21/7/15: Preliminary version

Inter-city specialization and trade in functions
versus sectors
Antoine Gervais
James R. Markusen
Anthony J. Venables
Abstract
Our model combines elements of several literatures including the new economic geography,
multinational firms, urban economics, and trade theory. A two-city country trades with thelarger world, and firms and workers within the country are mobile between the two cities.
Firms have two functions or occupations, such as headquarters and plant, which may belocated together (integrated firm) or in separate cities (fragmented firm), with fragmentation
incurring a cost. This element of the model is similar to Duranton and Puga (2005), but fromhere we move in a direction more linked to international trade theory. Industries differ infunction intensities, and cities differ in Ricardian comparative advantage in functions, or thefunctions (not industries) have location-specific agglomeration economies. Our approachcreates a distribution of fragmented and integrated firms across industries and across cities.
We generate a number of economic insights, several of which can be examined empirically.
First, as fragmentation costs fall, a city’s functional/occupational specialization rises and its
sectoral specialization falls. Second, as fragmentation costs fall, there is a fall in thecorrelation across industries between the share of workers employed in industry z who are
doing function i and industry z’s overall (integrated firm) i-function intensity. Put
differently, a city’s industrial mix becomes a weak predictor of its occupational mix,
consistent with A. Markusen and Barbour (2003).
Keywords:
JEL classification: F1, F23, R3
Authors’ Addresses:
Antoine Gervais
Department of Economics
University of Notre Dameagervais@nd.edu

James R. Markusen
Department of Economics
University of Colorado
james.markusen@colorado.edu

Anthony J. Venables
Department of Economics
University of Oxford
tony.venables@economics.ox.ac.uk

01. Introduction.
The production of final products and services typically requires a number of functions to be
performed. Manufactured goods require engineering, finance and marketing; construction
requires architects and lawyers, and so on. There may be spatial (e.g. inter-urban) differencesin the efficiency with which such functions can be supplied so, if the functions are not
perfectly tradable, efficiency differences in functions will translate into a pattern ofcomparative advantage in the final goods that use these functions. This paper investigates theimpact of such differences for firm organisation, city specialisation, trade in goods, and forthe associated gains from trade.
The concept of ‘function’ is fuzzy, depending on how narrowly it is defined. A rather
aggregate level is the distinction between headquarters and production, as developed in someof the literature on foreign direct investment (Markusen 2002) and more recent work in theurban context (Duranton and Puga 2005, Rossi-Hansburg et al. 2009). A much finer level isthat of a ‘task’, often thought of as a narrow stage of production and modelled as a continuum(Grossman and Rossi-Hansburg 2008, 2012, Autor 2013). Alternatively, functions could besynonymous with occupations. Indeed, a common statistical breakdown is to divide a firm’s
workforce into production (or blue-collar) and non-production (or white-collar) workers.
The sort of function we seek to model corresponds to quite broad aggregates, such asengineering, finance, or law. Such functions have several properties. First, most functions
are required in most sectors, though in different proportions, which could be referred to as thefunction intensity of a sector. Second, many large cities appear to have developed quite
broad functional specialisms. London and New York in business services: finance, but also
legal and advertising; the San Francisco area in both hardware and software; Los Angeles in arange of media and creative sectors. Some cities specialise in quite narrow ‘tasks’ but, atleast for large cities, the broader functional concepts seem more relevant. Third, we think offunctions as being associated with labour skills and firm capabilities, and suggest that these
may be the fundamental level at which city comparative advantage is based. Cities develop
the skill set – through learning or the composition of its labour force – that comes to define
what the city is good at. Finally, the different functions of a firm could be located together orgeographically separated. Many workers in London and New York may be in the same
occupation such as finance, accounting, law, or advertising, but they work for (under contractto) firms in different sectors and places.
The fundamental trade-off that we study arises from the facts that the efficiency with whichcities supply functions may vary, and that firms face additional costs if they source functions
from different cities: we call these fragmentation costs, arising if e.g. engineering has toacquired in one city, legal services another, and so on. We develop a simple model to show1

