The Comparative Advantage of Cities∗
Donald R. Davis†

Jonathan I. Dingel‡

Columbia University and NBER

Chicago Booth and NBER

September 23, 2019

Abstract
What determines the distributions of skills, occupations, and industries acrosscities? We develop a theory to jointly address these fundamental questions about
the spatial organization of economies. Our model incorporates a system of cities, theirinternal urban structures, and a high-dimensional theory of factor-driven comparativeadvantage. It predicts that larger cities will be skill-abundant and specialize in skillintensive activities according to the monotone likelihood ratio property. We test themodel using data on 270 US metropolitan areas, 3 to 9 educational categories, 22 occupations, and 19 industries. The results provide support for our theory’s predictions.
JEL Classification: F11, F14, R12, R13∗

Thanks to Rodrigo Adao, Arnaud Costinot, Gilles Duranton, Vernon Henderson, Corinne Low, Joan
Monras, Gianmarco Ottaviano, Keeyoung Rhee, John Romalis, Bernard Salanié, Kurt Schmidheiny, Will
Strange, Bruno Strulovici, Daniel Sturm, Matt Turner, Jonathan Vogel, and seminar audiences at Barcelona
GSE, Berkeley, CURE, NBER ITI, NBER URB, New York Fed, NYU Stern, Penn State, Philadelphia Fed,
Princeton IES Summer Workshop, UCLA, and the Urban Economics Association for helpful comments andsuggestions. We thank Dylan Clarke, Yuxiao Huang, and especially Luis Moreira da Costa and Antonio
Miscio for research assistance. Dingel thanks the Institute for Humane Studies, the Program for Economic
Research at Columbia University, and the Kathryn and Grant Swick Faculty Research Fund at the Universityof Chicago Booth School of Business for financial support.
†
drdavis@columbia.edu‡
jdingel@chicagobooth.edu

1Introduction

The level of economic activity varies considerably across cities. This motivates the studyof the agglomeration forces that give rise to concentration. The composition of economic
activity also varies considerably across cities. This is evident from Forbes ranking “America’s
smartest cities” and the use of place names like Wall Street and Silicon Valley as shorthandfor industries. Who lives in more productive cities and what they do there should inform
our understanding of how agglomeration economies work. But most theories of cities, to theextent that they consider the spatial distributions of skills and sectors, are dichotomous innature: individuals are either skilled or unskilled and cities either perform one function or all
functions. These low-dimensional accounts are not well suited to interpret data exhibiting
much richer variation.
What governs the distributions of skills and sectors across cities? Who lives where andwhat do they do there? These are challenging questions. In this paper, we develop ananalytic approach we term the comparative advantage of cities. Our aim is to provide thesimplest theoretical framework capable of providing unified answers to these questions andthen to take our model’s predictions to the data. We introduce a model of many cities, manyskills, and many sectors, and we show that it fits the cross-section of US cities well.
In our spatial-equilibrium model, the comparative advantage of cities is jointly governedby the comparative advantage of individuals and their locational choices. Cities are symmetric ex ante, so cross-city heterogeneity is an emergent outcome. Agglomeration economies
make cities with larger, more skilled populations exhibit higher total factor productivity(TFP). Locations within cities exhibit heterogeneity in their innate desirability, as is customary in land-use models (Fujita and Thisse, 2002, Ch 3). These cities are populated byheterogeneous individuals with a continuum of skill types, and these individuals may be
employed in a continuum of sectors. Comparative advantage causes more skilled individuals to work in more skill-intensive sectors, as in Ricardo-Roy models (Costinot and Vogel,2015). While those models of international trade assume exogenous endowments, we study
the comparative advantage of cities in an environment in which cities’ factor supplies areendogenously determined by the choices of mobile individuals. There is a complementarity
between individual income and locational attractiveness, so more skilled individuals are more
willing to pay for more attractive locations and occupy these locations in equilibrium, as in
the differential-rents model of Sattinger (1979).
In equilibrium, agglomeration, individuals’ comparative advantage, and locational heterogeneity within cities combine to deliver a rich set of novel predictions. Agglomeration
causes larger cities to have higher TFP, which makes a given location within a larger city1

more attractive than a location otherwise of the same innate desirability within a smallercity. For example, the best location within a larger city is more attractive than the best
location within a smaller city due to the difference in TFP. Since more skilled individuals occupy more attractive locations, larger cities are skill-abundant. The most skilled individuals
in the population live only in the largest city and more skilled individuals are relatively moreprevalent in larger cities. By individuals’ comparative advantage, the most skill-intensive
sectors are located exclusively in the largest cities and larger cities specialize in the production of skill-intensive output. Our model therefore predicts an urban hierarchy of skills andsectors. Under slightly stronger assumptions, larger cities will be absolutely larger in all
sectors.
As we discuss in Section 2, prior theories describing cities’ sectoral composition haveoverwhelmingly focused on the polar cases in which cities are either completely specialized“industry towns” or all produce the same composite output. The former is starkly at odds
with the data, while the latter makes no substantive prediction. Our theory predicts thatcities are incompletely specialized across sectors due to incomplete skill sorting: the tradeoff between city-level TFP differences and within-city locational desirability differences canmake individuals of a given skill level indifferent between locations in different cities. At thesame time, our model relates the pattern of specialization to cities’ observable characteristics.
It makes strong, testable predictions about the distributions of skills and sectors across cities.
We examine the model’s predictions about the distribution of skills and sectors across
US cities using data from the 2000 Census of Population, County Business Patterns, andOccupational Employment Statistics described in Section 4. We use two empirical approaches
to characterize cities’ skill and sectoral distributions. The first regresses a city’s log sectoral
employment on its log total population. More skilled groups and more skill-intensive sectors
should exhibit higher population elasticities. The second examines whether the distributions
exhibit the monotone likelihood ratio property, as per Costinot (2009), by comparing a pairof skills or sectors across a pair of cities.1 For example, comparing two cities and two
sectors, the skill-intensive sector should have relatively larger employment in the larger city.
To characterize sectoral size, we simply compare sectors’ employment levels across cities.
Section 5 reports the results, which provide support for our model’s predictions about
the pattern of skills and sectors across cities. Characterizing skills in terms of three or nine
educational groups, we find that larger cities are skill-abundant. Characterizing sectors interms of 19 industrial categories or 22 occupational categories, we find that larger cities
specialize in skill-intensive sectors. Our model’s predictions are generally borne out by thedata and are statistically significant. If deviations from our model’s predictions are idiosyn10fc (σ )The distribution fc (σ) likelihood ratio dominates fc0 (σ) if, for any σ > σ 0 , ffc0(σ)(σ) ≥ f 0 (σ 0 ) .
c2ccratic rather than systematic, the empirical success rate should increase with aggregation
over groups of cities. This is indeed the case. There is systematic variation in cities’ skill
and sectoral distributions that is consistent with the predictions of our theory.
Our account of the distributions of skills and sectors across cities has implications forboth research and policy. We find systematic sorting of finely differentiated educational
attainment levels across cities. This means that researchers examining spatial variation inbinary college/non-college terms report outcomes that partly reflect cross-city compositional
differences within these broad educational categories. Similarly, we find that a city’s sectoral composition is systematically related to its population size. This means that research
designs exploiting cross-city variation in sectoral composition, à la Bartik (1991), that do
not control for population size are relying upon size-driven variation, not just idiosyncratic
circumstances, for identification. Our model also has potentially interesting implications for
welfare. For example, our model of spatial sorting across locations means that biased technical change has asymmetric welfare consequences. In our setting, skill-biased technical change
raises attractive locations’ rental prices without increasing less attractive locations’ prices,while unskilled-biased technical change raises rental prices everywhere. Finally, due to themost skilled individuals residing only in the largest cities, the range of skills and thus therange of welfare is greater in larger cities. These differences likely shape local policymakers’
challenges and choices.
In sum, we develop a theory of the distributions of skills and sectors across cities of different sizes and show that it fits the US data well. The model combines elements of urban
land-use theory and assignment models of comparative advantage. These are a natural foundation for a theory of a system of cities because locational heterogeneity within and acrosscities leads the distribution of skills to exhibit the monotone likelihood ratio property. This
arises not by assumption but instead as an emergent property of the equilibrium that bringsin its wake similar implications for the distribution of sectors. This high-dimensional theory
provides a basis for what is, relative to the prior literature, a richer empirical examinationof the distributions of skills, occupations, and industries across cities of different sizes.

2

Related literature

Our contributions are related to a diverse body of prior work. Our focus on high-dimensional
labor heterogeneity is related to recent developments in labor and urban economics. Our theoretical approach integrates elements from the systems-of-cities literature, land-use theory,and international trade. Our model yields estimating equations and pairwise inequalities describing the comparative advantage of cities that are related to prior reduced-form empirical3

work in urban economics, despite a contrast in theoretical underpinnings.
Our theory describes a continuum of heterogeneous individuals. A large share of systemsof-cities theories describe a homogeneous population (Abdel-Rahman and Anas, 2004). Most
previous examinations of heterogeneous labor have only described two skill levels, typicallylabeled skilled and unskilled.2 Understanding the distribution of skills across cities withmore than two homogeneous types is valuable for a number of reasons. First, a prominent
literature in labor economics has emphasized the need to move beyond models with just two
homogeneous skill groups if we want to understand important empirical developments such as
wage polarization, job polarization, and simultaneous changes in between- and within-groupinequality (Acemoglu and Autor, 2011). Second, using only two skill groups is problematicfor understanding spatial variation in relative prices and quantities. The relative price ofskill varies across cities, as documented by Baum-Snow and Pavan (2013) and Davis andDingel (2019), and this is inconsistent with standard spatial-equilibrium models with two
homogeneous skill types and homothetic preferences.3 The relative quantity of skill varies
across cities, but whether this cross-city variation predicts city growth depends on how the
researcher partitions educational attainment into two skill groups.4
To overcome the difficulties of interpreting the world through a two-skill model, we assume
a continuum of skills, like the spatial models in Behrens, Duranton, and Robert-Nicoud(2014) and Davis and Dingel (2019). Relative to those papers, we introduce multiple tradable
sectors so that we can analyze the pattern of sectoral specialization, and we introduce intracity geographic heterogeneity that causes cities’ skill distributions to overlap rather thanexhibit strict sorting.
Our model integrates systems-of-cities theory with land-use theory. The canonical modelof land rents is one in which locations within a city have heterogeneous innate desirability(Fujita and Thisse, 2002, Ch 3). Prior models of a system of cities have only incorporated
urban structure as a city-level congestion mechanism, assuming that in equilibrium individuals are indifferent across all locations within a city (Abdel-Rahman and Anas, 2004;
2

A long line of empirical work describes cross-city variation in the share of residents who have a college
degree (Glaeser, 2008). Most closely related to our work is Hendricks (2011), who finds a weak relationshipbetween cities’ industries and college shares. We focus on theories in which labor is heterogeneous in an
asymmetric sense (e.g., more skilled individuals have absolute advantage in tasks or more skilled individuals
generate greater human-capital spillovers). There are also models describing matching problems, such as
Helsley and Strange (1990) and Duranton and Puga (2001), in which labor is heterogeneous in a horizontal
dimension.
3
Reconciling this fact with a two-skill model would require a departure from homotheticity such that
more skilled people find large cities less attractive than the less skilled. This is at odds with results on bothincome-specific price indices (Handbury, 2012) and endogenous amenities (Diamond, 2016).
4
Both Baum-Snow, Freedman, and Pavan (2017) and Diamond (2016) use two skill groups, but theyreport contrasting results about cities’ divergence in skills over time due to contrasting assignments of the“some college” population to “skilled” and “unskilled”, respectively.

4Behrens, Duranton, and Robert-Nicoud, 2014). Our model describes multiple cities withinternal geographies when individuals’ valuations of locations within cities differ systematically by income. The essential idea is that individuals choosing between living in Chicago or
Des Moines simultaneously consider in what parts of Chicago and what parts of Des Moines
they might locate.
Considering both dimensions simultaneously is more realistic in both the description of
the economic problem and the resulting predicted cross-city skill distributions. Large cities
like Chicago contain very heterogeneous neighborhoods, and within cities higher-income
individuals tend to choose more attractive, more expensive locations. The complementarityof skills and city size alone would lead to perfect sorting across cities by skill if there wereindifference within cities. Adding the dimension that agents sort by the innate desirabilityof locations within cities means that individuals of a given skill level are found in many cities
in equilibrium. This is more realistic than generating such overlap by simply perturbing
perfect sorting with idiosyncratic noise, since that approach would yield no relationshipbetween individuals’ incomes and their housing prices.
In our theory, we assume heterogeneity in the innate desirability of locations within eachcity without imposing a particular geography. We assume two restrictions on the relevant
heterogeneity. First, the distribution of innate locational desirability within each potential
city is ex ante identical, consistent with our aim of characterizing an emergent equilibrium.
Second, for our key theoretical results, this distribution must exhibit a regularity condition
introduced in subsection 3.4. Thus, we follow the long tradition of treating the distributionof the innate desirability of locations as exogenous, without assuming a functional form. We
show that our model makes predictions about the distribution of land prices consistent withavailable evidence while being sufficiently flexible to accommodate a number of potential
spatial configurations. While we think heterogeneous valuations of distinct locations withincities is an important subject in its own right, it also involves considerable complications, as
made clear by Duranton and Puga (2015), so in the present paper we limit our inquiry inthis dimension.
Our model belongs to a long theoretical tradition describing factor-supply-driven comparative advantage, as in the two-good-two-factor Heckscher-Ohlin theory formalized bySamuelson (1948). In international contexts, theorists have typically taken locations’ factor
supplies as exogenously endowed. Since individuals are mobile across cities, our theory endogenizes cities’ factor supplies while describing how the composition of output is governed bycomparative advantage. Our approach to comparative advantage with a continuum of factors
and a continuum of sectors follows a large assignment literature and is most closely related5to Costinot (2009).5 To obtain definite results about the distribution of outputs across countries, Costinot (2009) assumes that the exogenous distribution of factor endowments acrosscountries satisfies the monotone likelihood ratio property. Any economic mechanism thatmight generate that pattern is beyond the scope of his theory. By contrast, in the present
paper the fact that the distribution of skills across cities satisfies this property is a result
rather than an assumption. Similarly, while factor endowments, country size, and country
productivity are independent in Costinot (2009), they are closely linked in our model suchthat the distribution of endowments is tied to an observable characteristic, city size. Thus,
from a theoretical perspective, cities within a country constitute a natural setting to examine
these theories of comparative advantage. Moreover, the assumption of a common production
technology is likely more appropriate within than between economies, and data from a single
economy are likely more consistent and comparable than data combined across countries.6
The Heckscher-Ohlin model has been the subject of extensive empirical investigation ininternational economics. A pair of papers describe regional outputs using this framework.
Davis and Weinstein (1999) run regressions of regional outputs on regional endowments,
employing the framework of Leamer (1984), but they abstract from the issue of labor mobility across regions. Bernstein and Weinstein (2002) consider the two-way links between
endowments and outputs, concluding that if we know regions’ outputs, we know with considerable precision the inputs used, but not vice versa. We move beyond these papers in two
important respects. First, cities’ factor supplies are a feature to be explained. Second, ourmodel leads us to explore a dimension of the data not even contemplated in the prior work,
namely that the endogenously determined size of the city is itself systematically related toskill and sectoral structure.
Our theory predicts systematic variation in sectoral composition in the form of an urban
hierarchy of sectors. Prior systems-of-cities theories have overwhelmingly described sectoralcomposition in polarized terms (Abdel-Rahman and Anas, 2004).7 One class of models
5

Sattinger (1993) and Costinot and Vogel (2015) survey the assignment literature. While it might seemthat cities abundant in skilled labor must employ those factors in skill-intensive sectors, factor market
clearing alone does not imply such a result. There is considerable distance between factor supplies andproduction outcomes in a high-dimensional environment, and mild assumptions tend to deliver weak results,
as established by Jones and Scheinkman (1977) and Ethier (1984). Indeed, Costinot (2009) focuses precisely
on establishing conditions sufficient to tightly link factor supplies to production outcomes.
6
In our spatial-equilibrium setting, we make assumptions such that the sectoral assignment function iscommon across cities, as explained in our discussions of Lemma 5 and Proposition 1.
7
An exception is central place theory, and our model relates to that theory’s results in interesting ways.
Our model’s equilibrium exhibits a hierarchy of cities and sectors, as larger cities produce a superset of thegoods produced in smaller cities. Models in central place theory, dating from Christaller (1933) through Hsu,
Holmes, and Morgan (2014), have attributed this hierarchy property to the interaction of industry-specificscale economies and geographic market access based on the distance between firms located in distinct city
centers. Our model yields the hierarchy property in the absence of both. Our theory links the hierarchy of6

develops explicitly multi-sector economies in which each city specializes in a single traded
good due to external economies of scale that are sector-specific.8 In a complementary classof models, cities produce a single composite output, which may be interpreted as perfect
diversification without specifying sectors’ relative sizes. Neither complete specialization nor
perfect diversification provides a propitious starting point for empirical investigations.
The small empirical literature describing variation in cities’ sectoral composition is therefore only loosely informed by theory. For example, Holmes and Stevens (2004) survey thespatial distribution of economic activities in North America. They show that agriculture,mining, and manufacturing are disproportionately in smaller cities, while finance, insurance,
real estate, professional, and management activities are disproportionately in larger cities.
However, they do not reference a model or theoretical mechanism that predicts this pattern to be an equilibrium outcome. Similarly, Henderson (1983, 1991) reports regressions of
employment shares on population sizes. These are motivated by theories of specialization
linked to city size, but the regression specifications don’t follow directly from those models.
Our theory provides an explicit and distinct microfoundation for these regressions for an
arbitrary number of sectors. Moreover, it predicts that the estimated population elasticities
will be ordered by skill intensity.
A recent exception to the polarized view of sectoral specialization is Helsley and Strange(2014), who examine whether the equilibrium level of coagglomeration is efficient. Our papers
have quite distinct objectives. While Helsley and Strange (2014) make minimal assumptions
in order to demonstrate that Nash equilibria are generically inefficient when there are interindustry spillovers, we make strong assumptions that yield testable implications about
the distribution of sectoral activity across cities. This account of cities’ sectoral composition may inform a broader body of empirical work using sectoral composition as a source ofidentifying variation.9
In addition to sectoral composition, our theory describes sectoral size. Theories of localization and urbanization economies have contrasting predictions for cities’ absolute employment levels.10 In the canonical model of pure localization in Henderson (1974), specialized
sectors to a hierarchy of skills shaped by the internal geography of cities, neither of which has been consideredin central place theory.
8
Though Henderson (1974) characterizes the case of single-industry specialized cities, he suggests that“cities will probably specialize in bundles of goods, where, within each bundle, the goods are closely linked
in production.”9
A number of papers in urban economics posit theoretical models with no sectoral specialization, in
which all locations produce a homogeneous good, yet empirically estimate the model parameters exploiting
cross-city variation in industrial structure (Diamond, 2016; Notowidigdo, 2013; Yagan, 2014). Our multisector model links sectoral composition to city size and skill composition. We hope this framework may help
interpret such variation and accompanying identifying assumptions.
10
The literature traditionally distinguishes two types of external economies of scale (Henderson, 1987,

7

cities of different sizes host different sectors, yielding “textile cities” and “steel cities”. Sectoral specialization is the very basis for the city-size distribution, and one wouldn’t expect
large cities to be larger in all sectors. By contrast, urbanization models with a composite
output make no prediction about spatial variation in sectoral composition. Our paper both
introduces a multi-sector urbanization model in which larger cities are relatively larger inskill-intensive sectors and identifies conditions under which larger cities are absolutely larger
in all sectors.
A recent empirical literature has demonstrated significant agglomeration and coagglomeration of industries relative to the null hypothesis of locations being (uniformly) randomly
assigned in proportion to local population (Ellison and Glaeser, 1997; Duranton and Overman, 2005; Ellison, Glaeser, and Kerr, 2010). Our model’s predictions are consistent withthese findings. Since our theory says that sectors are ranked in terms of their relative employment levels, at most one sector could exhibit employment proportionate to total population.
All other sectors will exhibit geographic concentration. Similarly, since sectors more similarin skill intensity will exhibit more similar relative employment levels, the cross-city distribution of sectoral employment will be consistent with skill-related coagglomeration. Prior
studies have interpreted the agglomeration and coagglomeration of industries as evidenceof within-industry and industry-pair-specific interactions or spillovers. In our framework,
significant measured agglomeration and coagglomeration will arise even absent these forces.
In sum, we know of no prior spatial-equilibrium theory that makes the predictions yieldedby our model. Guided by our theoretical framework, our empirical investigation documentsa cross-city pattern of skills-driven comparative advantage not revealed by prior empirical
work.

