Cities’ Demand-Driven Industrial Compositions andRefined Market-Size Effects*
Takashi Onoda†
Japan Bank for International Cooperation and
TDB-CAREE, Graduate School of Economics, Hitotsubashi University

June 20, 2024
Abstract
Large cities specialize in income-elastic sectors. I develop a model of cities with sectorspecific trade costs and income elasticities of demand to explain this finding and derive potential implications. Higher productivity and better amenities make cities larger, command
agglomeration economy and higher income, and, consequently, spend more on income-elastic
sectors. This demand pattern translates into a more pronounced production pattern, explaining the observed employment distribution. The model refines the market-size effect on factor
prices, showing that wages rise with city (or country) size and expenditure shares in hightradability sectors. When sectors are gross complements or high-tradability sectors are incomeelastic, it suggests that large cities spend more on high-tradability sectors, raising the city-sizewage premium.
Keywords: Home-market effect, Market size, Trade cost, Non-homothetic preference, Cities* An earlier version of this paper was circulated under the title “Cities’ Demand-Driven Industrial Composition.”

This paper is based on a chapter of my dissertation at the University of Chicago. I am grateful to my advisors, Jonathan
Dingel, Rodrigo Adão, and Felix Tintelnot, for their invaluable guidance. I also thank Shota Fujishima, Kentaro
Nakajima, Nancy Stokey, Yoichi Sugita, Kensuke Teshima, Thomas Winberry, Dao-Zhi Zeng, and seminar participantsat the Hitotsubashi Trade/Urban Economics Workshop, Regional Science Workshop at Tohoku University, and thethird-year macro research seminar and International Trade Working Group at the University of Chicago.
†
Email: onoda.takashi@gmail.com

1

1

Introduction

Industrial composition varies significantly across cities. For example, Detroit and Silicon Valley are synonymous with cars and computers, respectively. Figure 1 shows that much of the variation in employment
composition across the U.S. Metropolitan Statistical Areas (MSAs) is linked to the income elasticity ofdemand for that industry’s output. Industries that employ relatively more workers in large MSAs have highincome elasticities of demand, such as air transport services and recreational and other services. Conversely,industries that employ relatively more workers in small MSAs, such as beverage and tobacco products, havelower income elasticities.

Income Elasticity of Demand

1.4

Insurance

Electronic equipment

1.2

Business services
Recreational and other services

Communication
Fin. Services nec

Trade

Wood prod.
Motor vehicles and parts

1.0

Transport equipment nec

Dairy prod.

Paper prod., publishing

Air transport

Water transport

Meat prod.
Manuf. nec

Leather prod.
Pub. Adm. and services
Water

Electricity

Machinery and equipment nec

β = 0.71

Chemical, rubber, plastic products

Food prod. nec

SE = 0.18

Construction
Transport nec

0.8

Petroleum, coal prod.

0.6

0.8

R2 = 0.29

TextilesBeverage and tobacco prod.

1.0

1.2

1.4

Population Elasticity of Employment

Figure 1: Elasticity of Employment with respect to MSA’s Population and Elasticity of Demandwith respect to Income
Notes: The upward-sloping line represents the weighted regression line, with total sectoral
employment in the sample used as weights. This graph omits the "Forestry" sector, whose verticaland horizontal axis values are 0.36 and 0.19, respectively, with a weight of less than 0.1 percent.
Standard errors are calculated using the bootstrap method. MSA population data are obtained fromthe 2016 Bureau of Economic Analysis. Sectoral employment is calculated by mapping the 2016
County Business Patterns data, based on NAICS, to GTAP sectors. The population elasticity of
employment is obtained while controlling for regions ∈ {Northeast, Midwest, South, West}. Income
elasticity estimates are from Caron et al. (2020). For data construction details, see Appendix A.

2

Motivated by this new finding, this study explores the implications of cross-location demand patterns
by developing a model of many equidistant cities with many sectors that feature sector-specific trade costs
and income elasticities of demand. My model builds on Matsuyama (2019), who theoretically studies international trade patterns with a continuum of differentiated-goods sectors under monopolistic competition andheterogeneous sectoral income elasticities. His model endogenously generates a demand pattern that a large
or productive country spends more on income-elastic sectors, which translates into employment and tradepatterns. I extend Matsuyama (2019)’s model by introducing worker mobility, location-specific amenity
levels, sector-specific trade costs, and more than two locations.
In my model, the fundamentally different productivity levels and amenities of cities generate a pattern ofcity size, eventually leading to the specialization pattern consistent with Figure 1. Cities with better fundamentals attract workers, generating an agglomeration economy. Owing to the fundamentals and agglomeration economy, residents in large cities have higher real income, spending relatively more on income-elastic
industries. This expenditure pattern yields an employment pattern, driven by the home-market effect, initially hypothesized by Linder (1961) and formally theorized by Krugman (1980). When the relative market
size of sectors varies across regions, regions export goods for which they have relatively large domestic markets. In the presence of trade costs and increasing returns, local firms are incentivized to operate in a sector
with a relatively larger home market, and this incentive is strong enough to amplify the demand pattern into
the production pattern. Hence, in equilibrium, regions with better fundamentals become larger and specialize
in income-elastic sectors.
As a new theoretical result, I show that sectoral expenditure composition influences wages in cities andcountries, refining the market-size effect on factor prices. Krugman (1980, 1991) demonstrates that thelarger country commands the higher wage because an advantage in market size—savings of trade costs—
must be offset by a disadvantage in production cost—a higher wage—due to firms’ free entry.1 Conversely,
given a wage level, a larger market generates stronger labor demand by attracting firms. In my model, thiseffect strengthens in sectors with lower trade costs (higher tradability), reflecting their intensive inter-city
1

The definitions and terminology of the market-size effect and the home-market effect have some variations in theliterature. This paper uses the “market-size effect on factor prices” and the “home-market effect on trade patterns” orsimply the “home-market effect.” It also calls them the “market-size effects.”

3

competition. Consequently, shifting expenditures toward higher-tradability sectors produces greater citylevel aggregate labor demand, raising the wage. A city-size wage premium potentially reflects two effects
through this newfound channel. First, when high-tradability sectors are income-elastic, large cities spend
more on them, raising the premium. Second, inter-sectoral substitution strengthens and weakens the marketsize effect on factor prices when sectors are gross complements and substitutes, respectively, affecting thepremium correspondingly. I discuss these effects on an economy consisting of manufacturing and services,which have substantially different trade costs and income elasticities.
Understanding cities’ industrial composition is important in its own right from several perspectives.
First, it is a critical factor for local economies because local economic performance is significantly affectedby the industries located in a city (e.g., Autor et al. 2013; Mian and Sufi 2014). Second, industrial composition is related to heterogeneous returns to experience across cities. Eckert et al. (2022) study a naturalexperiment of refugees arriving in Denmark and document 35% faster wage growth with each additional yearof experience for refugees. They find that a substantial part of this growth difference is attributable to cities’
industrial composition, suggesting that understanding the determinants of industrial composition is essential
in studying productivity growth in cities. Third, the mechanism that drives industrial composition is also
crucial for researchers who want to exploit regional variation in the size of industries. In this study, I showthat industrial composition is related to city size. Given this relationship, regressing dependent variables on
one or two independent variables (e.g., industrial size, city size, or wage levels) while not controlling for the
rest of the examined locations might lead to an omitted variable bias problem. Understanding the mechanism
can help researchers avoid this endogeneity issue.
To the best of my knowledge, this study is the first on cross-city inter-sectoral specialization patterns
within differentiated-goods sectors from a demand-side perspective. On intra-city issues, Hoelzlein (2023)
studies firm and labor sorting in his quantitative model with heterogenous income elasticities and skill intensities. Most of the existing works on cross-city specialization patterns focus on non-demand side factors(Behrens and Robert-Nicoud 2015; Davis and Dingel 2019; Duranton and Puga 2005; Gaubert 2018; Henderson and Ono 2008). A few studies focus on the demand side’s effect on cross-city differences (e.g.,
Handbury 2019). Among these studies, the one most closely resembling mine is by Dingel (2017). However, there are two significant differences between Dingel (2017) and this study. First, we focus on different

4

types of specialization and trade patterns. Dingel (2017) examines intra-sectoral trade, where different quality goods are gross substitutes. In contrast, this study examines inter-sectoral specialization, where goods
in different sectors can be gross complements or substitutes. Second, my model has mobile agents, unlike
Dingel (2017)’s model. These assumptions fit the urban economy environment, enabling the analysis of therelationship between the size of a city and its industrial composition.
The new result on factor prices contributes to studies of income inequality across regions and countries. Empirical works that analyze the market-size effect consider the proximity to and the size of othermarkets as well as home-market size. The aggregate measure is known as market potential (Harris (1954);
Hanson (2005)) or market access (Redding and Venables (2004)), and many have estimated the effect on
wages or economic development (Hanson (1997); Redding and Venables (2004); Hanson (2005); Head andMayer (2006); Redding and Sturm (2008); Head and Mayer (2011); Brülhart et al. (2012); Jacks and Novy
(2018); Jaworski and Kitchens (2019)); however, they assume a single goods-producing sector in constructing the market potential. Although it is practically reasonable, this study suggests that models with multiple
goods-producing sectors and location-specific expenditure composition (this variation is significant, especially across countries) can improve quantitative analysis in future research.
Furthermore, this work contributes to two strands of the international trade literature. One focuses on
demand variation as a driver of trade or industrial concentration (e.g., Flam and Helpman 1987; Stokey 1991;
Fajgelbaum et al. 2011; Caron et al. 2014; Matsuyama 2019; Costinot et al. 2019); the other examines thatof variation in tradability (Amiti (1998); Davis (1998); Hanson and Xiang (2004); Behrens (2005); Laussel
and Paul (2007); Erhardt (2017)). My model produces a condition for a location to become a net exporterin a given sector; it simultaneously reflects the two factors, refining our understanding of the home-market
effect.
This paper’s technical contribution is the simultaneously obtained analytical results on city size, wages,
and trade patterns in the presence of increasing returns, trade costs, worker mobility, and rich sectoral characteristics in many equidistant cities. I extend the technique of Zeng and Uchikawa (2014), who examine
a model of many equidistant countries, to prove the existence of an equilibrium.2 Subsequently, I obtain
2

Redding (2016) provides a condition for a spatial equilibrium to exist and be unique, as well as comparative statics

in a model with a rich geography and increasing returns, following Allen and Arkolakis (2014). However, multiple

5

sufficient conditions on cities’ productivity—jointly with amenities in the case of a common trade cost—for
a city to become larger and, separately, to command a higher wage than others. Finally, I analyze inter-city
trade patterns.
The rest of this paper is organized as follows. Section 2 introduces the model. Section 3 characterizes theequilibrium conditions by obtaining city-level labor demand and labor supply as functions of the city’s factor
price. The labor demand takes expenditure composition and population allocation as given and exhibits themarket-size effect on factor prices; thus, this part independently applies to a broad range of models. Section
4 obtains intuitions for city-size and factor-price patterns using these equilibrium conditions for one city.
Section 5 provides cross-city analysis in equilibrium with common tradability and replicates the employment
pattern of Figure 1. Subsequently, Section 6 completes the cross-city analysis with sector-specific tradability
and discusses potential implications for an economy consisting of manufacturing and services. Subsequently,
Section 7 presents two robustness checks of the employment pattern of Figure 1, one of which a model
prediction guides. Finally, Section 8 concludes the paper.

2

The Model

In this section, I introduce a model of N cities, whose set is N , and K sectors, whose set is K. Sectors
differ in relative real income elasticities ϵ(k) and iceberg trade costs τk , as well as preference shifters βk
and parameters for variable and fixed costs, vk and fk . Subscripts n, i, and j denote cities, and k and ℓdenote sectors. Cities fundamentally differ in productivity λn and amenity level an . A mass of L workers
are freely mobile and homogeneous except for an inherent preference for cities. Conditional on location,individual labor supply is inelastic. I start by explaining the workers’ problem. Appendix B provides detailed
derivations and proofs.
sectors with a non-Cobb-Douglas upper-tier preference prohibit us from applying his technique to my model.

