andLinTian FirstDraft: November2017 ThisVersion: …Stephen Redding, Stephanie Schmitt-Grohé, Bob Staiger, Martín Uribe, Eric Verhoogen, Jonathan Vogel, Shang-Jin Wei, David Weinstein

Geographic Fragmentation in a Knowledge Economy∗

Yang Jiao†and Lin Tian‡

First Draft: November 2017This Version: May 2019

Abstract

We investigate the role of information and communications technology (ICT) inshaping the spatial distribution of skills in the US, through the lens of cross-city jointproduction (e.g., sourcing, headquarter-subsidiary relation). Motivated by the stylizedfacts that big cities had become disproportionately more skill-intensive over the periodof 1980 to 2013, and industries that are more likely to fragment had seen a largerincrease in spatial skill dispersion during the same period, we propose a quantifiablespatial equilibrium model with fragmented cross-city production and heterogeneousskills. The model echoes that a nationwide communications cost reduction, throughimprovement in ICT, leads to skill reallocation into big cities due to the increase in cross-city joint productions. Consistent with model predictions, we find empirically—using anovel instrumentation strategy—that local Internet quality improvement in large citiesleads to skill inflows; while in small cities, it leads to skill outflows. Our quantitativeevaluation of the model shows that the improvement in Internet infrastructure accountsfor a significant share of the spatial redistribution of skills across US cities.

∗We are indebted to Treb Allen, Davin Chor, Donald Davis, Jonathan Dingel, Jonathan Eaton, PabloFajgelbaum, Teresa Fort, Juan Carlos Hallak, Rocco Macchiavello, Andrew H. McCallum, Antonio Miscio,Stephen Redding, Stephanie Schmitt-Grohé, Bob Staiger, Martín Uribe, Eric Verhoogen, Jonathan Vogel,Shang-Jin Wei, David Weinstein and seminar participants at Columbia, Fudan, Dartmouth, EconometricSociety China Meeting for helpful discussions and comments at various stages of this paper.†Dartmouth College and Fudan University. Email: [email protected].‡INSEAD. Email: [email protected].

1 Introduction

One of the most revolutionary developments in technology in the recent decades has been theadvances in information and communications technology (ICT). In the United States, thereported Internet usage has increased from non-existent in the 1980s to about 60% in 2000,and 80% in 2013 (see left panel of Figure 1). In particular, the fast adoption of ICT at work

1980 1990 2000 20100

20

40

60

80

100

120

140

160

Internet users/100 peopleFixed broadband subscriptions/100 peopleMobile cellular subscriptions/100 people

1995 2000 2005 2010 20155

10

15

20

25

30

Internet Use at Work/100 People

Figure 1: Increase in Internet Usage 1980 to 2013

Data source: left panel: World Development Indicators; right panel: authors’ calculation from CurrentPopulation Survey Internet and Computer Use Supplement

places—with less than 10% in 1995 to 25% in 2013, as shown in the right panel of Figure 1—alters what teams of economic agents can do at a distance, resulting in fundamental changesto the spatial organizations of productions. In particular, by reducing coordination frictions,more cross-region teams can be formed, giving rise to increasingly fragmented productionprocesses across geographic boundaries both internationally and domestically. While therehas been an extensive literature examining geographic fragmentation across internationalborders by means of offshoring (see, e.g., Hummels, Ishii and Yi, 2001; Antràs, Garicanoand Rossi-Hansberg, 2006; Grossman and Rossi-Hansberg, 2008), we know little about do-mestic production fragmentation and its associated impacts to the spatial distributions andaggregate levels of economic outcomes.

Production fragmentation across regions within the US is economically important. For

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example, as one prominent channel of production fragmentation, domestic sourcing acrossUS cities is found to be much more prevalent than international sourcing, as Fort (2017)documents. The key difference between these two types of production fragmentation lies inthe assumption on labor mobility. Individuals are relatively immobile across internationalborders, whereas they are generally assumed to have free mobility across cities, especially inthe long run. Domestic fragmentation of production may therefore lead to redistribution ofskills across different local labor markets, and have very different welfare and productivityimplications relative to the international context.

This paper develops a theoretical framework and presents empirical evidence showinghow improvement in ICT, through the formation of cross-region production teams, shapesthe spatial distribution of skills domestically across US cities. Intuitively, production of goodsinvolves two key tasks, knowledge inputs and standardized production (Garicano and Rossi-Hansberg, 2006; Arkolakis et al., 2018). Larger cities have a comparative advantage in themore knowledge-intensive tasks (Davis and Dingel, 2019), therefore attracting a greater shareof high-skilled workers who specialize in knowledge production. Workers performing tasksrelated to standardized production, on the other hand, tend to locate in smaller cities to savecosts. Cross-city production teams—with high-skilled workers in larger cities and low-skilledones in smaller cities—allow workers to take advantage of the differentiated locational benefitsoffered in cities of different sizes. Declining communications costs make such productionarrangement more viable, which reinforces the initial pattern of specialization across cities.1

As a result, larger cities become more specialized in high-skilled knowledge-intensive tasks.

Motivated by these theoretical insights, we first establish a set of stylized facts that connectspatial skill redistribution with the trend of rising production fragmentation across US cities.Using commuting zones to define cities, we show that larger cities specialize in skill intensiveactivities. This pattern of specialization had become more pronounced over the period of 1980to 2013, as high- and low-skilled workers are increasingly segregated spatially. Furthermore,the increase in the extent of spatial skill segregation is stronger for industries with greatertendency to fragment their production processes.

We then develop a theoretical framework that generates the observed stylized facts. Wepropose a spatial equilibrium model in a system-of-cities setting, with fragmentation costs,e.g., the costs of communicating and coordinating when the economic agents specializing

1In this paper, we use ‘firms’ and ‘production teams’ interchangeably, while allowing for cooperation inproductions to happen both intra- or inter-firms. For example, a furniture production team can be either anindividual firm, or consist of two firms with a furniture design firm and a furniture factory.

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in different tasks are not located in the same city. There are two types of agents: theskilled workers (or "managers"), who produce knowledge or “blueprint" of a product, and theunskilled workers (or "production workers"), who engage in the actual production. Agents aremobile across space. Managers choose the spatial organization of production—i.e., within-cityor cross-city production teams—to maximize profits. We show the existence and uniquenessof the equilibrium under regularity conditions. Equilibrium conditions determine the extentof production fragmentation and the distribution of skills, wages, housing prices and skillpremia.

We next provide empirical support for the proposed mechanism in our model by estimatingthe impact of local ICT improvement on the skill composition within cities. We instrumentfor quality of internet connectivity using ruggedness of the local terrain and local weatherconditions. Importantly, consistent with model predictions, we find that local ICT improve-ment in larger cities results in an increase in the share of high-skilled workers; whereas ICTimprovement in smaller cities leads to a decline in the share of high-skilled workers.

We finally parameterize our model to quantify the importance of ICT improvement inshaping the US labor market by facilitating cross-city joint production. We assemble a uniquedataset using the Orbis Database, which allows us to estimate the bilateral fragmentationcosts between city pairs. Through a counterfactual exercise, we show that improvement inICT since the 1980s, in particular, the Internet, is able to explain 16% of the increase in theskilled share in bigger cities.

This paper is related to several strands of literature. First, the geographic fragmentationof process in our project is similar to that in international offshoring. There is a large volumeof research on international offshoring, which arises when falling transportation or commu-nications costs motivate firms to disintegrate production and send certain jobs overseas totake advantage of comparative advantages. A consequence of this practice is growing verticalspecialization in which countries increasingly specialize in one part of a good’s productionprocess (Hummels, Ishii and Yi, 2001). Additionally, much research effort is devoted to anal-ysis of wage inequality, in response to the offshoring of unskilled labor-intensive tasks to lessdeveloped countries (see, e.g., Feenstra, 1998; Antràs, Garicano and Rossi-Hansberg, 2006;Grossman and Rossi-Hansberg, 2008).

Our work is closely connected to literature on cross-city analysis of production fragmen-tation. Duranton and Puga (2005) pioneers the theoretical research, for which they developa model with homogeneous labor who are mobile across cities and sectors. The model con-

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siders an endogenous relationship between local productivity and industrial agglomeration.The paper concludes that low communications cost facilitates separation of managerial andmanufacturing units in different cities. Liao (2012) extends the canonical model to includetwo types of workers and focuses specifically on business support services.2 The paper docu-ments that low-skill support workers tend to leave large cities and migrate to rural areas, andfinds that these low-skill workers are made better off as firm fragmentation allows supportworkers to benefit from the higher productivities in cities without bearing the high costs.Our paper is novel in several dimensions. Firstly, we documents new facts that link industrycharacteristic—the easiness of fragmentation—to the observed pattern of spatial skill redis-tribution. Secondly, we look beyond the role of national level ICT improvement and moveto the heterogeneous effect of ICT theoretically and empirically. Finally, we develop andestimate a spatial equilibrium model to quantify the role of internet infrastructure in shapingthe skill distribution of the US labor market.

There is a large empirical literature that supports our theoretical framework and results.Manufacturers often contract out specialized business services (Abraham and Taylor, 1996),and this propensity increases with city size (Ono, 2007). In particular, those with headquar-ters in large cities are more likely to contract out less important parts of the production pro-cesses (Ono, 2003). Determinants of firms’ decision to geographically separate headquartersfrom production include scale, with larger firms more likely to engage in spatial fragmenta-tion (Aarland et al., 2007), and proximity to production facilities (Holmes and Stevens, 2004;Henderson and Ono, 2008). In addition, this spatial specialization pattern has become morepronounced over time. Strauss-Kahn and Vives (2009) analyze that between 1996 and 2001,headquarters tend to move away from locations with relatively few other headquarters andbusiness service producers, and towards locations with a greater presence of them. Durantonand Puga (2005) document the pattern of increasing functional specialization in the US cities,with larger cities being more specialized in management functions whereas smaller cities inproduction through time. Our model is able to reproduce all these empirical results.