3

Model

We develop a general-equilibrium model that makes predictions about the distributions ofskills and sectors across cities of different sizes. The theory has few moving parts. Individuals
vary in skill levels, and skills govern comparative advantage across sectors. Cities are identicalex ante, but agglomeration forces produce asymmetric cities of different sizes that differ intheir equilibrium composition of skills and sectors of varying skill intensity.
Individuals freely choose their production sector, a city, and a location within that city.
The sector that a person of a given skill type chooses to work in depends on goods prices but
p.929). Localization economies are within-industry, reflecting the scale of activity in that industry in thatlocation. Urbanization economies are general, reflecting the scale of all economic activity in a location.
Beyond scale, Lucas (1988) has stressed the composition of a location’s human capital. The agglomeration
process generating city-level productivities in our theory incorporates both scale and composition effects.

8

is independent of locational choices. The pattern of locational choices reflects the fact thata location can be more attractive if it is in a city with higher TFP or in a more desirable
location within a given city. Facing this tradeoff, individuals are indifferent to the relative
contributions of these two margins to a scalar index of attractiveness. In equilibrium, alllocations of a given attractiveness are occupied by individuals of the same skill level, who are
all employed in the same sector. There is thus an isomorphism between the distribution oflocational attractiveness across cities and the distributions of skills and sectors across cities.
Since equally attractive locations can be found in multiple cities, cities’ skill and sectoral
distributions exhibit overlap rather than strict sorting.
We develop two propositions. Proposition 1 has three key elements. The first identifies
a regularity condition under which the distribution of locational attractiveness across cities
is log supermodular. The second shows that this regularity condition implies that skills andsectoral employment are also log supermodular. The third says the same for sectoral outputs
and revenues. In simple terms, Proposition 1 says that larger cities will be skill abundantand specialize in skill-intensive sectors. Proposition 2 addresses variation in the absolute(not only relative) size of skills and sectors across cities. It provides a (stronger) regularity
condition sufficient for all skills and all sectors to be absolutely larger in larger cities. These
two propositions provide the foundation for our empirical work in Section 5.

3.1

Preferences, production, and places

A measure L of heterogeneous individuals choose a city, a location within that city, anda sector in which to produce. There are discrete city sites, a continuum of skills, anda continuum of sectors. As in Davis and Dingel (2019), our model features symmetricfundamentals, but skill-biased agglomeration generates cities of heterogeneous sizes.
Individuals consume a freely traded final good. This final good is the numeraire andproduced by combining a continuum of freely traded, labor-produced intermediate goods
indexed by σ ∈ Σ ≡ [σ, σ̄]. These have prices p(σ) that are independent of city c because
trade costs are zero. Locations are characterized by their city c and their (inverse) innate
desirability τ ∈ T ≡ [0, ∞), so they have rental prices r(c, τ ). In the main text, desirability
stems from productivity benefits, while Appendix A.2 covers the case in which a location’s
desirability stems from its amenity value.11
11

These productivity and amenity interpretations of desirability differ slightly in functional form but yield
the same predictions for the quantities that we examine empirically. For expositional clarity, we employ
the productivity interpretation in the main text. As discussed further below, the canonical model of a
monocentric city with commuting costs is a special case of our very general framework. In the amenity
interpretation, higher-income individuals are more willing to pay for better amenities. Future work shouldseek to empirically identify which features within cities govern locational desirability.

9Final-goods producers have a CES production function
Q=

nZ

−1

B(σ)[Q(σ)]  dσ


o −1

,

(1)

σ∈Σ

where Q(σ) ≥ 0 is the quantity of intermediate good σ,  > 0 is the elasticity of substitution
between intermediates, and B(σ) > 0 is an exogenous technological parameter. The profits
of final-goods producers are given byZ
Π=Q−p(σ)Q(σ)dσ.

(2)

σ∈Σ

Heterogeneous individuals use their labor to produce intermediate goods. There is ameasure of L > 0 heterogeneous individuals with skills ω that have the cumulative distribution function F (ω) and density f (ω) > 0 on support Ω ≡ [ω, ω̄]. The productivity of anindividual of skill ω in sector σ who chooses location τ in city c isq(c, τ, σ; ω) = A(c)T (τ )H(ω, σ).

(3)A(c) ≥ 0 denotes city-level total factor productivity, which results from agglomeration andis taken as given by individuals. T (τ ) ≥ 0 reflects the productivity effects of location within
the city. We assume that T (τ ) is continuously differentiable and decreasing, so that higher-τ
locations are less desirable. We assume that the twice-differentiable function H(ω, σ) ≥ 0
is strictly log-supermodular in ω and σ and strictly increasing in ω.12 The former governs
comparative advantage, so that higher-ω individuals are relatively more productive in higherσ sectors.13 The latter says that absolute advantage is indexed by ω, so that higher-ω
individuals are more productive than lower-ω individuals in all sectors. Each individual
inelastically supplies one unit of labor, so her income is her productivity times the price ofthe output produced, q(c, τ, σ; ω)p(σ).
Locations within each city are heterogeneous, with the innate desirability of a locationindexed by τ ≥ 0. The most desirable location is denoted τ = 0, and higher values of τ denoteless desirable places.14 The supply of locations with innate desirability of at least τ is S(τ ).
This is a strictly increasing function, since the supply of available locations increases as one
In R2 , a function H(ω, σ) is strictly log-supermodular if ω > ω 0 , σ > σ 0 ⇒ H(ω, σ)H(ω 0 , σ 0 ) >H(ω, σ 0 )H(ω 0 , σ).
13
We refer to higher-ω individuals as more skilled and higher-σ sectors as more skill-intensive.
14
For example, in the canonical monocentric-city model in which everyone works in the central business
district and commuting to the center costs time, τ describes the residential location’s physical distance fromthe center, T (τ ) is labor time net of commuting, and the supply of locations is S(τ ) = πτ 2 . In our model,τ captures a wide variety of exogenous amenities that vary across locations within a city, but that provide
a common distribution of the innate desirability of locations across cities. While the main text takes the
supply of heterogeneous locations S (τ ) as exogenously given, Appendix A.4 endogenizes it.
1210lowers one’s minimum standard of innate desirability. S(0) = 0, since there are no locations
better than the ideal. We assume S(τ ) is twice continuously differentiable. Locations are
owned by absentee landlords who spend their rental income on the final good.15 The city has
sufficient land capacity that everyone can reside in the city and the least desirable locations
are unoccupied. We normalize the reservation value of unoccupied locations to zero, sor(c, τ ) ≥ 0.
Individuals choose their city c, location τ , and sector σ to maximize utility. An individual’s utility depends on their consumption of the numeraire final good, which is their income
after paying their locational cost:
U (c, τ, σ; ω) = A(c)T (τ )H(ω, σ)p(σ) − r(c, τ ).

(4)

Denote the endogenous quantity of individuals of skill ω residing in city c at location τ andworking in sector σ by L × f (ω, c, τ, σ).
City-level TFP, A(c), reflects agglomeration gains derived from both population size andcomposition. Note that city sites are ex ante identical: city-level TFPs differ in equilibrium
due to individuals’ locational choices. A(c) is higher when a city contains a larger and moreskilled population. Denote the endogenous quantity of individuals of skill ω residing in cityRR
c by L × f (ω, c) ≡ L × σ∈Σ τ ∈T f (ω, c, τ, σ)dτ dσ. Total factor productivity is ZA(c) = J L

j(ω)f (ω, c)dω ,

(5)

ω∈Ω

where J(·) is a positive, strictly increasing function and j(ω) is a positive, non-decreasing
function. Numerous agglomeration processes may generate such productivity benefits, andwe do not attempt to distinguish between them here.

3.2

Equilibrium

In a competitive equilibrium, individuals maximize utility, final-good producers and landowners maximize profits, and markets clear. Individuals maximize their utility by their choicesof city, location, and sector such that
15

Models in the systems-of-cities literature typically treat the distribution of rents as a secondary issue,
dispensed with in a variety of ways. Sometimes they allow for absentee landlords; sometimes they distributethe land rents on a per capita basis to local residents; sometimes it is assumed that all agents have a
proportional share in an economy-wide mutual fund in land. See, for example, Henderson (1987), Helpman(1998), Rossi-Hansberg, Sarte, and Owens (2010), or Diamond (2016). For simplicity we assume that rents
accrue to landlords who live outside the cities of interest and spend it on the numeraire final good.

11f (ω, c, τ, σ) > 0 ⇐⇒ {c, τ, σ} ∈ arg max U (c, τ, σ; ω).

(6)Profit maximization by final-good producers yields demands for intermediates
Q(σ) = I p(σ) −
B(σ)

,

(7)

P R R R
where I ≡ L c σ ω τ q(ω, c, τ, σ)p(σ)f (ω, c, τ, σ)dτ dωdσ denotes total income and theseproducers’ profits are zero. Profit maximization by absentee landlords engaged in Bertrand
competition causes unoccupied locations to have rental prices of zero,Z0r(c, τ ) × S (τ ) − LZf (ω, c, τ, σ)dωdσσ∈Σ= 0 ∀c ∀τ.

(8)

ω∈Ω

Market clearing requires the endogenous quantity of individuals of skill ω residing in city
c at location τ and working in sector σ, L × f (ω, c, τ, σ), to be such that the supply of alocation type is greater than or equal to its demand, the demand and supply of intermediate
goods are equal, and every individual is located somewhere. Denoting the set of city sites
by C:
Z

0

Z

S (τ ) ≥ L

f (ω, c, τ, σ)dσdωω∈Ω

Q(σ) =X

Q(σ, c) = L

f (ω) =

XZ
c∈C

c∈C

X

f (ω, c) =

c∈C

∀c ∀τ

(9)

σ∈Σ

Z
q(c, τ, σ; ω)f (ω, c, τ, σ)dωdτ

XZ
c∈C

∀σ

(10)

τ ∈T

ω∈Ω

Z
f (ω, c, τ, σ)dτ dσ

σ∈Σ

∀ω

(11)

τ ∈T

A competitive equilibrium is a set of functions Q : Σ → R+ , f : Σ × C × T × Ω → R+ ,r : C × T → R+ , and p : Σ → R+ such that conditions (6) through (11) hold.

3.3

A system of cities

Cities’ populations are endogenously determined in spatial equilibrium. Two equilibrium
arrangements are possible: either all cities are identical in their TFPs and population sizes,
or cities are heterogeneous. The latter is the empirically relevant case. Our predictions about
the distributions of skills and sectors across cities of different sizes apply to any equilibrium12

in which cities are heterogeneous.16
First, we solve for occupational assignments by exploiting the fact that locational andsectoral arguments are separable in individuals’ utility functions. Given goods prices p (σ),
individuals’ sectoral choices are independent of their locational choices:arg max A(c)T (τ )H(ω, σ)p(σ) − r(c, τ ) = arg max H(ω, σ)p(σ)σσ

Define the assignment function M (ω) = arg maxσ H(ω, σ)p(σ) so that we can define the income associated with each skill’s optimal occupational choice as G(ω) ≡ H(ω, M (ω))p(M (ω)).
By comparative advantage, M (ω) is increasing.17 By absolute advantage, more skilled individuals earn higher nominal incomes and G(ω) is a strictly increasing function.18
Second, we solve for locational assignments by introducing a notion of locational attractiveness. Individuals value the product A(c)T (τ ), which we dub attractiveness.
Definition 1. The attractiveness of a location in city c of (inverse) innate desirability τ isγ = A(c)T (τ ).
Note that individuals are indifferent to the relative contributions of endogenous city-level
TFP A (c) and a location’s innate desirability within the city T (τ ) to attractiveness.19 In
equilibrium, two locations of equal attractiveness must have the same price, so we can describe the rental price of a location of attractiveness γ as rΓ (γ). Thus, individuals’ locational
choices can be characterized in terms of attractiveness:
max γG(ω) − rΓ (γ)γ

In equilibrium, more skilled individuals occupy more attractive locations. More attractive
locations have higher rental prices. Since G(ω) is strictly increasing, locational attractiveness16

In fact, for a special case in which TFP A (c) depends on population size alone with constant elasticity
α and locational supply V (z), defined below, exhibits a constant elasticity of 1−α
α , it can be shown that any
collection of city sizes summing to L is indeed an equilibrium of the model.
17
Lemma 1 of Costinot and Vogel (2010) shows that M (ω) is continuous and strictly increasing in equilibrium, given equation (7). It is worth noting here, as well, the role played by our assumption of zero trade
costs. If trade between cities were costly, prices and sectoral assignments would be city-specific, hence p (σ, c)and M (ω, c). We follow a vast factor-price-equalization literature in assuming no trade costs. To the extent
that trade between cities is less costly than trade between countries, this is a weaker assumption than in theinternational trade literature, e.g. in Costinot and Vogel (2010). Intercity trade costs are one reason thedata might reject our model’s predictions making those predictions non-obvious.
18
Absolute advantage across all sectors is not necessary to insure income rises with skill. The weakerdH(ω, M (ω)) > 0, would
condition that productivity is increasing in skill at the equilibrium assignments, dωbe sufficient.
19This is why our model generates incomplete sorting across cities. Individuals of a given skill type will
be indifferent between equally attractive locations that are in different cities.

13complements individual skill. Since more skilled individuals are more willing to pay for more
attractive locations, equilibrium locational assignments feature positive matching between
skill ω and attractiveness γ.
We have considered a location’s attractiveness without regard to which city that location
belongs. If the entire population lived in a single city, then desirability τ would be a sufficient
statistic for attractiveness γ. In that case, equilibrium locational assignments and prices canbe characterized as in standard land-use models (Fujita and Thisse, 2002, Ch 3), as we show
in Appendix A.1. For a system of cities, we first characterize locational assignments andprices in terms of attractiveness γ using similar tools. We then translate these assignments
and prices into functions of c and τ to characterize the system of cities.
Within each city, more desirable (low τ ) locations are more attractive. Competition
among landlords ensures that the most desirable locations are those occupied, so the least
desirable occupied site τ̄ (c) ≡ supτ {τ : f (ω, c, τ, σ) > 0} in a city of population L(c) isdefined by L(c) = S(τ̄ (c)). Denote each city’s set of occupied locations by T̄ (c) ≡ [0, τ̄ (c)].
Less desirable locations have lower rental prices, and the least desirable occupied site has arental price of zero.
Rτ R
R
Lemma 1 (Populated locations). In equilibrium, S(τ ) = L 0 σ∈Σ ω∈Ω f (ω, c, x, σ)dωdσdx ∀τ ∈T̄ (c), r(c, τ ) is strictly decreasing in τ ∀τ < τ̄ (c), and r(c, τ̄ (c)) = 0.
These results suffice to allow us to make some cross-city comparisons prior to solving forlocational assignments and prices.
In equilibrium, cities with higher TFP have larger populations. Consider two cities, c
and c0 , that differ in productivity, with A(c) > A(c0 ). If c were less populous than c0 , thenits least desirable occupied location would be more desirable, τ̄ (c) ≤ τ̄ (c0 ), since the supplyof locations, S(τ ), is increasing (and common across cities). Since A(c) > A(c0 ), this would
make the least attractive occupied location in city c more attractive than the least attractive
occupied location in city c0 , A(c)T (τ̄ (c)) > A(c0 )T (τ̄ (c0 )). By lemma 1, each of these locations
has a rental price of zero. Every individual would strictly prefer the more attractive location
at the same price, so no one would choose to live in c0 at τ̄ (c0 ), contradicting the definitionof τ̄ (c0 ) as an occupied location. So the city with higher TFP must have a larger population.
A smaller city’s locations are a subset of those in a larger city in terms of attractiveness.
For every location in the less populous city, there is a location in the more populous citythat is equally attractive. The location in city c0 of innate desirability τ 0 is equivalent to alocation τ in city c, given by A(c)T (τ ) = A(c0 )T (τ 0 ). The equally attractive location in thelarger city has higher TFP but lower innate desirability. The more populous city also has
locations that are strictly more attractive than the best location in the less populous city;
there are locations of attractiveness γ ∈ (A(c0 )T (0), A(c)T (0)] found in c and not in c0 .
14Across and within cities, more skilled individuals occupy more attractive locations.
Without loss of generality, order and label positive-population cities from 1 to C so that
A(C) ≥ A(C − 1) ≥ · · · ≥ A(2) ≥ A(1). Denote the set of attractiveness levels occupiedin equilibrium by Γ ≡ [γ, γ̄], where γ ≡ A(C)T (τ̄ (C)) and γ̄ ≡ A(C)T (0). Individuals ofhigher skill have greater willingness to pay for more attractive locations, so in equilibrium
higher-ω individuals occupy higher-γ locations.
Lemma 2 (Locational assignments). In equilibrium, there exists a continuous and strictlyincreasing locational assignment function K : Γ → Ω such that (i) f (ω, c, τ, M (ω)) > 0 ⇐⇒A(c)T (τ ) = γ and K(γ) = ω, and (ii) K(γ) = ω and K(γ̄) = ω̄.
To obtain an explicit expression for K : Γ → Ω, we can denote the supply of locations acrossall cities combined with attractiveness γ or greater as SΓ (γ). The supply function isSΓ (γ) =X
c:A(c)T (0)≥γ




γ
−1
S T
.
A(c)

By definition SΓ (γ̄) = 0 and by the fact that the best locations are populated SΓ (γ) = L.
R γ̄
Lemmas 1 and 2 imply that SΓ (γ) = L γ f (K(x))K 0 (x)dx, so K(γ) = F −1 L−SLΓ (γ) .
These locational assignments yield an expression for equilibrium locational prices, which
are increasing in attractiveness and given by the upper envelope of individuals’ bid-rentschedules.
Lemma 3 (Locational prices). In equilibrium, the rent schedule rΓ (γ) is increasing andRγ
continuously differentiable on [γ, γ̄] and given by rΓ (γ) = γ G(K(x))dx.
This determination of locational assignments and prices within the system of cities in termsof attractiveness is analogous to determining these locational assignments and prices for asingle autarkic city with a supply of locations that is the sum of locations across the systemof cities. Next, we translate these assignments and prices stated in terms of attractivenessto locations within each city.