6

Workers’ Problem
The problem for a worker ζ is given bymaxn∈N ,Cn ,{Qnk }k∈K ,{qnk (ν)}ν∈Ωnk ,k∈K[

s.t. Cn =

∑

1
η

ϵ(k)
η

βk Cn

η−1
η

k∈K

∀k Qnk =qnk (ν)

σ−1
σ

(1)

] η

η−1

,

Qnk

[∫

Cn · an · δ(ζ, n),

] σ

σ−1

dν

,

Ωnk

En =

∑∫

pnk (ν)qnk (ν)dν,k∈K Ωnk

where n is the city chosen to reside in, pnk (ν) and qnk (ν) are the price and consumption, respectively, ofvariety ν in sector k in city n, Ωnk and Qnk are the set of available varieties and the composite consumption,
respectively, of sector k in city n, En is the nominal income in city n, an > 0, 0 < η < σ, η ̸= 1,
mink∈K {ϵ(k)/(1 − η)} > −1, and σ > 1. The utility comprises three factors: Cn , an , and δ(ζ, n).
The functional form of real income Cn captures the consumer’s non-homothetic preference, following
Comin et al. (2021), Hoelzlein (2023), and Matsuyama (2019). When ϵ(k) = 0 for all k ∈ K, this Cn
becomes a standard homothetic constant elasticity of substitution function. When ϵ(k) varies across sec1/η

ϵ(k)/η

tors, a higher ϵ(k) corresponds to a more income-elastic sector; the weight βk Cn

becomes relatively

more influential as real income grows. When solving the consumption optimization problem, a convenient
property of the functional form of Cn becomes clearer. The demand function derived from this preference
becomes−η 1−η η
Qnk = βk Cnϵ(k) Pnk
C n En ,where Pnk =[∫1−σ dν
ν∈Ωnk pnk (ν)(2)]1/(1−σ)is the price index for sector k in city n. This demand function

shows that the relative real income elasticity of demand ϵ(k) and price elasticity η at the sector level are
separated. I refer to ϵ(k) simply as the income elasticity throughout the paper. Price elasticities are common
among sectors, and price elasticity at the sector level is lower than that at the variety level (η < σ). Eq. (2)implies that sectors are gross complements when η < 1 and gross substitutes when η > 1. Additionally, the7expenditure share of sector k is obtained as follows:snk ≡ ∑

where Pn =

(∑

ϵ(k)

1−η
k∈K βk Cn Pnk

Pnk Qnk= βk Cnϵ(k)PQℓ∈K nℓ iℓ

(

Pnk
Pn

)1−η
,

(3)

) 1

1−η

, the price index of city n, is defined by Pn Cn = En . This ex-

penditure share is log-supermodular in Cn and ϵ(k) (∂ 2 log snk /∂Cn ∂ϵ(k) > 0).3 It shows that, holding
the price indices {Pk }k∈K constant, agents with higher real income Cn spend relatively more on sectors
with high ϵ(k). The expense of this convenient property is endogenously varying returns to scale. The∑
elasticity of Cn with respect to composite consumption Qnk is snk [1 + k∈K snk ϵ(k)/(1 − η)]−1 . Thus,a uniform proportional change of {Qnk }k∈K does not generate an equal proportional change of Cn unless∑k∈K snk ϵ(k)/(1 − η) = 0. Furthermore, it requires mink∈K {ϵ(k)/(1 − η)} > −1 to ensure the global
monotonicity of Cn .
Workers homogeneously appreciate amenities offered by city n, such as weather, landscape, and historic
heritage, the overall level of which ai measures. While “amenity” generally refers to access to local services
and consumer goods (e.g., restaurants)—referred to as consumption amenities—in this model, those local
services and goods contribute to Cn when consumed.4
Additionally, workers have heterogeneous preferences for cities (following Tabuchi and Thisse (2002),
Redding (2016), and others); the worker ζ and city n pair receives the idiosyncratic utility shock δ(ζ, n). I
assume δ(ζ, n) is independent and identically distributed (i.i.d.) across workers and cities according to the
Fréchet distribution with shape parameter 1/γ (P r[δ < x] = e−x−1/γ). Workers choose the city that offers

the highest utility, considering their consumption optimization. Thus, given the products of real income andthe utility from amenities in cities, {Cn an }n∈N , the probability of choosing city n for a given agent ζ is∑derived as P r[an Cn δ(ζ, n) = maxi ai Ci δ(ζ, i)] = (an Cn )1/γ / i∈N (ai Ci )1/γ . As the shock is i.i.d., city
n’s population is as follows:
Ln = ∑
3
4(Cn an )1/γ
1/γi∈N (Ci ai )

L.

(4)

A differentiable function f (x, y) is log-supermodular in x and y if and only if ∂ 2 log f (x, y)/∂x∂y > 0.
My model omits non-tradable sectors and classifies local services and goods as sectors with extremely high but notunbounded trade costs. They are arguably not completely non-tradable as travelers can visit restaurants, for example.

8Given real income and the utility from amenities in cities, the lower γ is, the greater the population dispersion;
therefore, the parameter γ measures the dispersion force from heterogeneous preferences. This dispersion
force is crucial in generating a cross-city demand pattern by allowing real income to vary across cities inequilibrium.5

Firms’ Problem
Production in my model is based on Krugman (1980). For all sectors, there are endogenous sets of varieties,
homogeneous firms, and monopolistic competition. I let wn denote the wage in city n. Cities differ inproductivity λn ; each firm in sector k in city n must employ fk /λn units of labor as a fixed cost and vk /λn
as a variable cost to produce a variety. A shipment of a variety from city n to city i requires an iceberg trade
cost τnik < ∞. To keep the model tractable for analytical results, I simplify geography. Cities’ locations are
symmetric, and the iceberg trade cost is constant and greater than one between different cities, conditionalon a sector; that is, τnik = τk > 1 when n ̸= i, and τnik = 1 when n = i. The problem for a firm that
produces variety ν in sector k in city n is

πnk (ν) =max∑[{pnik (ν),qnik (ν)}i∈N

i∈N]
vk qnik (ν)
fkpnik (ν)qnik (ν) − {τk + 1{i = n}(1 − τk )}
wn −
wn ,
λn
λn
(5)

σ
Qik ,
s.t.∀i, qnik (ν) = pnik (ν)−σ Pik

where πnk (ν) is the profit from optimized production, pnik (ν) is the market price for city i, qnik (ν) is themarket quantity for city i, and 1{i = n} is an indicator function taking the value of one when i = n. In thefollowing part, I omit ν unless necessary. If sector k in city n has non-zero production in equilibrium, πnk
must be zero such that there are no new entrants, which is the zero-profit condition. I focus on equilibria in
which all cities have non-zero production in all sectors, following Matsuyama (2019) and Hanson and Xiang
5

The model can incorporate workers’ land consumption paired with inelastic land supply as an additional disper-sion force using the same functional form as Hoelzlein (2023). In such a case, real income consists of goods and landconsumption. However, without heterogeneous preferences, real income equalizes across cities, eliminating demand
patterns. Thus, heterogeneous preferences are necessary even with the dispersion force from inelastic land supply.

9

(2004). The definition of such an equilibrium is as follows.

Definition of Equilibrium
An equilibrium is {Ln , Cn , wn , En , Pn }n∈N , {pnik , qnik }(n,i,k)∈N 2 ×K , and {Ωnk }(n,k)∈N ×K such that
1. workers optimize consumption and locational choice as eq. (1) for n ∈ N ,2. workers’ income is given by En = wn for n ∈ N ,3. producers optimize production as eq. (5) for all k ∈ K and n ∈ N ,4. the zero-profit condition holds such that πnk = 0 for all k ∈ K and n ∈ N ,5. the national labor market clearing condition∑n∈N Ln = L holds, and∑6. the local labor market clearing condition k∈K xnk = 1,)(∑∫
where xnk ≡ Ωnki∈N {τk − 1{i = n}(τk − 1)} vk qnik /λn + fk /λn dν/Ln is the employment
share of sector k in city n, holds for all n ∈ N .

3

Equilibrium Conditions

This section characterizes two equilibrium conditions: city-level labor demand and city-level labor supply. The city-level labor demand shows the effect of market size on factor prices and that of expenditurecomposition. The analysis of these effects is applicable to models with immobile workers (i.e., international
trade models) and models with homothetic preference because it takes population allocation and expenditurecomposition as given.

Sectoral Labor Demand
I derive sectoral labor demand from the zero-profit conditions. Given the well-known optimized prices
pnnk = σvk wn /(σ − 1)λn and pnik = τk vk wn /(σ − 1)λn for i ̸= n, the profit for a firm in sector k in city10

n becomes
πnk = 1 vk
qnnk + τk
σ − 1 λn

∑
i̸=n


1 vk
fk 
qnik −wn .
σ − 1 λn
λn
(

1
σ−1

I set vk = (σ −1)/σ and fk = 1/σ by replacing βk with β̃k = βk fk(6)vk σ σ/(σ−1) (σ − 1)−1)1−ηwithout

loss of generality, exploiting the isomorphism.6 Then, the zero-profit condition (πnk = 0) becomes(
1 = Ln

wn
λn

)−σ

σ−1
Pnk
En snk + τk

∑
i̸=n

(
Li

wn
τk
λn

)−σ

σ−1
Pik
Ei sik .

This zero-profit condition applies to N cities. Matrix operations yield(
σ−1
1 = (1 − ϕk )Ln En snk Pnk

wn
λn

)−σ

(

∑
ϕk N

+
1 + ϕk (N − 1)

i∈N

N

wi
λi

)σ 


(

wn
λn

)−σ
,

(7)

where ϕk (≡ τk1−σ < 1) measures the tradability of sector k and is often called “freeness of trade” in the“new trade theory” literature (Baldwin et al. 2003). Eq. (7) equates fixed cost, normalized to one, on theleft-hand side to gross profit, defined as profit before deducting fixed cost, on the right-hand side.
Eq. (7) decomposes a firm’s gross profit into two sources. The first source is the market access exclusive
to local firms, reflected in the first term on the right-hand side of eq. (7). 7 The size of this exclusive market
access increases with the sectoral market size Ln En snk ; thus, a larger sectoral market, ceteris paribus, raises
σ−1
profit for local firms. The factor Pnk
is an inverse measure of competition intensity; more competitors

and lower prices of other firms’ varieties reduce profit. The second term summarizes the gross profit fromthe “nationwide market”—the summation of a fraction ϕk of every market, which firms can “access” from
6This normalization is common in the “new trade theory” and “new economic geography.” See box 2.2. of Baldwin

et al. (2003) for an additional explanation of why it is without loss of generality.
σ−1
7(1 − ϕk )Ln En snk Pnk
corresponds to the market access (Redding and Venables (2004)) that only local firms
have and that of the real market potential (Head and Mayer (2004). Firms outside city n need to pay trade costs to sell
their varieties in the city n market; they raise the price by a factor of τk and receive profit that decreases by a factorof 1 − ϕk . One can interpret this as firms selling varieties at factory gate price (σvi wi /(σ − 1)λi ) have access to thefraction ϕk of the city n market. Conversely, local firms have the exclusive market access (1 − ϕk )Ln En snk .

11any city. A higher value of wn /λn reduces profit from both sources, as it translates into a higher price.
Throughout the paper, I refer to this variable as the factor price, interpreting effective labor (λn Ln ) as the
only factor in a city. This metric captures the productivity-adjusted compensation that firms face acrossdifferent locations. In the nationwide market, a factor price is evaluated against the nationwide average—
[∑
]1/σ
σ
the power mean
. Thus, eq. (7) implies advantages and disadvantages in these threei∈N (wi /λi ) /N
dimensions—market size, competition intensity, and factor prices—must offset each other in equilibrium.
Given a city’s factor price, a larger sectoral market attracts firms, intensifying competition. I translate
this intensive competition into labor demand. I rewrite eq. (7) as(
σ−1(1 − ϕk )Ln En snk Pnk

where

wn
λn

)−σ
=

1
,
1 + µnk]−1+ N − 1)(wn /λn )σ
(ϕ−1
k ∑
µnk =
−1
> 0.
σ
j∈N (wj /λj )

(8)

[

(9)

The right-hand side (1+µnk )−1 is the share of the exclusive market assess in gross profit. The normalization
1−σ= (λnk xnk Ln )·(wn /λn )1−σ +of fk = 1/σ makes labor demand by a firm equal 1/λn . Subsequently, Pnk∑
1−σ . After substituting this expression into eq. (8), matrix operations yieldn̸=i (λik xik Li ) · (τk wi /λik )xnk Ln wn = (1 + µnk )snk Ln wn −(wn /λn )σ ∑ Li wi(1 + µik )sik .
−1
ϕk + N − 1 i∈N (wi /λi )σ(10)Eq. (10) shows two forces on the relationship between sectoral market size snk Ln wn and sectoral labordemand in value xnk Ln wn .
First, sectoral labor demand increases with sectoral market size more than one-for-one. The difference
in expenditures, ceteris paribus, is amplified to that in labor demand because µnk > 0. Thus, the endogenous
variable µnk measures the amplification strength, and I call it the “home-market multiplier” hereafter. This
σ−1
amplification occurs because of inter-city competition. Eq. (8) implies that snk Pnk
is constant, given acity and tradability; thus, a higher expenditure share must intensify competition proportionally. However, a
proportional increase of local firms does not suffice because the market has competitors from other cities,
diluting the impact of local entrants on the price index. Consequently, a disproportionate amount of firms

12enter the market until the profit becomes zero. Furthermore, this excessive amount of firms increases net
exports, which is (xnk − snk )Ln wn in value. Thus, µnk also measures the strength of the sectoral homemarket as a net-export driver.
Second, higher tradability strengthens this amplification because high ϕk raises the home-market multiplier. This might sound counterintuitive because low trade costs weaken the incentive for firms to locate inlarger markets to save on trade costs—the incentive that is central to the new trade theory and new economic
geography. Indeed, a large ϕk shrinks the exclusive market access in eq. (8), reflecting this effect. However,there are two more effects. First, recall (1+µnk )−1 is the share of the exclusive home-market access in grossprofit, which decreases in µnk . Higher tradability improves firms’ access to other markets. Consequently,
firms make greater gross profit in the nationwide market, recovering a significant portion of fixed costs
there. Conversely, high tradability escalates competition in the home market, ensuring the aggregate gross1−σ—profit does not exceed the fixed cost. Consequently, intensifying the competition further—raising Pnk

requires more entrants. Second, higher tradability transforms local entrants into more significant competitors1−σ
1−σ
for firms in other cities. Thus, changing the relative intensity of competition, measured by Pnk
/Pik
, re-

quires more entrants. These two effects dominate the diminishing incentive to save trade costs, generatingstronger labor demand.8 This sector-specific amplification strength makes expenditure composition affectcities’ factor prices, which I analyze in the next subsection.

City-Level Labor Demand and Factor Price
The market-size effect on factor prices emerges when sector-level labor demand faces the local labor market∑∑
clearing condition. Aggregating eq. (10) over sectors with k∈K xnk = k∈K snk = 1 yields the “wageequation” of this model as(∑)(wn /λn )σk∈K µnk snk Ln λn (wn /λn )
(∑
)
∑
=∑
.
σ
j∈N (wj /λj )i∈Nk∈K µik sik Li λi (wi /λi )8(11)

The absence of (1 − ϕk ) and the presence of (1 + µnk ) in eq. (10) imply that the second additional effect offsets

the diminishing incentive to locate in larger markets. I analyze this mechanism in more detail in ?.

13

This equation shows that a factor price wn /λn increases with population Ln , productivity λn , and expenditure∑weighted mean of home-market multipliers k∈K µnk snk in equilibrium, reflecting the market-size effecton factor prices of this model. To fully analyze the market-size effect, I need one more step because µnk
is a function of {wi /λi }i∈N . I factorize the effect into that of overall market size Ln λn —a well-known
effect—and that of expenditure composition {snk }k∈K —a new effect. Proposition 1 states the former.
Proposition 1 (Overall-Market-Size Effect on Factor Prices). Given expenditure compositions in two cities
n wi{snk , sik }k∈K , the relative factor price w
λn / λi increases with the relative population Ln /Li and the relative

productivity λn /λi .
A greater population expands sectoral market size, requiring more entrants or a higher factor price as I
analyzed with eq. (7). However, at the city level, entrants do not appear because city-level labor supply isfixed; thus, the factor price must rise, making the city-level labor demand curve upward-sloping. Similarly,
a higher productivity, given an income (wage) per efficiency unit, increases expenditures per worker En ;
consequently, home sectoral markets grow in value, requiring a higher factor price.
Separately, market composition affects factor prices through the city-level home-market multiplier∑k∈K µnk snk .