Our paper is among the growing literature that develops models of a system of cities.Davis and Dingel (2019) incorporate Costinot and Vogel (2010) into a city system withexplicit internal urban structures. While previous literature generally assumes countries’factor endowments exhibit log-supermodularity, they obtain this property for cities skill dis-tributions endogenously. They show that larger cities are skill-abundant and specialize inskill-intensive industries. While agglomeration force is exogenous given in that paper, Davis

2Workers in business support services sector make up less than 1% of the total employment in the US.

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and Dingel (2012) endogenize this human capital externality due to idea exchange. Becauseof the stronger human externality in larger cities, they show that skill premia is larger inlarger cities. Our paper will not address the source of agglomeration while we think humancapital externality between high skilled managers who engage in more cognitive tasks is anatural assumption, possibly coming from the force described in Davis and Dingel (2012)or Duranton and Puga (2004). Behrens, Duranton and Robert-Nicoud (2014) have a modelwith a system of cities as well. While all the illuminating papers above construct modelswith system of cities as well, we study the endogenous choice of cross-city production teamswith heterogeneous agents, with explicit emphasis on production organizations. That is, thecross-city organization is a form of linkage between cities we would like to highlight.

Finally, our paper is closely related to the literature on quantitative spatial equilibriumanalysis, e.g., Allen and Arkolakis (2014) and Allen, Arkolakis and Takahashi (2014). Theexisting literature mostly focuses on transportation infrastructure that affects trade cost. Wediffer by considering the ICT infrastructure, in particular, the Internet. In our framework,the ICT improvement affects cross-city joint production cost instead of transportation cost.By doing so, our paper makes contact with a body of literature that studies the effect ofmodern technology improvement on production organizations (see, e.g., Fort, 2017; Tian,2019).

The rest of the paper is organized as follows. Section 2 presents the empirical findings.Section 3 and Section 4 introduce the model, provide theoretical analysis, and derive equi-librium properties. Section 5 investigates the heterogeneous effects of ICT improvement onskill composition across cities of different sizes. Section 6 provides a quantitative evaluationof our model and presents results from the counterfactual exercise. Section 7 concludes.

2 Data and Stylized Facts

2.1 Data Description

Our analysis draws on the Integrated Public Use Micro Samples (IPUMS, Ruggles et al.,2015). For 1980, we use 5% Census samples; for later years, we combine 2011, 2012 and 20131% American Community Survey (ACS) samples.3 Our worker sample consists of individuals

3The 2011-2013 sample, referred as the 2013 data henceforth, is the most recent 3-year ACS data whenthe paper is written.

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who were between the age 16 and 64, and who were working in the year preceding to thesurvey. Residents of institutional group quarters such as prisons and psychiatric institutionsare dropped along with unpaid family workers. Labor supply is measured by the product ofweeks worked times usual number of hours worked per week. All calculations are weightedby the Census sampling weight multiplied with the labor supply weight.

We define a city as a commuting zone (CZ), which is the geographic unit of analysisdeveloped by Tolbert and Sizer (1996) and applied in a number of papers, including Autorand Dorn (2013) and Burstein et al. (2017). Each CZ is a cluster of counties characterizedby strong commuting ties within and weak commuting ties across zones. For our analysis,we include 722 CZs in the mainland US. Following convention in the literature, we measurethe size of a commuting zone by its population.

Throughout the paper, we classify workers into high- and low-skill groups using theiroccupation wage in 1980. Following Acemoglu and Autor (2011), we rank skill levels ofdifferent occupations, approximated by the mean log wage of workers in each occupation in1980.4 We define skilled workers, also referred to as managers, as those whose occupationwage rank is higher than 75% of occupations in 1980. We vary the cutoff in robustness checksfor 67% and 80%. Further robustness checks using education information to classify high andlow skilled are also provided in the Appendix.

2.2 Stylized Facts on Skill Distribution

In this section, we establish three stylized facts: (1) Larger cities specialize in skill-intensiveactivities; (2) between 1980 and 2013, there had been substantial spatial segregation of thehigh and low skilled; and (3) this trend of spatial skill segregation are more concentrated inindustries with greater tendency to fragment their production processes.

The left panel of Figure 2 replicates a well-known fact: The largest cities, measured bytheir population sizes, are the ones that have the highest shares of high skilled employment intheir total employment in both 1980 and 2013.5 This pattern of skill specialization suggestscomparative advantage differences across cities of different sizes: Larger cities have a com-

4Examples of occupations in the lower, middle and upper wage-rank distributions are child-care workers,waiters and waitresses, housekeepers, and hotel clerks; machine operators, reception and information desk,typists, and carpenters; CEOs, engineers, architects, financial managers, and software developers, respec-tively.

5Figure 13 in the Appendix displays the relationship between log(1980 population) and log(2013 popula-tion) at the CZ level. The linear correlation is 0.99.

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parative advantage in more skill-intensive activities, probably due to stronger agglomerationforces, and smaller cities have a comparative advantage in less skill-intensive activities, aidedby the lower labor costs.

-.05

0.0

5.1

skille

d sh

are

6 8 10 12 14 16log(population) in 1980

1980 2013

.01

.02

.03

.04

chan

ge in

ski

lled

shar

e 19

80-2

013

6 8 10 12 14 16log(population) in 1980

Figure 2: Change in High Skilled Employment Share with respect to City Sizes

Notes: the left panel displays the regression line for the high skilled share (demeaned) in 1980 and 2013against city size (log of 1980 population) . The right panel displays the change in the skilled share from 1980to 2013. The skilled is defined as occupation rank above 75% using 1980 mean of log hourly wage.

Importantly, this pattern of specialization had become more pronounced over time. Theright panel of Figure 2 shows that larger cities had experienced a greater increase in theshare of high skilled employment between 1980 and 2013, thereby becoming even more skillintensive. For example, the share of high skilled employment in the largest city in the US,had risen by 4%; whereas it has only increased by less than 1% in the bottom percentileof the city-size distribution. Table 1 reports the formal statistical test, in which we regresschanges in the share of high-skilled employment at the city level onto city sizes measured bythe log of the 1980 city-level population. We find strong positive correlation between citysize and the magnitude of the change in the high-skilled employment share.6

6Table 6 in the Appendix report the robustness checks regarding the definition of the high skilled using 67%occupation wage rank cutoff, 80% occupation wage rank cutoff and education information (college educationand above).

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Dependent variable: change in high skilled employment share(1) (2)

City Size 0.004∗∗∗ 0.005∗∗∗(0.001) (0.001)

State fixed effect No YesObservations 722 722R2 0.037 0.357∗ p < 0.10, ∗∗ p < 0.05, ∗ ∗ ∗ p < 0.01

Table 1: Change in High Skilled Employment Share and City Size

Notes: City size is measured by log(population in 1980). Skilled workers are defined as workers whoseoccupation wage rank is higher than 75% of occupations in 1980. Column (1) reports results using robuststandard errors, and Column (2) reports results with standard errors clustered by state.

The spatial redistribution of skills, established in the previous set of results, suggests anincrease in spatial segregation of high- and low-skilled workers over the same period. Tostudy this spatial segregation more directly, we adopt a variant of the Kremer and Maskin(1996) measure of the degree of segregation, i.e.,

ρ = 1S

∑s

[∑cNcs · (πcs − πs)2

Ns · πs · (1− πs)

].

where s ∈ {1, 2, . . . , S} denotes a sector as defined by the Census ind1990 codes, Ncs is theemployment in sector s and city c, Ns is the total sectoral employment, πcs = Nskilled

cs

Ncsis the

high skilled employment share in sector s and city c, and πs = Nskilleds

Nsis the high skilled

employment share in sector s.7 As shown in Table 2, the Kremer and Maskin (KM) Index,denoted by ρ, had almost tripled from 1980 to 2013, indicating that the high skilled and thelow skilled had become increasingly more spatially segregated.

7This index measures how correlated the employment share of different occupations are within a city-sector. It is constructed as the ratio of the variance of share of the high skilled across cities to the variance ofan agent’s occupation status (i.e. the high skilled vs. the low skilled) of the a given sector, which is equivalentto the R2 value of a regression of share of the high skilled on a series of city dummies. When ρ = 0, thereis no segregation, i.e. the high skilled and the low skilled are always in the same cities; when ρ = 1, there iscomplete spatial segregation of the high skilled and the low skilled. We calculate the national average as anaverage value across sectors to account for possible changes in industry composition across time.

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Year ρ 95% Confidence Interval1980 0.00746 (0.00741, 0.00752)2010 0.0204 (0.0202, 0.0205)

Table 2: Segregation Index in 1980 and 2012

Notes: Skilled workers are defined as workers whose occupation wage rank is higher than 75% ofoccupations in 1980. 95% confidence interval of the index of segregation is:

F (N − J, J − 1)0.025

F (N − J, J − 1)0.025 + 1−ρρ

≤ ρ ≤ F (N − J, J − 1)0.975

F (N − J, J − 1)0.975 + 1−ρρ

,

where J = C + S (Kremer and Maskin, 1996).