3.4

The distributions of skills and sectors across cities

We can now characterize the distributions of rents, skills, and sectoral employment in asystem of cities. We first show how the distribution of locations across cities governs the distributions of skills and sectoral employment across cities through the locational and sectoral
assignment functions. We then identify a sufficient condition under which these distributions are log-supermodular. Finally, we identify conditions under which larger cities will
have larger populations of all skill types and employ more people in all sectors.
15

Since attractiveness is the product of city-level TFP and innate desirability, a city’s supplyof locations of a given attractiveness depends on both its endogenous TFP and the exogenous
supply of locations of particular desirability. The supply of locations with attractiveness γ
in city c is




γ
∂
−1
S (τ̄ (c)) − S T
if γ ≤ A(c)T (0)s(γ, c) ≡
∂γ
A(c)



γ
 1 V
if γ ≤ A(c)T (0)
A(c)
A(c)
=
,
0
otherwise(12)∂
where V (z) ≡ − ∂z
S (T −1 (z)) is the supply of locations with innate desirability T −1 (z). V (z)is a composite fundamental that depends only on the exogenous functions T (τ ) and S (τ ).
For example, if each city were a disc, S(τ ) = πτ 2 , and desirability reflected linear costs ofcommuting to the center, T (τ ) = d1 − d2 τ , this supply of locations within cities would be(d1 − z). The rental price of a location depends only on its attractiveness, so theV (z) = 2πd22
rental price of a location with innate desirability τ in city c is r(c, τ ) = rΓ (A(c)T (τ )).
The distribution of skills follows from s(γ, c) and locational assignments K : Γ → Ω.

Lemma 4 (A city’s skill distribution). The population of individuals of skill ω in city c is

L × f (ω, c) =K −10 (ω)s(K −1 (ω), c)

if A(c)T (0) ≥ K −1 (ω)

0otherwise.

Thus, the relative population of individuals of skill ω in two cities depends on the relative
supply of locations of attractiveness K −1 (ω). Since higher-ω individuals occupy more attractive locations and the most attractive locations are found exclusively in the larger city,
there is an interval of high-ω individuals who reside exclusively in the larger city. Individuals(ω,c)
of abilities below this interval are found in both cities, and their relative quantity ff(ω,c0 ) is−1
proportionate to the relative supply of locations of attractiveness K (ω).
The distribution of sectoral employment follows from s(γ, c), locational assignments K :Γ → Ω, and sectoral assignments M : Ω → Σ. As established in subsection 3.3, thesectoral assignment function M (ω) is common across c and τ because locations’ productivity
advantages are Hicks-neutral. Thus, the employment distribution f (σ, c) closely follows the
skill distribution f (ω, c).
Lemma 5 (A city’s sectoral employment distribution). The population of individuals em-

16ployed in sector σ in city c is
M −10 (σ) K −10 (M −1 (σ)) s (K −1 (M −1 (σ)) , c) if A(c)T (0) ≥ K −1 (M −1 (σ)).
L×f (σ, c) =0otherwise
As a result, two cities’ relative employment levels in sector σ depend on their relative supplies
of locations with attractiveness K −1 (M −1 (σ)).
We now identify the condition under which the distributions of rents, skills, and sectoral
employment across cities are log-supermodular functions. When the distribution of locational
attractiveness is log-supermodular, so are the distributions of skills and sectoral employment.
The first result follows from more skilled individuals occupying more attractive locations inequilibrium. The second result follows from the fact that sectoral assignments are common
across locations, so that sectoral composition is governed by skill composition.
Since the distribution of locations in terms of innate desirability τ is common acrosscities, cross-city differences in the distributions of locational attractiveness γ reflect differences in cities’ TFPs. Equation (12) demonstrates a hierarchy of locational attractiveness,since the most attractive locations are found exclusively in the highest-TFP city. Amonglevels of attractiveness that are supplied in multiple cities, equation (12) shows that cities’1) and a dilation of
TFPs shape the supply schedule s(γ, c) through both a scaling effect ( A(c)?

γV ( A(c)). Comparisons of relative supplies (s(γ, c)s(γ 0 , c0 ) > s(γ 0 , c)s(γ, c0 )) depend only on
the dilation.
Our main result, Proposition 1, states a sufficient condition for the ordering of city TFPs
to govern the ordering of locational supplies in any equilibrium. In turn, these govern the
distributions of skills and sectoral employment across cities in any equilibrium.

Proposition 1 (Cross-city distributions of attractiveness, skills, and sectors).
(a) The supply of locations of attractiveness γ in city c, s(γ, c), is log-supermodular if the
supply of locations with innate desirability T −1 (z) within each city, given by V (z), hasa decreasing elasticity.
(b) If V (z) has a decreasing elasticity, then f (ω, c) and f (σ, c) are log-supermodular.
(c) If V (z) has a decreasing elasticity, then sectoral output Q(σ, c) and revenue R(σ, c) ≡p(σ)Q(σ, c) are log-supermodular.
Proposition 1a links our assumption about each city’s exogenous distribution of locations,
V (z), to endogenous equilibrium locational characteristics, s(γ, c). The proof is in Appendix

17

B.20 Heuristically, note that a higher-TFP city is relatively abundant in more attractives(γ,c)is larger in the higher-TFP city. Equation (12) implies
locations when the elasticity ∂ ln∂ lnγγthat the γ-elasticity of s(γ, c) is the elasticity of V (z) at z = A(c). When this elasticity ishigher at lower values of z, an ordering of cities’ TFPs (and thus cities’ sizes) is an orderingof these elasticities, and thus an ordering of relative supplies at equilibrium. A number ofconceivable V (z) schedules satisfy this decreasing-elasticity condition.21 For example, in the
canonical monocentric-city model mentioned earlier, V (z) = 2π
(d1 − z) has an elasticity ofd22− d1z−z , which is decreasing in z.
An illustrative boundary case is when V (z) is constant.22 Per equation (12), there isa hierarchy of locational attractiveness in which the most attractive locations are foundonly in the largest cities, independent of V (z). For attractiveness levels that are attainedin both cities, there is no difference in relative supplies when V (z) is constant, becauses(γ,c)
is a constant when both s(γ, c) and s(γ 0 , c) are non-zero. Thus, s(γ, c) is weakly logs(γ 0 ,c)supermodular everywhere and strictly log-supermodular for the highest values of γ. For
the more general result, the decreasing-elasticity condition in Proposition 1 ensures that
intensive-margin variation in relative supplies aligns with the extensive margin of locational
attractiveness.
The distributions of skills and sectoral employment across cities in Proposition 1b follow the distribution of locational attractiveness. The skill distribution follows immediately
through the locational assignment function (lemma 4) and the employment distribution follows in turn through the sectoral assignment function (lemma 5). Note that these lemmas
show that the most skilled individuals working in the most skill-intensive sectors are foundonly in the most populous cities. The skills and sectors found in a smaller city are a strict
subset of those found in a larger city. This hierarchy of skills across cities at the top of theskill distribution does not depend on the shape of V (z). Since K : Γ → Ω and M : Ω → Σ
are strictly increasing functions, f (ω, c) and f (σ, c) are log-supermodular if and only if s(γ, c)
is log-supermodular. Thus, Proposition 1 links the log-supermodularity of these distributions
over the full range of skills and sectors to the properties of V (z). By equation (5), a citythat is larger and more skilled has higher TFP. In Section 3.3, we established that cities withhigher TFP have larger populations. Proposition 1b completes the circle by showing that if
one city has higher TFP than another, then its population is more skilled. TFP differences
20

Lemma 8 in Appendix B also shows that the sufficient condition is necessary for s(γ, c) to be logsupermodular for all possible values of A(c).
21
For example, if each city’s supply of locations with innate desirability T −1 (z) is the exponential, Weibull,gamma, or log-normal distribution, V (z) exhibits this decreasing-elasticity property. See Appendix A.3 fordetails.
22
For example, a linear city S (τ ) = 2τ with linear commuting costs T (τ ) = d1 − d2 τ makes V (z) aconstant.

18

driven by population size and skill composition are self-sustaining.
Given our assumptions about technologies and spatial equilibrium, the fact that larger
cities are skill-abundant and more skilled individuals work in more skill-intensive sectors
implies that larger cities produce relatively more in skill-intensive sectors. This pattern ofspecialization is closely related to the high-dimensional model of endowment-driven comparative advantage introduced by Costinot (2009), but in our setting cities’ populations andskill composition are endogenously determined and there is within-city heterogeneity in productivity.23 The distribution of output follows the distribution of skills because in spatial
equilibrium individuals employed in the same sector occupy locations of the same productivity.24 Thus, a city’s sectoral output is proportionate to its sectoral employment. Since that
output is freely traded, a city’s sectoral revenue is also proportionate to its sectoral employment. We therefore obtain part c of Proposition 1, which completes our characterization of
the pattern of comparative advantage across cities.25
When does the more productive city have a larger population of every skill type? Bylemma 4, whenever it has a larger supply of every attractiveness level, s(γ, c) ≥ s(γ, c0 ) ∀γ.
This is trivially true for γ > A(c0 )T (0). What about attractiveness levels found in bothcities? Proposition 2 identifies a sufficient condition under which a larger city has a larger
supply of locations of a given attractiveness. Its proof appears in Appendix B. Applying thisresult to the least-attractive locations yields a sufficient condition for larger cities to havelarger populations of all skill types and therefore employ more people in every sector.
Proposition 2 (City size and absolute size of local skills and sectors). For any A(c) > A(c0 ),γ, then s(γ, c) ≥ s(γ, c0 ). If
if V (z) has a decreasing elasticity that is less than -1 at z = A(c)−1γ

V (z) has a decreasing elasticity that is less than -1 at z = KA(c)(ω) = A(c) , then A(c) > A(c0 )implies f (ω, c) ≥ f (ω, c0 ) and f (M (ω), c) ≥ f (M (ω), c0 ) ∀ω ∈ Ω.
23

In particular, our Proposition 1c result that Q (σ, c) is log-supermodular is similar to Theorem 2 inCostinot (2009), but the economic environment and relevant assumptions differ. Assumption 2 in Costinot(2009)’s factor-endowment model is that countries’ exogenous endowments are such that countries can beranked according to the monotone likelihood ratio property. Our Proposition 1 identifies a sufficient condition
for cities’ equilibrium skill distributions to exhibit this property. Definition 4 of Costinot (2009) requiresthat factor productivity vary across countries (cities) in a Hicks-neutral fashion. Since productivity A(c)T (τ )varies both across and within cities, our production function q(c, τ, σ; ω) does not satisfy this requirement
for arbitrary locational assignments.
24
In the productivity interpretation of T (τ ), equilibrium productivity q(c, τ, σ; ω) = K −1 (ω)H(ω, σ) does
not vary across ω-occupied locations. In the amenity interpretation of T (τ ) described in Appendix A.2,
occupied locations’ productivities q(c, τ, σ; ω) = A(c)H(ω, σ) differ across cities in a Hicks-neutral fashion.
25
A traditional definition of comparative advantage refers to locations’ autarkic prices. In our setting,autarky means prohibiting both trade of intermediate goods and migration between cities. Since individuals
are spatially mobile, cities do not have “factor endowments”, and we must specify the autarkic skill distributions. If we consider an autarkic equilibrium with the skill distributions from the system-of-cities equilibrium,then larger cities have lower relative autarkic prices for higher-σ goods because they are skill-abundant, asshown by Costinot and Vogel (2010, p. 782).

19

Our two propositions characterize the distribution of skills and sectors across cities. If
V (z) has a decreasing elasticity, then larger cities are more skill-abundant and specialize inskill-intensive activities. If the elasticity is sufficiently negative, then larger cities are larger
in terms of all skill and sectors. We now turn to the data to see how well these predictions
describe US metropolitan areas.

4

Data description and empirical approach

Our model describes distributions within and across cities. While we have employed an
abstract idea of within-city heterogeneity in locational desirability, Proposition 1 makes concrete predictions about the distributions of skills and sectors across cities.26 We examine thepredictions of part b of Proposition 1 using two approaches. The first involves regression
estimates of the population elasticities of educational, occupational, and industrial populations. The second involves pairwise comparisons governed by the monotone likelihood ratio
property.
These tests require data on cities’ skill distributions, sectors’ skill intensities, and cities’
sectoral employment. We use public-use microdata from the US Census of Population to
identify the first two. The latter is described by data from County Business Patterns andOccupational Employment Statistics. The Census of Population describes individuals’ educational attainments, geographic locations, places of birth, occupations, and industries.
County Business Patterns describes cities’ industrial employment. Occupational Employment Statistics describes cities’ occupational employment. We combine these various dataat the level of (consolidated) metropolitan statistical areas (MSAs); see Appendix C for
details.
Subsections 4.1 through 4.3 describe the observable measures of skills, sectors, and skill
intensities that we employ in the two empirical approaches defined in subsection 4.4.
26

Proposition 1 also makes predictions about other economic outcomes, such as sectoral outputs (partc) and rental prices (implied by part a). We examine occupational employment levels, which are readily
available, since occupational output is not typically observed. The available evidence on urban costs, whichshows that the maximum, mean, and range of unimproved land prices are greater in larger cities (Combes,
Duranton, and Gobillon, 2012), is all consistent with our model’s predictions. We are not aware of a representative sample of (unimproved) land prices to examine stochastic or likelihood-ratio dominance in rental
prices across cities. Investigating the hedonic determinants of locational desirability, which our theoreticalmodel has abstractly treated as potentially production benefits or consumption amenities, is a first-orderquestion about the internal structure of cities that is quite beyond the scope of this paper.

20

4.1

SkillsFollowing a large literature, we use observed educational attainment as a proxy for individuals’ skills. Educational attainment is a coarse measure, but it is the best measure available
in data describing many people across detailed geographic locations.27 We do not assume
that individuals with the same educational attainment are equally skilled. We map the continuum of skills in our theory to the discrete set of educational levels observed in the databy assuming that the distribution of skills is increasing with educational attainment, suchthat the distribution of educational attainment across cities is log-supermodular if f (ω, c)
is log-supermodular.28 To describe cities’ skill distributions, we aggregate individual-level
microdata to the level of metropolitan statistical areas. A large literature in urban economics describes variation in terms of two skill groups, typically college and non-college
workers. Following Acemoglu and Autor (2011), we use a minimum of three skill groups.
The 2000 Census of Population microdata identify 16 levels of educational attainment, from“no schooling completed” to “doctoral degree”. We define three skill groups of approximately equal size among the working population: high-school degree or less; some college orassociate’s degree; and bachelor’s degree or more. In a more ambitious approach, we also
consider nine skill groups, ranging from individuals who never reached high school (4 percentof the population) to those with doctoral degrees (1 percent). Table 1 shows the population shares and percentage US-born for each of these skill groups in 2000.29 Foreign-born
individuals are disproportionately in the tails of the educational distribution.

4.2

Sectors

In our model, workers produce freely traded sectoral outputs indexed by σ that are used
to produce the final good. In the international trade literature, it is common to interpret
sectors in models of comparative advantage as industries. Recent work in both international
and labor economics has emphasized a perspective focused on workers completing tasks,
which empirical work has frequently operationalized as occupations (Grossman and RossiHansberg, 2008; Acemoglu and Autor, 2011). We will implement empirical tests using each.
We define sectors to be the 19 private-sector industries in the two-digit stratum of the North
27

In Appendix E.4, we infer skills from nominal wages and obtain similar results to those using educational
attainment.
28
Costinot and Vogel (2010, 774) show that log-supermodularity of factor supplies in an observed characteristic and unobserved skill ω is sufficient for mapping a theory with a continuum of skills to data withdiscrete observed characteristics.
29
This table describes labor-force participants 25 and older. See Appendix E for similar results using otherinclusion criteria.

21

Table 1: Skill groups by educational attainment
Population
Share
Population
Share
Skill (3 groups)
shareUS-born Skill (9 groups)
shareUS-bornHigh school or less.37
.78
Less than high school.04.29
High school dropout.08.73
High school graduate.25.88Some college.31.89College dropout.23.89
Associate’s degree.08.87Bachelor’s or more
.32
.85
Bachelor’s degree.20.86Master’s degree.08.84Professional degree.03.81
Doctorate.01.72
Notes: Sample is individuals 25 and older in the labor force residing in 270 metropolitan areas.
Data source: 2000 Census of Population microdata via IPUMS-USA

American Industry Classification System (NAICS) or the 22 occupational categories in thetwo-digit stratum of the Standard Occupational Classification (SOC).30
We measure industrial employment in a metropolitan area using data from the 2000
County Business Patterns, which counts paid employees in almost all non-farm, non-government
establishments. We measure occupational employment in a metropolitan area using estimates from the 2000 BLS Occupational Employment Statistics, which cover full-time andpart-time employees in all non-farm establishments. See Appendix C for details.

4.3

Skill intensities

Our theory makes the strong assumption that H(ω, σ) is strictly log-supermodular so that
sectors are ordered with respect to their skill intensities. In our empirical work, we infersectors’ skill intensities from the data using the observable characteristics of the workers
employed in them. We use microdata from the 2000 Census of Population, which contains
information about workers’ educational attainments, industries, and occupations. We use
the average years of schooling of workers employed in a sector as a measure of its skill
intensity.31 In doing so, we control for spatial differences by regressing years of schooling on
30

Per Costinot and Vogel (2010, 773-774), for mapping a continuum of sectors to coarse categories in thedata, it is sufficient that more skill-intensive tasks are relatively more prevalent in sectors employing moreeducated workers. Our industry results are not driven by our choice of NAICS aggregation: we find similar
empirical patterns when using three-digit industry definitions rather than two-digit definitions.
31
Autor and Dorn (2013) rank occupations by their skill level according to their mean log wage. Our
assumption of absolute advantage is consistent with such an approach. Using average log wages as ourmeasure of skill intensity yields empirical success rates comparable to and slightly higher on average thanthose reported in Section 5. We use years of schooling rather than wages as our measure of sectoral skill
intensities since nominal wages may also reflect compensating differentials or local amenities.

22

Table 2: Sectoral skill intensities
SOC
45
37
35
47
51
29
21
25
19
23

Skill
Occupational category
intensity
Farming, Fishing & Forestry
9.2
Cleaning & Maintenance
10.9
Food Preparation & Serving Related
11.5
Construction & Extraction
11.5
Production
11.5
Healthcare Practitioners & Technical
15.6
Community & Social Services
15.8
Education, Training & Library
16.3
Life, Physical & Social Science
17.1
Legal Occupations
17.3

NAICS
11
72
23
56
48
52
51
55
54
61

Industry
Forestry, fishing, hunting & agriculture support
Accommodation & food services
Construction
Admin, support, waste mgt, remediation
Transportation & warehousing
Finance & insurance
Information
Management of companies & enterprises
Professional, scientific & technical services
Educational services

Skillintensity
10.9
11.8
11.9
12.2
12.6
14.1
14.1
14.5
15.3
15.6

Notes: Skill intensity is average years of schooling for individuals 25 and older after
removing metropolitan-area fixed effects. Data source: 2000 Census of Population
microdata via IPUMS-USA

both sectoral and city fixed effects, but we have found that omitting the city fixed effects
has little effect on the estimated skill intensities. Table 2 reports the five least skill-intensive
and five most skill-intensive sectors among both the 19 industrial categories and the 22
occupational categories. There is considerably greater variation in average years of schoolingacross occupational categories than across industries.32

4.4

Empirical tests

Proposition 1b says that the distribution of skills across cities, f (ω, c), and the distribution of sectoral employment across cities, f (σ, c), are log-supermodular functions. Logsupermodularity has many implications; we focus on two that are amenable to empiricaltesting. If the function f (ν, c) is log-supermodular, then
1. a linear regression ln f (ν, c) = αν + βν ln L(c) + ν,c in which αν are fixed effects andL(c) is city population yields βν ≥ βν 0 ⇐⇒ ν ≥ ν 0 ;2. if C and C 0 are distinct sets and C is greater than C 0 (inf c∈C L(c) > supc0 ∈C 0 L(c0 )) andnC (nC 0 ) is the number of elements in C (C 0 ),
1 X
1 X
1 X
1 X
ln f (ν, c) +ln f (ν 0 , c0 ) ≥ln f (ν 0 , c) +ln f (ν, c0 ) ∀ν > ν 0 .
nC c∈CnC 0 c0 ∈C 0
nC c∈CnC 0 c0 ∈C 0
The first implication, which we will refer to as the “elasticity test,” says that the citypopulation elasticity of the population of a skill type in a city f (ω, c) is increasing in skill
32

The standard deviations of average years of schooling across occupational and industrial categories are
2.3 and 1.2, respectively.