I can obtain Theorem 1 from eq. (11) and that ∂µnk /∂(wn /λn ) < 0.
Theorem 1 (Sectoral-Market-Composition Effect on Factor Prices). Order sectors according to their tradabilities such that ϕk increases with k. Let high-tradability sectors account for greater shares in an expenditure composition {sa (k)}k∈K than in {sb (k)}k∈K in the sense that the cumulative sum of expenditure shares∑∑is smaller for all k (i.e., ∀k ∈ K, kx=1 sa (x) ≤ kx=1 sb (x)). Then, a city with {sa (k)}k∈K commands ahigher factor price (wn /λn ) than an equal-sized and equally productive city with {sb (k)}k∈K .
Theorem 1 coupled with Proposition 1 refine the market-size effect on factor prices, considering thesector-specific power of market size. Labor demand is more responsive to expenditures in high-tradability
sectors. Consequently, when a city modifies its demand towards these sectors, it generates greater aggregate
labor demand, resulting in a higher factor price. This subsection’s analysis is agnostic on mechanisms behindcity-specific market compositions. The next subsection shows what determines market compositions in mymodel.

14

Market Composition
I solve for an expenditure share after substituting eq. (3) for Pnk in eq. (7) with Pn = En /Cn as


snk =

1
 (1 − ϕk )(1 + µnk )

(

1
1−η

β̃k

ϵ(k)
1−η

Cn

·

1−η
)σ−1  σ−ηCn1

λn (λn Ln ) σ−1



.

(12)

This equation provides two observations. First, a general equilibrium force attenuates and amplifies expenditure patterns when sectors are gross complements (η < 1) and gross substitutes (η > 1), respectively. 9ϵ(k)To see this, recall that β̃k Cnϵ(k)of β̃k Cn

is the weight of sector k in city n’s expenditure as eq. (3) shows. The productand (Pnk /Pn )1−η becomes the expenditure share, and the logarithmic derivative of which withϵ(k)respect to log β̃k Cn

is one. In contrast, that of an expenditure share in eq. (12) is∂ log snk=ϵ(k)∂ log β̃k Cnϵ(k)A high value of β̃k Cn




> 1

σ−1
σ−η 

< 1

if η > 1,
if η < 1.

generates a high expenditure share for that sector, attracting firms and lowering

the sectoral price index relative to other sectors.10 This relative price change attenuates the high expenditure share by substitution when η < 1. In contrast, it amplifies the expenditure share when η > 1.
The same observation holds when I explicitly focus on the effect of heterogeneous income elasticities as∂ 2 log snk /∂ log Cn ∂ϵ(k) = (σ − 1)/(σ − η). A city with high real income spends relatively more on
income-elastic sectors, and whether it is amplified depends on η.
Second, factor prices influence cities’ expenditure shares along the tradability dimension. The logarith9
10

Matsuyama (2019) discusses this effect in detail.
This demand-driven change in price indices can be interpreted as the Schmookler effect (Schmookler (1966)).

Relabeling varieties as intermediate inputs and consumption composites as final products like Matsuyama (2019), a
low price index corresponds to high sectoral productivity in final good production.

15

mic cross derivative of eq. (12) with respect to factor price and tradability is



< 0

∂ 2 log snk∂(wn /λn )∂ϕk > 0if η > 1,(13)
if η < 1,because ∂ 2 log(1 + µnk )/∂(wn /λn )∂ϕk < 0. A high factor price tilts expenditures towards sectors withhigh and low tradabilities when η < 1 and η > 1, respectively. This force originates from the zero profit
condition. A disadvantage in factor price requires mild competition to offset in equilibrium. In other words,
firms exit from the sectoral market. Higher tradability aggravates this disadvantage, accelerating firms’ exit
and bidding up the sectoral price index.11 The outcome depends on the inter-sectoral price elasticity η.

City-Level Labor Demand with Inter-Sectoral Substitution
Result (13) enables us to characterize the relationship between the city size and the factor price—the citylevel labor demand curve—considering endogenous expenditure shifts across the tradability spectrum. Suppose all sectors have the same income elasticity (∀k, ϵ(k) = ϵ). Then, city size increases the factor price,
following eq. (11), which in turn shifts expenditures towards sectors with high and low tradabilities when
sectors are gross complements and gross substitutes, respectively, following result (13). Theorem 1 suggests that this shift of expenditures amplifies and dampens the rise of the factor price, respectively; clearly,
it steepens the upward-sloping labor demand curve with a factor price at the vertical axis when sectors aregross complements (Figure 2). When sectors are gross substitutes, it flattens the curve and, possibly, theslope becomes downward with strong inter-sectoral substitutions (Figure 2).
11 2∂ log(Pnk /wn )/∂wn ∂ϕk > 0, following from eq. (21) in Appendix B.3.

16

wn
λn

η<1
η>1

0

Ln

Figure 2: Market-Size Effect on Factor Prices with Inter-Sectoral Substitution

City-Level Labor Supply
This subsection obtains an equation that expresses city-level labor supply. Plugging the sector-level gooddemand function (2) into the definition of real income, expressing the price indices using employment sharesand city size, and substituting (an /a1 )(Ln /L)γ1 C1 for Cn following eq. (4) yield{( )1−(σ−1)( ϵ(k) +1)γ)(1−η)( ϵ(k) +1) [
1−η
1−η
Ln
σ−1
1=
β̃k
· λn
(1 − ϕk )xnk L1
L1
k∈K
1−η
( )1−σ }] 1−σ( )σ−1 ( )−(σ−1)γ( ϵ(k) +1) ∑
1−η
Ln
wi
wn
xik Li
+ ϕk
,
λn
L1
λi
∑

(

an
a1

(14)

i∈N

which uses city 1 as a base. To interpret this equation, I impose Assumption 1.
1
Assumption 1. γ > σ−1
maxk∈K θ(k) where θ(k) = [1 + ϵ(k)/(1 − η)]−1 .

Assumption 1 ensures that the dispersion force from the heterogeneous inherent preference for cities isstronger than the agglomeration force, preventing a city from attracting all workers.1213 The factor 1/(σ −1)
is the elasticity of the agglomeration economy or the positive externality in Krugman-type models with homothetic preference; the mass of varieties in a location increases with market size, and consumers have a
12

Without the term maxk∈K θ(k), this condition corresponds to that for the existence of a unique equilibrium of

Redding (2016) without residential land use.
13
It does not contradict Assumption 4. When η > 1, these assumptions can be rewritten as ∀k, (σ − 1)−1 <γθ(k)−1 < (η − 1)−1 . Given the assumption of η < σ, this can be simultaneously satisfied.

17

love-of-variety preference.14 The additional factor maxk∈K θ(k) reflects that Cn has endogenously varying returns to scale in terms of {Qnk }k∈K . The product of these factors is the upper limit of the positive
externality from a greater population.
Given Assumption 1 and common terms among cities (a1 , L1 , and∑i∈N xik Li (wi /λi )1−σ ), eq. (14)shows that a city-level labor supply Ln increases with the factor price wn /λn conditional on a level ofamenity an , a productivity λn , and employment composition {xnk }k∈K . A high factor price raises purchasing power for varieties produced in other cities, elevating real income. A caveat is that this observation
ignores endogenous changes in cities’ employment compositions. Nevertheless, it enables us to obtain intuitions for general equilibrium results by the partial equilibrium analysis in the next section.

4

Partial Equilibrium Analysis

Given the two equilibrium relationships between city size and the factor price, this section briefly illustrates
how cities’ productivities and amenities shape patterns of city size and factor prices. Figure 3a depicts thelabor-demand curve (LD curve) from the wage equation (11) and the labor-supply curve (LS curve) from eq.
(14), holding expenditure and employment compositions constant and taking the common terms as given. 15
14

Another interpretation is a positive externality on productivity, as adopted by Matsuyama (2019). When labelingvarieties as intermediate inputs and consumption composites as final products, the agglomeration economy works on
the productivity of final products.
15
How the curves intersect is not easily identifiable from the equations in this part. The depiction here is based on
the theoretical results in the general equilibrium. The LS curve intersecting the LD curve from below is consistent withthe cross-city analysis in the next section. Moreover, it suggests the stability of an equilibrium.

18

wn
λn

wn
λn

LS

LS LD

wn
λn

LS
LD

LD

Ln
(a) Original State

Ln(b) Higher Productivity (λn ↑)Ln(c) Better Amenities (an ↑)Figure 3: Partial Equilibrium Analysis
When city n becomes more productive (λn ↑), the LD curve shifts up, and the LS curve shifts to the right,
as in Figure 3b. The LD curve shifts because higher productivity implies higher income for a given factor
price. Consequently, the home market expands in value, and the factor price must rise to maintain the zeroprofit condition. The LS curve shift reflects that real income increases for two reasons. First, higher income
improves purchasing power for varieties produced in other cities. Second, higher productivity increases themass of locally-produced varieties, lowering sectoral price indices. Owing to the shifts of the two curves,
the new intersection occurs at higher levels of city size and the factor price compared to before.
When city n has better amenities (an ↑), the LS curve shifts to the right, as in Figure 3c, because thebetter amenities attract more people. However, the LD curve is unaffected because amenities do not affectproduction. Consequently, the intersection moves along the LD curve, and the population and factor price
are higher than before.16
This partial equilibrium analysis predicts that cities with better fundamentals become larger and pay
higher factor prices. Although this analysis ignores endogenous changes in expenditure and employment
compositions and other cities’ populations, these results carry over to general equilibrium as the following
sections detail.
16

Even when the model incorporates inelastic land supply, the same result holds as long as the agglomeration force

is stronger than the dispersion force from the inelastic land supply. While this result is contrary to what a Rosen-Roback
model (Rosen 1979; Roback 1982) implies, it is not new in the literature. For example, Glaeser and Gottlieb (2009)
point out that rising amenities can increase wages because of agglomeration economy.

19

5

Cross-City Analysis with Only Sector-Specific Income Elasticities

This section demonstrates how differences in the fundamentals generate cities’ patterns of population, factor
prices, expenditures, and employment in general equilibrium, replicating the employment pattern in Figure
1. To this end, I strip the model of heterogeneous tradability by assuming ϕk = ϕ for all k in this section andfocus on the effects that heterogeneous income elasticities alone generate. In the next section, I reconsider
heterogeneous tradability to explore the potential implications of the demand-side mechanism.
The analysis of equilibrium existence precedes the cross-city analysis. To prove the existence, I adapt
the technique used by Zeng and Uchikawa (2014), who study a model of many equidistant countries. They
construct a map of wages and apply Brouwer’s fixed-point theorem to prove the existence. Extending their
method to my model requires identifying the city with the highest factor price in equilibrium. An additional
weak assumption makes this identification feasible, which I impose as Assumption 2 after defining an income
fundamental.
γ+θ(k)

Definition 1 (Income Fundamental). City n has a better income fundamental than city i if only if an λnγ+θ(k)

a i λi>for all k.

Assumption 2. There exists a city that has a better income fundamental than any other city.
In the rest of this section, I designate the city with the best income fundamental as city 1, setting w1 =λ1 = a1 = 1 without loss of generality. Subsequently, Assumption 2 ensures that city 1 has the highest
factor price in equilibrium, although I discuss the order of factor prices later in detail.17 Assumption 2, in
addition to the dispersion force being more potent than the agglomeration force (Assumption 1), suffices toshow the existence of an equilibrium as Lemma 1.
Lemma 1 (Existence of Equilibrium). Given Assumptions 1 and 2 and common tradability, there exists anequilibrium.
17

The inequality in Definition 1 is strict because it simplifies the exposition. A weak inequality suffices for the

existence of an equilibrium.

20

The endogenous expenditure composition coupled with more than two cities makes it infeasible to provethe uniqueness and stability generally.18 Nevertheless, all the following cross-city comparisons hold in all
the possible equilibria in which all sectors have non-zero production in all the cities. Further, Appendix C
proves that the equilibrium is stable and unique for the case of two cities. The rest of this section comparescities in an equilibrium with Assumptions 1 and 2 and a common trade cost.

City Size and Factor Prices
The first cross-city comparison concerns city size. The partial equilibrium analysis (Figures 3b and 3c)
suggests that cities with better fundamentals become larger. This result remains the same, and Proposition 2
considers the two fundamentals jointly.
(1+(σ−1)−1 )θ(k)Proposition 2 (Pattern of City Size). Suppose that a city has higher values of an λnθ(k)and an λnthan another city for all k. Then, the city has a greater population.
The condition in Proposition 2 ensures that city n offers a higher utility before considering the Fréchet
utility shock (Cn an > Ci ai ). A higher productivity generates the agglomeration economy by increasing
the mass of varieties that given labor produces. As discussed with Assumption 1, (σ − 1)−1 θ(k) measures(1+(σ−1)−1 )θ(k)the strength of the agglomeration economy in this model. However, the condition on an λndoes not suffice in Proposition 2 for two reasons. First, a higher productivity contributes to other cities’
agglomeration economies by trade though it does so with attritions. Second, residents consume imports,
the prices of which local productivity does not directly affect. In sum, inter-city trade attenuates crosscity variations in the agglomeration economy, lowering the elasticity of a relative agglomeration economy
with respect to relative productivity. The condition of Proposition 2 ensures that the aggregate effect from a
relative amenity level and relative productivity is greater in city n, regardless of the degree of the attenuation.
The second cross-city comparison concerns factor prices. The result of the partial equilibrium analysis
remains the same again as Proposition 3.
Proposition 3 (Pattern of Factor Prices). A city with a better income fundamental commands a higher factor
price.
18

Cities are not atomic in this model, so the proof of stability requires the analysis of all possible cases of migrations.