All together, these results suggest that there has been a growing spatial segregation ofthe high skilled and the low skilled, with larger cities increasingly specializing in high-skilledoccupations and smaller cities in low-skilled occupations. We find that this pattern of seg-regation across space is closely related to the production fragmentation activities in the USeconomy. Fort (2017) documents that firms’ adoption of communications technology facili-tates firms’ sourcing, particularly from domestic suppliers. If our story on the segregation islinked with sourcing tasks after improvement in ICT over the past three decades, one wouldexpect that industries that experience more sourcing would also undergo larger skill segre-gation. Figure 3 confirms this hypothesis by illustrating the relationship between the KMIndex and the fraction of plants that engage in sourcing activities—measured by the pur-chases of contract manufacturing services (CMS) from other plants (within its company orfrom another company)—in each of the 86 four-digit NAICS manufacturing industries. Forexample, computers and related equipment industry has a very high sourcing index (50% ofplants source from another plant) features a relatively large increase in the KM index; whilebakery product industry has a very low sourcing index (8% of plants source from anotherplant) exhibits a negative increase in the KM index.

In summary, we establish three stylized facts. Larger cities have a comparative advantagein skill-intensive activities (Fact 1). This pattern of specialization had become stronger overthe past three decades as high- and low-skilled workers become more segregated geographi-cally (Fact 2). The extent of segregation varies across industries systematically, matching thecross-industry heterogeneity in the extent of production fragmentation (Fact 3). These formthe empirical basis for our central hypothesis: Improvements in ICT technology over the last

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-.50

.51

1.5

0 .2 .4 .6Fort (2017) Firm Sourcing Index

95% CI Fitted valuesChange in KM Segregation Index

Figure 3: KM Skill Segregation Index and Fort (2017) Index on Firm Fragmentation

Notes: Each point denotes an industry. The correlation between KM skill segregation index and Fort (2017)plant sourcing index is 0.51.

three decades had allowed firms to better leverage comparative advantage differences acrosscities by engaging in cross-city productions, which allow them to locate different segmentsof production activities in different geographic regions. This reinforces the initial pattern ofspecialization and drives the redistribution of skills and incomes.

In the next section, we propose a quantitative model with cross-city production teams thatcaptures these forces in a rich geographic setting, to study the impact of communicationscost reduction on the skill and income distribution across cities.

3 The Model

3.1 Set-up

We consider an economy with a finite number of cities, indexed by n ∈ N . There is acontinuum of agents, distinguished by their skill levels, each of whom inelastically suppliesone unit of labor. The measures of high-skilled workers (which we refer to as managers) and

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low-skilled workers (which we refer to as production workers) are Lm and Lp, respectively.

Individuals consume two goods: a homogeneous tradable good and housing. The utilityfunction follows a standard Cobb-Douglas form:

U(c, h) = α−α(1− α)−(1−α)cαh1−α, (1)

where c is the consumption of the tradable good and h is the consumption of housing.Managers and production workers choose their living locations to maximize their utilities.8

The homogeneous tradable good is produced anywhere in the economy with varying pro-ductivity levels as specified below. A manager living in city n can choose to set up theproduction team in any city c ∈ N .9 Managers living in n that produce in c incur a pro-ductivity loss that we model as iceberg bilateral fragmentation costs, τnc ≥ 1, with τnn

normalized to 1. These costs reflect the costs of managing off-site workers, e.g., communi-cations or coordination costs between managers and production workers located in differentcities.

A manager living in n and managing workers in c has the following production technology:

ync = anclβ. (2)

Each production team is comprised of a single manager and l homogeneous production work-ers.10 The production technology, which follows Lucas (1978), involves two elements: First,anc denotes the “manager’s productivity”; second, β < 1 is an element of diminishing returnsto scale, or the “span of control”. Formally, a manager in n is characterized by a productivityvector an = (an1, an2, . . . , anN). These productivity vectors are origin-city specific and areallowed to vary across managers, leading managers in n to make different choices on theproduction locations. In doing so, we assume that the productivity heterogeneities originatefrom managers, who take the role of developing blueprint for the products and providingmanagement capital for the production process. It is easy to extend the model to allow for

8There is a number of papers that study the various forms of mobility cost in reality, see, e.g., Enrico(2011), Baum-Snow and Pavan (2012), and Ferreira, Gyourko and Tracy (2011). This paper focuses on thelong-run impact of ICT improvement through the lens of cross-region joint production, and we thus take aposition that in the long run, individuals are very much mobile.

9To the extent possible, we use n to denote the manager’s living location (the source of the blueprint ormanagement capital) and c to index the location of the production.

10Note that this production setup is equivalent to any constant returns to scale production function. Boththe skilled input M and the unskilled input L can be equivalently translated to M production teams, each ofwhich consists of 1 single manager and L

M production workers.

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worker productivity to vary for the production in such a way that none of the results thatwe focus are affected.

The manager’s productivity, anc, has two components: (1) local agglomeration forcef(Lmn ), which is an increasing function of the total mass of managers in city n, Lmn ; and(2) a random draw, denoted by anc. The two components are assumed to enter the man-ager’s productivity function multiplicatively:

anc = f(Lmn )anc. (3)

In particular, A manager who lives in city n draws her productivity anc from N ∈ N citiessimultaneously. Each anc is drawn independently from a Fréchet distribution with cumulativedistribution function:

G(a) = exp(−Tna−θ

),

where Tn is an exogenous technology parameter representing city n fundamental, and θ

represents the dispersion of the draws: a higher value θ > 0 decreases the dispersion of themanager’s productivity across locations.11

3.2 Manager’s Optimization

In this environment, managers face a three-step optimization problem. First, managerschoose where to live (i.e., headquarter locations). Second, the manager then chooses thespatial production organization (i.e., locations of production teams). Finally, the managerdecides on the production scale (i.e., how many workers to hire). We consider the optimizationproblem in a backward order, starting from the last step.

3.2.1 Production Scale

Managers are the residual claimants of the firm’s profit. The income of a manager who livesin city n and manages workers in city c is:

πnc = ancτnc

lβ − wcl. (4)

11The assumption of having i.id draws across all locations is observationally equivalent to a joint Fréchetdistribution assumption. See Eaton and Kortum (2002), footnote 14, for a discussion.

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Recall τnc ≥ 1 is the iceberg cost which reflects the cost of managing workers remotely, e.g.,the communications cost between city n and city c.

Given anc, a manager chooses the size of her production team, l, to maximize her income.Taking the first order condition of (4) with respect to l, we obtain the optimal productionscale l∗,

l∗ = ( βancτncwc

)1

1−β . (5)

Notice that a more productive manager, i.e., high anc, manages a larger production team.

Combining (4) and (5), a manager living in city n with a production team in city c hasan income of:

π∗nc = ββ

1−β (1− β)( anc

τncwβc

)1

1−β . (6)

Note both a higher iceberg fragmentation cost τnc and a higher worker wage wc would reducethe manager’s income.

3.2.2 Production Locations

A manager who lives in city n would choose to locate the production team in a city thatmaximizes her income π∗nc, as specified in (6). The assumption on the idiosyncratic componentof manager’s productivity allows us to derive the following result, which we call the joint-production gravity equation.

Proposition 1 The probability of a manager who lives in city n and locates production incity c is

Tn(τncwβc )−θΦn

, (7)

where Φn is city n’s “fragmentation potential,” defined by

Φn ≡∑k

Tn(τnkwβk )−θ. (8)

Note that the summation of all probabilities is 1.

Proof. See Appendix.

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Based on this proposition, it is easy to see that an ICT development that drives downcross-city fragmentation cost, τnc, increases the possibility of cross-city production teamsrelative to domestic production teams, holding everything else equal. By the WLLN, theabove gravity equation also gives the share of managers living in city n and locating theirproduction teams in city c:

xnc ≡LmncLmn

= Tn(τncwβc )−θΦn

. (9)

3.2.3 Living Locations

Individuals choose their living locations to maximize their utility. From (1), we derive theindirect utility function for an agent with income πn facing rent pn in city n:

V (pn, πn) = πnp1−αn

. (10)

Additionally, given the Cobb-Douglas preference, the equilibrium housing rent in city n isgiven by:

pn = (1− α)Wn

Hn

, (11)

where Wn is the total income in city n, including both city n manager’s and productionworker’s income, and Hn is the exogenously given housing supply in city n.

Given the distribution of the productivity draws and the profit function π∗nc in (6), we canderive the distribution for managers’ income.

Proposition 2 The income of a manager who lives in city n follows the following Fréchetdistribution with cumulative distribution function:

G(π) = exp(−[ββ(1− β)1−β]−θ (f(Lmn ))θ Φnπ

−θ(1−β)). (12)

Proof. See Appendix.

By properties of Fréchet distributions, the expected income of a manager living in city nis thus:

E[πn] = ζ[[f(Lmn )]θΦn]1

θ(1−β) ,

where ζ ≡ θβ−β

1−β∫+∞

0 exp(−x−θ(1−β)

)x−θ(1−β)dx.

15

Managers choose their living locations to maximize the their indirect utility in (10). De-noted by Ψn, a manager’s natural logarithm of the expected utility function is given by:

Ψn = log(E[πn]p1−αn

)= const+ 1

1− β log[f(Lmn )] + 1θ(1− β) log Φn − (1− α) log pn. (13)

A manager’s problem is therefore to maximize Ψn. In equilibrium, managers are indifferentbetween living in city n and n′ (conditional on that there are non-zero managers in bothcities), so that Ψn = Ψ′n, ∀n, n′, or

11− β log[f(Lmn )] + 1

θ(1− β) log Φn − (1− α) log pn (14)

= 11− β log[f(Lmn′)] + 1

θ(1− β) log Φn′ − (1− α) log pn′ .