23ω.33 Similarly, the population elasticity of sectoral employment f (σ, c) is increasing in skill
intensity σ. Our theory thus provides a structure to interpret previous work describing the
population elasticities of sectoral employment, such as Henderson (1983) and Holmes andStevens (2004).34 Standard econometric tests are available to assess whether our estimatedpopulation elasticities exhibit the property that ν ≥ ν 0 ⇒ βν ≥ βν 0 .
The second implication, which we will refer to as the “pairwise comparisons test”, saysthat if cities are divided into bins ordered by population sizes, then in any pairwise comparison of two bins and two skills/sectors, the bin containing more populous cities will haverelatively more of the more skilled type. Our theory therefore implies thousands of pairwise
comparisons for skills and millions for sectors. Appendix D shows that, in the presenceof additive random errors to ln f (ν, c), the likelihood of a successful pairwise comparisonincreases with the difference in population size, the difference in skill (intensity), and thenumber of cities assigned to each bin. To summarize this test, we report the fraction of
pairwise inequalities matching the predicted sign, weighted by the product of the two cities’
difference in log population and two sectors’ difference in skill intensity.35 When comparing
narrow educational categories, we also weight by the product of the educational categories’
population shares.36 To assess the statistical significance of the fraction of these pairwise
comparisons that yield the expected inequality, we compute the probability of obtaining
the observed success rate under the null hypothesis that skills and sectors are uniformly
distributed across cities.37
These two empirical tests are not independent, since they are both implied by log33

The linear regression may understood as a first-order Taylor approximation: ln f (ν, c) ≈ ln f (ν, c∗ ) +∂ ln f (ν,c)∂ ln f (ν,c)∗∂ ln L(c) |c=c∗ (ln L(c) − ln L(c )) +  = αν + βν ln L(c) + ν,c , where βν = ∂ ln L(c) |c=c∗ is increasing in ν bylog-supermodularity of f (ν, c).
34Henderson (1983) regresses employment shares on population levels, but reports “percent ∆ share /percent ∆ population”, which is equal to βσ −1 in our notation. Similarly, Holmes and Stevens (2004) describe
how location quotients, a city’s share of industry employment divided by its
P share of total employment, varyf (σ,c)/ c0 f (σ,c0 )with city size. In our notation, a location quotient is LQ(σ, c) =, so the L(c)-elasticity ofL(c)/L
LQ(σ, c) is βσ − 1.
35
We report unweighted success rates in Appendix E. An unweighted statistic assigns equal weight tocorrectly predicting that Chicago (population 9 million) is relatively more skilled than Des Moines (456thousand) and correctly predicting that Des Moines is relatively more skilled than Kalamazoo (453 thousand).
Given the numerous idiosyncratic features of the real world omitted from our parsimonious theory, the former
comparison is much more informative about the relevance of our theory than the latter. Stated differently,
a failure to correctly order Chicago and Des Moines should be much more damning for our theory than afailure to correctly order Des Moines and Kalamazoo. Weighting accomplishes this.
36
An unweighted average, reported in Appendix E, treats comparisons involving high school graduates (25percent of the workforce) and those involving PhDs (1 percent) equally, while these differ in their economic
import.
37
Appendix D provides details of a permutation test in which we shuffle our observations 1,000 times tocompute the distribution of success rates under the null. Power to reject the null decreases as the numberof bins decreases.

24supermodularity. Appendix D describes how they are related. In short, success of one test
implies success of the other, to the extent that the first-order approximations of ln f (ν, c) fit
the data well. We also implement a test for systematic deviations from log-supermodularity
proposed by Sattinger (1978).38

5

Empirical results

In this section, we test our predictions relating cities’ sizes to their distributions of skill, occupational employment, and industrial employment. First, we examine whether populations
are log-supermodular in educational attainment and city size. This prediction is a muchstronger characterization of cities’ skill distributions than the well-known fact that larger
cities typically have a greater share of college graduates. Second, we examine whether thepattern of sectoral employment is strongly ordered by this pattern of skills. Our theory’s
predictions are more realistic than those which say cities will be completely specialized in
an industry or produce in fixed proportions across all industries, as implicit in models witha single output. They are also more specific than theories allowing arbitrary patterns ofinterindustry spillovers. Finally, we examine whether larger cities are larger in all skill andsectoral categories or whether, as might be suggested by theories of industrial localization,different skills and sectors attain their maximal presence at different points in the city-size
distribution.
The data are broadly consistent with our predictions. Larger cities are more skilled thansmaller cities when comparing narrowly defined educational categories, although in the lowest
educational category, international migrants play an important role that is omitted from ourmodel. More skill-intensive sectors are relatively larger in more populous cities: sectoral
population elasticities rise with skill intensity, and pairwise comparisons yield statisticallysignificant results in the direction predicted by our theory. We show that there are notsystematic violations of our predicted pattern of comparative advantage. Consistent with
our model, there is a strong tendency for larger cities to be larger in all skills and sectors.

5.1

Larger cities are relatively more skilled

This subsection tests our prediction that larger cities have relatively more skilled populations.
We empirically describe skill abundance using the two tests described in subsection 4.4. We
38

These cross-sectional results should be thought of as examining a long-run equilibrium. Such a focus isvaluable as well if one wants to think about comparative statics for long-run outcomes. Of course, if one
wants to think explicitly about transitional dynamics, one would need to augment our model with adjustment
frictions appropriate to the shock contemplated.

25

Table 3: Population elasticities of three skill groups

Dependent variable: ln f (ω, c)βω1 High school or less × log populationβω2 Some college × log populationβω3 Bachelor’s or more × log population(1)(2)
Population
AllUS-born
share
0.954
0.895
0.37
(0.0108) (0.0153)
0.996
0.969
0.31
(0.0105) (0.0122)
1.086
1.057
0.32
(0.0153) (0.0162)

Share
US-born
0.78
0.89
0.85

Notes: Standard errors, clustered by MSA, in parentheses. Sample is individuals 25 and older in thelabor force residing in 270 metropolitan areas.

first do these exercises using three skill groups defined by educational attainment levels andthen repeat them using nine very disaggregated skill groups. In Appendix E.4, we infer skills
from nominal wages and obtain similar results.
5.1.1

Three skill groups

The elasticity test applied to the three skill groups across 270 metropolitan areas is reported
in Table 3. The results match our theory’s prediction that larger cities will have relatively
more people from higher skill groups. The population elasticities are monotonically increasing in educational attainment and the elasticities differ significantly from each other.39 In
anticipation of issues related to international immigration that arise when we examine nineskill groups, the second column of the table reports the population elasticities of US-born
individuals for these three educational categories. The estimated elasticities are slightly
lower, since foreign-born individuals are more concentrated in larger cities, but the differences between the elasticities are very similar. Larger cities’ populations are more skilled,on average.
The pairwise comparison test examines ordered groups of cities to see if the relative population of the more skilled is greater in larger cities. Following subsection 4.4, implementing
this test involves defining bins of cities. Ordering cities by population, we partition the
270 metropolitan areas in our data into 2, 3, 5, 10, 30, 90, and 270 bins of cities. Making
pairwise comparisons between three skill groups and as many as 270 metropolitan areas
involves computing up to 108,945 inequalities.40 Note that prior work typically describes a
39

Younger cohorts have higher average educational attainment. The results in Tables 3 and 5 are robustto estimating the elasticities for educational groups within 10-year age cohorts. Thus, our results are notdue to the young being both more educated and more likely to live in large cities.
m(m−1)
3×2
40
With n city bins and m skill groups, we make n(n−1)
comparisons. For example, 270×269
2
2
2
2 =

26

contrast between large and small cities for skilled and unskilled, whereas our most aggregatedcomparison is between large and small cities for three skill groups.
Table 4 reports the success rates for these pairwise comparisons for both the population asa whole and those individuals born in the United States. When making pairwise comparisonsacross all 270 metropolitan areas, the success rates are 67 and 69 percent, respectively, andhighly statistically significant. As we decrease the number of bins (increase the number ofcities per bin), the success rates increase, consistent with binning reducing the influence ofidiosyncratic errors. When using five bins or fewer, these success rates exceed 97 percent.41
These results are all statistically significant at the 1-percent level, except for the 2-bin casethat involves only three comparisons, severely reducing the test’s power.42
Both our empirical tests show a clear central tendency in the data across three skill groups.
More skilled individuals are relatively more prevalent in more populous cities. Moreover,these patterns are difficult to reconcile with a “skilled-unskilled” dichotomy. Individuals
with “some college” are distinct from both those with high school or less education andthose with a bachelor’s or higher education.

108, 945.
41
Our comparisons of two or five bins of cities are analogous to the empirical exercises presented in
Eeckhout, Pinheiro, and Schmidheiny (2014) and Bacolod, Blum, and Strange (2009).
42
Under a uniform null hypothesis, the probability that three tossed coins all turn up heads is 1/8, so theminimum possible p-value is 0.125. Similarly, our 1,000 simulations of the three comparisons under the nullyield three correct comparisons in 17.8% of simulations, bounding the p-value.

27

Table 4: Pairwise comparisons of three skill groups
Total
Success rate Success rate
Bins comparisonsAll
US-born
270

108,94590

12,015

30
10

1,305
135

5

30

3

9

2

3

0.67

0.69

(0.00)(0.00)

0.74

0.77(0.00)(0.00)

0.82

0.86(0.00)(0.00)

0.89

0.91(0.00)(0.00)

0.97

0.99(0.00)(0.00)

1.00

1.00(0.00)(0.00)

1.00

1.00(0.17)(0.17)

Notes: P-values for uniform null hypothesis inparentheses. Sample is individuals 25 and older in thelabor force residing in 270 metropolitan areas. Outcomes
weighted by log-population differences.

5.1.2

Nine skill groups

We next examine our tests for the case with nine skill groups. Per Lemma 4, our predictions
about the top of the skill distribution are stronger in the sense that the skills found in alarger city are a superset of those found in a smaller city, independent of the shape of thelocational supply function V (z). The results of the elasticity test are presented in Table
5. The more educated skill groups generally have higher population elasticities, as would
be expected from the three-skill-group results. This pattern is very clear for high school
graduates through professional degree holders, an interval that accounts for 87 percent of
the US population. The extreme tails of the distribution, however, do not conform to theprediction of our model.
The population elasticity of PhDs is somewhat below that of other college-educatedcategories. This is problematic for all theories in which skill-biased agglomeration shouldcause the most skilled to concentrate in larger cities. On the other hand, only one percent
of US workers possess a doctoral degree, so their spatial distribution has limited impact on
sectoral employment patterns and aggregate outcomes. The lower-than-expected population
elasticity reflects the cross-city distribution of institutions of higher education. The 27
percent of PhDs whose occupation is “postsecondary teacher” exhibit a population elasticity
28

Table 5: Population elasticities of nine skill groups

Dependent variable: ln f (ω, c)βω1 Less than high school × log populationβω2 High school dropout × log populationβω3 High school graduate × log populationβω4 College dropout × log populationβω5 Associate’s degree × log populationβω6 Bachelor’s degree × log populationβω7 Master’s degree × log populationβω8 Professional degree × log populationβω9 PhD × log population

(1)
(2)
All
US-born
1.089
0.858(0.0314) (0.0239)
1.005
0.933
(0.0152) (0.0181)
0.925
0.890
(0.0132) (0.0163)
0.997
0.971
(0.0111) (0.0128)
0.997
0.965
(0.0146) (0.0157)
1.087
1.059
(0.0149) (0.0164)
1.095
1.063
(0.0179) (0.0181)
1.113
1.082
(0.0168) (0.0178)
1.069
1.021
(0.0321) (0.0303)

Population
share
0.04

Share
US-born
0.29

0.08

0.73

0.25

0.88

0.23

0.89

0.08

0.87

0.20

0.86

0.08

0.84

0.03

0.81

0.01

0.72

Notes: Standard errors, clustered by MSA, in parentheses. Sample isindividuals 25 and older in the labor force residing in 270 metropolitan areas.

of 1.01, while the remainder of PhDs exhibit a population elasticity of 1.14. Thus, outside ofhigher education, which may be poorly described by a competitive model with agglomeration
economies, the distribution of PhDs across cities is well predicted by our theory.43
Those with less than a high-school education, who account for 12 percent of the working
population, exhibit higher-than-expected population elasticities. Within the context of ourmodel, this could be attributed to the locational supply function V (z) failing to exhibita decreasing elasticity, causing s (γ, c) to not be log-supermodular everywhere. As noted
previously, while our predictions about the top of the skill distribution are independent of
the shape of V (z), our predictions about lower skills depend on V (z) exhibiting a decreasingelasticity. Other assumptions about this function can deliver non-monotonic population
elasticities. For example, it is possible to make assumptions on V (z) such that population
elasticities decrease with skill among the least-skilled segment of the population and increase
43

About two-thirds of faculty in higher education are employed by public institutions, which are notprofit-maximizing and for practical and political reasons are likely to be proportionate to local population.
Furthermore, universities are conventionally land-intensive enterprises.

29

with skill among the most-skilled segment of the population.44
Looking outside our model’s mechanics, foreign-born individuals are more concentrated
in larger cities, regardless of their educational attainment. The high population elasticity of
the least-educated group is attributable to 71 percent of those with less than a high school
education being immigrants. The second column of Table 5 shows that if we restrict attention
to US-born individuals, this population elasticity drops from 1.089 to 0.858, below that of
all other skill groups.45 An alternative to restricting attention to US-born individuals is toestimate the population elasticities using data from 1980, when immigrants were a muchsmaller share of the US population. In 1980 data, the least skilled group has the lowest
population elasticity, and the difference between the 1980 and 2000 population elasticities
is almost entirely attributable to the rising share of the foreign-born in this least-skilled
population. We discuss mechanisms – mechanisms omitted from our model – that may
cause immigrants of all skill levels to concentrate in larger cities in Appendix E.1.46
We now turn to the pairwise comparisons for the case with nine skill groups in 2000.
These comparisons, presented in Table 6, check the predicted inequalities for as many as
36,315 city pairs for each pairing of the nine skill groups, separately for the population
as a whole and for those individuals born in the United States. These comparisons ask a
44

In particular, suppose that V (z) had an increasing elasticity at low values of z and a decreasing elasticity
V (z)
V (z)
at high values of z, ∂ ln
> 0 if z < z ∗ and ∂ ln
< 0 if z > z ∗ . It can then be shown that s (γ, c)∂ ln z∂ ln z∗would be log-submodular for γ < A (1) z and log-supermodular for γ > A (C) z ∗ . Hence, f (ω, c) would be
log-submodular for ω < K (A (1) z ∗ ) and log-supermodular for ω > K (A (C) z ∗ ).
45
Interestingly, among US-born individuals, the nine estimated elasticities naturally break into the three
more aggregate educational attainment categories that we used: βω1 , βω2 , βω3 ∈ (0.85, 0.94); βω4 , βω5 ∈(0.96, 0.98); βω6 , βω7 , βω8 , βω9 ∈ (1.02, 1.09).
46
The fact that the extreme tails of the skill distribution do not conform to our predictions invites acomparison to Eeckhout, Pinheiro, and Schmidheiny (2014). Eeckhout, Pinheiro, and Schmidheiny (2014)
introduce a model in which larger cities’ skill distributions have the same median skill but exhibit thicker tails.
In particular, their model predicts that skill types’ population elasticities will be U-shaped, with the medianskill type exhibiting the lowest population elasticity. The population elasticity of a skill type should increase
with its distance from the median skill. By symmetry, percentiles equally distant from the median should
exhibit the same population elasticity. For example, the 1st and 99th percentile skill types should exhibit the(same) highest population elasticity. In their empirical work, Eeckhout, Pinheiro, and Schmidheiny (2014)use cost-of-living-adjusted wages to measure skill and identify thicker tails in larger cities at the 10th and 90th
percentiles. However, their predictions do not hold throughout the skill distribution. Any partition of skills
into three groups with equal-sized low and high groups should yield U-shaped population elasticities. Table
3 rejects this prediction for a benchmark case of three groups of approximately equal size: the population
elasticities of the high-school-or-less and bachelors-or-more groups are not close to equal. Instead, population
elasticities are monotonically increasing in educational attainment, consistent with our prediction. Looking
at the nine-skill case in Table 5, we see that the median skill in the United States is a college dropout,
which has a considerably higher population elasticity than high school graduates (0.997 vs. 0.925). This
is at odds with the Eeckhout, Pinheiro, and Schmidheiny (2014) prediction. Moreover, Table 5 shows thatthe elasticities are monotonically increasing for high school graduates through professional degrees, which
jointly account for 87 percent of the labor force. While the population elasticity of professional degrees isgreater than that for PhDs (1.113 vs. 1.069), neither their theory nor ours predicts this modest decline.

30

lot of the data, so we should not expect perfection. They predict, for example, that the
number of associate’s degree divided by the number of college dropouts will be higher in Des
Moines than Kalamazoo, because the former’s population is 3,000 residents larger. Whenmaking pairwise comparisons across all 270 metropolitan areas, the success rates are 61 and64 percent, respectively, and highly statistically significant. Aggregating raises the successrates, consistent with binning reducing the influence of idiosyncratic errors. When using fivebins or fewer, these success rates exceed 75 percent. As suggested by the elasticity test, thesuccess rate is higher when restricting attention to US-born individuals. These results are all
statistically significant at the 1-percent level, except for the 2-bin case with all individuals,
which is significant at the 10-percent level.
Both our empirical tests demonstrate strong support for our theory’s predictions. In general, more educated individuals are relatively more prevalent in larger cities. The nine-skillgroup predictions are far more detailed than prior descriptions of cities’ skill distributions.
The high population elasticity of those not reaching high school does not match our model;
however, these individuals are a small fraction of the population and are overwhelmingly
foreign-born. International immigrants are particularly attracted to large cities; we showthat these outliers are absent in 1980, when US-born individuals were a much larger shareof the least-skilled group. The outliers in 2000 dampen the success rate of the pairwise comparisons, but our model’s predictions are an apt description of the broad pattern and highly
statistically significant. Thus, our theory provides stronger and more detailed predictions
about skill patterns across cities than prior work and, on the whole, the US data strongly
support those predictions.

31

Table 6: Pairwise comparisons of nine skill groups
Bins

Total
Success ratecomparisons
All

270

1,307,340

90

144,18030
1015,660
1,620

5

360

3

108

2

36

Success rate
US-born

0.61

0.64

(0.00)(0.00)

0.66

0.71(0.00)(0.00)

0.72

0.77(0.00)(0.00)

0.74

0.81(0.00)(0.00)

0.76

0.85(0.00)(0.00)

0.75

0.88(0.02)(0.00)

0.75

0.90(0.08)(0.01)

Notes: P-values for uniform null hypothesis in parentheses.
Sample is individuals 25 and older in the labor force residing in
270 metropolitan areas. Outcomes weighted by product of
log-population differences and educational population shares.

5.2

Larger cities specialize in skill-intensive sectors

This subsection examines the pattern of sectoral employment across cities. In our theory,larger cities are relatively more skilled and the sectoral assignment function is common
across cities, so larger cities employ relatively more labor in skill-intensive sectors. Section 5.1
established that larger cities are relatively more skilled, but it need not be the case that larger
cities specialize in skill-intensive sectors, since our theory relied on a number of assumptions
to obtain this result. We now examine whether larger cities are relatively specialized in skillintensive sectors. Since employment levels in both industries and occupations are readily
available in the data, we test the employment implications of Proposition 1b.
5.2.1

The distribution of occupations across cities

We first implement the elasticities test and the pairwise comparisons test interpreting sectors
as occupations.47 We begin with a visualization of the elasticity results. Define the skill
47

While there is a long line of literature looking at cities’ industrial composition, occupational composition
has been little studied. A notable exception is Duranton and Puga (2005), who document that larger
cities have more managers per production worker. Our investigation extends their inquiry to many more
occupational categories.