21

Assumption 1 implies (1 + (σ − 1)−1 )θ(k) < γ + θ(k); thus, relative productivity weighs more in
the factor-price pattern (Proposition 3) than in the city-size pattern (Proposition 2). This result reflects thatthe home-market effect couples city size with a productivity to determine a factor price (Proposition 1). A
higher value of γ makes productivity have greater weights in the order of income fundamentals; a strong
dispersion force weakens the relationship between amenities and city size, making productivities relatively
more influential in wage determination.
Thus, generally, cities with better fundamentals—productivities and amenities—become larger and command a higher factor price. Particularly, when I condition one of the fundamentals, the order of city size
perfectly coincides with that of nominal income. Recall nominal income En equals wn in this model.
Corollary 1 (City-Size and Nominal-Income Patterns Conditional on One Fundamental). A city with a better
fundamental conditional on the other becomes larger and the residents earn higher nominal income.
Workers in large cities tend to have higher nominal income in equilibrium. This result is consistent
with the stylized fact that nominal income is higher in larger cities, even when we control for observable orunobservable workers’ characteristics (Behrens and Robert-Nicoud 2015; Glaeser and Mare 2001).

Expenditure, Trade, and Employment
A city becomes large by offering a high utility before considering the Fréchet utility shock (Cn an > Ci ai ).
Subsequently, a cross-city expenditure pattern is clear when we condition on cities’ amenity level. Residents
in larger cities spend more on the income-elastic sectors, as we saw ∂ 2 log snk /∂ log Cn ∂ϵ(k) = (σ−1)/(σ−η) > 0 in Section 3. City-specific amenity levels possibly complicate this expenditure pattern. Nevertheless,I can obtain Proposition 4.
Proposition 4 (Expenditure Pattern). A sectoral expenditure share ratio of a larger city to a smaller city
conditional on one fundamental is greater in a more income-elastic sector; that is, ∀(n, i) ∈ {(n, i) ∈N 2 | Ln > Li ∧ an = ai } ∪ {(n, i) ∈ N 2 | Ln > Li ∧ λn = λi }, ∀(k, ℓ) ∈ {(k, ℓ) ∈ K2 |ϵ(k) > ϵ(ℓ)},
snk /sik > snℓ /siℓ .
Thus, residents in larger cities spend more on income-elastic sectors even when conditioned on cities’
productivity. This result reflects that better amenities attract workers, generating the agglomeration economy.

22

The home-market effect amplifies these expenditure patterns into trade patterns as Lemma 2, reflecting theforce in eq. (10).
Lemma 2 (The Home-Market Effect). There is a relationship between expenditure shares and employment
shares such that
xnk = snk + µn (snk − sk ),

(15)

where
[

(ϕ−1 + N − 1)(wn /λn )σ
∑
µn =
−1
σ
j∈N (wj /λj )

]−1
> 0,

sk =

∑

∑

i∈N

Li w i µi
sik .
j∈N Lj wj µj

Accordingly, a city is a net exporter in each sector with expenditure share snk higher than the nationwide
weighted mean sk .
Employment shares increase more than one-for-one with expenditure shares. Figure 4 depicts eq. (15).

xnk

wn
λn

′
wn
wn
<
′
λn
λn

′
wn
λ′n

45◦ line

mk

mk

mnk

Figure 4: Home-Market Effect on Trade with Common Trade Cost

The slope—one plus a home-market multiplier µn —depends on the factor price (wn /λn ), but it always
intersects the 45-degree line from below at the point where the expenditure share equals the nationwide levelsk . When an expenditure share is greater than sk , city n becomes a net exporter.19 A higher factor priceattenuates this amplification as the slope becomes flatter; in other words, a higher factor price erodes the19

Net exports increasing with demand corresponds to the “strong home-market effect,” defined by Costinot et al.

(2019).

23

competitiveness of local firms, diminishing the power of a large market. This result implies that large cities
tend to have less production specialization than small cities, which is consistent with the stylized fact thatlarge cities have more industrial diversification (Duranton and Puga (2000)).20 Corollary 2 describes cities’
exporter-importer statuses, according to Lemma 2 and the expenditure pattern of Proposition 4.
Corollary 2 (Trade Pattern with Heterogeneous Income Elasticities). Suppose that there exist two cities thatshare the level of either productivity or amenities, and the large one is a net exporter while the small one isa net importer in a sector. Then, in a more income-elastic sector, the larger one is a net importer only if thesmaller one is a net importer, and the smaller one is a net exporter only if the larger one is a net exporter.
In short, large and small cities tend to be net exporters in income-elastic and income-inelastic sectors,
respectively, because they tend to have expenditure shares higher than the nationwide weighted mean forcorresponding sectors. This is an N -equidistant-city version of the home-market effect on trade patterns.
Variations in relative market size generate comparative advantages, translating into the employment patternwith amplification. Unlike Krugman (1980), who assumes an exogenous preference difference to generate
the heterogeneous relative demand, and similar to Matsuyama (2019), differences in relative market size
arise endogenously from the non-homothetic preference.
Cities’ production pattern replicates Figure 1. Consider two cities according to eq. (15) or Figure 4 bydivision into cases. First, suppose one of cities n and i is a net exporter, and the other is a net importer.
Then, snk > sik implies xnk > xik . Second, suppose both cities are net exporters. Then, snk > sik andwn /λn < wi /λi imply xnk > xik . Finally, suppose both cities are net importers. Then, snk < sik andwn /λn < wi /λi imply xnk < xik . In equilibrium, small cities tend to have lower wn /λn , and they tend tohave higher and lower snk in income-inelastic and income-elastic sectors, respectively, following Corollary 1
and Proposition 4. Thus, small cities tend to have relatively large employment (high xnk ) in income-inelastic
sectors, replicating Figure 1.21
20

In two-location models, goods market clearing conditions imply less specialization in the larger location because

one country’s trade surplus equals the other’s trade deficit. Unlike those models, goods market clearing conditions donot, at least not directly, imply this result in this many-location model.
21
This analysis left out two cases: one where both cities are net exporters and snk > sik and wn /λn > wi /λi , andanother where both cities are net importers and snk < sik and wn /λn > wi /λi . The weaker amplification from the246

Cross-City Analysis with Added Sector-Specific Trade Costs

This section explores potential implications of the demand-side mechanism using the model with sectorspecific income elasticities and trade costs. The additional heterogeneity makes the model more complex.
Owing to this complexity, I assume that cities differ only in productivities to keep the analysis simple, and I
let city 1, without loss of generality, be the fundamentally most productive city.22
Assumption 3. ∀n an = 1, and ∀n ≥ 2 λn < λ1 .
Additionally, I impose Assumption 4.
Assumption 4. η < max{1, 1 + mink∈K ϵ(k) + 1/γ}.
As discussed in Section 3, inter-sectoral substitution with η > 1 can attenuate the market-size effecton factor prices through market composition. Assumption 4 ensures the positive slope of city-level labordemand, which is crucial in obtaining analytical results. Assumption 4 is reasonable to impose when thenumber of sectors is not too large because Comin et al. (2021) estimated the inter-sectoral price elasticity to
be 0.07–0.13 using ten sectors. Subsequently, one can prove the existence of an equilibrium as Lemma 3.
Lemma 3 (Existence of Equilibrium). Given Assumptions 1, 3, and 4, there exists an equilibrium.
All the following cross-city comparisons hold in all the possible equilibria in which all sectors have
non-zero production in all the cities. For the remainder of this section, we compare cities in an equilibrium,
under Assumptions 1, 3, and 4.23

City Size and Factor Prices
Similar to Corollary 1 for the common tradability case, the order of productivities becomes that of factor
prices and, consequently, that of nominal income as Lemma 4.
higher factor price makes it infeasible to tell which city has a greater employment share. Two-location models do nothave this issue because the two locations cannot simultaneously be net exporters or importers.
22
I assume that city 1 is strictly more productive than any other city to make the following exposition clear, but thestrictness is by no means necessary.
23
The analysis of the uniqueness and stability is left for future research. An earlier version of this paper assumes
that sectors are either tradable or untradable and shows that the equilibrium is unique and stable for the two-city case.

25

Lemma 4 (Patterns of Factor Prices and Nominal Income). A fundamentally more productive city commands
a higher factor price, and the residents earn higher nominal income.
The order of city size, unlike factor prices, is generally infeasible to pin down because heterogeneous
tradabilities provide cities with a way to raise real income, not relying on good fundamentals—specialization
in low-tradability sectors. To understand this possibility, suppose some sectors are extremely tradable, andthe others are hardly tradable. Then, the price indices of the extremely tradable sectors do not differ much
across cities, whereas the price indices of the hardly tradable sectors are relatively inexpensive in cities that
produce rich varieties in these sectors; therefore, cities specializing in the hardly tradable sectors enjoy the
inexpensive price indices while not inflating the price indices of the extremely tradable sectors, raising real
income.24 Assumption 5 weakens this mechanism and eliminates the possibility that any other city becomesthe largest as Lemma 5.
Assumption 5. (ϕ̄−1 − 1)/(ϕ−1 − ϕ̄−1 ) ≥ (1 − ληn )/(ληn − λσn ) for all n ≥ 2 where ϕ̄ = maxk∈K ϕk andϕ = mink∈K ϕk .
Lemma 5 (Largest City). Given Assumption 5 in addition, the most fundamentally productive city becomesthe largest city.
The more modestly tradabilities differ (i.e., the smaller ϕ−1 − ϕ̄−1 is) and the lower tradabilities generally
are (i.e., the greater ϕ̄−1 − 1 is), the more likely Assumption 5 holds. Separately, a larger intra-sectoral priceelasticity σ and a smaller inter-sectoral price elasticity η makes it more likely. The former makes a price index
insensitive to the mass of variety, dampening the effect of specialization. The latter prevents inter-sectoral
substitutions from facilitating the supply-side specialization in low-tradability sectors. Consequently, thelargest city has the highest factor prices as Corollary 3.
Corollary 3. Given Assumption 5 in addition, the largest city commands the highest factor price, and theresidents earn the highest nominal income.
24

This analysis holds the income level constant. Specializing in low-tradability sectors has another force: it reduces

nominal income (Theorem 1).

26

Expenditure, Trade, and Employment
The model has two expenditure patterns. The first one is the same as Section 5 except that I need to condition
on tradability in this case. The sectoral expenditure share ratio of the larger city to the smaller city is greaterin a more income-elastic sector conditional on tradability.25 Thus, the force from cross-city real income
inequality is still operative. Given Theorem 1, it suggests that cross-city nominal income inequality widens
when high-tradability sectors have high income elasticities.
Heterogeneous trade costs generate the second expenditure pattern. Inequalities (13) showed that ahigh factor price tilts expenditures towards sectors with high and low tradability when η < 1 and η > 1,
respectively. I can isolate this force in equilibrium by conditioning income elasticity as Proposition 5.
Proposition 5 (Expenditure Pattern across Tradability Spectrum). The sectoral expenditure share ratio of ahigher-factor-price city to a lower-factor-price city is greater in a sector with a higher and lower tradability
conditional on income elasticity when sectors are gross complements and substitutes, respectively; that is,∀(n, i) ∈ {(n, i) ∈ N 2 | wn /λn > wi /λi }, ∀(k, ℓ) ∈ {(k, ℓ) ∈ K2 |ϕk > ϕℓ , ϵ(k) = ϵ(ℓ)}, snk /sik >snℓ /siℓ for η < 1, and snk /sik < snℓ /siℓ for η > 1.
Remark. Given Lemma 4, a higher factor price implies a higher wage and income.
Recall that the largest city commands the highest factor price when Assumption 5 holds. Given Theorem 1, this result suggests that endogenous expenditure composition amplifies and dampens a city-sizewage premium (dwn /dLn ) by the market-composition effect when sectors are gross complements and grosssubstitutes, respectively.
Trade patterns reflect sector-specific amplification strengths as well as relative market size. In thecommon-tradability case, expenditure shares determine the net exporter-importer status. Heterogeneous tradability makes it weighted relative market size—a product of snk and µnk , divided by the sum of products—asstated in Lemma 6 and illustrated in Figure 5.
25

To verify this result, notice that a common value of ϕk implies the same value for the factor (1 − ϕk )(1 + µnk )in expenditure shares of eq. (12).

27Lemma 6 (Home-Market Effect with Sector-Specific Trade Costs). Given an equilibrium, there is a relationship between expenditure shares and employment shares such thatx̃nk = s̃nk + µnk (s̃nk − s̃k ),

(16)

where
µnk xnkx̃nk = ∑,ℓ∈K µnℓ snℓµnk snks̃nk = ∑,ℓ∈K µnℓ xnℓ

s̃k =

∑
i∈N

∑
Li wi ℓ∈K µiℓ siℓ∑∑s̃ik .
j∈N Lj wjℓ∈K µjℓ sjℓAccordingly, a city is a net exporter in each sector whose weighted relative market size s̃nk is higher thanthe corresponding nationwide weighted mean s̃k (xnk > snk ⇐⇒ s̃nk > s̃k ).

x̃nk45◦ line

m̃k

m̃k

m̃nk

Figure 5: Home-Market Effect with Sector-Specific Trade Costs

Cities have comparative advantages in sectors with high s̃nk in the way analogous to the commontradability case (Lemma 2). The formula of s̃nk shows the importance of tradabilities of other sectors forwhich the city has relatively large markets. For example, consider sectors k and ℓ whose relative market
sizes snk and snℓ are equally large in city n, and recall that µnk increases with ϕk . When sector k has a hightradability, s̃nℓ becomes low, making city n, ceteris paribus, less likely to be a net exporter in sector ℓ, andvice versa. Thus, the tradability of sector k affects the net-exporter-importer status in sector ℓ. This result is
intuitive because high tradability reinforces the power of market size, drawing more workers into sector k.
Consequently, the factor price rises, making firms in sector ℓ less competitive in the nationwide market.