3.3 Worker’s Optimization

Similar to managers, production workers also choose the city to live to maximize their indirectutility in (10), given their income wn and housing price pn. In equilibrium, production workersare indifferent and thus receive the same indirect utility across the cities, i.e., V w

n = V wn′ =

v ∀ n, n′,. We therefore obtain the following equilibrium condition:

wn/p1−αn = wn′/p

1−αn′ . (15)

4 Equilibrium Analysis

We characterize the spatial equilibrium in this section. We first show the existence anduniqueness of the spatial equilibrium. We then focus on a simplified two-city model to deriveanalytic results for the effects of changes in fragmentation costs, τnc, on distribution of skillsand wages. We finally perform numerical simulation on an eight-city model.

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4.1 Definition

In a spatial equilibrium, managers and workers are indifferent across locations.12 With ex-ogenous parameters {Tn, τnc, Hn}, ∀n, c ∈ {1, 2, ..., N}, and a mass of managers LM andworkers LP , an equilibrium is a vector of labor allocations {Lmnc, LPnc}, and prices {pn}, {wn}such that:

1. Production workers maximize their utility in (10);

2. Housing prices pn are determined by (11);

3. Managers maximizes their expected utility in (13);

4. Labor markets clear for both managers and workers:

Lm =∑n

Lmn =∑n,c

Lmnc, (16)

andLp =

∑n

Lpn =∑n,c

Lpnc, (17)

where Lpnc refers to the mass of production workers hired by managers from n and livingin city c and is given by:

Lpnc = ηw−1c

(Tn(τncwβc )−θ

)Φ

1θ(1−β)−1n [f(Lmn )]

11−βLmn .

13 (18)

4.2 Equilibrium Properties

In this section, we show, using Banach Fixed Point Theorem, that the equilibrium exists andis unique. For derivation of analytic results, we follow conventional literature and adopt thefollowing parametric assumption for the city-level agglomeration forces for managers (see,e.g., Allen and Arkolakis, 2014):

f(L) = Lγ, where γ > 0.12Given the unbounded Frechét distribution draws, we can show that all cities have non-zero mass of

production workers. Additionally, to be consistent with data, we further assume that city fundamentals alsoensure all cities have non-zero mass of managers.

13See Appendix A.4 for details on derivation of the demand for production workers.

17

Combining this assumption and the equilibrium housing prices in (11), we can re-writethe indifference conditions for workers and managers in (14) and (15) as:

γ

1− β log Lmn

Lmc+ 1θ(1− β) log Φn

Φc

= (1− α) log pnpc

= log wnwc, (19)

and

(wnwc

) 11−α

=ζ(Lmγθ

n Φn)1

θ(1−β)Lmn +∑k η(Tk(τknwβn)−θ

)Φ

1θ(1−β)−1k [Lmk ]

γ1−β+1

ζ(Lmγθc Φc)

1θ(1−β)Lmc +∑

k η(Tk(τkcwβc )−θ

)Φ

1θ(1−β)−1k [Lmk ]

γ1−β+1

Hc

Hn

. (20)

We can solve for wnwc

and LmnLmc

from the two equations above. Furthermore, the relativenumber of production workers in city n and city c is given by:

LpnLpc

=∑k ηw

−1n

(Tk(τknwβn)−θ

)Φ

1θ(1−β)−1k [Lmk ]

γ1−β+1

∑k ηw−1

c

(Tk(τkcwβc )−θ

)Φ

1θ(1−β)−1k [Lmk ]

γ1−β+1

. (21)

Finally, the skill premia, defined as the log difference between the manager’s and pro-duction worker’s expected income, is given by:

logE[πn]− logwn = γ

1− β logLmn + 1(1− β)θ log Φn − logwn. (22)

Denoted ∆nc = τ−θnc , we obtain the following equation from the definition of Φn in (8):

∆11T1 ∆12T1 ∆13T1 . . . ∆1NT1

∆21T2 ∆22T2 ∆23T2 . . . ∆2NT2... ... ... . . . ...

∆N1TN ∆N2TN ∆N3T2 . . . ∆NNTN

w−βθ1

w−βθ2...

w−βθN

=

Φ1

Φ2...

ΦN

. (23)

From equation (19), we get:

Lm γ

1−βn ∝ wnΦ

− 1θ(1−β)

n . (24)

Therefore, up to a constant, and using results from equations (20), we can rewrite the

18

matrix as:ζΦ

1θ(1−β)1 + ηT1∆11w

−βθ1 Φ

1θ(1−β)−11 ηT2∆21w

−βθ1 Φ

1θ(1−β)−12 . . . ηTN∆N1w

−βθ1 Φ

1θ(1−β)−1N

ηT1∆12w−βθ2 Φ

1θ(1−β)−11 ζΦ

1θ(1−β)2 + ηT2∆22w

−βθ2 Φ

1θ(1−β)−12 . . . ηTN∆N2w

−βθ2 Φ

1θ(1−β)−1N

......

. . ....

ηT1∆1Nw−βθN Φ

1θ(1−β)−11 ηT2∆2Nw

−βθN Φ

1θ(1−β)−12 . . . ζΦ

1θ(1−β)N + ηTN∆NNw

−βθN Φ

1θ(1−β)−1N

×

w

1−βγ

+11 Φ

− 1θ(1−β)−

1θγ

1

w1−β

γ+1

2 Φ− 1

θ(1−β)−1

θγ

2...

w1−β

γ+1

N Φ− 1

θ(1−β)−1

θγ

N

=

w

11−α

1 H1

w1

1−α

2 H2...

w1

1−α

N HN .

(25)

Multiply both sides by wauxn , where aux ∈ R is an auxiliary parameter, we get:ζΦ

1θ(1−β)1 waux1 + ηT1∆11w

−βθ1 Φ

1θ(1−β)−11 waux1 . . . ηTN∆N1w

−βθ1 Φ

1θ(1−β)−1N waux1

ηT1∆12w−βθ2 Φ

1θ(1−β)−11 waux2 . . . ηTN∆N2w

−βθ2 Φ

1θ(1−β)−1N waux2

.... . .

...

ηT1∆1Nw−βθN Φ

1θ(1−β)−11 wauxN . . . ζΦ

1θ(1−β)N wauxN + ηTN∆NNw

−βθN Φ

1θ(1−β)−1N wauxN

×

w

1−βγ

+11 Φ

− 1θ(1−β)−

1θγ

1

w1−β

γ+1

2 Φ− 1

θ(1−β)−1

θγ

2...

w1−β

γ+1

N Φ− 1

θ(1−β)−1

θγ

N

=

w

11−α

+aux1 H1

w1

1−α+aux

2 H2...

w1

1−α+aux

N HN

(26)

Denote xn =(

11−α + aux

)logwn and x = (x1, x2, ...xn)′. Then

Fi(x) = log[∑j

1Hi

η exp(xi)−βθ+aux

1/(1−α)+auxTj∆ij exp(xj)1−βγ +1

1/(1−α)+aux(Φaj

)−1− 1θγ

+ ζ1Hi

exp(xi)1−βγ +1+aux

1/(1−α)+aux (Φai )− 1θγ ],

whereΦan = Tn∆nk exp(xn)

−βθ1/(1−α)+aux .

We can show that:

d(F (x), F (y)) = maxi|...| ≤ ... ≤ ρ ·maxk|xk − yk| = ρd(x,y).

19

where

ρ = | −βθ + aux

1/(1− α) + aux|+ |

1−βγ

+ 11/(1− α) + aux

|+ |(1 + 1θγ

)( −βθ1/(1− α) + aux

)|

If there exists an aux such that ρ < 1, using Banach Fixed Point Theorem, the equilibriumexists and is unique.

4.3 Equilibrium with Infinite Fragmentation Cost

When the bilateral fragmentation cost τnc = +∞, ∀n 6= c, the system of equilibrium condi-tions reads as follows:

γ(logLmn − logLmn′) = (logwn − logwn′)− (log T1θn − log T

1θn′), (27)

(1

1− α + β

1− β

)(logwn−logwn′) =

(γ

1− β + 1)

[logLmn −logLmn′ ]+1

1− β [log T1θn −log T

1θn′ ].

(28)

In this case, the cross-city fragmentation cost is prohibitively high such that all managerswill hire production workers in the same city as the one she lives in. We can show that underregularity conditions, i.e., γ + 1 > γ

1−α , cities with high technology parameters are not onlylarger, but also more have a greater share of the high-skilled labor.14 More formally,

Proposition 3 Given f(Lmn ) = Lmγn and γ + 1 > γ

1−α , when τnc = +∞, ∀n 6= c, the spatialequilibrium exists and is unique. The number of managers in each city Lmn and the numberof production workers in each city Lpn satisfy that

Lmn ∝ T κn , (29)

Lpn ∝ T κn . (30)

Proof. See Appendix.14The assumption that γ + 1 > γ

1−α implies that the elasticity of agglomeration, which is positivelycorrelated with γ is smaller than the elasticity of urban costs, which is positively correlated with 1−α. Thisensures that cities have a finite size in equilibrium. See Behrens, Duranton and Robert-Nicoud (2014) for anexcellent discussion.