32

intensity of an occupation as the average years of schooling of individuals employed in thatoccupation. Figure 1 plots the 22 occupational categories’ estimated population elasticities
of employment against these occupational skill intensities.48 Our theory says the population
elasticity of occupational employment should rise with skill intensity and indeed we see aclear positive relationship.49 Outliers in the figure include close-to-unitary elasticities forthe relatively skilled occupations in education, healthcare, and social services, which may
reflect non-traded status. On the other side, computer and mathematical occupations havean elasticity that is quite high relative to their average years of schooling.
We can also look at this more formally. With the population elasticities of occupationsin hand, the hypothesis that βσ ≥ βσ0 ⇐⇒ σ ≥ σ 0 involves 231 (= 22 × 21/2) comparisonsof the estimated coefficients.50 This hypothesis is rejected at the 5-percent significance levelin only 46 of the 231 such comparisons. The occupational elasticity results are broadly
consistent with our prediction that larger cities specialize in skill-intensive activities.
The results for pairwise comparisons for occupations appear in Table 7.51 Using 276
cities and 22 occupational categories to make more than 8 million pairwise comparisons
yields an average success rate of 57 percent.52 While far from perfect, our model’s predictive
power over these millions of pairs of cities and occupations is highly statistically significant.
As we increase the number of cities per bin, the success rate increases up to 75 percent,as idiosyncratic errors are averaged out to reveal the central tendency of the data. These
results are statistically significant across all levels of aggregation.
Thus, both the estimated population elasticities and pairwise comparisons reveal a broadand strong tendency for more populous metropolitan areas to employ relatively more individuals in skill-intensive occupations.

48

These elasticities are estimated without including zero-employment observations. The results obtained
when including those observations are similar.
49
Studies of international trade have characterized an analogous relationship by regressing country-sectoral
exports on country fixed effects, sectoral fixed effects, and the interaction of measures of sectoral factor
intensity and country factor abundance (e.g., Romalis 2004; Nunn 2007). Estimating ln f (σ, c) = ασ + αc +β (skill (σ) × ln L(c)) + σ,c in our setting yields a β̂ coefficient of 0.030 with a standard error of 0.003.
50
The elasticity estimates appear in Appendix Table E.2.
51
With 276 metropolitan areas and bins that are factors of 270, the number of cities per bin may differ byone.
52
These pairwise comparisons omit zero-employment observations. The results obtained when including
those observations are similar.

33

1.4

Figure 1: Occupations’ population elasticities and skill intensities

Parameter estimate
1
1.2

Computer & MathematicalBusiness & Financial Operations

Architecture & Engineering
Legal
Life, Physical, & Social Science
Arts, Design, Entertainment, Sports, & Media

Protective Service
Office & Administrative Support
Management
Transportation & Material Moving
Personal Care & Service
Cleaning & MaintenanceConstruction & Extraction

Production

Sales & Related
Education, Training, & Library
Installation, Maintenance, & Repair
Healthcare Practitioners & Technical
Healthcare Support
Community & Social Services

Food Preparation & Serving Related.8

Farming, Fishing, & Forestry

6

8
10
12
14
16
18
Skill intensity (employees' average years of schooling)

Table 7: Pairwise comparisons of occupations
Total
Bins comparisons Success rate276

8,073,382

0.57(0.00)

90

925,15530

100,485

10

10,3950.62(0.00)

0.63(0.00)

0.65(0.00)

5

2,310

0.71
(0.00)

3

693

0.74
(0.00)

2

231

0.75
(0.03)

Notes: P-values for uniform null hypothesis in parentheses.
Outcomes weighted by product of log-population differences andskill-intensity differences.

34

20

5.2.2The distribution of industries across cities

We again implement the elasticities test and the pairwise comparisons test, now interpreting
sectors as industries. Define the skill intensity of an industry as the average years of educationof those employed in that industry. A visualization of the elasticity test appears in Figure 2.53
Again, as predicted by our theory, there is a clear positive relationship so that the population
elasticity of industrial employment is rising with the skill intensity of the industry.54 Testing
the hypothesis that βσ ≥ βσ0 ⇐⇒ σ ≥ σ 0 for the 19 industries involves 171 (= 19 ×
18/2) comparisons of these estimated elasticities.55 This hypothesis is rejected in only 33
comparisons, so the elasticity implication holds true for industries about 80 percent of thetime.
The results for pairwise comparisons for industries appear in Table 8. Making more than
6 million pairwise comparisons yields a statistically significant success rate of 61 percent.
Raising the number of cities per bin raises the success rate monotonically and always at ahigh level of statistical significance. The success rate rises to 77 percent when contrasting
two bins of large and small cities.
Thus, both the estimated population elasticities and pairwise comparisons reveal a precise and systematic pattern in which more populous metropolitan areas employ relatively
more individuals in skill-intensive industries.56 Prior work, including Henderson (1983) andHolmes and Stevens (2004), estimated industries’ population elasticities, although without
having a theoretical foundation for interpreting them. Our theory provides a basis for such
estimation and predicts the ordering of the elasticities based on skill intensities. Other prior
work, namely Henderson (1997), contrasted industrial patterns among large- and mediumsize cities. Our theory implies such comparisons between two or many more groups definedby cities’ sizes, and our results show that there is precise and systematic empirical content
even as we look at these finer comparisons between (groups of) cities.

53

As for occupations, these elasticities are estimated without including zero-employment observations. The
results obtained when including those observations are similar.
54
A regression on the interaction of skill intensity and log population, as described in footnote 49, yields
a coefficient of 0.080 with a standard error of 0.008.
55
The elasticity estimates appear in Appendix Table E.3.
56
We have found broadly similar results when examining industries at a finer level of disaggregation andwhen restricting attention to manufacturing industries.

35

1.6

Figure 2: Industries’ population elasticities and skill intensities

Parameter estimate
1.2
1.4

Management of companies and enterprises

Professional, scientific and technical services
Admin, support, waste mgt, remediation services
Educational services
Finance and insurance Information
Transportation and warehousing
Real estate and rental and leasing
Wholesale tradeArts, entertainment and recreation
Construction

1

Other services (except public administration)
Utilities
Manufacturing
Retail trade

Accommodation and food services

Health care and social assistance.8

MiningForestry, fishing, hunting, and agriculture support

11

12
13
14
15
Skill intensity (employees' average years of schooling)

Table 8: Pairwise comparisons of industries
Total
Bins comparisons Success rate276

6,469,758

0.61(0.00)

90

684,85530

74,385

10

7,695

0.64
(0.00)

0.66(0.00)

0.71(0.00)

5

1,710

0.74
(0.00)

3

513

0.74
(0.01)

2

171

0.77
(0.05)

Notes: P-values for uniform null hypothesis in parentheses.
Outcomes weighted by product of log-population differences andskill-intensity differences.

36

16

5.3

Testing for systematic failures of comparative advantage

Our results for the cross-city distributions of skills, industries, and occupations demonstrate
systematic patterns in line with our theory’s predictions. While demonstrating predictive
power, the pairwise comparisons also fall well short of 100 percent success. This is not
surprising, given that our model’s parsimony stems from making strong assumptions that
omit various features that influence the real world.
An important question is whether our theory’s unsuccessful pairwise predictions are
merely idiosyncratic deviations from the pattern of comparative advantage or are systematic
violations of our predicted pattern. The fact that our pairwise comparisons success rates
increase with the number of cities per bin is consistent with aggregation over idiosyncraticerrors, as shown in Appendix D. We also implement a formal test for systematic violationsof the predicted patterns.
Sattinger (1978) develops an approach to test for such systematic violations in the formof systematic intransitivity in the pattern of comparative advantage. It is possible for thef (σ,c0 )f (σ 0 ,c0 )f (σ 0 ,c00 )(σ,c)
data to exhibit, for c > c0 > c00 and σ > σ 0 > σ 00 , ff(σ0 ,c) ≥ f (σ 0 ,c0 ) and f (σ 00 ,c0 ) ≥ f (σ 00 ,c00 )00

f (σ,c )without exhibiting ff(σ(σ,c)
00 ,c) ≥ f (σ 00 ,c00 ) . With hundreds of metropolitan areas and dozens ofsectors, it is easy to find three cities and three sectors in the data exhibiting such intransitivity. But do intransitivities arise systematically? Sattinger (1978) shows that if ln f (σ, c)
is a polynomial function of β̂σ and ln L(c), then there can be systematic intransitivity only
if ln f (σ, c) is a function of higher-order interactions of β̂σ and ln L(c). We therefore added
quadratic terms and their interactions to our elasticity regressions. These did little to improve the regression’s adjusted R2 , and F-tests yielded p-values that did not come close to
rejecting the null that these additional terms were uninformative. There is no evidence ofsystematic intransitivity in comparative advantage. While our theory’s predictive successes
are systematic, the empirical departures from our theory appear to be idiosyncratic.

5.4

Larger cities are larger in all skills and sectors

As described in Section 2, different agglomeration theories have different implications for the
relationship between city size and sectoral employment levels. Localization theories makethe trade-off between industry-specific agglomeration economies and general congestion costs
the foundation of the city-size distribution. Formally, steel cities would come in one size,
textile cities in another, and so on. Localization theories militate against the idea thatlarger cities will be larger in all sectors. Our theory, in its baseline form, does not requirethat larger cities are larger in all sectors. However, by focusing on urbanization economies,
our theory allows that large cities may be the largest site of economic activity for all sectors
37

(Proposition 2). Our empirical exercise in this sub-section asks what weight should be placedon the predictions flowing from the localization and urbanization archetypes.
We have already shown that larger cities have relatively larger numbers of skilled workers.
We now investigate whether larger cities also tend to have larger populations of all skill types.
They certainly do, as all the population elasticities reported in Tables 3 and 5 are strongly
positive. Among the nine educational categories, the prediction that c > c0 ⇒ f (ω, c) ≥f (ω, c0 ) is true in 88 percent of 326,835 cases. The largest metropolitan area, New York,has the largest population in all educational categories, except the least skilled “less thanhigh school” category, which is most populous in the second-largest metropolitan area, Los
Angeles. Among US-born individuals, New York has the largest population in all nine
educational categories.
Turning to sectors, larger cities tend to have larger sectoral employment in all activities. This tendency is clear from the population elasticities plotted in Figures 1 and 2, as
they are all strongly positive. Among the 19 2-digit NAICS industries, the prediction thatc > c0 ⇒ f (σ, c) ≥ f (σ, c0 ) is true in 86 percent of 721,050 cases. Sixteen industries attaintheir maximal size in the largest metropolitan area (New York). The three exceptions are
manufacturing (second-largest Los Angeles), mining (tenth-largest Houston), and forestry,
fishing, hunting, and agriculture support (thirteenth-largest Seattle). The analogous resultsfor occupational categories show a similar tendency for larger cities to have higher employment levels in all occupations. The c > c0 ⇒ f (σ, c) ≥ f (σ, c0 ) prediction holds true in89 percent of occupational comparisons, and 19 of the 22 occupations attain their maximal
size in the largest metropolitan area, New York. The exceptions are production occupations(Los Angeles), architecture and engineering occupations (fifth-largest San Francisco), andfarming, fishing, and forestry occupations (51st-largest Fresno).
These findings are more consistent with urbanization economies than localization mechanisms at the city level. While particular examples such as San Francisco’s concentrationof architecture and engineering occupations may be consistent with localization economies,
the very large majority of sectors exhibit larger employment levels in more populous cities.
Our theory, which parsimoniously assumes only urbanization economies, matches the dataon cities’ sectoral composition and sectoral sizes quite well relative to existing models.

6

Discussion and conclusions

In this paper, we introduce a model that simultaneously characterizes the distribution ofskills and sectors across cities. We describe a high-dimensional economic environment thatis a system of cities in which cities’ internal geographies exhibit substantive heterogeneity and38

individuals’ comparative advantage governs the distribution of sectoral employment. Ourmodel achieves two aims. First, we obtain “smooth” predictions, in the sense that cities’skill and sectoral distributions will be highly overlapping. These are more realistic thanprior theories describing cities that are perfectly sorted along skills or polarized in termsof sectoral composition. Second, we obtain “strong” predictions, in the sense that cities’skill and sectoral distributions will exhibit systematic variation according to the monotone
likelihood ratio property. These are much finer than the predictions of many prior theoriesof the spatial organization of economy activity and guide our empirical investigation.
Examining data on US metropolitan areas’ populations, occupations, and industries inthe year 2000 reveals systematic variation in the cross-city distribution of skills and sectorsthat is consistent with our theory. Larger cities are skill-abundant. Our results using threeroughly equal-sized categories of educational attainment are quite strong. Even disaggregated to nine educational categories, the cross-city distribution of skills is broadly consistent
with our theory. Larger cities specialize relatively in skill-intensive activities. More skillintensive occupations and industries tend to have higher population elasticities of employment, and in pairwise comparisons, the more skill-intensive sector tends to employ relatively
more individuals in the larger city. Consistent with our approach based on urbanization
economies, larger cities tend to have larger absolute employment in all sectors.
We believe that our framework is amenable to both theoretical and empirical applications and extensions. The “strong” character of our predictions and their demonstrated
relevance for describing US cities in 2000 suggest that their examination in other settings, such as economies at different stages of development or in different historical periods, would be interesting. The “smoothness” resulting from the simultaneous consideration of cross- and within-city heterogeneity in a continuum-by-continuum environment
would make our model amenable to theoretical analyses of the consequences of commuting
costs, globalization, and skill-biased technical change. For example, skill-biased technological change has consequences for both the wage distribution and the land-rent distribution in
our model. Technological change, in the form of a change from B(σ) to B 0 (σ), is skill-biased
if B 0 (σ 0 )B(σ) ≥ B 0 (σ)B(σ 0 ) ∀σ 0 ≥ σ. Such a shift increases both wage inequality and rent
0)
0)
0
0)
r0 (γ 0 )≥ G(ω∀ω 0 ≥ ω and rΓ0 (γ) ≥ rrΓΓ(γ∀γ 0 ≥ γ.57 In models
inequality, in the sense that GG0(ω
(ω)
G(ω)
(γ)
Γ
in which individuals are indifferent across all locations within a city, the rent gradient isinvariant to the wage distribution. In our assignment model of differentiated locations, thecomposition of income growth governs the consequences for less-skilled individuals’ housing
57

Analogous to our result linking rent inequality to wage inequality, Clemens, Gottlieb, Hemous, and Olsen(2018) propose an assignment model in which greater income inequality among consumers causes greater
income inequality among their producers of vertically differentiated services.

39

costs.58
In short, our theoretical approach provides a new perspective on the comparative advantage of cities that aligns well with the patterns of specialization documented in our empirical
work.

58

Couture, Gaubert, Handbury, and Hurst (2019) quantify such welfare implications of income growth byintroducing idiosyncratic preferences shocks to a model of urban spatial sorting. In line with our logic, theyfind that income growth at the top of the distribution raises low-income households’ rents much less thanbroad-based income growth.

40References
Abdel-Rahman, H. M., and A. Anas (2004): “Theories of systems of cities,” in Handbookof Regional and Urban Economics, ed. by J. V. Henderson, and J. F. Thisse, vol. 4, chap. 52,
pp. 2293–2339. Elsevier.
Acemoglu, D., and D. Autor (2011): “Skills, Tasks and Technologies: Implications for
Employment and Earnings,” in Handbook of Labor Economics, ed. by O. Ashenfelter, and
D. Card, vol. 4, pp. 1043–1171. Elsevier.
Albert, C., and J. Monras (2018): “Immigration and Spatial Equilibrium: the Roleof Expenditures in the Country of Origin,” CEPR Discussion Papers 12842, C.E.P.R.
Discussion Papers.
Autor, D. H., and D. Dorn (2013): “The Growth of Low-Skill Service Jobs and thePolarization of the US Labor Market,” American Economic Review, 103(5), 1553–97.
Bacolod, M., B. S. Blum, and W. C. Strange (2009): “Skills in the city,” Journal of
Urban Economics, 65(2), 136–153.
Bartik, T. J. (1991): Who Benefits from State and Local Economic Development Policies?
W.E. Upjohn Institute for Employment Research.
Bauer, T., G. Epstein, and I. Gang (2005): “Enclaves, language, and the location
choice of migrants,” Journal of Population Economics, 18(4), 649–662.
Baum-Snow, N., M. Freedman, and R. Pavan (2017): “Why Has Urban Inequality
Increased?,” mimeo.
Baum-Snow, N., and R. Pavan (2013): “Inequality and City Size,” The Review of Economics and Statistics, 95(5), 1535–1548.
Behrens, K., G. Duranton, and F. Robert-Nicoud (2014): “Productive Cities: Sorting, Selection, and Agglomeration,” Journal of Political Economy, 122(3), 507 – 553.
Bernstein, J. R., and D. E. Weinstein (2002): “Do endowments predict the locationof production?: Evidence from national and international data,” Journal of International
Economics, 56(1), 55–76.
Christaller, W. (1933): Die zentralen Orte in Suddeutschland. Gustav Fischer, Jena.

41

Clemens, J., J. Gottlieb, D. Hemous, and M. Olsen (2018): “The Spillover Effectsof Top Income Inequality,” .
Combes, P.-P., G. Duranton, and L. Gobillon (2012): “The Costs of Agglomeration:
Land Prices in French Cities,” CEPR Discussion Paper 9240.
Costinot, A. (2009): “An Elementary Theory of Comparative Advantage,” Econometrica,
77(4), 1165–1192.
Costinot, A., and J. Vogel (2010): “Matching and Inequality in the World Economy,”
Journal of Political Economy, 118(4), 747–786.
(2015): “Beyond Ricardo: Assignment Models in International Trade,” Annual
Review of Economics, 7(1).
Couture, V., C. Gaubert, J. Handbury, and E. Hurst (2019): “Income Growth andthe Distributional Effects of Urban Spatial Sorting,” .
Davis, D. R., and J. I. Dingel (2019): “A Spatial Knowledge Economy,” American
Economic Review, 109(1), 153–170.
Davis, D. R., and D. E. Weinstein (1999): “Economic geography and regional production structure: An empirical investigation,” European Economic Review, 43(2), 379–407.
Diamond, R. (2016): “The Determinants and Welfare Implications of US Workers’ Diverging Location Choices by Skill: 1980-2000,” American Economic Review, 106(3), 479–524.
Duranton, G., and H. G. Overman (2005): “Testing for Localization Using MicroGeographic Data,” Review of Economic Studies, 72(4), 1077–1106.
Duranton, G., and D. Puga (2001): “Nursery Cities: Urban Diversity, Process Innovation, and the Life Cycle of Products,” American Economic Review, 91(5), 1454–1477.
(2005): “From sectoral to functional urban specialisation,” Journal of Urban Economics, 57(2), 343–370.
(2015): “Urban Land Use,” in Handbook of Regional and Urban Economics, ed.
by G. Duranton, J. V. Henderson, and W. S. Strange, vol. 5A, chap. 8, pp. 467–560.
North-Holland.
Edin, P.-A., P. Fredriksson, and O. Aslund (2003): “Ethnic Enclaves and the Economic Success of Immigrants—Evidence from a Natural Experiment,” The Quarterly Journal of Economics, 118(1), 329–357.
42

Eeckhout, J., R. Pinheiro, and K. Schmidheiny (2014): “Spatial Sorting,” Journalof Political Economy, 122(3), 554 – 620.
Ellison, G., and E. L. Glaeser (1997): “Geographic Concentration in U.S. Manufacturing Industries: A Dartboard Approach,” Journal of Political Economy, 105(5), 889–927.
Ellison, G., E. L. Glaeser, and W. R. Kerr (2010): “What Causes Industry Agglomeration? Evidence from Coagglomeration Patterns,” American Economic Review, 100(3),1195–1213.
Ethier, W. (1984): “Higher dimensional issues in trade theory,” in Handbook of International Economics, ed. by R. W. Jones, and P. B. Kenen, vol. 1, chap. 03, pp. 131–184.
Elsevier, 1 edn.
Fujita, M., and J.-F. Thisse (2002): Economics of Agglomeration. Cambridge University
Press.
Gabszewicz, J. J., A. Shaked, J. Sutton, and J. Thisse (1981): “International Tradein Differentiated Products,” International Economic Review, 22(3), pp. 527–534.
Gibson,
C.,
and
K.
Jung
(2006):“Historical
CensusStatistics on the Foreign-Born Population of the United States:
18502000,”
Population Division Working Paper 81,
US Census Bureau,
http://www.census.gov/population/www/documentation/twps0081/twps0081.html.
Glaeser, E. L. (2008): Cities, Agglomeration, and Spatial Equilibrium, The Lindahl Lectures. Oxford University Press.
Grossman, G. M., and E. Rossi-Hansberg (2008): “Trading Tasks: A Simple Theoryof Offshoring,” American Economic Review, 98(5), 1978–97.
Handbury, J. (2012): “Are Poor Cities Cheap for Everyone? Non-Homotheticity and theCost of Living Across U.S. Cities,” mimeo.
Helpman, E. (1998): “The size of regions,” in Topics in Public Economics: Theoreticaland Applied Analysis, ed. by D. Pines, E. Sadka, and I. Zilcha, pp. 33–54. Cambridge
University Press Cambridge.
Helsley, R. W., and W. C. Strange (1990): “Matching and agglomeration economies
in a system of cities,” Regional Science and Urban Economics, 20(2), 189–212.