28

Nevertheless, standard relative market size is still influential. Sectors with the same tradability have thesame home-market multiplier in a city, implying s̃nk /s̃nℓ = snk /snℓ ; thus, similar to Proposition 4, larger
cities have comparative advantages in income-elastic sectors, conditional on tradability: ∀(k, ℓ) ∈ {(k, ℓ) ∈K2 |ϵ(k) > ϵ(ℓ), ϕk = ϕℓ }, s̃nk /s̃ik > s̃nℓ /s̃iℓ . 26

Discussion on Manufacturing-Services Economy
To obtain concrete implications of the market-composition effect, I consider a specific example: an economy
of manufacturing and services. Services are characterized by higher income elasticity and lower tradability
relative to manufacturing (Comin et al. (2021), Anderson et al. (2014)).27 These characteristics suggestthat a city-size wage premium possibly attenuates, owing to the sectoral market-composition effect. Twoforces influence cities’ expenditure composition. On the one hand, high real income in large cities drives
expenditures toward services. On the other hand, large cities tend to feature high factor prices, which is a
disadvantage especially for manufacturing because of its high tradability, raising their sectoral price index.
Consequently, as Proposition 5 shows, expenditures shift toward manufacturing when η < 1. However,when η is close to one, the second force becomes negligible. These considerations suggest that the net effectmakes large cities spend more on services, which have low tradability, leading to a smaller city-size wage
premium (dwn /dLn ).
Conversely, this analysis suggests that an improvement in services’ tradability potentially raises a city26

Another force makes cities with low and high factor prices tend to be net exporters in high- and low-tradability

sectors, respectively. I refer the readers to ? for a detailed discussion of this result. A short explanation is that,as Section 3 showed, high tradability, ceteris paribus, makes relative market size more powerful in attracting firms
captured by that ∂µnk /∂ϕk > 0. An inexpensive factor price reinforces this effect as µnk is log-supermodular in(wn /λn )−1 and ϕk (∂ 2 log µnk /∂(wn /λn )−1 ∂ϕk > 0). Thus, high-tradability sectors have even higher µnk in cities
with lower factor prices. This pattern of µnk dominates the pattern of relative market size snk across the tradability
spectrum in determining s̃nk , regardless of the level of η. Hence, the pattern of net exports can be opposite to that ofexpenditure shares. This result highlights the importance of exploiting exogenous demand factors, instead of relying
on expenditure shares, in tests of the home-market effect, similar to Costinot et al. (2019).
27
Using Canada’s provincial data from 1997 to 2007, Anderson et al. (2014) estimate that geography reduces services trade some seven times more than goods trade overall.

29

size wage premium. Tradability is not constant over time, and trade costs in services have steadily fallen
since 1980 because of the fall of communication costs (Head et al. (2009); Eckert (2019)). When tradabilities
become uniform across sectors, expenditure compositions no longer affect wages, nullifying the attenuation
effect on a city-size wage premium.28

7

Robustness Checks

In this section, I provide robustness checks that address two supply-side alternative explanations of cities’
specialization pattern in Figure 1. The first one is skilled-labor supply in cities. Skilled workers, simultaneously high-income earners, tend to reside in large cities, and those cities tend to host skill-intensive sectors(Davis and Dingel 2020). The second one is sector-specific strengths of agglomeration externalities; firms
in sectors that benefit more from agglomeration economy locate in larger cities (Gaubert (2018)). These
supply-side mechanisms can generate the employment pattern in Figure 1 if sectors’ income elasticities correlate with their supply-side characteristics. Indeed, as is well known, there is a positive correlation between
skill intensities and income elasticities (Caron et al. 2014, 2020). I address these concerns in two ways. First,
I test if the positive relationship in Figure 1 is robust to controlling for the skilled-labor supply in MSAs.
Second, I test a model prediction of an employment pattern conditional on city size. I implement regressions
for multiple years separately in each test, which serve as additional robustness checks.

Controlling for Skilled-Labor Supply
The first robustness check modifies the two-step regression analysis that produced Figure 1. The first step
estimates the elasticity of employment with respect to MSA’s population conditional on skilled-labor supply;
the regression model is given by
28

The insight in this subsection also applies to international income inequality. If anything, the absence of workers’

mobility makes the analysis more straightforward. Productive or large countries command high factor prices and highreal income, becoming high-income countries and spending more on services—the sector with high income elasticity
and low tradability. Services’ trade cost reduction can widen income inequality between countries.

30

(
Lik = αk · exp ξkP log(P opulationi ) + γ1k log(Collegei ) + γ2k log(1 + Collegei )+∑)γ3k Di,k,region· eik ,

regionwhere Lik , P opulationi , and Collegei are the employment of sector k, the population, and the college
employment ratio, respectively, of MSA i, Di,k,region is a region dummy variable for sector k taking a valueof one if MSA i belongs to that region∈ {N ortheast, M idwest, South, W est}, and eik is the error
term. I use the population data from the Bureau of Economic Analysis (BEA) and the employment datafrom the County Business Patterns (CBP). I calculate the college employment ratios for full-time workers in
MSAs from U.S. Census data via Integrated Public Use Microdata Series (IPUMS) (Ruggles et al. 2023).
The model includes two terms with Collegei to purge the skill-supply effect from estimated conditional
population elasticities of employment ξˆkP .
As the regression model shows, I implement level regressions by the Poisson Pseudo Maximum Likelihood (PPML) estimation. As discussed in Silva and Tenreyro (2006), log-linear regressions require a considerably specific condition on error terms to obtain consistent estimators. Moreover, it is problematic in
log-linear estimations when zeros are in the data. In contrast, PPML provides consistent estimators that do
not require this, and it is efficient with various error term patterns. For this reason, PPML is extremely common in gravity equation estimations in international trade where zeros are prevalent and error terms show
heteroskedasticity. In my dataset, 9.5% of the sample is zero. To address these zeros and obtain consistentestimators, I use PPML.
The second-step regression estimates the relationship between income elasticities of demand and theestimated conditional population elasticities of employment with the following model.

ϵˆk = α + β ξˆkP + ek ,where ϵ̂k is an estimated income elasticity of demand for sector k output, ξˆkP is the estimate from the first step,and ek is the error term for sector k. I borrow estimates of income elasticities from Caron et al. (2020), who

31obtained them through a structural estimation with international trade data. Using an estimate as a regressor
generally causes an attenuation bias from sampling errors, but ξˆkP has negligible standard errors from the firststep, alleviating the concern.29 This second-step regression weights sectors by their total employment in thesample to prevent small sectors like forestry from influencing the result excessively. Table 1 summarizes theresults of this cross-sectional regression for three years.

Table 1: Results with Population Elasticity of Employment

Pop. elasticity of emp. (β̂)

College ratio controls (1st st.)Observations
R2
Adjusted R2

2006

Income elasticity of demand
2011
2016
2006
2011

2016

(1)

(2)

(3)

(4)

(5)

(6)

0.64∗∗∗(0.18)

0.69∗∗∗(0.17)

0.71∗∗∗(0.19)

0.41∗(0.23)

0.63∗∗∗(0.20)

0.71∗∗∗(0.23)

29
0.23
0.20

29
0.32
0.29

29
0.29
0.26

✓
29
0.06
0.03

✓
29
0.15
0.12

✓
29
0.17
0.14

Notes: ∗∗∗ ,∗∗ , and ∗ denote significance at the 1%, 5%, and 10% levels, respectively. Bootstrap standard errors
are in parentheses. This table displays the coefficients of regressing sectors’ income elasticities of demand on
their elasticities of employment with respect to MSA’s population. The first-stage regression, which obtains theelasticities of employment, control for region ∈ {Northeast, Midwest, South, West} for (1)-(6) and log(college
ratio) and log(1 + college ratio) for (4)-(6). Employment data are from Country Business Pattern data, income
elasticities are from Caron et al. (2020), and MSAs’ population data are from the Bureau of Economic Analysis.

The results show that the positive relationship is robust to the skilled-labor controls. The three left
columns contain results without the controls of the college employment ratio, the 2016 result of which I
used to create Figure 1. β̂ is significantly positive in all three years. The right three columns display the
results when controlling for the college employment ratio. It reduced the size of β̂ in 2006, downgrading thesignificance to the 10% level. However, β̂ is significantly positive in the other two years at the 1% level.
29

The median of standard errors of ξˆkP in the first-step regression is 0.0012, while ξˆkP ranges from 0.19 to 1.52;

whereas Caron et al. (2020) report that standard errors for their estimates are between 0.05 and 0.2 for most sectors,and ϵ̂k in my sample ranges from 0.36 to 1.41.

32

To gauge the size of the effect, suppose two cities have populations in the first and third quartiles and thesame college-worker ratio. The coefficient size of 0.71 in 2016 means that a sector with a 0.1 higher income
elasticity boasts a 10 percent higher employment share in the larger city.30 Thus, the demand-side effect is
econometrically and economically significant even after controlling for skilled-labor supply.

Employment Pattern Conditional on City Size
The theoretical model navigates the second robustness check, which aims to mute the supply-side mechanisms. City size is the common source of comparative advantages in the theories of Davis and Dingel2020 and Gaubert (2018), generating agglomeration externalities for workers and sectors. To cancel thesemechanisms, I focus on a cross-city employment pattern conditional on city size. In my model, given two
equal-sized cities, one must have a higher productivity, and the other must have a higher level of amenities,
similar to a Rosen-Roback model, unless they have the same levels for both. The one with higher productivity
commands higher nominal and real income, making it equally attractive to workers despite the worse amenities. Subsequently, different real income generates different sectoral demand. Formally, I obtain Lemma
7.
Lemma 7 (Expenditure Pattern Conditional on City Size). Given Assumptions 1 and 2 and a common tradecost, suppose that two cities are equal-sized in equilibrium. Then, the sectoral expenditure share ratio of thehigher-income city to the lower-income city is higher in a more income-elastic sector; that is, ∀(n, i, k, ℓ) ∈{(n, i, k, ℓ) ∈ N 2 × K2 | Ln = Li , En > Ei , ϵ(k) > ϵ(ℓ)}, snk /sik > snℓ /siℓ .
Households in the higher-income city enjoy higher real income, spending more on income-elastic sectors.
As discussed in Section 5, this expenditure pattern translates into an employment pattern, according to eq.
(15), generating a pattern that a high-income city tends to have high employment shares in income-elastic
sectors.
To test this prediction, I implement another two-step regression analysis. The first-step regression estimates the income elasticity of employment with respect to MSA’s income per capita conditional on MSA’s
population ξkI . The model is given by
30

This description interprets the result according to the reverse regression formula ξˆkP = β̂ −1 α + β̂ −1 ϵˆk + β̂ −1 ek .

33

(
Lik = αk · exp

ξkI log(Incomei ) + ξkP log(P opulationi ) +∑)γk Di,k,region· eik ,

regionwhere Incomei is per capita personal income of MSA i, data of which are from the BEA. The second step
regresses income elasticities on these ξˆkI byϵ̂k = α + β ξˆkI + ek .
The theoretical model predicts β > 0.

Table 2: Results with Conditional Income Elasticity of Employment

Income elasticity of demand
2006
2011
2016
(1)

(2)

(3)

Cond. income elasticity of emp. (β̂)

0.134∗∗∗(0.038)0.111∗∗∗(0.039)0.129∗∗∗(0.043)Observations
R2
Adjusted R2

29
0.316
0.291

29
0.229
0.200

29
0.246
0.219

Notes: ∗∗∗ ,∗∗ , and ∗ denote significance at the 1%, 5%, and 10% levels, respectively. Bootstrap standard
errors are in parentheses. This table displays the coefficients of regressing sectors’ income elasticities on theirelasticities of employment with respect to MSA’s income per capita. The first-stage regression, which obtains
the elasticities of employment, control for MSA’s population and region ∈ {Northeast, Midwest, South, West}.
Employment data are from Country Business Pattern data, income elasticities are from Caron et al. (2020), andMSAs’ population data are from the Bureau of Economic Analysis.

Table 2 shows that the regression results are consistent with this prediction; β̂ is significantly positive inall three years. Sectors that employ relatively more workers in cities with higher income, conditional on city
size, tend to be income-elastic. Thus, the specialization pattern is robust to the alternative explanation by
sector-specific strengths of agglomeration externalities. In theory, the effect of skilled-labor supply is alsoabsent in this test. In reality, it survives if skilled workers cluster in some cities for reasons that city size

34

alone cannot explain (e.g., Diamond (2016)). Nevertheless, the two robustness tests jointly demonstrate thatthe demand-side effect in cities’ specialization pattern is robust to the supply-side explanations.

8

Conclusion

Beyond explaining cities’ specialization patterns from the demand-side perspective, my model implies that
endogenous expenditure composition can amplify or attenuate cross-location income inequality. As expenditure patterns substantially vary across cities and countries, using multiple-sector models with sector-specifictrade costs can improve quantitative analysis in future research.
I expect supply-side comparative advantages to also affect factor prices through the newfound channel.
On the one hand, relatively higher productivity in a high-tradability sector reduces the sectoral price, lowering the expenditure share when sectors are gross complements. On the other hand, I conjecture relative
productivity and relative tradability are complementary in generating labor demand. This channel deserves
further theoretical and empirical investigation.