20

4.4 Equilibrium with Finite Fragmentation Costs

We next analyze the equilibrium with finite fragmentation costs. We start with a simple two-city case to elucidate the mechanism of skill relocation after a reduction in fragmentationcosts. We then extend the analysis to a multi-city scenario.

4.4.1 A Two-City Analysis

We start with a two-city case with quasi-symmetric communications cost and fixed housingsupply.15 Using the simple model, we highlight the mechanism behind the skill relocationafter cross-city communications cost reduction.

First, it is easy to see that when the cross-city fragmentation cost is infinite, the city withthe greater technology parameter is larger and more skill intensive. We can solve for Lm1

Lm2and

w1w2

explicitly using (27) and (28):

log Lm1

Lm2=

α1−α

γ + 1− γ1−α

[log T1θ

1 − log T1θ

2 ]. (31)

log w1

w2= 1γ + 1− γ

1−α(log T

1θ

1 − log T1θ

2 ), (32)

Suppose that T1 > T2, then city 1 is the city with larger population.

If there is a reduction in communications cost, we can show that a small reduction in frag-mentation cost, e.g., improvement in ICT that facilitates cross-city communication, resultsin a spatial reallocation of skills. Specifically, the share of the high skilled employment inthe initially larger city will increase, whereas the share of the high skilled employment in theinitially smaller city will decrease.

Proposition 4 In the two-city case, if τ12 = τ21 goes down around the neighborhood of theinfinite communication cost, and suppose T1 > T2, Lm1 would go up and Lm2 would go down.

This proposition essentially supports that if ICT improvement reduces cross-city commu-nications cost, then our model is able to replicate the observed fact that bigger cities are

15We assume Hn = Hc = 1. The analysis can be easily extend to the case with different city-level housingsupplies.

21

attracting a larger proportion of the high skilled in recent decades. Through a numericalsimulation of the two-city equilibrium, we confirm—as shown in Figure 4—the proposition’sprediction on skill flows after the ICT improvement. Furthermore, Figure 5 shows thatthe skill premium increases with ICT openness. The intuition is that the skill reallocationstrengthens the agglomeration force, which allows the managers to raise their productivitymore after the ICT improvement.

0 0.2 0.4 0.6 0.8 10.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

ICT Openness ∆

Sha

re o

f Man

ager

s in

City

1

0 0.2 0.4 0.6 0.8 10.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

ICT Openness ∆

Sha

re o

f Man

ager

s in

City

2

Figure 4: Two-City Equilibrium: Share of Managers in City 1 and City 2

4.5 An Eight-City Simulation

We now turn to a multi-city analysis to explore the heterogeneous effect of a reduction in thefragmentation cost on the city-level skill composition. The objective is to have a relativelylarge number of cities to mimic the fact that the population of every single city, even thelargest city, constitutes only a small fraction of the total population so that a single city’sICT improvement would not affect the other cities significantly. At the same time, we wantto avoid simulating a too-many city case, which could be too demanding computationally. Tothis end, we choose an eight-city scenario, in which we consider the case that all cities havethe same housing supply and there are four big cities and four small cities with technology

22

0 0.2 0.4 0.6 0.8 11.9

1.92

1.94

1.96

1.98

2

2.02

2.04

2.06

2.08

2.1

ICT Openness ∆

Ski

ll P

rem

ium

Figure 5: Two-City Equilibrium: the Skill Premium

parameters given as follows:

T1 = T2 = T3 = T4 > T5 = T6 = T7 = T8.

Figure 6 shows that if there is an ICT improvement in a small city, say City 8, whichreduces the bilateral fragmentation costs between City 8 and all the other seven cities, thenthe share of managers decreases in City 8. The intuition is that as a small city, the low skilledwage here is relatively lower. When the city gets more connected with the rest of the nation,some managers in those bigger cities will find it profitable to locate their production teamsin the smaller city, which increases the local demand for production workers there, givingrise to an inflow of the low-skilled workers.

In contrast, Figure 7 shows that if there is an ICT improvement in a big city, say City4, which reduces the bilateral fragmentation costs between City 4 and all the other sevencities„ then the share of managers increases here. The intuition is that as a big city, the lowskilled wage here is relatively higher. When the city gets more connected with the rest ofthe nation, some managers will find it more profitable to relocate to that city to leverage thestrong agglomeration externalities, while keeping their production teams in other low-costsmall cities. In doing so, larger cities will attract an inflow of high-skilled labors.

23

0 0.2 0.4 0.6 0.8 10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ICT Openness ∆ of City 8

Sha

re o

f Man

ager

s in

City

8

Figure 6: Eight-City Equilibrium: Share of Managers

0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

ICT Openness ∆ of City 4

Sha

re o

f Man

ager

s in

City

4

Figure 7: Eight-City Equilibrium: Share of Managers

24

5 Heterogeneous Effects of ICT on City Skill Compo-sition: the Evidence

Guided by our theoretical model, we aim to empirically investigate the heterogeneous effectsof fragmentation cost reduction on the share of skilled employment across cities of differentsizes. We focus on one channel: Improvement in ICT reduces bilateral fragmentation cost,e.g., by facilitating cross-city communications between the managers and production workers.In particular, our model predicts that local ICT improvement drives up the high skilledemployment share in bigger cities, while reducing the high skilled employment in smallercities. We examine these predictions by presenting evidence from the US cities using the USInternet infrastructure data. The empirical specification we employ follows a long differenceexercise:

∆skilled sharei = β0 +β1city sizei +β2∆interneti +β3city sizei ∗∆interneti + γXi + εi, (33)

where ∆skilled sharei is the change in the high skilled employment share in city i between1980 and 2013, city size denotes the log (population) in 1980, ∆interneti is the change inInternet quality in city i between 1980 and 2013, and Xi is other controls including statedummies.16

Our key coefficients of interest are β2 and β3. The model predicts that β2 < 0 and β3 > 0,which imply that ICT improvement in a small city will reduce the skilled employment sharelocally, while ICT improvement in a big city will increase the local skilled employment share.

5.1 Internet Data

We supplement our baseline dataset with the Internet infrastructure data, which is drawnfrom the US Federal Communications Commission (FCC) Fixed Broadband DeploymentDatabase. Fixed Broadband providers are required to provide the lists of census blocks towhich they offer service in at least one location in the block. The database also providesadditional information about the service, including the download and upload bandwidths.The data available is from December 2014. We identify the maximum bandwidth at theblock level and then compute the population-weighted download and upload bandwidths

16Note that in 1980, internet technology was not made available for commercial uses in the economy. Asa result, ∆interneti is computed using the ending year’s Internet quality.

25

to the commuting zone level.17 Figure 8 shows the Internet quality map across the CZs,measured by the average of download and upload bandwidths at the CZ level.

96.0 − 984.264.1 − 96.048.5 − 64.137.0 − 48.526.0 − 37.011.5 − 26.0

Figure 8: Internet Quality in US Commuting Zones

Notes: Upload and download speeds are measured in Megabytes per second.

5.2 Internet Quality, City Size and the Skilled Employment Share

Using the internet data, we run the specification in (33). The first two columns of Table 3report the key results. In columns (1) and (2) show the results without and with state fixedeffects. In contrast to results in Table 1, when we take into account the effect of Internetquality and its interaction term with city size, the positive relationship between the changein skilled employment share and city size observed earlier disappears. The coefficient oncity size becomes negative (but not robustly significant). Importantly, consistent with themodel predictions, we find that β2 < 0 and β3 > 0, as shown in both column (1) and column(2). These two results jointly imply that better Internet quality reduces small city’s high

17The 15-digit census block ID comes from the 2010 census. In computing the population-weighted av-erage internet quality measures, we use the 2010 population information at the PUMA level—the smallestgeographic unit in the 2010 census. We aggregate data from the more finely divided census block level to thePUMA level using simple averages.

26

skilled employment share, while increasing big city’s high skilled employment share, therebyconfirming the model predictions. It is also worth-noting that the point estimates on β2 andβ3 do not vary regardless of whether state fixed effect is included or not.

One concern that arises when estimating Equation (33) is that internet quality is endoge-nous, which include concerns for (1) the presence of long-run local employment trends, (2)unobserved local shocks affecting both internet improvement and changes in skill share overtime, and (3) reverse causality, i.e., increases in share of high skilled workers driving theimprovement in internet. To address the first concern, in column (3) and column (4), weperform a falsification test by replacing the left hand side variable by the change in the highskilled employment share from an earlier period, between 1950 and 1980. The estimatesshow that there is indeed no role of later development of ICT on the change of skilled em-ployment share in earlier years. To address the second and the third concern, we consideran instrumental-variable approach. We instrument internet quality with a measure of theruggedness of the local terrain and the average amount of snow a given CZ receives in a year.

Utilizing the instruments for Internet and recover the total effect of ICT requires thetwo instruments satisfying the following exclusion restriction: The predicted internet qualitymust be uncorrelated with the change in skilled share not induced by internet quality. Thisis likely to hold for two reasons. First, while terrain and local weather conditions are likelycorrelated with other factors that may affect the level in the share of skilled workers throughother channels, it is unlikely that they affect the flow of skilled workers between the twoperiods. For this to happen, the correlation between the instruments and these other factorswill have to become stronger over time. Next, the correlations will also have to be system-atically correlated with city size, i.e., the correlations are positive for larger cities (therebyattracting an inflow of high skilled workers) and negative for smaller cities (thereby resultingin an outflow of high skilled workers). Furthermore, as shown in Table 1, the J-test foroveridentifying restrictions generates a p-value of 0.2792, offering further reassurance on thevalidity of the instruments.