43

(2014): “Coagglomeration, Clusters, and the Scale and Composition of Cities,”
Journal of Political Economy, 122(5), 1064 – 1093.
Henderson, J. V. (1974): “The Sizes and Types of Cities,” American Economic Review,
64(4), 640–56.
(1983): “Industrial Bases and City Sizes,” American Economic Review, 73(2),164–68.
(1987): “General equilibrium modeling of systems of cities,” in Handbook of Regionaland Urban Economics, ed. by E. S. Mills, vol. 2, chap. 23, pp. 927–956. Elsevier.
(1991): Urban Development: Theory, Fact, and Illusion. Oxford University Press.
(1997): “Medium size cities,” Regional Science and Urban Economics, 27(6), 583–
612.
Hendricks, L. (2011): “The Skill Composition Of U.S. Cities,” International Economic
Review, 52(1), 1–32.
Holmes, T. J., and J. J. Stevens (2004): “Spatial distribution of economic activities inNorth America,” in Handbook of Regional and Urban Economics, ed. by J. V. Henderson,
and J. F. Thisse, vol. 4, chap. 63, pp. 2797–2843. Elsevier.
Hsu, W.-T., T. J. Holmes, and F. Morgan (2014): “Optimal city hierarchy: A dynamic
programming approach to central place theory,” Journal of Economic Theory, 154(0), 245– 273.
Jones, R. W., and J. A. Scheinkman (1977): “The Relevance of the Two-Sector Production Model in Trade Theory,” Journal of Political Economy, 85(5), 909–35.
Leamer, E. E. (1984): Sources of International Comparative Advantage: Theory and Evidence. MIT Press.
Lucas, R. E. (1988): “On the mechanics of economic development,” Journal of Monetary
Economics, 22(1), 3–42.
Notowidigdo, M. J. (2013): “The Incidence of Local Labor Demand Shocks,” mimeo.
Nunn, N. (2007): “Relationship-Specificity, Incomplete Contracts, and the Pattern ofTrade,” The Quarterly Journal of Economics, 122(2), 569–600.

44

Romalis, J. (2004): “Factor Proportions and the Structure of Commodity Trade,” American Economic Review, 94(1), 67–97.
Rossi-Hansberg, E., P.-D. Sarte, and R. Owens (2010): “Housing Externalities,”
Journal of Political Economy, 118(3), 485–535.
Ruggles, S., J. T. Alexander, K. Genadek, R. Goeken, M. B. Schroeder,and M. Sobek (2010): “Integrated Public Use Microdata Series: Version 5.0 [Machinereadable database],” Minneapolis, MN: Minnesota Population Center.
Samuelson, P. A. (1948): “International trade and the equalisation of factor prices,” The
Economic Journal, 58(230), 163–184.
Sattinger, M. (1978): “Comparative Advantage in Individuals,” The Review of Economics
and Statistics, 60(2), 259–67.
(1979): “Differential Rents and the Distribution of Earnings,” Oxford Economic
Papers, 31(1), 60–71.
(1993): “Assignment Models of the Distribution of Earnings,” Journal of Economic
Literature, 31(2), 831–80.
Yagan, D. (2014): “Moving to Opportunity? Migratory Insurance over the Great Recession,” mimeo.

A

Appendix: Theory

A.1

Autarkic equilibrium

Suppose the entire population lives in a single city, denoted c. It has an exogenous population
L(c) and skill distribution F (ω). With fixed population, autarky TFP is fixed by equation(5).
Lemma 6 (Autarky locational assignments). In autarkic equilibrium, there exists a continuous and strictly decreasing locational assignment function N : T̄ (c) → Ω such thatf (ω, c, τ, M (ω)) > 0 ⇐⇒ N (τ ) = ω, N (0) = ω̄ and N (τ̄ (c)) = ω.

45

This assignment function is obtained by equating supply and demand of locations:Z τZZf (ω, c, x, σ)dωdσdx
L(c) − S(τ )−1⇒ N (τ ) = F
L(c)S(τ ) = L

0

σ∈Σ

ω∈Ω

Given individuals’ equilibrium locations within the city, the schedule of locational rental
prices supporting these assignments comes from combining individuals’ utility-maximizing
decisions and the boundary condition r(c, τ̄ (c)) = 0.59
Lemma 7 (Autarky locational prices). In autarkic equilibrium, r(c, τ ) is continuously difR τ̄ (c)ferentiable on τ ≥ 0 and given by r(c, τ ) = −A(c) τ T 0 (t)G(N (t))dt for τ ≤ τ̄ (c).
The properties of interest in a competitive equilibrium are characterized by the assignment functions M : Ω → Σ and N : T̄ (c) → Ω. In the autarkic equilibrium of a single citywith an exogenous population, more skilled individuals work in more skill-intensive sectors
and occupy more desirable locations.

A.2

Amenity interpretation of desirability

The productivity and amenity interpretations of desirability yield very similar results but
differ slightly in notation. In the amenity interpretation, an individual’s productivity andutility areq(c, τ, σ; ω) = A(c)H(ω, σ)(A.1)

U (c, τ, σ; ω) = T (τ ) [A(c)H(ω, σ)p(σ) − r(c, τ )](A.2)

where amenity T (τ ) determines the value of the individual’s disposable income after paying
his or her locational price.60 In this interpretation, preferences are non-homothetic in amanner akin to that of Gabszewicz, Shaked, Sutton, and Thisse (1981). Higher-income
individuals are more willing to pay for higher-amenity locations because a more desirable
location complements their higher consumption of tradables.
In this case, instead of γ = A(c)T (τ ) = A(c0 )T (τ 0 ) ⇐⇒ r(c, τ ) = r(c0 , τ 0 ) = rΓ (γ),
the appropriate equivalence between two locations is their “amenity-amplified price”, which
is T (τ )r(c, τ ). So the equivalence statement is now γ = A(c)T (τ ) = A(c0 )T (τ 0 ) ⇐⇒59

Note that this boundary condition implies that in equilibrium every individual’s final-good consumption
is strictly positive, provided that H (ω, σ) > 0.
60
Recall that the final good is the numeraire.

46

T (τ )r(c, τ ) = T (τ 0 )r(c0 , τ 0 ) = rΓ (γ). The results in lemma 1 are unaltered, though the proof
is modified to use the relevant U (c, τ, σ; ω). The expressions for K : Γ → Ω, γ̄, and γ
are unaltered. This leaves the conclusions of lemmas 2 and 3 intact. The locational price(τ ))schedule is given by r(c, τ ) = rΓ (A(c)T= A(c) rΓγ(γ) .
T (τ )
These locational prices do not appear in the endogenous definition of A(c) nor the proofs
of Lemma 4 and subsequent results. When evaluated at equilibrium, occupied locations’productivities q(c, τ, σ; ω) = A(c)H(ω, σ) differ across cities in a Hicks-neutral fashion. Thus,
Proposition 1c holds true. As a result, the predictions about cities’ population, sectors, andproductivities described in subsection 3.4 are unaltered by interpreting T (τ ) as describing
consumer amenity benefits rather than productivity benefits.
The wage distribution does differ between the productivity and amenity interpretationsof desirability. In the productivity interpretation, productivity q(c, τ, σ; ω) does not varyacross ω-occupied locations in equilibrium, as reported in footnote 24, and p (σ) is common
across locations, so individuals of the same skill earn the same income everywhere. In theamenity interpretation, an individual’s income p (σ) q(c, τ, σ; ω) is equal to A (c) G (ω), so
individuals of the same skill earn higher nominal incomes in larger cities in equilibrium.

A.3

Cities’ internal geographies

A number of conceivable V (z) schedules satisfy the decreasing-elasticity condition of Proposition 1. We provide some examples here:
Monocentric-city model: For the monocentric city’s disc geography, S(τ ) = πτ 2 ,with linear transportation costs, T (τ ) = d1 − d2 τ , the supply of locations within cities(d1 − z) has an elasticity of − d1z−z , which is decreasing in z.
V (z) = 2πd22Exponential family: The exponential family of distributions has PDFs that can be
written in the (canonical) form V (z|η) = v1 (z) exp (η · v2 (z) − v3 (η)), where η and v2 (z)may be vectors. Thus, if V (z) is a member of the exponential family, we are interested inits elasticity∂ ln V (z) ∂ ln v1 (z)=+ η · ∇ln z v2 (z)∂ ln z∂ ln zV (z)• Exponential: v1 (z) = 1 and v2 (z) = z. Therefore ∂ ln
= ηz and the elasticity is∂ ln zdecreasing because η < 0 for the exponential distribution.
V (z)= k − 1 + ηkz k and the• Weibull: v1 (z) = z k−1 and v2 (z) = z k . Therefore ∂ ln
∂ ln zelasticity is decreasing because η = −1
< 0 in the standard expression of Weibull
λkparameters.

47

V (z)• Gamma: v1 (z) = 1 and v2 (z) = [ln z, z]. Therefore ∂ ln
= η1 + η2 z and the elasticity∂ ln zis decreasing because n2 = −β < 0 in the standard expression of Gamma parameters.
V (z)= −1 + η1 +• Log-normal: v1 (z) = √12π z1 and v2 (z) = [ln z, (ln z)2 ]. Therefore ∂ ln
∂ ln z1
2η2 ln z and the elasticity is decreasing because η2 = − 2σ2 < 0 in the standard expression of log-normal parameters.

A.4

Endogenous supply of heterogeneous locations

In the main text, the supply of locations with innate desirability of at least τ is S (τ ). Thissection relaxes the assumption of an inelastic supply schedule and shows that our main
results still hold.
Let the cost (in units of the numeraire good) of building s units at a location of (inverse)
innate desirability τ be C (s; τ ). Profit-maximizing, perfectly competitive landlords build
additional units until rent equals marginal cost.
π (s; τ, c) = rΓ (A (c) T (τ )) s − C (s; τ )π 0 (s; τ, c) = 0 ⇒ rΓ (A (c) T (τ )) = C0 (s; τ )βs, where β > 1 and h (τ ) is a supply shifter. When h (τ ) is greater, the
Let C (s; τ ) = βh(τ)
cost of building a given number of units at τ is lower. Given this functional form, the supplyof units in city c of attractiveness γ is

 


1
1
γ
−1
ln s (γ, c) =ln rΓ (γ) +ln h T
.
β−1
β−1
A (c)
Thus, s (γ, c) is log-supermodular if and only if h (T −1 (γ/A (c))) is log-supermodular in(γ, A). Note that Aγ is submodular and log-modular in (γ, A) on R2+ . Thus, this function is
log-supermodular if and only if h (T −1 (z)) has a decreasing elasticity by lemma 8.
Thus, with endogenous supplies of locations, the decreasing-elasticity sufficient conditionof Proposition 1 applies to the schedule of exogenous supply shifters rather than the inelastic
supply schedule. A sufficient condition to obtain the result in Proposition 2 is that h (τ ) is
increasing.

B

Appendix: Proofs

Proof of Lemma 1:

48

R τ0 RR
Proof. Suppose that ∃τ 0 < τ̄ (c) : S(τ 0 ) > L 0 σ∈Σ ω∈Ω f (ω, c, x)dωdσdx. Then ∃τ ≤RR
τ 0 : S 0 (τ ) > L σ∈Σ ω∈Ω f (ω, c, τ )dωdσ. Then r(c, τ )) = 0 ≤ r(c, τ̄ (c)), so U (c, τ, σ; ω) >U (c, τ̄ (c), σ; ω) ∀ω∀σ since T (τ ) is strictly decreasing. This contradicts the definition ofτ̄ (c), since τ̄ (c) is a location that maximizes utility for some individual. Therefore S(τ ) =RRτ R
L 0 σ∈Σ ω∈Ω f (ω, c, x)dωdσdx ∀τ ≤ τ̄ (c).
Suppose that ∃τ 0 , τ 00 : τ 0 < τ 00 ≤ τ̄ (c) and r(c, τ 0 ) ≤ r(c, τ 00 ). Then U (c, τ 0 , σ; ω) >U (c, τ 00 , σ; ω) ∀ω∀σ since T (τ ) is strictly decreasing. This contradicts the result that τ 00
maximizes utility for some individual. Therefore r(c, τ ) is strictly decreasing in τ ∀τ ≤ τ̄ (c).
Suppose r(c, τ̄ (c)) > 0. Then by its definition as a populated location, ∃ω : A(c)T (τ̄ (c))G(ω)−
r(c, τ̄ (c)) ≥ A(c)T (τ̄ (c) + )G(ω) ∀ > 0. This inequality is false for all ω for sufficiently
small , by the continuity of T (τ ). Therefore r(c, τ̄ (c)) = 0.
Proof of Lemma 2:This proof is analogous to the proof of lemma 6 below.
Proof of Lemma 3:
Proof. By utility maximization
γG(K(γ)) − rΓ (γ) ≥(γ + dγ)G(K(γ)) − rΓ (γ + dγ)(γ + dγ)G(K(γ + dγ)) − rΓ (γ + dγ) ≥γG(K(γ + dγ)) − rΓ (γ)
Together, these inequalities imply(γ + dγ)G(K(γ)) − γG(K(γ))(γ + dγ)G(K(γ + dγ)) − γG(K(γ + dγ)) rΓ (γ + dγ) − rΓ (γ)
≥
≥
dγ
dγ
dγ
Γ (γ)
Taking the limit as dγ → 0, we obtain ∂r∂γ
= G(K(γ)). Integrating from γ to γ andRγusing the boundary condition rΓ (γ) = 0 yields rΓ (γ) = γ G(K(x))dx.

Proof of Lemma 4:
Proof. In city c, the population of individuals with skills between ω and ω + dω isZ ω+dω
L
ω


 −1


 −1

K (ω)
K (ω + dω)−1−1
f (x, c)dx = S T
−S T
.
A(c)
A(c)

Taking the derivative with respect to dω and then taking the limit as dω → 0 yields the
population of ω in c. Using the definition of s(γ, c) yields the desired expression.
Proof of Lemma 5:49

Proof. In city c, the population of individuals employed in sectors between σ and σ + dσ is
Z σ+dσLσ −1 −1
 −1 −1K (M (σ))
K (M (σ + dσ))−1
−1f (x, c)dx = S T
−S T
.
A(c)
A(c)

Taking the derivative with respect to dσ and then taking the limit as dσ → 0 yields the
population employed in σ in c. Using the definition of s(γ, c) yields the desired expression.

Proof of Lemma 6:
Proof. Nearly all of our argument follows the proof of Lemma 1 in Costinot and Vogel
R(2010). Define f (ω, c, τ ) ≡ σ∈Σ f (ω, c, τ, σ)dσ. Define Ω(τ ) ≡ {ω ∈ Ω|f (ω, c, τ ) > 0} andT (ω) ≡ {τ ∈ [0, τ̄ (c)]|f (ω, c, τ ) > 0}.
1. T (ω) 6= ∅ by equation (11) and f (ω) > 0. Ω(τ ) 6= ∅ ∀τ ≤ τ̄ (c) by lemma 1.
2. Ω(τ ) is a non-empty interval for τ ∈ [0, τ̄ (c)]. Suppose not, such that ω < ω 0 < ω 00 withω, ω 00 ∈ Ω(τ ) and ω 0 ∈/ Ω(τ ). ∃τ 0 : ω 0 ∈ Ω(τ 0 ). Suppose τ 0 > τ. By utility maximization
A(c)T (τ 0 )G(ω 0 ) − r(c, τ 0 ) ≥ A(c)T (τ )G(ω 0 ) − r(c, τ )A(c)T (τ )G(ω) − r(c, τ ) ≥ A(c)T (τ 0 )G(ω) − r(c, τ 0 )These jointly imply (T (τ 0 ) − T (τ ))(G(ω 0 ) − G(ω)) ≥ 0, contrary to τ 0 > τ and ω 0 > ω.
The τ 0 < τ case is analogous, using ω 0 and ω 00 . Therefore Ω(τ ) is a non-empty interval.
The same pair of inequalities proves that for τ < τ 0 ≤ τ̄ (c), if ω ∈ Ω(τ ) and ω 0 ∈ Ω(τ 0 ),then ω ≥ ω 0 .
3. Ω(τ ) is a singleton for all but a countable subset of [0, τ̄ (c)]. Since Ω(τ ) ⊂ Ω is a
non-empty interval for any τ ∈ [0, τ̄ (c)], Ω(τ ) is measurable for any τ ∈ [0, τ̄ (c)]. Let
T0 denote the subset of locations τ such that µ[Ω(τ )] > 0, where µ is the Lebesgue
measure over R. T0 is a countable set. For any τ ∈ T0 , define ω(τ ) ≡ inf Ω(τ ) andω̄(τ ) ≡ sup Ω(τ ). Because µ[Ω(τ )] > 0, we know ω̄(τ ) > ω(τ ). Thus, for any τ ∈ T0 ,there exists a j ∈ N such that ω̄(τ ) − ω(τ ) ≥ (ω̄ − ω)/j. From the last result in step2, we know that for any τ 6= τ 0 , µ[Ω(τ ) ∩ Ω(τ 0 )] = 0. Thus, for any j ∈ N, there canbe at most j elements {τ1 , . . . , τj } ≡ T0j ⊂ T0 for which ω̄(τi ) − ω(τi ) ≥ (ω̄ − ω)/jfor i = 1, . . . , j. By construction, T0 = ∪j∈N T0j , where T0j is a countable set. Since
the union of countable sets is countable, T0 is a countable set. The fact that Ω(τ ) isa singleton for all but a countable subset of [0, τ̄ (c)] follows from the fact that T0 is a50

countable set and the fact that only the nonempty intervals of Ω with measure zeroare singletons.
4. T (ω) is a singleton for all but a countable subset of Ω. This follows from the same
arguments as in steps 2 and 3.
5. Ω(τ ) is a singleton for τ ∈ [0, τ̄ (c)]. Suppose not, such that there exists τ ∈ [0, τ̄ (c)] forwhich Ω(τ ) is not singleton. By step two, Ω(τ ) is an interval, so µ[Ω(τ )] > 0, where µ
is the Lebesgue measure over R. By step four, we know that T (ω) = {τ } for µ-almost
all ω ∈ Ω(τ ). Hence condition (11) implies
f (ω, c, τ ) =f (ω)δ Dirac [1 − 1Ω(τ ) ] for µ-almost all ω ∈ Ω(τ ),(A.3)

where δ Dirac is a Dirac delta function. Combining equations (9) and (A.3) withµ[Ω(τ )] > 0 yields S 0 (τ ) = +∞, which contradicts our assumptions about S(τ ).
Step 5 means there is a function N : T → Ω such that f (ω, c, τ ) > 0 ⇐⇒ N (τ ) = ω.
Step 2 says N is weakly decreasing. Since Ω(τ ) 6= ∅ ∀τ ≤ τ̄ (c), N is continuous and satisfies
N (0) = ω̄ and N (τ̄ (c)) = ω. Step 4 means that N is strictly decreasing on (0, τ̄ (c)).
Proof of the explicit expression of N (τ ) that follows Lemma 6:
Z τZ

Z

S(τ ) = Lf (ω, c, x, σ)dωdσdx
0

σ∈Σ

Z τZ

f (ω)δ Dirac [x − N −1 (ω)]dωdx

=L
0

ω∈Ω

Z τZ

f (N (τ 0 ))δ Dirac [x − τ 0 ]N 0 (τ 0 )dτ 0 dx

=L
0

ω∈Ω

0

Z ττ
= −L
f (N (x))N 0 (x)dx = L(1 − F (N (τ )))0
L − S(τ )−1⇒ N (τ ) = F
L
Proof of Lemma 7:
Proof. By utility maximization
A(c)T (τ )G(N (τ )) − r(c, τ ) ≥ A(c)T (τ + dτ )G(N (τ )) − r(c, τ + dτ )A(c)T (τ + dτ )G(N (τ + dτ )) − r(c, τ + dτ ) ≥ A(c)T (τ )G(N (τ + dτ )) − r(c, τ )51

Together, these inequalities implyr(c, τ + dτ ) − r(c, τ )A(c)T (τ + dτ )G(N (τ )) − A(c)T (τ )G(N (τ ))≤dτdτ
A(c)T (τ + dτ )G(N (τ + dτ )) − A(c)T (τ )G(N (τ + dτ ))≤dτ)Taking the limit as dτ → 0, we obtain ∂r(c,τ= A(c)T 0 (τ )G(N (τ )). Integrating from τ to∂τR τ̄ (c)τ̄ (c) and using the boundary condition r(c, τ̄ (c)) = 0 yields r(c, τ ) = −A(c) τ T 0 (t)G(N (t))dt.