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AppendixA

Data

To create Figure 1, I borrow estimates of income elasticities from Caron et al. (2020). Using 1997 international trade data for 109 countries, they estimate the elasticities for 49 sectors. The elasticity varies
from 0.137 for “Processed rice” to 1.311 for “Financial services nec.” I use datasets from County Business
Patterns (CBP) for employment data. CBP provides employment data of MSAs annually for sectors classified according to the North American Industry Classification System (NAICS). Caron et al. (2020) use
Global Trade Analysis Project (GTAP) sector classification, which differs from NAICS. In most cases, one
GTAP sector code corresponds to multiple 3-digit or 4-digit NAICS codes. Following Carrico et al. (2012)and mapping NAICS data to GTAP sectors, I create employment data by GTAP sectors. CBP datasets do
not contain employment-size information for MSA-NAICS pairs with fewer than three establishments from2017. CBP provides employment size class data (e.g., “25,000–49,999”) for these pairs before 2017. Small
employment values are informative in detecting specialization patterns; therefore, I use employment data up
to 2016 throughout the study and the midpoint of the employment size class for pairs only with employment
size class information. I drop sectors whose employment is zero in more than 90 percent of MSAs in 2016.
The regression procedure for Figure 1 is the same as the first robustness check in Section 7 except that it
does not control for the college employment ratio. It uses PPML and calculates standard errors by a bootstrap

40

procedure that resamples 500 times with replacement over MSAs and sectors separately before the first stage;
each sample has a different set of MSAs and a different set of sectors, but all MSAs, in a given sample, havethe same set of sectors for employment data. I calculate the standard errors in the robustness checks in thesame way.
The first robustness check in Section 7 uses college employment ratios that I measure by college workers
to non-college workers working in a given MSA. I use the U.S. census data via IPUMS on 25–55-year-old
workers whose “Usual hours worked per week” are at least 35 hours and exclude individuals living in groupquarters (Ruggles et al. 2023). I classify workers who have completed at least four years of college as college
workers and all other workers as non-college workers.
Table 3 displays the distribution of population, the college-employment ratio, and per capita income in2016. The largest MSA in 2016 in this sample is New York-Newark-Jersey City (NY-NJ-PA), which had apopulation of 19,943,198, while the smallest is Parkersburg-Vienna, WV, which had a population of 91,940.
The college employment ratio and per capita personal income have substantial variations across MSAs. The
table also shows their correlations to population. The high value of the college employment ratio validates
the concern of the alternative explanation by skill-labor supply in the robustness checks.

Table 3: Distribution of Population, College-Employment Ratio, and Per Capita Personal Income
across MSAs in 2016Min
Q1MedianPopulation
91,940 174,538 380,010
College-employment ratio
0.19
0.40
0.54
Per capita personal income 25,593 38,812 43,289

Q3847,8350.69
49,803

Max
19,943,198
1.68
10,6272

Corr to pop.
1.00
0.41
0.43

Notes: Correlation is calculated after taking logs of both variables.

Table 4 summarizes the elasticities of employment with respect to the MSA’s population I estimated andthe income elasticities I borrow from Caron et al. (2020).

41

Table 4: Estimates for Sectors
GTAP code

Sector

atp
ppp
obs
cmn
ofi
tex
ros

Air transport
Paper prod., publishing
Business services
Communication
Fin. Services nec
TextilesRecreational and otherservices
Water transport
Insurance
Construction
Electronic equipment
Transport nec
Trade
Pub. Adm. andservices
Electricity
Manuf. nec
Petroleum, coal prod.
Machinery andequipment nec
Meat prod.
Chemical, rubber,plastic products
Water
Food prod. nec
Leather prod.
Transport equipmentnec
Beverage and tobacco
prod.
Motor vehicles andparts
Wood prod.
Dairy prod.
Forestry

wtp
isr
cns
ele
otp
trd
osg
ely
omf
p_c
ome
omt+cmt
crp
wtr
ofd
lea
otn
b_t
mvh
lum
mil
frs

Elasticity of
employment toMSA’s
population
1.52
1.19
1.17
1.16
1.16
1.14
1.10

Income elasticity
of demand

1.10
1.05
1.03
1.02
1.02
1.01
1.00

1.05
1.41
0.83
1.21
0.83
1.09
0.99

0.96
0.95
0.93
0.92

0.94
1.02
0.78
0.93

0.92
0.91

1.06
0.91

0.90
0.89
0.86
0.85

0.97
0.83
1.02
1.03

0.70

0.76

0.69

1.06

0.65
0.58
0.19

1.07
1.02
0.36

1.06
1.07
1.21
1.17
1.13
0.77
1.17

Notes: Adm., bev., fin., equip., mach., manuf., prod., and pub., stand for administration, beverage,
financial, equipment, machinery, manufactures, products, and public, respectively.

42

B

Derivation of Theoretical Results

B.1

Workers’ Problem

Given a city to reside in, the consumption optimization problem is given by
C n + λ C n −maxCn ,{Qnk }k∈K ,{qnk (ν)}ν∈Ω[

nk ,k∈K

+

∑

1
η

ϵ(k)
η

βk Cn

η−1
η

Qnk

] η 
η−1


k∈K

(
ξk

∑

[∫

Qnk −

qnk (ν)

σ−1
σ

] σ )
σ−1

dν

Ωnk

k∈K

(

+ω En −∑∫

pnk (ν)qnk (ν)dν),

k∈K Ωnk

where λ, {ξk }k∈K , and ω are the Lagrange multipliers. The first order conditions (FOCs) are as follows:) η −1 (
)
(
η−1
η−1
∑ ϵk η1 ϵ(k)∑ η1 ϵ(k)− η1 η−1ηηηηηCn : 1 + λ = λβk Cn Qnk
β Cn Cn Qnk
,
η−1
η k
k∈K
k∈K
) η −1 (
(
)
η−1
ϵ(k)
ϵ(k)
η−1
1
∑ η1 η η−1−1
Qnk : λβk Cn Qnkηβkη Cn η Qnkη= ξk ,k∈K[∫
qnk (ν) :ξkqnk (ν)σ−1
σ] σ −1
σ−1

dν

Ωnk

[

λ :Cn =

∑

1
η

ϵ(k)
η

βk Cn

η−1
η

σ−1

] η

η−1

Qnk

k∈K

[∫
ξk :Qnk =qnk (ν) σ −1 = ωpnk (ν),

qnk (ν)

σ−1
σ

] σ

σ−1

dν

Ωnk

ω :En =∑∫

pnk (ν)qnk (ν)dνk∈K Ωnk

First, I derive the usual result with constant elasticity of substitution from the FOC with respect to (w.r.t.)qnk (ν) and that w.r.t. ξk .
1

1

σPnk Qnk= pnk (ν)qnk (ν) σ ,43

where Pnk =

[∫

1−σ
dν
Ωnk (pnk (ν))]1/(1−σ). It follows from this result, the FOC w.r.t Qnk , and that w.r.t

qnk (ν),
1

λ(

ϵ(k)

η−1

βkη Cn η Qnkη
∑

1
η

ϵ(ℓ)
η

ℓ∈K βℓ Cn

−1

ϵ(k)

1

= ωPnk =⇒
1
) η−1
η−1

Qnℓη

η−1

βkη Cn η Qnkη
ϵ(ℓ)
η

1
η

η−1
η

=

βℓ Cn Qnℓ

Pnk Qnk.
Pnℓ Qnℓ(17)

Next, I obtain the expenditure by using eq. (17).

En =

∑∫

pnk (ν)qnk (ν)dν =k∈K Ωnk∑Pnk Qnk

k∈K

=

∑

1

Pnℓ Qnℓϵ(k)η−1βkη Cn η Qnkη
1

ϵ(ℓ)

η−1

βℓη Cn η Qnℓηη−1∑ η1 ϵ(k)Pnℓ Qnℓβk Cn η Qnkη
= 1 ϵ(ℓ) η−1βℓη Cn η Qnℓη k∈K
η−1
Pnℓ Qnℓ= 1 ϵ(ℓ) η−1 Cn η ,βℓη Cn η Qnℓηk∈K

where the third equality follows from (17) and the last from the FOC w.r.t. λ. The demand function immediately follows.
−η 1−η η
Qnk = βk Cnϵ(k) Pnk
C n En .

Subsequently, the expenditure share (eq. (3)) follows.
ϵ(k)snk = ∑1−ηβk Cn Pnkϵ(ℓ)

1−ηk∈K βℓ Cn Pnℓ

44

.

(18)

Substituting eq. (18) for Qnk in the FOC w.r.t. λ yields indirect real income.
[
Cn =

∑

1
η

ϵ(k)
η

βk Cn

(

ϵ(k)
η

1
η

βk Cn

k∈K

(
⇐⇒ Cn =

∑

1−η
η

−1
Pnk
Un

η
)η−1 ] η−1

En

)− 1

1−η

1−η
βk Cnϵ(k) Pnk

En

k∈K

B.2

Firms’ Problem and Labor Demand

I begin by substituting the optimized production into the zero-profit condition. As for the optimized production, the optimized price for a firm in city n to sell in city i is as follows.

∀k, ∀n, pni,k =




 σ

i=n



 τk σi ̸= n.

vk
σ−1 λn wnvk
σ−1 λn wn

Subsequently, the profit becomesπnk =∑ 1 vkfk
1 vk
qnn,k + τk
qni,k −.
σ − 1 λnσ − 1 λnλni̸=nThe zero-profit condition requires this πnk to be zero, yielding the labor demand per firm as follows:qnn,k + τk∑qni,k 

i̸=n

vk
fk
fk
+
=σ .
λ n λnλn(19)

The fixed cost and productivity levels determine the labor demand. Now, I use normalization. It can be1

1

shown that βk , fk , and vk affect the equilibrium values of endogenous variables only through βk1−η fkσ−1 vk .
I set vk = (σ − 1)/σ and fk = 1/σ so that pnk = wn /λn and the labor demand per firm is 1/λn . I replace
( 1
)1−η
σ−1
σ/(σ−1)−1
βk by β̃k = βk fk vk σ(σ − 1)so that this change does not affect the equilibrium valuesof endogenous variables. Thus, it follows from eq. (19) and the normalization that the aggregate supply of45

goods by a firm is given by∀k, qnn,k + τk∑

qni,k = 1.

i̸=nThis supply is equal to the demand in equilibrium as follows:

σ−1
1 = Ln p−σnn,k Pnk En snk + τk

(
= Ln

wn
λn

)−σ

∑

σ−1
Li (τk pni,k )−σ Pik
Ei sik

i̸=n
σ−1
Pnk
En snk + τk

∑

(
Li

i̸=n

wn
τk
λn

)−σ

σ−1
Pik
Ei sik .

(20)

This equation reflects the zero-profit condition and holds for all n for each sector k. I use a matrix Rk thatis defined by
Rk = (1 − ϕk )I + ϕk S,
where I is the identity matrix of size N , and S is a square matrix of size N in which all the entries equal to
1. The inverse matrix is
Rk−1 =

1
1I−
S.
1 − ϕk(1 − ϕk )(1/ϕk + N − 1)

Given Rk , eq. (20) for all n can be summarized asσ−1
L1 P1k
E1 s1k





σ
λ−σ
1 w1




 


 

 L2 P σ−1 E2 s2k   λ−σ wσ 
2

  2

2k
Rk 
=
.
..
..

 

.
.

 


 

σ−1
−σ σ
L N PN k E n s N k
λN w N
Premultiplying Rk−1 yields
σ−1
Ln En snk · Pnk=∑
1(wn /λn )σ−(wi /λi )σ ,1 − ϕk(1 − ϕk )(1/ϕk + N − 1)i∈N46from which eq. (7) in the main text follows. Substituting (λnk xnk Ln ) · (wn /λn )1−σ +∑n̸=i (λik xik Li ) ·1−σmakes the matrix form be(τk wi /λik )1−σ for Pnk













σ
λ−σ
1 w1





1−σ
σ
 −σ

 w1 s1k L1   x1k λ1 L1 w1

σ



 
 −1  λ2 w2 .
.

 = diag Rk 
 · R 
..
..
,


 
 k 
..



 

.


1−σ
σ


w N s N k LN
x N k λN L N w N
σ
λ−σ
w
N
N
where I used En = wn . Further manipulations yield
















1−σ
σ


 −σ
 x1k λ1 L1 w1
 w1 s1k L1  λ w σ 




2
2
..
.







 = diag R−1 ..
  Rk 
.
k



.




.




.



1−σ



xN k λσN LN wN
w N s N k LN
σ
λ−σ
N wN

∑


sik Li wi
s1k L1 w1
−σ
1
λ−σ
∑
−σ wσ − λ1σ1−σ1i −σ σ ∑n λ−σ
1/ϕ
+N
−1
λ
k
−σ σ

n
n
n wn
n
x
L
w
λ1k w1 − (1/ϕ +N −1)
λi wi − 1/ϕ +N−1  1k 1 1
k
k
 

..
..
=
=⇒ 
.
.
 

 

∑
 −σ
sN k L N w N
sik Li wi
1−σ−σ1xN k LN w N
λN
−
λ
∑
∑
−σ
σ
i
N
1/ϕ
+N
−1
λ
w
λ−σ wσ
k
σ
λ−σ
1 w1

n
σ
n n
λ−σ
N wN − (1/ϕ +N −1)





.




n
σ
n n
λ−σ
i wi − (1/ϕ +N −1)kkEq. (10) in the main text immediately follows from this matrix form. Premultiplying the matrix above by∑∑
the row matrix [1, 1, . . . , 1] with k∈K xnk = k∈K snk = 1 yields
(

∑

)

∑

(wn /λn )σ ∑ Li wi(1 + µik )sik
(wi /λi )σ(ϕ−1
k + N − 1) i∈N
k∈K
k∈K
∑
∑
∑ Li w i1⇐⇒ (µnk snk )Ln wn = (wn /λn )σ(1 + µik )sik−1
(wi /λi )σ
(ϕ
+
N
−
1)
k
i∈N
k∈K
k∈K
(∑
)
σ
µnk snk Ln λn (wn /λn )(wn /λn )(∑)⇐⇒ ∑= ∑ k∈K
.
σ
(w
/λ
)
j
j
j∈N
i∈N
k∈K µik sik Li λi (wi /λi )Ln w n =1+µnk snk

Ln w n −This is the city-level labor demand (11).

47

B.3

Derivation of Expenditure Shares and Labor Supply

I start with dividing both sides of the zero-profit condition (7) by wσ ,(Ln snk ·

Pnk
wn

)σ−1

∑
σ
λ−σ
1
i∈N (wi /λi )
n
=
− σ
,
1 − ϕkwn (1 − ϕk )(1/ϕk + N − 1)(21)

where I used En = wn . I solve for an expenditure share after substituting eq. (3) for Pnk in eq. (21) with
Pn = En /Cn .


snk =

1
 (1 − ϕk )(1 + µnk )

(

1
1−η

β̃k

ϵ(k)
1−η

1−η
)σ−1  σ−η


Cn

Cn

,



1

λn (λn Ln ) σ−1

which is eq. (12).
I obtain an equation expressing city-level labor supply without employment shares to prove propositions.
Aggregating eq. (12) over sectors yields

1 
Cn = λn (λn Ln ) σ−1 1 1−η − 1−η · σ−1
σ−η

β̃k

.

