Figure 1 and the
sharp bends in the lines in Figure A-1 is striking. We find a similar pattern in Figure A-2,
with a clear contrast between the curves in this figure at lower ranks and the approximately
linear relationship of Figure 2.
We also recreate Table 1 using the lower population threshold of 0.1%. The first three
columns in Table A-1 show that the mean squared errors for U.S. White and Black popula-
tions using this lower threshold are much larger than the errors reported in Table 1 using the
1% population threshold. This result is consistent with the highly non-linear relationships
between log population share and log rank shown in Figures A-1 and A-2.
Finally, Figures A-3-A-8 show the average growth rates of different size-ranked cities for
each decade from 1910-2020 for U.S. White and Black populations. The decade-by-decade
growth rates shown in these figures are broadly similar to the multi-decade growth rates
shown in Figures 5-7.
29
All White Black All White Black
1910 8.7 9.8 95.3 204 202 160
1920 7.6 12.1 94.5 205 205 172
1930 7.1 9.9 96.7 207 207 183
1940 8.5 9.1 96.0 209 209 185
1950 5.1 5.6 77.1 209 209 189
1960 5.1 4.9 41.8 209 209 181
1970 6.7 6.6 40.3 243 243 208
1980 4.3 5.3 23.5 331 330 251
1990 4.6 5.7 26.3 366 365 274
2000 4.6 6.0 31.6 366 365 294
2010 5.4 7.3 35.7 363 362 314
2020 7.2 9.5 40.0 384 384 344
Table A-1: First three columns: Mean squared errors (%) relative to estimated power law
distributions for U.S. White and Black populations using all cities with populations equal to
at least 0.1% of the largest city population, 1910-2020. Last three columns: Number of cities
with populations greater than or equal to 0.1% of largest city population for U.S. White and
Black populations, 1910-2020.
30
Rank
Population Share
12510 20 50 100 200
0.001 0.01 0.1
1910
1940
1980
2020
Figure A-1: U.S. city size distribution using all cities with populations equal to at least 0.1%
of the largest city population, 1910-2020.
Rank
Population Share
12510 20 50 100 200
1e-04 0.001 0.01 0.1
1910 White
1910 Black
1950 White
1950 Black
Figure A-2: U.S. White and Black city size distributions using all cities with populations
equal to at least 0.1% of the largest city population, 1910-1950.
31
050 100 150
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1910-1920
1920-1930
1930-1940
1940-1950
Figure A-3: Growth rates of different size-ranked U.S. White cities, 1910-1950.
050 100 150 200
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1950-1960
1960-1970
1970-1980
1980-1990
Figure A-4: Growth rates of different size-ranked U.S. White cities, 1950-1990.
32
050 100 150 200 250 300
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1990-2000
2000-2010
2010-2020
Figure A-5: Growth rates of different size-ranked U.S. White cities, 1990-2020.
020 40 60 80 100 120
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1910-1920
1920-1930
1930-1940
1940-1950
Figure A-6: Growth rates of different size-ranked U.S. Black cities, 1910-1950.
33
020 40 60 80 100 120
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1950-1960
1960-1970
1970-1980
1980-1990
Figure A-7: Growth rates of different size-ranked U.S. Black cities, 1950-1990.
050 100 150
-0.10
-0.05
0.00
0.05
0.10
Rank
Log Growth Rate
1990-2000
2000-2010
2010-2020
Figure A-8: Growth rates of different size-ranked U.S. Black cities, 1990-2020.
34