In the course of proving Proposition 1, we use the following lemma.
Lemma 8. Let f (z) : R → R++ and g(x, y) : R2+ → R++ be C 2 functions. If g(x, y) is
submodular and log-modular, then f (g(x, y)) is log-supermodular in (x, y) if and only if f (z)has a decreasing elasticity.
Proof. f (g(x, y)) is log-supermodular in (x, y) if and only if∂ 2 ln f (g (x, y))∂ ln f (z)∂ 2 ln f (z)
=
gxy +
gx gy
≥0
∂x∂y
∂z
∂z 2
|z=g(x,y)
If g(x, y) is submodular (gxy < 0) and log-modular (g = ggxxygy ), this condition can be written
as




∂ ∂ ln f (z)∂ ln f (z) ∂ 2 ln f (z) gx gy
+
=
≤ 0.
∂z
∂z 2
gxy |z=g(x,y) ∂z∂ ln z

Proof of Proposition 1:
Proof. Recall that the supply of locations with attractiveness γ in city c is



γ
 1 V
if γ ≤ A(c)T (0)
A(c)
A(c)
s(γ, c) =.
0otherwise0
It is obvious that γ > γ 0 , c > c0 ⇒ s(γ, c)s(γ 0 , c0 ) ≥ s(γ, c0 )s(γ 0 , c) is true when γ > A(c )T (0).
γ
For γ ≤ A(c0 )T (0), the inequality holds true if and only if V A(c)is log-supermodular inγ
2(γ, c). Note that A is submodular and log-modular in (γ, A) on R+ . Therefore, by lemma28, V Aγ is log-supermodular in (γ, A)on R+ if and only if V (z) has a decreasing elasticity.
γ
Since A(c) is increasing in c, V A(c)is log-supermodular in (γ, c) on R+ × C if V Aγ

52

is log-supermodular in (γ, A). Thus, s(γ, c) is log-supermodular if V (z) has a decreasingelasticity.
Proof of Proposition 2:
Proof. s(γ, c) ≥ s(γ, c0 ) is trivially true for γ > A(c0 )T (0). For γ ≤ A(c0 )T (0),0s(γ, c) ≥ s(γ, c ) ⇐⇒ ln V

γ
A(c)




− ln Vγ
A(c0 )

≥ ln A(c) − ln A(c0 )This condition can be rewritten as
Z ln A(c)Z ln A(c)−∂ ln V (z)d ln xd ln x ≥∂ ln z |z= γx
ln A(c0 )ln A(c0 )Z ln A(c) −∂ ln V (z)− 1 d ln x ≥0
∂ ln z |z= γx
ln A(c0 )Thus, a sufficient condition for the larger city to have more locations of attractiveness γ
V (z)γwhen V (z) has a decreasing elasticity is ∂ ln
≤ −1 at z = A(c).
∂ ln zC

Appendix: Data

Data sources: Our metropolitan population data are from the US Census website (2000).
Our data on individuals’ demographics, educational attainments, geographic locations, andsectors of employment come from the 5 percent sample of the 2000 US Census and the
1 percent metro sample of the 1980 US Census made available by IPUMS-USA (Ruggles,
Alexander, Genadek, Goeken, Schroeder, and Sobek, 2010). Our data on industrial employment come from the 2000 County Business Patterns, available from the US Census Bureau
website. Our data on occupational employment come from the 2000 Occupational Employment Statistics, available from the Burea of Labor Statistics website.
Geography: We use (consolidated) metropolitan statistical areas as defined by the OMB
as our geographic unit of analysis.
The smallest geographic unit in the IPUMS-USA microdata is the public-use microdata
area (PUMA), which has a minimum of 100,000 residents. We map the PUMAs to metropolitan statistical areas (MSAs) using the MABLE Geocorr2K geographic correspondence engine
from the Missouri Census Data Center. In some sparsely populated areas, a PUMA is larger
than a metropolitan area. We drop six MSAs in which fewer than half of the residents of
the only relevant PUMA live within the metropolitan area. As a result, there are 270 MSAs
when we use these microdata.
53

The 1980 Census of Population IPUMS-USA microdata do not identify PUMAs, so we
use the “metarea” variable describing 270 consolidated MSAs for the regressions in Table
E.1.
The County Business Patterns data describe 318 metropolitan statistical areas. These
correspond to a mix of OMB-defined primary and consolidated metropolitan statistical areas outside New England and New England county metropolitan areas (NECMAs). We
aggregate these into OMB-defined (consolidated) metropolitan statistical areas to obtain
276 MSAs.
The Occupational Employment Statistics data describe 331 (primary) metropolitan statistical areas. We aggregate these into OMB-defined (consolidated) metropolitan statisticalareas to obtain observations for 276 MSAs.
Skill distribution: Our sample of individuals in the IPUMS data includes those 25
and older in the labor force. We exclude individuals living in group quarters. Using the“educd” variable from IPUMS, we construct nine levels of educational attainment: less thanhigh school (educd values 2-24), high school dropout (30-61), high school graduate (62),college dropout (65, 71), associate’s degree (81), bachelor’s degree (101), master’s degree(114), professional degree (115), and doctorate (116). There is at least one observation in
every educational category in every metropolitan area. In Appendix E, we report robustness
checks using a narrower sample of IPUMS observations: full-time full-year (FTFY) workers,
defined as individuals 25 and older who reported working at least 35 hours per week and 40
weeks in the previous year.
In Appendix E, we report robustness checks using aggregate tabulations from the Census
of Population, Summary File 3, available from the US Census website. These tabulations
are less noisy than the IPUMS observations, because they come from the 1-in-6 Census longform rather than the 1-in-20 public-use microdata. Because they are aggregate tabulations,
we cannot condition on individual characteristics like labor-force participation or birthplace.
Sectoral skill intensity: Using the same sample of individuals 25 and older in the labor
force, we measure a sector’s skill intensity by calculating the average years of schooling of its
employees after controlling for spatial differences in average schooling. We calculate yearsof schooling using the educational attainment “educd” variable from IPUMS at its finest
level of disaggregation. For instance, this means that we distinguish between those whose
highest educational attainment is sixth grade or eighth grade. We use the “indnaics” and“occsoc” variables to assign individuals to their 2-digit NAICS and 2-digit SOC sectors of
employment. Aggregating observations to the MSA-sector level, weighted by the IPUMSprovided person weights, we regress the average years of schooling on MSA and sectoral
dummies. The sectoral dummy coefficients are our measure of skill intensities.
54

Industrial employment: There are 19 2-digit NAICS industries covered by both the
Census of Population microdata and the County Business Patterns data (the latter omits
public administration, NAICS 92). The County Business Patterns data are an almost exhaustive account of US employer establishments. When necessary to protect the confidentiality of
individual establishments, employment in an industry in a location is censored and reported
as falling within an interval rather than its exact number. In our empirical work, we use the
midpoints of these intervals as the level of employment. There are four (C)MSAs that havezero establishments in mining; the remaining 5240 (= 19 × 276 − 4) industry-metropolitan
pairs have at least one establishment. The County Business Patterns data omit self-employedindividuals and employees of private households, railroads, agriculture production, the postal
service, and public administrations. See the CBP methodology webpage for details.
Occupational employment: There are 22 2-digit SOC occupations. Across 331 (P)MSAs,
there should be 7282 metropolitan-occupation observations. The 2000 BLS Occupational
Employment Statistics contain employment estimates for 7129 metropolitan-occupation observations, none of which are zero. The 153 omitted observations “may be withheld frompublication for a number of reasons, including failure to meet BLS quality standards or theneed to protect the confidentiality of [BLS] survey respondents.”

D

Appendix: Empirical Tests

D.1

Population elasticities and pairwise comparisonsThis section describes the relationship between our two empirical tests in more detail.
If f (ν, c) is log-supermodular and f (ν, c) > 0 ∀ν∀c,
ν > ν 0 , c > c0 ⇒ ln f (ν, c) + ln f (ν 0 , c0 ) ≥ ln f (ν 0 , c) + ln f (ν, c0 ).
If C and C 0 are distinct sets and C is greater than C 0 (inf c∈C L(c) > supc0 ∈C 0 L(c0 )) and nC is thenumber of elements in C while nC 0 is the number of elements in C 0 , then log-supermodularityof f (ν, c) implies

1 X
1 X
1 X
1 X
ln f (ν, c) +ln f (ν 0 , c0 ) ≥ln f (ν 0 , c) +ln f (ν, c0 ) for ν > ν 0
nC c∈CnC 0 c0 ∈C 0
nC c∈CnC 0 c0 ∈C 0
Suppose that the world is noisy. Consider the following form for f (ν, c), which is a55

first-order approximation for any form,ln f (ν, c) =αν + βν ln Lc + ν,cwhere ν,c is an error term with E(ν,c ) = 0. The probability of obtaining the expected
inequality when ν > ν 0 , C > C 0 is

P = Pr

= Pr

!
X
X
1 X
1
1
1 X
ln f (ν, c) +ln f (ν 0 , c0 ) ≥ln f (ν 0 , c) +ln f (ν, c0 )nC c∈CnC 0 c0 ∈C 0
nC c∈CnC 0 c0 ∈C 0
!


1 X
1 X
[ν 0 ,c − ν,c ] +[ν,c0 − ν 0 ,c0 ] ≤ (βν − βν 0 ) ln L̄C − ln L̄C 0
nC c∈CnC 0 c0 ∈C 0

P
where ln L̄C 0 denotes an average log population, ln L̄C 0 ≡ n1 0 c0 ∈C 0 ln Lc0 . This probabilityCis higher when there is a larger difference in population size between the two bins and when
the difference in population elasticities, βν − βν 0 , is larger. Since log-supermodularity implies
that βν is increasing in ν, this probability is higher when the difference ν − ν 0 is larger.
To illustrate the properties of this probability, consider the special case in which thePiiderror term is normally distributed, ν,c ∼ N (0, σ 2 ). Then n1C c∈C [ν 0 ,c − ν,c ] is distributed


h

i 
2
1
1
.
Thus,
the
left
side
of
the
inequality
insidetheprobability
isdistributedN0,2
N 0, 2σ
+
σ2 .
nC
nC
nC 0
r h
i
Therefore, we can write that the left side divided by 2 n1C + n1 0 σ 2 is distributed standardCnormal, N (0, 1). Thus, the probability of obtaining the expected inequality is








(βν − βν 0 )0 r
P =Ω 
ln
L̄
−
ln
L̄
C
C
h
i


2 n1C + n1 0 σ 2Cwhere Ω(·) denotes the cumulative distribution function of N (0, 1). If nC equals nC 0 , theexpression for P simplifies to
√



nC
P =Ω √ · (βν − βν 0 ) · ln L̄C − ln L̄C 0 .
2 σ2The probability of obtaining the inequality depends on the difference in population sizeln L̄C − ln L̄C 0 , the difference in population elasticities (βν − βν 0 ), the noisiness (σ 2 ) of therelationship, and the number of cities aggregated (nC ). When the deterministic function is56

log-supermodular (c > c0 ⇒ Lc ≥ Lc0 ; ν > ν 0 ⇒ βν ≥ βν 0 ), P → 1 as σ 2 → 0 (and P → 1/2 as
σ → ∞). When the function is log-modular, P → 1/2 as σ 2 → 0, and when the function is
log-submodular, P → 0 as σ 2 → 0. As nC increases (as we aggregate cities into fewer bins),
it becomes more likely that we obtain the expected inequality. However, using fewer bins
also decreases the number of times that we evaluate this inequality, so we will tend to havelower power to reject the null hypothesis that f (ν, c) is log-modular.
Our finding that βν is increasing in ν when estimated in the population elasticity test
implies that this pairwise comparison test will tend to have the correct inequality, and its
success rate will increase with differences in city size and aggregation. The success of theelasticity test implies success of the pairwise comparison test (with aggregation) to the extent
that the log-linear approximation of f (ν, c) is a good approximation. At the same time,σ 2  0, so we should not expect the pairwise comparison test to have a 100-percent successrate. We use an exact test to compute the probability of obtaining our observed successrates under the null hypothesis that the deterministic function is log-modular (βν = β ∀ν).

D.2

P-values for pairwise comparisons test

To assess the statistical significance of the fraction of these pairwise comparisons that yield
the expected inequality, we compute the probability of obtaining a success rate at leastas high as the observed success rate under the null hypothesis that skills and sectors are
uniformly distributed across metropolitan areas. To do so, we employ a permutation test,
shuffling the microdata observations to construct cities of the true population size withrandomly assigned skill and sectoral distributions. For example, we shuffle the 4.4 million
individuals in our IPUMS microdata sample to randomly assign them to metropolitan areas.
We then perform the relevant pairwise comparisons test and record its success rate. We repeat this process 1000 times, yielding a cumulative distribution for the pairwise comparisonstest’s success rate under the null hypothesis. This distribution is centered around 0.5, andthe p-value assigned to a success rate is the fraction of success rates in the cumulative distribution simulated under the null hypothesis that exceed the observed rate. When there are
more observations (more bins, more educational categories, etc), the cumulative distribution
exhibits less dispersion, yielding a more powerful test.

57

E

Appendix: Empirical results

E.1

US-born and foreign-born skill distributions

Table 5 shows that more educated skill groups generally have higher population elasticities:
the population elasticity of professionals exceeds that of bachelor’s degree holders, which
exceeds that of associate’s degree holders, and so forth. In this appendix subsection, weinvestigate outliers that deviate from this broad pattern.
In particular, the population elasticity of those never reaching high school is 1.089. Thisskill group constitutes about 4 percent of the population and is overwhelmingly foreign-born.
The second column of Table 5 reveals that its high population elasticity is attributable tothe presence of foreign-born individuals with low educational attainment in larger cities.
How should we interpret the difference between the spatial distribution of skills among the
population as a whole and among US-born individuals? One possibility is that immigrants
strongly prefer larger cities for reasons omitted from our model, causing less-skilled foreignborn individuals to disproportionately locate in larger cities. This would be consistent withan established literature that describes agglomeration benefits particular to unskilled foreignborn individuals, such as linguistic enclaves (Edin, Fredriksson, and Aslund, 2003; Bauer,
Epstein, and Gang, 2005).61
Eeckhout, Pinheiro, and Schmidheiny (2014) articulate another possibility, in which an
economic mechanism they term “extreme-skill complementarity” causes less skilled individuals, foreign-born or US-born, to disproportionately reside in larger cities. Since this theory
is silent with regard to birthplace, it predicts that in the absence of foreign-born low-skilledindividuals, US-born low-skilled individuals would disproportionately locate in larger cities.
We attempt to distinguish between these hypotheses by looking at the skill distributionsof US cities two decades earlier. In 2000, foreign-born individuals were 11 percent of the
US population, while in 1980 they constituted about 6 percent (Gibson and Jung, 2006).
More importantly, in 2000, foreign-born individuals constituted 71 percent of the least-skilledgroup, while in 1980 they were only 27 percent. If our hypothesis that less-skilled foreignborn individuals are particularly attracted to larger cities is correct, then the population
elasticity of less-skilled types should be lower when foreign-born shares are lower. Table
E.1 demonstrates that this is the case in 1980.62 It does not provide any evidence that the61

Another potential mechanism is that immigrants may find larger cities’ combination of higher nominal
wages and higher housing prices more attractive than natives (Diamond, 2016), possibly because they remit
their nominal incomes abroad or demand less housing than US-born individuals. Albert and Monras (2018)
study this mechanism in detail.
62
The educational categories in Table E.1 differ from prior tables because Census microdata collected prior
to 1990 identify coarser levels of educational attainment in terms of years of schooling rather than highest

58least skilled were overrepresented in larger cities in 1980, among either the population as awhole or US-born individuals. Reconciling these results with the birthplace-neutral modelof Eeckhout, Pinheiro, and Schmidheiny (2014) would require that the production functionchanged from top-skill complementarity in 1980 to extreme-skill complementarity in 2000.
The contrast in the least-skilled population elasticities between 1980 and 2000 for the
population as a whole overwhelmingly reflects the increasing foreign-born share in the leastskilled groups. For foreign-born individuals with less than high school education, the population elasticities were 1.46 in 1980 and 1.43 in 2000. For US-born individuals, these population
elasticities were 0.89 in 1980 and 0.86 in 2000. That is, these birthplace-specific elasticities
hardly budged over twenty years. What changed was the foreign-born share: the vast majority of the difference in population elasticities for the less-than-high-school skill group in 1980
and 2000 is due to its increasingly foreign-born composition. If the total less-than-high-school
population were its year-2000 size and exhibited the foreign-born and US-born population
elasticities estimated for the year 2000, but the share of this group that was US-born wereits 1980 share (73%) rather than its 2000 share (29%), the less than high school group’s
population elasticity would be 0.944. This is close to the population elasticity of 0.974 estimated using the 1980 data. Thus our model’s poor prediction of the less-than-high-school
population elasticity in 2000 is attributable to large cities being particularly attractive to
foreign-born individuals.