 (1 − ϕk )(1 + µnk ) 


∑
k∈K

σ−η

σ−1
1−η

ϵ(k) σ−1
Cn 1−η

(22)

This equation, unlike eq. (14) in the main text, expresses real income in city n as a function of the city
size and wages because the zero-profit conditions removed employment shares. The first and second factors
in the right-hand side reflect productivity in producing a variety and the mass of available varieties fromefficiency units of labor, respectively. 31 Combining eq. (22) with population allocation with Fréchet utility
shocks (eq. (4)) yields1=

 σ−1 ϵ(k)(+1)(σ−1) [()∑  β̃ 1−η C 1−η
Ln γ
1

k

k∈K

 (1 − ϕk )(1 + µnk )L1

L1

a−1
n

]( ϵ(k) +1)(σ−1) (
1−η

Ln
L1

)−1

λ−σ
n

1−η
 σ−η




,

(23)

which uses city 1 as the base. Recall ∂µnk /∂(wn /λn ) < 0. Given Assumption 1, a city size Ln and factor
31

ϵ(k) σ−1
1−ηThe right-hand side has Cn, but mink∈K {ϵ(k)/(1 − η)} > −1 ensures that Cn in the left-hand side domi-

ϵ(k) σ−1
ϵ(k) σ−1nates Cn 1−η , allowing me to ignore Cn 1−η when evaluating the directions of changes.

48

price wn /λn are in an inverse relationship in eq. (23).32

B.4

Proofs of Theorem, Propositions and Lemmas

Proof of Proposition 1. It follows from equations (9) and (11) for two cities.
Proof of Theorem 1. Given wages,∑ak∈K µnk snk ≥∑bk∈K µnk snk . Subsequently, equations (9) and (11)for the two cities imply Theorem 1.
Proof of Lemma 1. Given common tradabiliy, the city-level labor demand (11) becomes∑[
where µi =(wn /λn )σµn Ln λn (wn /λn )
=∑
,
σ
(w
/λ
)
j
j
j∈N
i∈N µi Li λi (wi /λi )
]−1

σ
(ϕ−1
∑+N −1)(wi /λσi ) − 1
(w
/λ
)
j
j
j∈N

and

∑

(24)

k∈K mk = 1. This equation holds for each city, but one ofthem is redundant because of Walras’s law. Dividing each one by that of city 1 yields
[

] ( )
σ
(w
/λ
)
j
j
wn 1−σ Ln
j∈N
∑
λ
1=
.
n
(ϕ−1 + N − 1)(wi /λi )σ − j∈N (wj /λj )σλnL1(ϕ−1 + N − 1) −

∑

(25)

I show that there exists {C1 , L1 , {wn , Ln }n=2,...N } that solves the system of eq. (23) for n = 1, . . . , N and∑(25) for n = 2, . . . , N , and Ln = L by Brouwer’s fixed point theorem. To do so, I use Lemma 8.
(

Lemma 8. Suppose mink (a−1n)( −1 )γ+ ϵ(k) +1 −1∑1−η
λn> 1. Then, given (L1 , Ψ) ∈ [L/ n∈N λ−1
n , L] × [1 +−1 +ϕ(N − 1), N ], there exits unique (C1 , Ln /L1 , wn ) ∈ R++ × (0, λ−1n ) × (wn , λn ) where wn = [Ψ(ϕ32Although this relationship sounds counterintuitive, eq. (22) considers endogenous changes in price indices that awage change entails; namely, a higher input cost necessitates more-than-proportional increases of the price indices tooffset the disadvantage for local firms, hurting households.

49

N − 1)−1 ]1/σ λn that solves the system of equations:1=∑

Wk (C1 , L1 , Ψ),

(26)

k∈K

1=

∑

Wk (C1 , L1 , Ψ)

k∈K

·


(
)
ϵ(k)
+1 −1
( L )γ(σ−1) 1−η
n

 L1
[

1=

ϕ−1 + N − 1 −−(σ−1)
an

(

ϵ(k)
+1
1−η

)

λ−σ
n

∑

[

σ
j∈N (wj /λj )∑ϕ−1 + N − 1 − (wn /λn )−σ j∈N (wj /λj )σ


 σ−1
1−η
where Wk (C1 , L1 , Ψ) =



β̃k

(
(σ−1)

ϵ(k)
1−η +1)C1(1−ϕ)L1[

ϕ−1 + N − 1 − Ψ(wn /λnϕ−1 + N − 1 − Ψ

]

(
λn

Ψ
1 − ϕ−1 +N
−1

wn
λn

)1−2σ

)−σ

 1−η
] σ−η


Ln
,
L1

,

(27)

(28)

 1−η
] σ−η
.



Proof. Eq.(26) has a unique solution C1 . Given this C1 and wn ∈ (wn , λn ], I let Ln /L1 = f1n (wn ) be thesolution to eq. (27) and Ln /L1 = f2n (wn ) be that to eq. (28). I show that there exists unique wn ∈ (wn , λn )that solves f1n (wn ) = f2n (wn ). Given Assumption 1, limwn →wn f1n (wn ) = ∞ > limwn →wn f2n (wn ) = 0.
At wn = λn , f2n (λn ) = λ−1
n , and eqs. (26) and (27) imply(
γ(σ−1)min (f1n (λn ))
k

ϵ(k)
+1
1−η

γ(σ−1)=⇒ min (λn f1n (λn ))
k

(

)

−1

ϵ(k)
+1
1−η

)

−(σ−1)
an
−1

(

ϵ(k)
+1
1−η

)

λ−σ
n ≤1
(

< 1 (∵ min a−1
nk)( −1 )γ+ ϵ(k) +1 −1
1−η
> 1)
λn

⇐⇒ f1n (λn ) < λ−1⇐⇒ f1n (λn ) < f2n (λn ).
n
f1n (wn ) and f2n (w) are continuous on (wn , λn ). Thus, there exists wn ∈ (wn , λn ) such that f1n (wn ) =f2n (wn ) by the intermediate value theorem, and it is unique because f1n (wn ) and f2n (wn ) decreases andincreases, respectively, in wn . Further, 0 < f1 (λnn ) < Ln /L1 < f2n (λn ) = λ−1
n .
∑
′
′
Given Lemma 8, I let f3 map (L1 , Ψ) ∈ [L/ n∈N λ−1
n , L] × [1 + ϕ(N − 1), N ] to (L1 , Ψ ) such that∑∑
L′1 = L/ n∈N (Ln /L1 ) and Ψ′ = n∈N (wn /λn )σ , where Ln /L1 and wn form the unique solution in∑′′
Lemma 8. Lemma 8 implies L/( n∈N λ−1n ) < L1 < L and 1 + ϕ(N − 1) < Ψ < N . Thus, f3 is a map∑
from [L/ n∈N λ−1
n , L] × [1 + ϕ(N − 1), N ] to itself, and it has a fixed point by Brouwer’s fixed point50

theorem. Given the fixed point (L1 ,∑n∈N (wn /λn )σ ), the solution to the system of equations in Lemma

8 provide C1 , {Ln /L1 , wn }n=2...N satisfying eq. (23) for n = 1, . . . , N and (25) for n = 2, . . . , N , and∑n∈N Ln = L. □Proof of Proposition 2. I prove Ln > Li by contradiction. Suppose Ln < Li . Then, eqs. (29), (30), and(31) imply
(

)

]
∑
) [ −1σ−σ
λi −σ ϕ + N − 1 − (wi /λi )j∈N (wj /λj )∑1 > min,kλnϕ−1 + N − 1 − (wn /λn )−σ j∈N (wj /λj )σ
]
[
∑
(
)
ϕ−1 + N − 1 − (wn /λn )−σ j∈N (wj /λj )σ λi wi wn 1−2σ∑/.
1>
ϕ−1 + N − 1 − (wi /λi )−σ j∈N (wj /λj )σ λn λi λn
(

ai
an

)−(σ−1) ϵ(k) +1 (1−ηGiven mink (an /ai ) (λn /λi )(1+(σ−1)−1 )θ(k)> 1, the first inequality implies wi /λi < wn /λn . Canceling out

the term in square brackets yields(
1 > min
k

(
> min
k

(

ai
an
ai
an

)

)−(σ−1) ϵ(k) +1 (
1−η

)

(

)−(σ−1) ϵ(k) +1 (
1−η

λi
λn
λi
λn

)1−σ (
)1−σ

wi wn
/
λi λ n)1−2σ,where the second inequality follows from wi /λi < wn /λn . This inequality contradicts mink∈K (an /ai ) (λn /λi )θ(k) >1.
Proof of Proposition 3. Eqs. (23) and (25) for cities n and i can be rewritten as

1=

∑

Wnk ,

k∈K

1=

∑

Wnk

k∈K

{[
·

(29)

 1−η

)
)
(
(
ϵ(k)
)−(σ−1) ϵk +1 ( )−σ  σ−η
+1 −1 (
( L )γ(σ−1) 1−η
1−η
a
λ
i

i

i

 Ln

an

λn

ϕ−1 + N − 1 − (wi /λi )−σ∑j∈N (wj /λj )

∑
−σ

σ



]} 1−ησ−η

,σϕ−1 + N − 1 − (wn /λn )j∈N (wj /λj )
[
]
∑
(
)
ϕ−1 + N − 1 − (wn /λn )−σ j∈N (wj /λj )σ λi wi wn 1−2σ Li∑
1=/,
ϕ−1 + N − 1 − (wi /λi )−σ j∈N (wj /λj )σ λn λi λn
Ln

51

(30)
(31)


 σ−1
1−η
where Wnk =



β̃k

(
(σ−1)

ϵ(k)
1−η +1)Cn(1−ϕ)Ln

[
1−

∑
−σ

σ
(wn /λn )j∈N (wj /λj )ϕ−1 +N −1

1, the same steps as Lemma 8 lead to [(ϕ−1 + N − 1)−1

∑

 1−η
] σ−η
j∈N (wj /λj ). Given that mink∈K (an /ai )(λn /λi )γ+θ(k) >σ ]1/σ λi < wi < λi wn /λn .

Proof of Proposition 4. Substituting eq. (4) for real income in the sectoral expenditure share ratio that follows from eq. (12) yields (snk /sik ) / (snℓ /siℓ ) = (Ln /Li )σ−1)(ϵ(k)−ϵ(ℓ))γ( σ−η

.

Proof of Corollary 2. It follows from Proposition 4 and Lemma 2.
Proof of Lemma 3. The equilibrium conditions are eq. (12) for each city-sector pair, eqs. (11) and (23) for∑
each city, and N Ln = L. Eq. (11) for one city is redundant because of Walras’s law. I divide eq. (11) bythat of city 1 and substitute eq. (12) for {snk }n∈N ,k∈K ; then, for city n
(

wn
λn

)2σ−1

−
λn

(

σ−1
σ−η

)
η

∑

=


Xk C1 , L1 ,∑ ( w j )σ
λj

j∈N

k∈K



σ−1
σ−η

 (1 + µnk ) σ−1(1 + µ1k ) σ−η

(

Cn
C1

) 1−η (σ−1+ ϵ(k) ) (
σ−η

1−η

Ln
L1

) σ−1
σ−η

,

where

X k C 1 , L 1 ,∑ ( w j )σ
λj

j∈N

[
∑

=
k∈K

 = ∑ µ1k s1kk∈K µ1k s1kϕ−1
k +N −1σ − 1j∈N (wj /λj )

[

∑



∑


]−1 


ϕ−1
k +N −1σ − 1j∈N (wj /λj )((1 − ϕk )−1


]−1 


β̃

ϵ(k)
1
1−η +11−η C11k
L1σ−1((1 − ϕk )−1β̃)σ−1 (ϵ(k)
11−η +11−η C1
1
k
L1σ−1

1−

∑

j∈N (wj /λj )ϕ−1
k +N −1)σ−1 (
1−

∑

σ

 1−η
) σ−η
j∈N (wj /λj )ϕ−1
k +N −1σ 1−η .
) σ−η


Substituting eq. (4) for Cn /C1 yields
(

wn
λn

)2σ−1

−

λn

(

σ−1
σ−η

)
η

=

∑
k∈K

(
·


Xk C1 , L1 ,∑ ( w j )σ
j∈N

λj

∑
σ
ϕ−1
j∈N (wj /λj )k +N −1−∑−σσ
ϕ−1j∈N (wj /λj )k + N − 1 − (wn /λn )) σ−1 (
σ−η

[

Ln
L1

Given Assumption 4, wn increases with Ln /L1 , as analyzed in Section 3.

52]) γ( ϵ(k) +1)(1−η)+1 σ−1
1−η

σ−η

.

(32)

I show that there exists {C1 , L1 , {wn , Ln }n=2,...N } that solves the system of eq. (23) for all n and (32)∑for n = 2, . . . , N , and n Ln = L by Brouwer’s fixed point theorem. I start with Lemma 9.
Lemma 9. Given (L1 , Ψ) ∈ [L/1
1− 1−η
mink ϵ(k)1
γ( 1−η +1)+ 1−η∑n∈N ℓ̄n , L] × [1 + maxk ϕk (N − 1), N ] where ℓ̄n= λχn , and χ =, there exits unique (C1 , Ln /L1 , wn ) ∈ R++ ×(0, ℓ̄n ]×(wn , λn ) where wn = [Ψ((maxk ϕk )−1 +N − 1)−1 ]1/σ λn that solves the system of equations:1=∑

Wk (C1 , L1 , Ψ),

k∈K

1=

∑

Wk (C1 , L1 , Ψ)

k∈K

(
1=
·

wn
λn

∑

)1−2σ

(

λn

σ−1
σ−η

(33)


(
)
ϵk
+1 −1
( L )γ(σ−1) 1−η
n

 L1

[
λ−σ
n

−σ
ϕ−1
k + N − 1 − Ψ(wn /λn )ϕ−1
k +N −1−Ψ

1−η
] σ−η


,



(34)

)
η

(
Xk (C1 , L1 , Ψ)

) σ−1 (

ϕ−1
k +N −1−Ψ

σ−η

−σ
ϕ−1
k + N − 1 − Ψ(wn /λn )

k∈K

[

Ln
L1

]

) γ( ϵ(k) +1)(1−η)+1 σ−1
σ−η

1−η

,

(35)

where for all k

Wk (C1 , L1 , Ψ) =


(
)
ϵk
1
[

 β̃ 1−η C (σ−1) 1−η +1



k

1

(1 − ϕk )L1

1−

Ψ

 1−η
] σ−η


ϕ−1
k +N −1.