how the interaction between fragmentation costs, the function intensity of different sectors,
and efficiency differences between cities cause firms in some sectors to integrate production
in one place, and in others to fragment it between cities. Firms’ choices have implications forcities’ production structures; to what extent are cities able to specialise in the functions inwhich they are most efficient, and how does this map into the final product (sectoral)
specialisation of cities?
Following from this, we investigate the effects of changes in fragmentation costs (arisingperhaps from communication or transport improvements) on the production structure of cities
and on their size and the real incomes. Real income gains are particularly large if there are
increasing returns to functional concentration in a city, and fragmentation of firms allows
cities to develop their functional specialisms. Changes in the production structure of each city
may also change the production structure of the economy as a whole. How do changes in thecosts of trading functions within an economy shape the external trade of the country?
In order to investigate these questions we develop a model that has elements of economic
geography, the literature on vertical multinationals, urban economics, and external economies
of scale with some novel twists. There are two regions or cities, with identical workers who
are mobile between jobs within and between cities. There are many final products (sectors)
and just two functions, each final product requiring the functions in different proportions.
There is free trade in final products, capturing the idea that the cities under study areembedded in an integrated market. Firms in each sector may perform both functions in one
location, referred to as integrated firms, or one function in each location, referred to as
fragmented firms. However, splitting the production of a good between two locations incurs
a ‘fragmentation cost’. This may be the cost of transporting ‘functions’ between cities, but isbetter thought of as coordination costs and the communication costs of maintaining links with
suppliers in different cities.
The efficiency with which functions are produced is city specific, and we start with thesimplest case in which there are Ricardian differences in the productivity of functions
between cities. This provides a very clean example of how reducing fragmentation costs
causes firms (in some sectors) to fragment, and causes cities to move from sectoral towards
functional specialisation. Sectors with extreme function intensities are more likely to containintegrated firms, concentrating production in the city with the advantage in the function inwhich they are intensive. Sectors which draw more equally on both functions will containfirms that are fragmented, performing each in the city with respective efficiency advantage.
The Ricardian model provides a simple introduction, but functional comparative advantage
is, we think, more likely to arise endogenously from cities’ acquired skills and consequent
increasing returns to scale. Economies of scale are, we assume, external to the firm and2sector, occurring at the city-function level. In this case city specialisation turns out to be a
discontinuous function of fragmentation costs and there is a range of fragmentation costs at
which there are multiple equilibria. This arises because of the interaction between firm’s
location decisions and scale economies: economies of scale large enough to overcome
fragmentation costs are achieved only if a wide range of sectors fragment. If firms in all
sectors are fully integrated neither city has a large enough comparative advantage to induce
fragmentation; but if firms are fragmented then cities are functionally specialised, creating
the scale and productivity differences that support fragmentation. Welfare gains fromreductions in fragmentation costs can be particularly large if they induce spatial
reorganisation and the move from sectoral to functional specialisation.
The questions we pose and the model we develop touches on many strands of international
and urban economics. The division of firms’ activities (at least, HQ and production) has beenstudied in the literature on foreign direct investment (see Markusen 2002). Perhaps closest in
spirit to this paper is the urban model of Duranton and Puga (2005), the focus of which isprecisely the move from sectoral to functional specialisation, although again in the context ofthe division of HQ and production.1 The international trade literature has analysed trade intasks, both in constant returns models (Grossman and Rossi-Hansburg 2008) and underincreasing returns (Grossman and Rossi-Hansburg 2012). As well as its international focus,
this literature works in a framework of many tasks and few final sectors. This does not
capture the ubiquity which, we argued above, distinguishes functions from tasks; ourapproach therefore works with few functions (tasks) and many final sectors. The present
paper also draws on economic geography modelling (Fujita et al. 1999), particularly in its
analysis of the multiplicity of equilibria occurring at intermediate levels of spatial frictions.
Finally, there are literatures on the impact of internal geography on external trade. Uneven
distribution of factors of production within a country is studied by Courant and Deardorff(1992) and following literature (Brakman and van Marrewijk 2013), and the physicalgeography of proximity to ports is studied by Limao and Venables (2002). In the present
paper the internal geography arises from city variation in efficiency in the production offunctions.
We generate a number of economic insights, several of which we can examine empirically.
First, as fragmentation costs fall, a city’s functional specialization rises and its sectoral
specialization falls. A corollary is that a city’s industrial mix becomes a weak predictor of its
functional mix. Interpreting function mix as occupational mix, this is consistent with thefindings of A. Markusen and Barbour (2003). So, for example, a city may be relatively

1

See Rossi-Hansburg et al. (2009) for intra-urban separation of HQ and production.