The results from the 2SLS estimation are shown in Column (5) of Table 1. The resultsremain qualitatively unchanged from the OLS estimates. The point estimates of the coeffi-cients on Internet quality and Internet quality × City Size are greater in value than the OLSestimates. This could be due the classical measurement errors in the regressors which wouldresults in attenuation bias. Crucially, the model predictions that β2 < 0 and β3 > 0 continueto hold.

27

Dependent variable: change in the high skilled employment shareOLS 2SLS

(1) (2) (3) (4) (5)1980-2013 1950-1980 1980-2013

City size -0.006∗ -0.003 0.012∗∗ 0.006 -.025∗∗(0.003) (0.004) (0.006) (0.007) (0.01)

Internet quality -0.024∗∗∗ -0.022∗ 0.012 -0.001 -0.084∗∗(0.008) (0.012) (0.015) (0.018) (.035)

Internet quality × city size 0.002∗∗∗ 0.002∗ -0.00 -0.000 .007∗∗∗(0.001) (0.001) (0.001) (0.002) (0.003)

State fixed effect No Yes No Yes YesObservations 722 722 722 722 722

R2 0.045 0.362 0.047 0.283 0.321Sargan over-identification P-value 0.2792S-W F-stats (First Stage)

Internet quality 10.14Internet quality × city size 20.63

Table 3: Internet Quality, City Size and the Skilled Employment

Notes: City size is measured by log(population in 1980). Internet quality is the log of population weightedaverage of upload and download bandwidths measured in year 2014. Robust standard errors are inparentheses, and column (2) and column (4) are clustered by state. We report Sanderson-Windmeijer(S-W) F- statistics for the first stage regressions. )∗ p < 0.10, ∗∗ p < 0.05, ∗ ∗ ∗ p < 0.01

28

6 Quantitative Analysis

We next calibrate the model parameters, to quantify the contribution of the US Internetinfrastructure in shaping the skill redistribution of the US labor market, through our proposedchannel of cross-city joint production.

6.1 Parametric assumptions

We maintain the functional form assumption for the agglomeration force, i.e., f(Lmn ) = Lmγn .

Recall that the parameter γ governs the extent of regional agglomeration. Moreover, weparametrize the bilateral fragmentation cost as follows:

log τnc = λnc + δd log dnc + δIqnc. (34)

The fragmentation cost between two cities n and c is assumed to take a semi-parametricform, i.e., a power function of the bilateral geographic distance dnc between the two citiesand quality of the Internet connection between the two cities qnc, in addition to a term λnc

that summarizes all other associated costs, e.g., if the two cities are in located in the samestate, and if the two cities share a common border. We refer to δd as the distance elasticityof joint-production and δI as the Internet elasticity of joint-production.

For the bilateral Internet connection, we assume that it adopts a quasi-symmetric formsuch that

qnc = qn ∗ qc, (35)

where qn denotes city n Internet quality. Note that by using the interaction term qn ∗ qc, weallow for potential complementarity in both cities’ Internet quality. For instance, if there isno Internet in city c, then no matter how good city n Internet is, the bilateral communicationcost would remain very high.

6.2 Calibration of Parameters

6.2.1 Parameters from Existing Literature

There are several parameters in our model that are commonly used in the literature and weadopt these values directly from the literature. We use existing estimates for the values of

29

the share of spending on housing, 1−α, the strength of agglomeration forces, γ and the spanof control, β. We set 1− α at 0.24 (Davis and Ortalo-Magné, 2011; Behrens, Duranton andRobert-Nicoud, 2014); γ at 0.05 (Combes and Gobillon, 2015), and β at 0.53 (Buera andShin, 2013). See Table 5 for details.

6.2.2 Estimation of θ

The Frechét distribution parameter θ is related to the income distribution of managers. Fromthe cumulative distribution function of manager’s income in (12), we obtain:

− log[− logG(π)] = θ(1− β) log π + log[(Lmn )γθΦn] + constant (36)

We use the 3-year ACS 2011-2013 data to obtain the high skilled hourly wage distribution.Using the individual hourly wage information, we run an OLS regression with city fixed effect,which absorbs the log[(Lmn )γθΦn] term. The OLS estimation then gives θ(1 − β) = 1.93,implying a value of 4.11 for θ.

6.2.3 Estimation of Fragmentation Cost

Recall that the bilateral fragmentation cost is assumed to take the following functional form:

log τnc = λnc + δd log dnc + δIqnqc. (37)

The key parameter of interest is δI , the elasticity of fragmentation with respect to internetquality. The estimation challenge is that the aggregate cross-city fragmentation cost τnc isnot directly observed. To overcome the challenge, we rely on the gravity equation derivedin (9). We first compute Xnc, the number of occurrences of the joint productions in city cthat originate from city n by multiplying both sides of equation (9) by the total number ofmanagers in the origin city Lmn :

Xnc = LmnTnτ

−θnc w

−βθc

Φn

. (38)

Normalizing by Xnn and utilizing the assumption that τnn = 1, we obtain the following

30

function that links τnc with city-level worker wages and Xnc:

τnc =(wβθc Xnc

wβθn Xnn

)−1/θ

. (39)

Since wn is directly observed our data set, we can calculate τnc using additional informationon cross-city joint productions, i.e., Xnc. We rely on data on multi-locational production tomeasure Xnc. The data is constructed using the proprietary Orbis Database, which reportsownership information for subsidiary plants. We define a headquarter-subsidiary pair if aheadquarter has strictly more than 50% of the ownership of a given subsidiary. Moreover,the database reports the locations of the subsidiary and the headquarter, which allows us tocount the number of headquarter-subsidiary pairs at the city-pair level. Specifically, for eachcity c, we calculate Xnc by counting the number of subsidiaries belonging to headquarterslocated in a given commuting zone n.

Admittedly, these headquarter-subsidiary pairs by no means capture all the cross-cityjoint-production forms, e.g., firms contracting with each other remotely or firm’s domes-tic outsourcing are not included. However, given the data limitation, we view that thisheadquarter-subsidiary as a reasonable starting point to study this question for three rea-sons: First, the headquarter-subsidiary relationship fits the high skilled and the low skilledjoint-production setting well in the theory part; second, this identifies one specific channelthrough which firms can achieve fragmented production, and; third, it helps to establish alower bound for the contribution of the increasing cross-city joint production, enabled byreduction in communications costs, in the spatial skill redistribution across the US cities.

Using the estimates of τnc ∀n 6= c from (39), we can estimate (37) using the followingspecification:

log τnc = χn + ιc − δd log dnc + δIqnqc + ΘHnc + εnc (40)

where χn and ιc are origin and destination fixed effects, respectively, dnc is the distancebetween two cities n and c, qn denotes the internet quality in city n and is measured bylog(1 + internet bandwidthn), and Hnc is a vector of city-pair controls, including dummiesfor being in the same states and shared border.

We estimate Equation (40) via OLS. The coefficients estimates δd and δI are reportedin Table 4. We find that as expected, long geographical distance reduces the number of

31

headquarter-subsidiary pairs. Nevertheless, the high bilateral Internet quality induces moreheadquarter-subsidiary pairs.

Estimates OLSδd 0.264***

(.0026)δI 0.011***

(.0020)Fixed Effects Yes

N 44,203

Table 4: Gravity Equation EstimatesNotes: Robust standard errors in parentheses. Significance levels: * 10%, ** 5%, ***1%.

6.2.4 Housing Supply Hn

Another set of parameters in the model is the exogenous housing supply, which is estimatedusing the information on the city-level housing rents pn and total incomeWn from the IPUMS.To get city-specific housing supply, we use average city-level wage for workers and total incomeof the city, i.e.,

Hn

Hn′= Wn/w

1(1−α)n

Wn′/w1

(1−α)n′

.

We normalize HCZ=100 = 1 and then compute Hn for other cities using the ratio above.18

6.2.5 City Technology Tn

The fifth and final step requires estimating the city-specific technology parameter Tn. Com-bining the definition for the city-level fragmentation potential Φn in (8) and the equilibriumcondition (19), we obtain:

γ

1− β log Lmn

Lmc+ 1θ(1− β) log Tn

Tc+ 1θ(1− β) log

∑k(τnkwβk )−θ∑k(τckwβc )−θ

= log wnwc.

18CZ = 100 denotes a commuting zone with a CZ code 100.

32

We normalize TCZ=100 = 1 and then pick Tn so that the model implied values for log TnTc

exactly match the corresponding values estimated.

Parameter Value Description Moment / Source1− α 0.24 Share of spending on housing Literatureγ 0.05 Knowledge spillover within a city Literatureθ 4.11 Frechét parameter High skilled wage distributionβ 0.53 Span of control Literatureδd 0.264 Distance elasticity of sourcing Implied from gravity estimatesδI 0.011 Internet elasticity of sourcing Implied from gravity estimates

Table 5: Calibrated Model Parameters and Data Targets

6.3 Calibration Results

We next assess the fit of the calibrated model. We first look at the city-specific technologyparameters, Tn. Figure 9 shows the model-generated technology parameters for 1980 (upperpanel) and 2013 (lower panel), respectively. We can see that CZs with greater technologyparameters are concentrated in large cities (e.g., New York, San Francisco and Chicago) andother denser areas along the coasts. Interestingly, while the set of places with greater technol-ogy parameters remain largely unchanged, over time, commuting zones that start with bettertechnology have grown to be even more technologically advanced. This is consistent withour proposed hypothesis that ICT improvement had enabled cities to be more specialized,reinforcing the initial patterns of the comparative advantage of cities.