Proof. Eq. (33) has a unique solution C1 . Given this C1 and wn ∈ (wn , λn ], I let Ln /L1 = g1n (wn )be the solution to eq. (34) and Ln /L1 = g2n (wn ) be that to eq. (35). I show that there exists uniquewn ∈ (wn , λn ) that solves g1n (wn ) = g2n (wn ) by the intermediate value theorem. Given Assumption 4,
limwn →wn g1n (wn ) > 0 = limwn →wn g2n (wn ). At wn = λn , eqs. (33) and (34) imply
{
min
k

(
γ(σ−1)(g1n (λn ))

ϵ(k)
+1
1−η

)

−1

λ−σ
n

}
≤ 1 =⇒ g1n (λn ) < 1.

As for g2n (λn ), eq. (35) implies(

1 ≤ λn

σ−1
σ−η

)
η

[

max (g2n (λn ))

]
ϵ(k)
σ−1
γ( 1−η +1)(1−η)+1 σ−ηk53=⇒ g2n (λn ) > 1.

(36)

Therefore, g1n (λn ) < g2n (λn ), and there exists an intersection at wn ∈ (wn , λn ) by the intermediate value
theorem because g1n and g2n are continuous on (wn , λn ]. Further, the intersection is unique because g1n (wn )and g2n (wn ) decreases and increases, respectively, in wn ∈ (wn , λn ]. It also follows that 0 < g1n (wn ) =g2n (wn ) < g2n (λn ). Eq. (35) implies(

1 ≥ λn

σ−1
σ−η

)
η

[
]
ϵ(k)
σ−1
γ( 1−η +1)(1−η)+1 σ−ηnmin (g2 (λn )).
k

It follows that1
1− 1−η
mink ϵ(k)1
γ( 1−η +1)+ 1−ηLn< g2n (λn ) ≤ λn
L1

= ℓ̄n .

This completes the proof of Lemma 9.
Given Lemma 9, I let g3 map (L1 , Ψ) ∈ [L/∑′′n∈N ℓ̄n , L] × [1 + maxk ϕk (N − 1), N ] to (L1 , Ψ ) suchthat
L′1 = ∑L,n∈N (Ln /L1 )′Ψ =∑ ( w n )σ
n∈N

λn

,

where Ln /L1 and wn form the unique solution in Lemma 9. Lemma 9 implies ∑ 1 ℓ̄ < L′1 < L and
n∈N n
∑
′
1 + maxk ϕk (N − 1) < Ψ < N . Thus, g3 is a map from [L/ n∈N ℓ̄n , L] × [1 + maxk ϕk (N − 1), N ] to∑itself, and it has a fixed point by Brouwer’s fixed point theorem. Given the fixed point (L1 , n (wn /λn )σ ),the solution to the system of equations in Lemma 9 provide C1 , {Ln /L1 , wn }n=2...N satisfying eq. (23) for∑n = 1, . . . , N and (32) for n = 2, . . . , N , and n∈N Ln = L. □Proof of Lemma 4. Let city n be the more productive city (λn > λi ). Equilibrium conditions (23) and (32)can be rewritten as

54

1=

∑

1=

(wj /λj )σ ),

(37)

j∈N

k∈K

∑

∑

Wk (Cn , Ln ,∑Wk (Cn , Ln ,(wj /λj )σ )

j∈N

k∈K

( )σ  1−η


(
)
wj
ϵ(k)
)−σ ϕ−1 + N − 1 − (w /λ )−σ ∑+1 −1 (
( L )γ(σ−1) 1−η
 σ−η
i
i
j∈N λjkλii( )σ 
,
·
∑
wj
 Ln

λn
−σ
ϕ−1
+
N
−
1
−
(w
/λ
)
n
n
j∈N λj
k
(
1=

·

wi wn
/
λi λn

)1−2σ (

(

λi
λn

)

) σ−1 η ∑σ−η

Xk (Cn , Ln ,

k∈K

−σ
ϕ−1
k + N − 1 − (wn /λn )ϕ−1
k + N − 1 − (wi /λi )

∑

∑
−σ

j∈N
j∈N

(
(

∑

(38)

(wj /λj )σ )

j∈N

wj
λj

wj
λj

)σ  σ−1
σ−η

)σ 

(

[

Li
Ln

]

) γ( ϵ(k) +1)(1−η)+1 σ−1
σ−η

1−η

,

(39)

where functions Wk and Xk are defined in the proof of Lemma 3. Applying the same steps as the proof of∑
Lemma (9) lead to [((max ϕk )−1 + N − 1)−1 j∈N (wj /λj )σ ]1/σ < wi /λi < wn /λn .
Proof of Lemma 5. I prove L1 > Li for all i ≥ 2 by contradiction. Suppose L1 ≤ Li in an equilibrium.
Then, eqs. (37) and (38) where n = 1 imply
∑
σ
−σ
ϕ−1
j∈N (wj /λj )k + N − 1 − (wi /λi )
≤1
∑
σ
ϕ−1
j∈N (wj /λj )k +N −1−[]( )−σϕk )−1 + N − 1wiσ (max∑⇐⇒+ λσi ,≥ (1 − λi )σλij∈N (wj /λj )min λ−σikwhere the second inequality used wi /λi < 1. Further, eq. (39) and∑k∈K Xk (C1 , L1 ,

(40)

∑

j∈N (wj /λj )

imply
(
1≥

wi
λi

(

> λi

)1−2σ

σ−1
σ−η

(

λi
(

)
η

σ−1
σ−η

(

)
η

min
k

ϕ−1
k +N −1−∑σj∈N (wj /λj )
∑
−σ
σ
ϕ−1
j∈N (wj /λj )k + N − 1 − (wi /λi )(min ϕk )−1 + N − 1 −∑σj∈N (wj /λj )∑(min ϕk )−1 + N − 1 − (wi /λi )−σ j∈N (wj /λj )σ

55

) σ−1
σ−η

) σ−1
σ−η

(∵ wi /λi < 1.)σ) = 1

Substituting eq. (40) for (wi /λi )−σ yields
 σ−1
∑
σ−η
(min ϕk )−1 + N − 1 − j∈N (wj /λj )σ[]
1 > λi∑(min ϕk )−1 − (max ϕk )−1 + λσi (max ϕk )−1 + N − 1 − j∈N (wj /λj )σ
(
) (
σ−1
) σ−η
σ−1
∑
η
(min ϕk )−1 − 1
σ−η
> λi
(∵
(wj /λj )σ < N )(min ϕk )−1 − (max ϕk )−1 + λσi [(max ϕk )−1 − 1]
(

σ−1
σ−η



)

η

j∈N

(

= λi

σ−1
σ−η

)



σ−1


 σ−η
(
)
1−ληi
(max ϕk )−1 −1
σ−1
1
+
η
1
+
η
η
λi −λσ(min ϕk )−1 −(max ϕk )−1i[]  > λi σ−η 
= 1.
−1 −1
1−ληi
(max
ϕ
)
k
σ
σ
1 + λi λη −λσ
1 + λi (min ϕk )−1 −(max ϕk )−1iiThis is a contradiction. Thus, L1 > Li for all i ≥ 2.
Proof of Proposition 5. The sectoral expenditure share ratio follows from eq. (12) as](wi /λi )−σ − (wn /λn )−σ∑1+(1/ϕk + N − 1)/ j∈N (wj /λj )σ − (wi /λi )−σ]−1 } 1−ησ−η(wi /λi )−σ − (wn /λn )−σ
∑
· 1+
.
σ
−σ
(1/ϕℓ + N − 1)/ j∈N (wj /λj ) − (wi /λi )snk snℓ/=sik siℓ[{[Given the same amenity level, wn > wi implies wn /λn > wi /λi following Proposition (4). Subsequently,
the product inside the curly brackets is greater than one.
Proof of Lemmas 2 and 6. Multiplying eq. (10) by µnk /(Ln wn∑∑ℓ∈K µnℓ snℓ ) yields∑ Li w iµnk xnk(1 + µnk )µnk snkµnk(wn /λn )σ= ∑−∑(1 + µik )sik−1
(wi /λi )σℓ∈K µnℓ snℓℓ∈K µnℓ snℓℓ∈K µnℓ snℓ (ϕk + N − 1)Ln wn
i∈N(1 + µnk )µnk snkµnk(wn /λn )σ ∑Lwµ∑ i i ik σ sik ,= ∑−∑Ln w nℓ∈K µnℓ snℓℓ∈K µnℓ snℓj∈N (wj /λj )i∈Nwhere the second equality follows from the definition of µnk (9). Substituting eq. (11) for56∑j∈N (wj /λj )σyields∑∑µnk xnk(1 + µnk )µnk snk
Lwµ(∑ i i ik )
∑
= ∑
− µnk
sik
µ
s
µ
s
µ
s
L
w
j
j
jℓ
jℓ
ℓ∈K nℓ nℓℓ∈K nℓ nℓ
j∈N
ℓ∈K
i∈N
(∑
)
∑
µ s
(1 + µnk )µnk snkℓ∈K µjℓ sjℓ Li wi(∑)(∑ ik ik ) .
∑
= ∑
− µnk
µ
s
µ
s
L
w
j j
ℓ∈K nℓ nℓj∈Nℓ∈K jℓ jℓℓ∈K µjℓ sjℓi∈NThis completes the proof of Lemma 6. Lemma 2 is a special case of µnk = µn , and immediately follows
from this.
Proof of Lemma 7. I prove that the higher-income city is more productive than the lower-wage city by contradiction. Suppose the higher-income city has a lower factor price; that is, wn > wi , and wn /λn < wi /λi .
Then, it implies λn > λi , which in turn implies wn /λn > wi /λi , according to the city-level labor demandeq. (24). This is a contradiction; therefore, wn /λn > wi /λi . Subsequently, eq. (24) implies λn > λi .
Dividing the expenditure share ratio for sector k by that of sector ℓ yields
snk snℓ/=sik siℓ(Cn
Ci) ϵ(k)−ϵ(ℓ) (σ−1) 1−η
1−η

(

σ−η

=

Lγn /anLγi /ai) ϵ(k)−ϵ(ℓ) (σ−1) 1−η
1−η

(

σ−η

=

ai
an

) ϵ(k)−ϵ(ℓ) (σ−1) 1−η
1−η

σ−η

,

where the second equality follows from (4). Proposition (2) implies that when two cities have the same
population and one has a higher productivity, the other must have a higher level of amenities. Thus, ai /an >1.

CEquilibrium Stability and UniquenessThis appendix shows the stability and uniqueness of an equilibrium for the case of two cities and common
tradabilities. I start by defining a stable equilibrium in this model of two cities.
Definition 2 (Stable Equilibrium). A competitive equilibrium in the model of two cities is a stable equilibrium if and only ifd (C1 a1 /C2 a2 )1/γ
< 1.
d (L1 /L2 )This inequality guarantees that, with the migration of infra-marginal agents who are indifferent between two

57

cities (C1 a1 δ(ζ, 2) = C2 a2 δ(ζ, 2)), the expanding city would not experience a sufficient relative gain in real
income to support the post-migration population allocation. Given this definition, I obtain Proposition 6.
Proposition 6 (Uniqueness and Stability). Given Assumptions 1 and 2, common tradability, and N = {1, 2},
the equilibrium in which all sectors have non-zero production in all cities is unique and stable.
Proof. I introduce new variables, V1 and V2 , defined as−γ
V1 = a1 C1 L−γ
1 and V2 = a2 C2 L2 .

Subsequently, the condition for a stable equilibrium becomesd (V1 /V2 )1/γ (L1 /L2 )<1
d (L1 /L2 )d (V1 /V2 )=⇒< 0,
d (L1 /L2 )where I used V1 = V2 at an equilibrium. Given V1 , V2 , and normalization of λ1 = a1 = w1 = 1, the systemof equations that determines real income in the two cities can be rewritten as follows:

1=

)
(
 1
ϵk
∑  β̃ 1−η (V1 )(σ−1) 1−η +1
k∈K

1=



k

1−ϕ

(

L1

(
)

)(σ−1) ϵk +1
1 (

1−η
1−η
V

2
∑  β̃
k


k∈K 

[

a2

1−ϕ

γ(σ−1)

ϵk
+1
1−η

(
γ(σ−1)

L2

)

ϵk
+1
1−η

−1

)

[

−1

1−η
 σ−η
σ ]
(w
/λ
)
j
j
j∈N
1 − −1.
ϕ +N −1 

∑

∑

[
1−

j∈N (wj /λj )

1−η
 σ−η


]

(ϕ−1 + N − 1)(w2 /λ2 )σ 



] ( )
σ
(w
/λ
)
j
j
w2 1−2σ L2
j∈N
∑
.
1=
λ
2
ϕ−1 + N − 1 − (w2 /λ2 )−σ j∈N (wj /λj )σλ2
L1
ϕ−1 + N − 1 −

σ

(41)

,

(42)

∑

(43)

This system maps (L1 , L2 ) to (V1 , V2 ), and (L1 , L − L1 ) that makes V1 = V2 is an equilibrium. We know
1from Proposition 1 that there exits such (L1 , L − L1 ). Eq. 43 implies ∂w2 /∂ L
L2 < 0. Subsequently, eqs. 41

and 42 with L1 + L2 = L imply
dV1< 0,
d (L1 /L2 )

dV2
> 0.
d (L1 /L2 )58

Thus, V1 decreases and V2 increases with L1 /L2 , implying the intersection (V1 = V2 ) is unique. Further,
these signs imply
d (V1 /V2 )< 0.
d (L1 /L2 )Therefore, it is a stable equilibrium.

59