3

specialized in blue-collar-intensive industries, but the workers in these industries are doing
primarily white-collar functions.
2. The model
The ingredients of the model are locations, focussing on two cities; a primary factor, labour,which is mobile between cities; sectors, which we model as a continuum; and two functions,each using labour and being used as input by sectors. We build the model in stages, initially
focusing on sectors and functions. In section 3 we use this to draw out results on
fragmentation and specialisation, whilst keeping the general equilibrium side of the model inthe background; we are able to do this by making sufficient assumptions to ensure that thetwo cities are symmetric, with the same wages. Section 4 then adds the general equilibriumside of the model enabling analysis of a richer set of possibilities.
The wage rate in city j is wj, j = 1, 2. The functions are labelled A and B, and we assume that
producing one unit of function i in city j requires λij (i = A, B, j = 1, 2) units of labour. We
look at cases where these productivity differences are Ricardian and where they are
endogenous due to increasing returns. Functions are used in the production of freely traded
final goods. There is a continuum of such goods, indexed z ε [0, 1], with price p(z) the same
in both cities. Each final goods sector contains a number of firms each of which produces
one unit of output using as inputs a(z) units of function A and b(z) of B. These input
coefficients are fixed, the same in each city and, for simplicity we assume that firms use nolabour. Internal returns to scale are constant, so setting firm scale at unity is without loss ofgenerality. The input of each function varies across sectors, and we rank sectors such thatlow z sectors are A-intensive, a’(z) < 0, b’(z) > 0.
Since the technology with which functions are combined into final goods, a(z), b(z), is thesame in both cities, urban comparative advantage is determined entirely by the efficiency
with which cities use labour to produce functions, λij. Cities are labelled such that
productivity differences (if any) give city 1 a comparative advantage in function A, i.e.
λA1/λB1 < λA2/λB2. Since low z sectors are A-intensive, city 1 will be attractive (other things
equal) for firms in low z sectors, and city 2 attractive for high z sectors.
Firms in each sector can source functions from either city, but if the two functions come fromdifferent cities then a fragmentation cost is incurred.2 Each firm can therefore operate in oneof three modes, choosing to operate entirely in city 1, entirely in 2, or to locate one function
in city 1 and the other in city 2.3 Firms that produce in a single city are ‘integrated’ and will2

We think of functions as being produced within the organisational boundaries of each firm, although they
could just as well be outsourced and purchased through an arms-length relationship.

3

The assignment of which function to which city will become clear, and does not merit additional4

be labelled by subscript 1, 2 according to city of operation; those operating in both are‘fragmented’ (subscript F). Fragmented firms incur additional cost, T, of operating in two
locations. The profits of a firm in sector z for each of the three production modes are
therefore

1 z   p( z )  a( z )A1  b( z )B1 w1 , F  z   p( z )  a( z )A1w1  b( z )B 2 w2   T ,(1)

 2 z   p( z )  a( z )A2  b( z )B 2 w2 .
The term in square brackets is unit production cost. Thus, a firm in sector z uses a(z) units offunction A and b(z) units of B. The functions use labour, with input per unit output given by
the λij, depending on the city (j = 1, 2) in which the sector performs the function (i = A, B).
Wage costs depend on where the functions are performed, and hence on the sector’s function
intensity and chosen mode.
Firms’ choice of mode partitions sectors into three groups. First is a range of z in whichfirms are integrated and produce both functions in city 1; as we show below, these will be
low z sectors, intensive in function A. Second is a range of sectors in which firms are
fragmented producing function A in city 1 and function B in city 2; if such sectors exist they
will be those with intermediate values of z (i.e. using both functions in similar proportions).
Third are high z (B-intensive) sectors in which firms are integrated and operate only in city 2.
The boundaries between these ranges are denoted z1, z2, and are the sectors for which

different modes of operation are equi-profitable, i.e. 1  z1    F  z1  ,  F  z2    2 z2  .
Using (1), these mode-boundaries are implicitly defined by F  z1   1 z1   b( z1 )w1B1  w2B 2   T  0 ,(2)

 F  z2    2 z2   a( z2 )w2A2  w1A1   T  0 .
The relationship between sectors, functions, and chosen modes of production is illustrated on
figure 1, where the horizontal axis is the range of sectors, z ε [0, 1], and the vertical is inputof each function per firm. This is illustrated for an example in which a( z )  1   (1  2 z ) / 2

and b( z )  1   (1  2 z ) / 2 , so that a(z) + b(z) = 1. The inequalities at the bottom indicate
the relative profitability of operating each mode, with mode-boundaries z1, z2, indicated by
the vertical dashed lines. The shaded area gives total city 1 use (and hence production) offunction B (with output denoted XB1) and function A (output XA1) under the assumption that

notation.