Figure 10 shows the change in the high skilled share against initial city size. We see a clearpositive relationship between the change in the share of high skilled employment and the sizeof the city. The linear regression gives a coefficient of 0.008 and it is statistically significantat 1% level. This number is slightly higher than that reported in Table 1. Note that the skillshares were not directly targeted by our calibration procedure. The figure shows that thatthe model is able to deliver similar results than the ones observed in the data.

Figure 11 plots the 1980 population, model vs data. These variables in the data were notused in the model’s calibration directly. The figure reveals that the calibrated model overalldoes a good job in matching the population-size distribution, especially at the upper tail. Themodel captures close to 80% of the variation observed in the data in city-size distribution.

33

Figure 9: City-Specific Technology Parameters

Notes: The Upper Panel shows the model generated technology parameter (Tn) in 1980, and the LowerPanel shows the model generated technology parameter (Tn) in 2013.

34

9 10 11 12 13 14 15 16 17−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

chan

ge in

the

high

ski

lled

shar

e

city size

regression line

Figure 10: Simulated Relationship between Skilled Worker Share and City Size

Notes: we first solve our model using the estimated parameters and also the data in 1980 and 2013,respectively. We then calculate the change of the skilled employment share for each city and plot it againstthe model implied city size.

68

1012

1416

Actu

al C

Z Po

pula

tion

in 1

980

6 8 10 12 14 16Simulated CZ Population in 1980

Figure 11: Actual City Size and Simulated City Size in 1980

Notes: we first solve our model using the estimated parameters and data in 1980. We then calculate themodel implied equilibrium city sizes for each CZ and plot them against the actual 1980 population.

35

6.4 Counterfactual Analysis

Equipped with our calibrated model, we perform a counterfactual analysis to quantify howInternet infrastructure shapes the skill redistribution across US cities through cross-city jointproductions. The idea is to simulate the counterfactual change in the skilled employmentshare assuming there was no Internet and compare with the actual data.

To implement this, we first solve the initial equilibrium system of 722 commuting zonesfor 1980 and 2013, taking calibrated parameters in Table 5 and plugging in the actual dataincluding the Internet quality data. Under this scenario, the implied correlation betweenchange in the share of high skilled workers and city size is 0.0077. Next, we simulate acounterfactual scenario in which we remove the effect of internet improvement by setting theInternet quality to 0 for all cities in 2013. Resolving the system, we then compute the changein the skilled employment share ∆skilled share in this counterfactual case without Internet,i.e., qn = 0 for all n. In doing so, we capture other forces that may drive the observed spatialskill redistribution, while removing the effect of Internet improvements. In this counterfac-tual scenario, the correlation between change in the share of high skilled workers and city sizeis reduced to 0.0065 and remains significant at 1% level. The fitted line graphs are shown inFigure 12 for both the simulation based on actual data and the counterfactual scenario with-out internet. These results imply that the improvement in internet infrastructure, throughfacilitating cross-city joint productions, can explain 16% of the skill redistribution across UScities.

36

-.02

0.0

2.0

4C

hang

e in

the

Hig

h Sk

illed

Shar

e

-8 -6 -4 -2Log(Population) in 1980

With internet Without Internet (counterfactual)

Figure 12: City Size and Internet

Notes: we first solve our model using the estimated parameters and also the Internet quality data. Then wesolve our model by making the Internet quality equal 0 and re-solve the model. Then we compare the citysize (log of population) under these two scenarios.

7 Conclusion

In conclusion, this paper documents empirical facts on the changing spatial distribution ofU.S. labor force. During 1980 and 2013, there has been more pronounced segregation of thehigh skilled workers and the low low skilled workers across the US cities. Big cities attractmore skilled workers. This trend is more salient in industries that are easier to fragment.Based on these facts, we develop a model of production fragmentation in a system-of-citiessetting with heterogeneous agents. Our model differentiates from other literature in similarsystem-of-cities setting given our explicit emphasis on production organization structure, andcross-city production team formations. The model reveals the role of falling communicationscost in shaping production fragmentation decisions and generates novel predictions on theskill spatial distribution. We investigate the heterogeneous effect of US Internet infrastructureimprovement on the skill composition of cities of different size, fully consistent with themodel prediction. Finally, our quantitative evaluation of the model using US data echoesthat Internet quality is important for the skill spatial distribution.

37

References

Aarland, Kristin, James C Davis, J Vernon Henderson, and Yukako Ono. 2007.“Spatial organization of firms: The decision to split production and administration.” TheRand Journal of Economics, 38(2): 480–494.

Abraham, Katharine G, and Susan K Taylor. 1996. “Firms’ Use of Outside Contractors:Theory and Evidence.” Journal of Labor Economics, 394–424.

Acemoglu, Daron, and David Autor. 2011. “Skills, tasks and technologies: Implicationsfor employment and earnings.” Handbook of labor economics, 4: 1043–1171.

Allen, Treb, and Costas Arkolakis. 2014. “Trade and the Topography of the SpatialEconomy.” The Quarterly Journal of Economics, 129(3): 1085–1140.

Allen, Treb, Costas Arkolakis, and Yuta Takahashi. 2014. “Universal gravity.” NBERWorking Paper, , (w20787).

Antràs, Pol, Luis Garicano, and Esteban Rossi-Hansberg. 2006. “Offshoring in aKnowledge Economy.” The Quarterly Journal of Economics, 121(1): 31–77.

Arkolakis, Costas, Natalia Ramondo, Andrés Rodríguez-Clare, and StephenYeaple. 2018. “Innovation and Production in the Global Economy.” American EconomicReview, 108(8): 2128–73.

Autor, David H., and David Dorn. 2013. “The Growth of Low-Skill Service Jobs andthe Polarization of the US Labor Market.” American Economic Review, 103(5): 1553–97.

Baum-Snow, Nathaniel, and Ronni Pavan. 2012. “Understanding the city size wagegap.” The Review of economic studies, 79(1): 88–127.

Behrens, Kristian, Gilles Duranton, and Frédéric Robert-Nicoud. 2014. “ProductiveCities: Sorting, Selection, and Agglomeration.” Journal of Political Economy, 122(3): 507– 553.

Buera, Francisco J, and Yongseok Shin. 2013. “Financial frictions and the persistenceof history: A quantitative exploration.” Journal of Political Economy, 121(2): 221–272.

Burstein, Ariel, Gordon Hanson, Lin Tian, and Jonathan Vogel. 2017. “Tradabil-ity and the Labor-Market Impact of Immigration: Theory and Evidence from the U.S.”National Bureau of Economic Research Working Paper 23330.

38

Combes, Pierre-Philippe, and Laurent Gobillon. 2015. “The empirics of agglomerationeconomies.” In Handbook of regional and urban economics. Vol. 5, 247–348. Elsevier.

Costinot, Arnaud, and Jonathan Vogel. 2010. “Matching and Inequality in the WorldEconomy.” Journal of Political Economy, 118(4): 747–786.

Davis, Donald R., and Jonathan I. Dingel. 2012. “A Spatial Knowledge Economy.”National Bureau of Economic Research, Inc NBER Working Papers 18188.

Davis, Donald R., and Jonathan I. Dingel. 2019. “A Spatial Knowledge Economy.”American Economic Review, 109(1): 153–70.

Davis, Morris A, and François Ortalo-Magné. 2011. “Household expenditures, wages,rents.” Review of Economic Dynamics, 14(2): 248–261.

Duranton, Gilles, and Diego Puga. 2004. “Micro-foundations of urban agglomerationeconomies.” In Handbook of Regional and Urban Economics. Vol. 4 of Handbook of Regionaland Urban Economics, , ed. J. V. Henderson and J. F. Thisse, Chapter 48, 2063–2117.Elsevier.

Duranton, Gilles, and Diego Puga. 2005. “From sectoral to functional urban specialisa-tion.” Journal of urban Economics, 57(2): 343–370.

Eaton, Jonathan, and Samuel Kortum. 2002. “Technology, geography, and trade.”Econometrica, 70(5): 1741–1779.

Enrico, Moretti. 2011. “Local labor markets.” Handbook of labor economics, 4: 1237–1313.

Feenstra, Robert C. 1998. “Integration of Trade and Disintegration of Production in theGlobal Economy.” Journal of Economic Perspectives, 12(4): 31–50.

Ferreira, Fernando, Joseph Gyourko, and Joseph Tracy. 2011. “Housing busts andhousehold mobility: An update.” National Bureau of Economic Research.

Fort, Teresa C. 2017. “Technology and production fragmentation: Domestic versus foreignsourcing.” The Review of Economic Studies, 84(2): 650–687.

Garicano, Luis, and Esteban Rossi-Hansberg. 2006. “Organization and Inequality ina Knowledge Economy.” The Quarterly Journal of Economics, 121(4): 1383–1435.

39

Grossman, Gene M., and Esteban Rossi-Hansberg. 2008. “Trading Tasks: A SimpleTheory of Offshoring.” American Economic Review, 98(5): 1978–97.

Henderson, J Vernon, and Yukako Ono. 2008. “Where do manufacturing firms locatetheir headquarters?” Journal of Urban Economics, 63(2): 431–450.

Holmes, Thomas J, and John J Stevens. 2004. “Spatial distribution of economic activ-ities in North America.” Handbook of regional and urban economics, 4: 2797–2843.

Hummels, David, Jun Ishii, and Kei-Mu Yi. 2001. “The nature and growth of verticalspecialization in world trade.” Journal of international Economics, 54(1): 75–96.