E.2

Sectoral population elasticities

Tables E.2 and E.3 report the population elasticities estimates depicted in Figures 1 and 2,
along with the accompanying standard errors. Table E.3 also reports elasticities estimatedusing only city-industry employment levels that are not censored by being reported as fallingwithin an interval in the County Business Patterns data.

degree attained.

59

Table E.1: Population elasticities of seven skill groups, 1980
(1)
(2)
All
US-born
0.974
0.890(0.0244) (0.0269)
1.017
0.995
(0.0165) (0.0187)
0.993
0.974
(0.0101) (0.0119)
1.055
1.041
(0.0162) (0.0166)
1.093
1.074
(0.0154) (0.0155)
1.106
1.088
(0.0171) (0.0177)
1.134
1.114
(0.0216) (0.0215)

Dependent variable: ln f (ω, c)βω1 Less than high school × log populationβω2 High school dropout × log populationβω3 Grade 12 × log populationβω4 1 year college × log populationβω5 2-3 years college × log populationβω6 4 years college × log populationβω7 5+ years college × log population

Population
Share
share
US-born
0.09
0.73
0.13

0.92

0.34

0.93

0.10

0.94

0.12

0.92

0.11

0.92

0.11

0.90

Notes: Standard errors, clustered by MSA, in parentheses. Sample is individuals 25
and older in the labor force residing in 270 metropolitan areas in 1980.

Table E.2: Occupational employment population elasticities
βσ1 Farming, Fishing, and Forestry Occupations
βσ2 Building and Grounds Cleaning and Maintenance Occupations
βσ3 Food Preparation and Serving Related Occupations
βσ4 Construction and Extraction Occupations
βσ5 Production Occupations
βσ6 Transportation and Material Moving Occupations
βσ7 Installation, Maintenance, and Repair Occupations
βσ8 Healthcare Support Occupations
βσ9 Personal Care and Service Occupations
βσ10 Office and Administrative Support Occupations
βσ11 Protective Service Occupations

0.807(0.0470)1.038(0.0103)
0.984(0.0104)
1.035(0.0138)1.040(0.0250)1.058(0.0135)1.012(0.0110)0.977(0.0130)1.064(0.0170)1.079(0.00999)
1.120(0.0140)

βσ12 Sales and Related Occupations
βσ13 Management Occupations
βσ14 Arts, Design, Entertainment, Sports, and Media Occupations
βσ15 Business and Financial Operations Occupations
βσ16 Computer and Mathematical Occupations
βσ17 Architecture and Engineering Occupations
βσ18 Healthcare Practitioners and Technical Occupations
βσ19 Community and Social Services Occupations
βσ20 Education, Training, and Library Occupations
βσ21 Life, Physical, and Social Science Occupations
βσ22 Legal Occupations

Notes: Standard errors, clustered by MSA, in parentheses.

60

1.035
(0.00917)
1.080(0.0143)
1.157(0.0189)
1.202(0.0177)
1.393(0.0335)
1.205(0.0255)
0.998(0.0139)
0.982(0.0199)1.010(0.0168)
1.166(0.0295)
1.198(0.0221)

Table E.3: Industrial employment population elasticities
βσ1 Forestry, fishing, hunting, and agriculture support
βσ2 Accommodation and food services
βσ3 Construction
βσ4 Admin, support, waste mgt, remediation services
βσ5 Transportation and warehousing
βσ6 Other services (except public administration)
βσ7 Retail trade
βσ8 Manufacturing
βσ9 Wholesale trade
βσ10 Real estate and rental and leasing
Observations
Only uncensored observations

(1)
0.774

(2)
0.660

(0.0546)(0.142)

0.992

0.991

(0.0125)(0.0129)

1.059

1.057

(0.0158)(0.0160)

1.200

1.207

(0.0206)(0.0195)

1.178

1.175

(0.0253)(0.0264)

1.032

1.032

(0.0109)(0.0110)

0.961

0.962

(0.00739)(0.00713)

0.979

0.973

(0.0293)(0.0296)

1.165

1.159

(0.0193)(0.0190)

1.162

1.160

(0.0142)(0.0145)

5,240

4,295
Yes

βσ11 Utilities
βσ12 Arts, entertainment and recreation
βσ13 Mining
βσ14 Health care and social assistance
βσ15 Finance and insurance
βσ16 Information
βσ17 Management of companies and enterprises
βσ18 Professional, scientific and technical services
βσ19 Educational services

Observations
Only uncensored observations

Notes: Standard errors, clustered by MSA, in parentheses.

61

(1)
0.999

(2)
0.917

(0.0338)(0.0545)

1.130

1.125

(0.0220)(0.0251)

0.829

0.633

(0.0621)(0.122)

0.946

0.946

(0.0127)(0.0127)

1.170

1.167

(0.0206)(0.0215)

1.171

1.184

(0.0216)(0.0248)

1.506

1.406

(0.0435)(0.0469)

1.263

1.267

(0.0174)(0.0177)

1.203

1.205

(0.0346)(0.0391)

5,240

4,295
Yes

E.3

Skill distribution: Alternative inclusion criteria

In the main text, we describe the skill distribution using labor-force participants 25 andolder. Tables E.4 through E.6 demonstrate that we obtain similar results when using alternative inclusion criteria. First, we use a broader sample of individuals, counting everyone 25
and older regardless of labor-force participation. For this, we use aggregate tabulations of
population over 25 by educational attainment from the Census of Population, Summary File
3. Second, we use a narrower sample of individuals, restricting attention to the populationof full-time full-year (FTFY) workers 25 and older in the IPUMS microdata.
Table E.4: Skill groups by educational attainment
Summary File 3
population share.46

IPUMS FTFY
population share
.35

Share
US-born.79

Some college

.28

.32

.89

Bachelor’s or more

.27

.34

.85

Skill (3 groups)
High school or less

Skill (9 groups)
Less than high school
High school dropout
High school graduate
College dropout
Associate’s degree
Bachelor’s degree
Master’s degree
Professional degree
Doctorate

Summary File 3
population share
.07
.12
.27
.21
.06
.17
.06
.02
.01

IPUMS FTFY
population share
.03
.07
.24
.24
.08
.21
.08
.03
.01

Share
US-born
.27
.73
.88
.90
.88
.87
.84
.82
.72

Notes: “Summary File 3” denotes 2000 Census of Population Summary File 3 tabulations of
populations 25 and older by educational attainment for 270 metropolitan areas. “IPUMSFTFY” denotes full-time, full-year employees 25 and older residing in 270 metropolitan areas in
IPUMS Census of Population microdata. The “share US-born” columns are computed using the
IPUMS FTFY microdata.

Table E.5: Population elasticities of three skill groups

Dependent variable: ln f (ω, c)

(1)
SF3
All

(2)
IPUMS FTFYAll

(3)
IPUMS FTFY
US-bornβω1 High school or less × log population

0.976

0.955

0.899

(0.00984)(0.0110)(0.0160)

βω2 Some college × log population

1.010

0.999

0.973

(0.00770)(0.0105)(0.0123)βω3 Bachelor’s or more × log population

1.093

1.097

1.068

(0.0125)(0.0152)(0.0163)

Notes: Standard errors, clustered by MSA, in parentheses. Column 1 sample is Summary File
3 tabulations of populations by educational attainment. Column 2 sample is full-time, full-year
employees 25 and older residing in 270 metropolitan areas in IPUMS Census of Population
microdata. Column 3 is the column 2 sample restricted to individuals born in the United States.

62

Table E.6: Population elasticities of nine skill groups

Dependent variable: ln f (ω, c)

(1)
SF3
All

(2)
IPUMS FTFYAll

(3)
IPUMS FTFY
US-bornβω1 Less than high school × log population

1.026

1.119

0.877

(0.0221)(0.0336)(0.0270)βω2 High school dropout × log populationβω3 High school graduate × log populationβω4 College dropout × log populationβω5 Associate’s degree × log populationβω6 Bachelor’s degree × log populationβω7 Master’s degree × log populationβω8 Professional degree × log populationβω9 PhD × log population

1.012

1.014

0.941

(0.0146)(0.0156)(0.0193)

0.953

0.927

0.893

(0.0111)(0.0135)(0.0168)

1.010

1.002

0.977

(0.00838)(0.0113)(0.0131)

1.013

0.999

0.968

(0.0121)(0.0142)(0.0155)

1.096

1.098

1.070

(0.0119)(0.0148)(0.0165)

1.094

1.117

1.085

(0.0155)(0.0178)(0.0184)

1.115

1.105

1.076

(0.0139)(0.0170)(0.0180)

1.079

1.080

1.031

(0.0282)(0.0334)(0.0315)

Notes: Standard errors, clustered by MSA, in parentheses. Column 1 sample is Summary File
3 tabulations of populations by educational attainment. Column 2 sample is full-time, full-year
employees 25 and older residing in 270 metropolitan areas in IPUMS Census of Population
microdata. Column 3 is the column 2 sample restricted to individuals born in the United States.

63

E.4

Skill distribution: Inferring skills from wages

In the main text, we use educational attainment as a proxy for skill, assuming that the distribution of skills increases with educational attainment. In this section, we infer skills fromnominal wages. The theoretically appropriate wage measure differs between the productivity
and amenity interpretations of a location’s desirability.
In the productivity interpretation of the desirability of a location, individuals of the sameskill earn the same income, K −1 (ω) G (ω), everywhere they locate in equilibrium. Since both
locational assignments K and the skill component of income G are strictly increasing functions, there is a one-to-one mapping between wages and skill. Thus, if the skill distributionf (ω, c) is log-supermodular, the number of people earning a given wage in a city is alsolog-supermodular in the wage and city population size. If we divide individuals into twenty
wage ventiles, there should be relatively more people in the higher wage ventiles in larger
cities.
In the amenity interpretation of desirability, individuals of the same skill earn different
incomes in different cities. As shown in Appendix A.2, an individual of skill ω in city c has
a nominal income of A (c) G (ω), which obviously varies with city c. Thus, nominal wages
alone cannot be used to infer an individual’s skill. Within a city, the relative income of two
G(ω)A(c)G(ω)
skill types is independent of the city, A(c)G(ω
0 ) = G(ω 0 ) . In our model, the least skilled type ωis present in all cities. Thus, if we normalize all observed incomes by the least skilled type’s
income in each city, we can compare the distributions of these normalized incomes acrosscities. To do so empirically, we normalize wages in each city relative to the fifth percentile
wage in that city and compute national ventiles of this normalized wage distribution. There
should be relatively more people in the higher normalized-wage ventiles in larger cities.
Table E.7 reports the estimated population elasticities of these wage ventiles for the
productivity and amenity interpretations for both the entire population and US-born individuals. These population elasticities are almost always monotonically increasing acrosswage ventiles. The sole statistically significant exception is the first ventile. The five percentof the population that earns the lowest nominal wages are overrepresented in larger cities
relative to our model’s predictions. But the overwhelming majority of the wage distribution
exhibits log-supermodularity, in line with our theory.

E.5

Pairwise comparisons: Alternative weighting schemes

In the main text, we report the success rate for the pairwise comparisons test as a weighted
average of the share of pairwise comparisons that yield the predicted inequality. As shown inAppendix D, in the presence of idiosyncratic errors, the predicted inequality should hold with
64

Table E.7: Population elasticities of wage ventiles(1)(2)
productivity interpretationAllβω1 wage ventile 1 × log populationβω2 wage ventile 2 × log populationβω3 wage ventile 3 × log populationβω4 wage ventile 4 × log populationβω5 wage ventile 5 × log populationβω6 wage ventile 6 × log populationβω7 wage ventile 7 × log populationβω8 wage ventile 8 × log populationβω9 wage ventile 9 × log populationβω10 wage ventile 10 × log populationβω11 wage ventile 11 × log populationβω12 wage ventile 12 × log populationβω13 wage ventile 13 × log populationβω14 wage ventile 14 × log populationβω15 wage ventile 15 × log populationβω16 wage ventile 16 × log populationβω17 wage ventile 17 × log populationβω18 wage ventile 18 × log populationβω19 wage ventile 19 × log populationβω20 wage ventile 20 × log population

US-born(3)(4)
amenity interpretation
AllUS-born

0.898

0.816

1.013

0.930

(0.0171)(0.0164)(0.00843)(0.0172)

0.869

0.780

0.958

0.875

(0.0167)(0.0180)(0.0108)(0.0204)

0.868

0.795

0.951

0.882

(0.0133)(0.0166)(0.0121)(0.0217)

0.878

0.814

0.955

0.899

(0.0127)(0.0182)(0.0136)(0.0214)

0.916

0.862

0.962

0.911

(0.0111)(0.0162)(0.0154)(0.0229)

0.929

0.881

0.968

0.925

(0.0123)(0.0176)(0.0153)(0.0216)

0.945

0.902

0.982

0.942

(0.0120)(0.0168)(0.0163)(0.0218)

0.975

0.934

0.997

0.959

(0.0111)(0.0154)(0.0145)(0.0196)

0.993

0.958

1.006

0.971

(0.0119)(0.0160)(0.0135)(0.0181)

1.009

0.972

1.014

0.982

(0.0110)(0.0146)(0.0121)(0.0158)

1.023

0.992

1.014

0.984

(0.0124)(0.0157)(0.0117)(0.0149)

1.035

1.004

1.020

0.991

(0.0110)(0.0138)(0.0125)(0.0148)

1.046

1.020

1.021

0.994

(0.0115)(0.0140)(0.0136)(0.0164)

1.062

1.036

1.035

1.008

(0.0116)(0.0138)(0.0123)(0.0136)

1.073

1.046

1.020

0.994

(0.0129)(0.0152)(0.0124)(0.0133)

1.103

1.078

1.040

1.014

(0.0134)(0.0154)(0.0114)(0.0121)

1.123

1.097

1.057

1.031

(0.0142)(0.0155)(0.0134)(0.0136)

1.178

1.157

1.076

1.052

(0.0155)(0.0167)(0.0151)(0.0149)

1.238

1.215

1.115

1.091

(0.0183)(0.0192)(0.0173)(0.0165)

1.268

1.250

1.163

1.143

(0.0175)(0.0181)(0.0165)(0.0157)

Notes: Standard errors, clustered by MSA, in parentheses. Sample is individuals 25 andolder earning hourly wages greater than $2/hour residing in 270 metropolitan areas. Wages
in columns 3 and 4 are normalized by city as described in the text.

65

greater probability when there are larger differences in population size and skill intensity.
There is more at stake when evaluating our model’s prediction comparing the skill distributions of Chicago (population 9 million) and Des Moines (population 456 thousand) than its
prediction comparing Des Moines and Kalamazoo (population 453 thousand). Nonetheless,
we report success rates with alternative weighting schemes, including no weights at all. Since
there is significant variation in differences in populations and skill intensities, as depicted in
Figure E.1, the unweighted success rates may differ substantially from those weighted by therelevant criteria.
Tables E.8 through E.11 present the pairwise comparisons for skill groups, occupations,and industries with a variety of weights for the same set of bins presented in the main text.
The unweighted success rates show patterns similar to those of the weighted success rates interms of birthplace and number of bins. The unweighted success rates are generally lower
than the weighted success rates, demonstrating that our model’s predictions are borne out inthe data more frequently when the relevant differences in population size and skill intensity
are larger. Consistent with the results of subsection 5.3, our theory’s predictive successes are
systematic and the empirical departures from our theory appear to be idiosyncratic. These
results are highly statistically significant, confirming the model’s predictive power.

024
Difference in log population

37,950 pairwise comparisons
between 276 MSA populations
Data source: Census 2000
PHC-T-3

6

Density
.3
.1
0

0

0

.05

.2

.2

.1

DensityDensity
.15

.4

.4

.2

.5

.6

.25

Figure E.1: Differences in populations and skill intensities

02

4
Difference in schooling

6

8

0

1

2
3
Difference in schooling45

231 pairwise comparisons between 171 pairwise comparisons between
22 2-digit SOC occupations
19 2-digit NAICS industries
Data source: 2000 Census ofPopulation microdata via
IPUMS-USA

66

Data source: 2000 Census ofPopulation microdata via
IPUMS-USA

Table E.8: Pairwise comparisons of three skill groups
Pairwise comparisons of three skill groups
Total
Unweighted
Pop-diff-weighted
Bins comparisons All US-born All
US-born
2
3
1.00
1.00
1.00
1.00
3
5

9
30

10

135

30

1305

90

12015

270

108945

(.17)

(.17)

(.17)

(.17)

1.00

1.00

1.00

1.00

(.00)

(.00)

(.00)

(.00)

.93

.97

.97

.99

(.01)

(.00)

(.00)

(.00)

.77

.81

.89

.91

(.00)

(.00)

(.00)

(.00)

.70

.74

.82

.86

(.00)

(.00)

(.00)

(.00)

.64

.67

.74

.77

(.00)

(.00)

(.00)

(.00)

.60

.61

.67

.69

(.00)
(.00)
(.00)
(.00)
Notes: P-values for uniform null hypothesis in parentheses.
Sample is individuals 25 and older in the labor force residing in
270 metropolitan areas.

Bins
2
3
5
10

Table E.9: Pairwise comparisons of nine skill groups
Pairwise comparisons of nine skill groups
Total
Unweighted
Pop-diff-weighted Pop-diffxedu-sharecomparisons All US-born All
US-bornAll
US-born
36
.61
.86
.61
.86
.75
.90
108
360
1620

30

15660

90

1441802701307340

(.38)

(.02)

(.38)

(.02)

(.08)

(.01)

.58

.80

.61

.83

.75

.88

(.33)

(.00)

(.26)

(.00)

(.02)

(.00)

.58

.76

.62

.80

.76

.85

(.21)

(.00)

(.12)

(.00)

(.00)

(.00)

.58

.67

.63

.75

.74

.81

(.07)

(.00)

(.02)

(.00)

(.00)

(.00)

.57

.62

.62

.70

.72

.77

(.01)

(.00)

(.00)

(.00)

(.00)

(.00)

.55

.59

.59

.66

.66

.71

(.00)

(.00)

(.00)

(.00)

(.00)

(.00)

.54

.56

.57

.61

.61

.64

(.00)
(.00)
(.00)
(.00)
(.00)
(.00)
Notes: P-values for uniform null hypothesis in parentheses. Sample is individuals 25
and older in the labor force residing in 270 metropolitan areas.

67

Bins
2

Table E.10: Pairwise comparisons of occupations
Total
Unweighted
Pop-diff weighted
Pop-diff x skill-diffcomparisons success rate weighted success rate weighted success rate
231
.70
.70
.75

3

693

5

2,310

10

10,3953090
276100,485
925,155
8,073,382

(.02)

(.02)

(.03)

.66

.68

.74

(.00)

(.00)

(.00)

.64

.67

.71

(.00)

(.00)

(.00)

.59

.62

.65

(.00)

(.00)

(.00)

.56

.60

.63

(.00)

(.00)

(.00)

.55

.59

.62

(.00)

(.00)

(.00)

.54

.55

.57

(.00)
(.00)
(.00)
Notes: P-values for uniform null hypothesis in parentheses.

Bins
2

Table E.11: Pairwise comparisons of industries
Total
Unweighted
Pop-diff weighted
Pop-diff x skill-diffcomparisons success rate weighted success rate weighted success rate
171
.65
.65
.77

3

513

5

1,710

10

7,695

30

74,38590684,855276

6,469,758

(.09)

(.09)

(.05)

.64

.64

.74

(.02)

(.03)

(.01)

.62

.64

.74

(.00)

(.00)

(.00)

.59

.62

.71

(.00)

(.00)

(.00)

.57

.60

.66

(.00)

(.00)

(.00)

.56

.59

.64

(.00)

(.00)

(.00)

.54

.57

.61

(.00)
(.00)
(.00)
Notes: P-values for uniform null hypothesis in parentheses.

68