5there is one firm active in each sector. This framework of sectors, functions, and firms
provides the basis for investigating patterns of firm organisation and urban specialisation.

Figure 1: Sectors, functions, firm types and employment
a(z)+b(z)=1
B: b(z)

XB1 = z1[1-γ(1-z1)]/2

XB2 = (1 – z1)(1 + γ z1)/2(1+γ)/2a(z) = [1+γ(1 – 2z)]/2

A: a(z)(1-γ)/2
XA1 = z2[1+γ(1 - z2)]/2

Function
intensity
Sectors, z 

0 Integrated in 1:
π1(z) > πF(z)z1Fragmented:
πF (z) > π2(z)πF (z) > π1(z)XA2 = (1-z2)(1-γz2)/2

Integrated in 2:
π2 (z) > πF(z)1z23. Sectoral and functional specialisation in symmetric equilibria.
We start by looking at the way in which firms’ mode of operation and the consequent
location of sectors and functions depend on technology and fragmentation costs. Throughout
this section we make a number of strong assumptions which make cities and sectors
symmetrical. Function intensity of sectors is linear in z so, as in figure 1,a( z )  1   (1  2 z ) / 2 and b( z )  1   (1  2 z ) / 2 . This form is symmetric, with middle

sector, z = ½, equally intensive in A and B; parameter γ measures the heterogeneity offunction intensities across sectors. The labour input requirements of functions, λij, aredescribed below, and will be constructed to be symmetrical (so city 1’s productivity
advantage in A will be equal to city 2’s advantage in B). Together with the assumption ofsymmetry of cities (developed explicitly in section 4), these conditions imply equality ofwages in each city, w1 = w2, with common value denoted w. These assumptions enable us toderive a number of key results in this section. The full general equilibrium is set out insection 4 and asymmetric cases analysed in section 5.

6

3.1: Ricardian functional advantage.
Ricardian productivity differentials are captured by assuming that the labour input
coefficients λij, (i = A, B, j = 1, 2) are exogenous. City 1 has a productivity advantage infunction A and city 2 has an equal advantage in function B, so we define Δλ ≡ λA2 – λA1 = λB1– λB2 > 0. This supports full symmetry of cities and functions, so equilibrium will havez1  1  z2 , i.e. the mode-boundaries are equi-distant above and below the mid-sector,
z2  1 / 2  1 / 2  z1 (see figure 1). Explicit values for the mode-boundaries, z1, z2, then comefrom eqns. (2),bz1   T / w ,az2   T / w .

We assume that fragmentation costs are incurred in labour, so T = tw.4 Using our specificfunctional forms for a(z), b(z) this givesz1 

1  1  2t 
1  1  ,2     z2 1  1  2t 
1  1  .
2     (3)

Integration to fragmentation: Equations (3) capture the way in which the sourcing offunctions by firms in each sector depends on fragmentation costs, technological differences,and function intensities. If t   / 2 then z1 = z2 = ½; this is the highest value of t at which
any sector fragments; we refer to it as the critical value and denote it t*. For t  t* all firms
are integrated and sectors are partitioned between cities; city 1 has sectors z < ½, i.e. sectors
intensive in function A, and city 2 has sectors z > ½.
If t < t*, fragmented firms emerge, first in sectors that have similar use of both functions, i.e. z
in an interval around ½ and of width z2  z1 

1  2t 1
, wider the smaller is t and the   

larger are productivity differences,  . This and eqns. (3) are illustrated on figure 2, which
has sectors on the vertical axis and fragmentation costs, t, on the horizontal. Thus, at t < t*
the most A-intensive sectors operate with integrated firms in city 1, the most B-intensive areintegrated in city 2, and those with intermediate function intensities are fragmented.

4

In general this could be a combination of labour from both cities. With symmetric cities, the sourcing of thislabour is irrelevant for profits.