Kremer, Michael, and Eric Maskin. 1996. “Wage inequality and segregation by skill.”National bureau of economic research.

Liao, Wen-Chi. 2012. “Inshoring: The geographic fragmentation of production and inequal-ity.” Journal of Urban Economics, 72(1): 1–16.

Lucas, Robert. 1978. “On the size distribution of business firms.” the Bell Journal ofEconomics, 508–523.

Ono, Yukako. 2003. “Outsourcing business services and the role of central administrativeoffices.” Journal of Urban Economics, 53(3): 377–395.

Ono, Yukako. 2007. “Market thickness and outsourcing services.” Regional Science andUrban Economics, 37(2): 220–238.

Ruggles, Steven, Katie Genadek, Ronald Goeken, Josiah Grover, and MatthewSobek. 2015. “Integrated Public Use Microdata Series: Version 6.0 [Machine-readabledatabase].” Minneapolis: University of Minnesota.

Strauss-Kahn, Vanessa, and Xavier Vives. 2009. “Why and where do headquartersmove?” Regional Science and Urban Economics, 39(2): 168–186.

Tian, Lin. 2019. “Division of Labor and Productivity Advantage of Cities: Theory andEvidence from Brazil.” Unpublished Manuscript.

Tolbert, Charles M, and Molly Sizer. 1996. “US commuting zones and labor marketareas: A 1990 update.”

40

A Appendix

A.1 Proofs

Proof for Proposition 1

Proof. Denote Xnc = ancτncw

βc, then

Gnc(x) = Pr(Xnc ≤ x) = Pr(anc ≤ τncwβc x) = e−Tn(τncwβc )−θx−θ .

DefineX = max

cXnc.

ThenGn(x) = Pr(X ≤ x) = ΠN

c=1Gnc(x) = e−Φnx−θ .

The probability that city c provides highest x to n is:

Pr[Xnc ≥ max{xns; s 6= c}] =∫ ∞

0Πs 6=c[Gns(x)]dGnc(x) = Tn(τncwβc )−θ

Φn

.

Proof for Proposition 2

Proof.

Pr(πn ≤ k) = Pr

[β

β1−β (1− β)[f(Lmn )]

11−β max

c{( anc

τncwβc

)1

1−β } ≤ k

]

= Pr[maxcanc ≤ ββ(1− β)1−βτncw

βc k

1−β/[f(Lmn )]]

= e−[f(Lmn )]θΦn[ββ(1−β)1−β ]−θk−θ(1−β).

41

A.2 Additional Figures and Tables

68

1012

1416

ln(p

opul

atio

n in

201

3)

6 8 10 12 14 16ln(population in 1980)

Figure 13: log(population in 2013) against log(population in 1980) across CZs

Notes: Each dot represents a commuting zone. The linear correlation between log(population in 2013) andlog(population in 1980) is 0.99.

42

Dependent variable: change in high skilled employment share(1) (2) (3) (4) (5) (6)

high skilled defined by 67% cutoff 80% cutoff college and abovelog(population in 1980) 0.001 0.002∗∗ 0.005∗∗∗ 0.006∗∗∗ 0.004∗∗∗ 0.004∗∗∗

(0.001) (0.0007) (0.0006) (0.0007) (0.0007) (0.0006)state fixed effect No Yes No Yes No YesObservations 722 722 722 722 722 722R2 0.003 0.298 0.112 0.372 0.119 0.355∗ p < 0.10, ∗∗ p < 0.05, ∗ ∗ ∗ p < 0.01

Table 6: Change in High Skilled Employment Share and City Size: Robustness Checks

Notes: Column (1)-(2) define the high skilled as the occupations whose rank are above 67% of alloccupations in 1980. Column (3)-(4) define the high skilled as the occupations whose rank are above 80% ofall occupations in 1980. Column (5)-(6) define the high skilled as workers who have college education orabove. Column (1), (3) and (5) leave out the state fixed effect and report the robust standard errors.Column (2), (4) and (6) use the state fixed effect and report the standard errors clustered at the state level.

43

A.3 Proof of Proposition 4

Proof. If ∆ is very small, we can do a first order expansion with respect to ∆

1θ(1− β) [∆wβθ − βθ logw −∆w−βθ] + 1

1− β [log T1θ

1 Lmγ1 − log T

1θ

2 Lmγ2 ] = logw

11− α logw = − β

1− β logw +(

γ

1− β + 1)

[logLm1 − logLm2 ]

+ 11− β [log T

1θ

1 − log T1θ

2 ]+(1

θ(1− β)(wβθ − w−βθ) + η

η + ζ(w−βθ − wβθ + 1

x− x)

)∆

where

x =(T1

T2

) 1θ(1−β)

w−β

1−β+βθLmγ

1−β+1

That islogw = γ logLm + log T

1θ

1 − log T1θ

2 + 1θ

∆(wβθ − w−βθ)

(1

1− α + β

1− β

)logw =

(γ

1− β + 1)

logLm + 11− β [log T

1θ

1 − log T1θ

2 ]+(

1θ(1− β)(wβθ − w−βθ) + η

η + ζ(w−βθ − wβθ + 1

x− x)

)∆.

They lead to

(γ + 1− γ

1− α) logLm = α

1− α [log T1θ

1 − log T1θ

2 ] +(

11− α + β

1− β

)1θ

(wβθ − w−βθ)∆

−(

1θ(1− β)(wβθ − w−βθ) + η

η + ζ(w−βθ − wβθ + 1

x− x)

)∆

44

That is

(γ + 1− γ

1− α) logLm = α

1− α [log T1θ

1 − log T1θ

2 ] + α

1− α1θ

(wβθ − w−βθ)∆

− η

η + ζ(w−βθ − wβθ + 1

x− x)∆

We finally arrive at

(γ + 1− γ

1− α) logLm = α

1− α [log T1θ

1 − log T1θ

2 ] + [ α

1− α1θ

+ η

η + ζ](wβθ − w−βθ)∆

+ (x− 1x

)∆

where

x =(T1

T2

) 1θ

(1−β)(1+βθ+ α1−α )

> 1.

and

w =(T1

T2

) 1θ

1γ+1− γ

1−α > 1

It is then easy to see that logw increases and log p increases locally with ICT improvement.

Lp1Lp2

= ηw−11 T1(wβ1 )−θΦ

1θ(1−β)−11 L

m γ1−β+1

1 + ηw−11 T2(wβ1 )−θ∆Φ

1θ(1−β)−12 L

m γ1−β+1

2

ηw−12 T2(wβ2 )−θΦ

1θ(1−β)−12 L

m γ1−β+1

2 + ηw−12 T1(wβ2 )−θ∆Φ

1θ(1−β)−11 L

m γ1−β+1

1

Then

Lp1Lp2

= T11/T22w−1(wβ)−θΦ

1θ(1−β)−112 (Lm1 /Lm2 )

γ1−β+1 + T−θ2 /T−θ2 w−1(wβ)−θ∆

1 + T1/T2∆Φ1

θ(1−β)−112 (Lm1 /Lm2 )

γ1−β+1

lp = T1/T2w−1(wβ)−θΦ

1θ(1−β)−112 (Lm1 /Lm2 )

γ1−β+1 + T2/T2w

−1(wβ)−θ∆

1 + T1/T2∆Φ1

θ(1−β)−112 (Lm1 /Lm2 )

γ1−β+1

lp =

(T1T2

)w−1−βθ

(Φ1Φ2

) 1θ(1−β)−1

Lmγ

1−β+1 + w−1−βθ∆

1 +(T1T2

) (Φ1Φ2

) 1θ(1−β)−1

Lmγ

1−β+1∆

45

When the cross-city communication cost is infinite,

lp =(T1

T2

)w−

11−β

(T1

T2

) 1θ(1−β)−1

lγ

1−β+1m =

(T1

T2

) 1θ

α1−α

γ+1− γ1−α

That gives

log lp =α

1−αγ + 1− γ

1−α[log T

1θ

1 −log T1θ

2 ]−(T1

T2

) 1θ(1−β)

w−β

1−β+βθLmγ

1−β+1 −(T1

T2

)− 1θ(1−β)

wβ

1−β−βθLm−γ

1−β−1

∆

which leads to

log lp =α

1−αγ + 1− γ

1−α[log T

1θ

1 − log T1θ

2 ]−[w1+βθ lp − w−1−βθ l−1

p

]∆

It is clear that ∆ increase will drive out production workers from big city to small city.

A.4 Demand for Production Workers

We derive the labor demand for production workers given by equation (18).

Denote Xnc = ancτncw

βc, then from Proposition 1:

Gnc(x) = Pr(Xnc ≤ x) = Pr(Anc ≤ τncwβc x) = e−Tnc(τncw

βc )−θx−θ

Joint distribution that a manager from city n locates her production team in city c and thatancτncw

βc

= x is:

Pr(argmaxkgnk

τnkwβk

= c ∩ anc

τncwβc

= x) = θTnc(τncwβc )−θx−θ−1e−Φnx−θ

46

Given lnc = β1

1−βw−1c [f(Lmn )]

11−β

[ancτncw

βc

] 11−β

, we have:

Lpnc = β1

1−βφncw−1c [f(Lmn )]

11−βLmn

[∫ ∞0

(θx−θ−1e−Φnx−θ

)x

11−β dx

]= ηw−1

c

(Tnc(τncwβc )−θ

)Φ

1θ(1−β)−1n

[[f(Lmn )]

11−βLmn

]

where η = β1

1−β∫∞

0 y−1

θ(1−β) e−ydy.

47