7

Figure 2: Sectoral mode of operation.
Sectors,z

Integrated: A and B in 2z2

z1= z2 = 1/2

Fragmented:
A in 1, B in 2

z1

Integrated: A and B in 1tt *= Δλ/2

The figure is constructed with γ = 1 and Δλ = 0.4. The critical value t* is proportional to Δλand, for a given value of t/Δλ the range of fragmented firms is larger the smaller is γ, theparameter that measures the range of function intensities.5
Sectoral to functional specialisation: Preceding paragraphs established where firms in eachsector locate their activities. The dual question is: what activities take place in which cities?
There are two aspects of this; what sectors are present in each city and, more importantly,
what is the output of each function in each city, XAj, XBj?
The number of firms of each mode in sector z is denoted nk(z) ≥ 0, k = 1, 2, F. Only one
mode is active in each range of z, so sectors are active (the number of firms non-zero) in city
1, in both, or in city 2 according as:n1( z) > 0 for z < z1;nF( z) > 0 for z1 < z < z2;

n2( z) > 0 for z > z2.

(4)

Output levels of each function in each city, Xij, depend on demand from firms in each sector
and city, their function intensity and their mode. They are given by

X A1  0 a( z )n1 ( z )  nF ( z )dz ,1

X B1  0 b( z )n1 ( z )dz ,1

X B 2  b( z ) 0 n2 ( z )  nF ( z )dz

X A2  0 a( z )n2 ( z )dx ,

1

(5)

1

5

The figure has γ = 1, this being the special case in which all sectors become fragmented (z1 = 0 and
z2 = 1) at t = 0. If sectors are more similar in function intensity, γ < 1, then all sectors become
fragmented at some positive value of t; if γ > 1 then extreme sectors use only one function (see fig 1).

8

For the remainder of this section we assume that the number of firms is constant and the samefor all sectors, denoted N , so for each z, n1 ( z )  nF ( z )  n2 ( z )  N . The partition of sectors
between cities is given by (3) so this, with (4) and (5) gives the following output levels. (Seefig. 2 above for easy read-off; derivatives come from routine calculation and hold for t ≤ t*).

X A1  N 0 2 a( z )dz  z 2 1   (1  z 2 )N / 2 ,dX A1
N 1   (1  z2 )
0.
2
dt

X B1  N 0 1 b( z )dz  z1 1   (1  z1 )N / 2 ,dX B1
N 1   (1  z1 )
0.
2
dt

z

z

1

X A2  N z a( z )dz  (1  z 2 )(1  z 2 ) N / 2 ,
2

(6)

dX A2
N
 1   (1  z2 )
 0.
dt
2
dX B 2N 1   (1  z1 )
0.
dt
2

1

X B 2  N z b( z )dz  (1  z1 )(1  z1 ) N / 2 ,1

Given the dependence of mode-boundaries {z1, z2} on t (eqn. 3), equations (6) indicate howvarying fragmentation costs changes the pattern of activity in the economy. We summarise
results in proposition 1.
Proposition 1:i) If t  t*   / 2 then z2  z1  1/ 2 :a) Mode: All firms in all sectors are integrated.
b) Sectors: Each sector operates in a single city.
c) Functions: Fraction1  
1   [0.5, 0.75] of each function is produced in the
2  2 city where it has comparative advantage.
ii) If t  t*   / 2 then z2  z1 

1  2t 1 0 : reductions in t bring,   a) Mode: An increase in the range of sectors z2  z1 in which firms are fragmented.
b) Sectors: A decrease in sectoral specialisation, i.e. an increase in the range ofsectors, to which each city contributes at least one function.
c) Functions: An increase in functional specialisation as outputs of functions movefurther in line with cities’ comparative advantage (see derivatives in (6)).
3.2. External economies of scale.
Ricardian efficiency differences might be due to differences in cities’ history or physicalgeography but are exogenous. We now suppose that productivity is endogenous, determinedby the scale of activity of each function in each city. Given the substantial evidence base on9the presence of urban agglomeration economies this case is empirically relevant. It is also
more complex although, since economies of scale are assumed to be external to the firm, we
can keep the description of firms simple, as above.
Labour input coefficients are function and city specific, and are now assumed to be based onan endogenous part deriving from productivity spillovers in the same function and city, as
well as a possible Ricardian component, which we now denote, ΛA1, ΛB1, ΛA2, ΛB2, and   A2   A1   B1   B 2  0 . These productivity spillovers are denoted, sA1, sB1, sA2,
sB2, with parameter σA, σB measuring the impact of spillovers in each function. The
endogenous and Ricardian components of labour input coefficients are additive, giving

A1   A1   As A1 , A2   A2   A s A2 ,

B1   B1   B sB1 ,(7)

B 2   B 2   B sB 2 .

We assume that the spillovers generated by each function in each city are equal to output inthe function/ city pair, so sij  X ij , i = A, B, j = 1, 2. Hence, productivity differentials are,using eqns. (6) in (7),B1  B 2     B N  1 / 2  z11   (1  z1 ) ,(8a)

 A2   A1     A N 1 / 2  z2 1   (1  z2 ) ,(8b)

Thus, if z2 is large a relatively small range of sectors undertake function A in city 2, thisreducing city 2 productivity in A, i.e. raising  A2   A1 . If these spillovers are equally
powerful in both functions, σ ≡ σA = σB > 0, and there is symmetry so w = w1 = w2, then themode-boundaries (2) become, F z1   1 z1   1   (1  2 z1 )B1  B 2 w / 2  tw  0 ,(9a)

 F  z 2    2 z 2   1   (1  2 z 2 ) A2   A1 w / 2  tw  0 .

(9b)

To analyse these relationships, we focus on (9a) and (9b), two equations in z1 and B1  B 2
(the other pair being symmetric). Figure 3 illustrates these equations. The labour-input
relationship (9a, dashed line) is downward sloping as a higher value of z1 increases city 1
output of function B, thereby reducing B1 , the labour input requirement in 1 (and increasingB 2 ; with constant returns and Ricardian differences, this line would be horizontal). The
mode-boundary relationship is also downwards sloping, because higher productivity infunction B will enlarge the set of sectors operating in integrated mode in city 1. The modeboundary depends on fragmentation costs, and is drawn for two values of t. Given labour10

input requirements, a higher t means that more sectors are integrated, hence the curve lies tothe right.
Looking at the mode boundary for the higher value of t, there are three equilibria. The leftmost, labelled S, is ‘stable’, as sectors to the left of this point have profits in integrated mode
higher than in fragmented mode; starting to the left of this point sectors become integrated,
increasing z1 . Equilibrium U is unstable, and the right-most equilibrium is S*, wherez1  1 / 2  z2 , the boundary at which all sectors are integrated.

Figure 3: Productivity and mode boundaries

B1  B 2

Mode-boundary(9b): high t

SLabour -input (9a)U

Mode-boundary(9b): low t

S*

z1
From this figure we can see how varying t changes equilibrium outcomes. At very high t the
curves do not intersect, so the unique equilibrium is at S*, with all sectors integrated. At
somewhat lower levels there are three equilibria, as discussed above. There is a critical value
t** at which points U and S* merge. This is easily found analytically; it is the value of t at
which both (9a) and (9b) hold at z1  1 / 2 , giving t * *    N / 4 / 2 . This reduces to
the Ricardian case if σ = 0, while σ > 0 implies a strictly higher critical point t**. At valuesof t < t** full integration ceases to be an equilibrium, and there is a unique equilibrium valueof z1 . This tracks to the left – fewer sectors integrated – as t falls, possibly reaching theboundary with all sectors fragmented, z1  0 .

11This is illustrated in figure 4, giving mode-boundaries for the fully integrated equilibrium S*
and for the equilibrium with fragmentation, S. Solid lines on the figure are equilibrium
values of {z1, z2}. The appendix gives further details on parameter values at which various
outcomes occur, and results are summarised in proposition 2.

Figure 4: Sectoral mode of operation with increasing returns to scalezz2

Integrated: A and B in 2

z2

S*

t**

Fragmented:
A in 1, B in 2

z1= z2=1/2

S
z1

Integrated: A and B in 1t

Proposition 2:i) If t  t * *    N / 4 / 2 there is an equilibrium in which all sectors areintegrated.
ii) If t  t * * there is a unique equilibrium, in which a range of firms are fragmented.
iii) There is a range of values of t  t * * at which there are multiple equilibria. In thisrange integration of all sectors is an equilibrium, and so too is fragmentation of anintermediate range of sectors.
iv) Increasing returns (σ > 0) means that, should fragmentation occur, the range of sectorsthat are fragmented is wider, at each t and for each ΔΛ, than if σ = 0.

4. General Equilibrium: wages, prices, and industry scale
To this point we have assumed that product prices are constant, that there is a fixed and equal
number of firms in all sectors, and that there is sufficient symmetry for wages to be the same12

in both cities. We now lay out the general equilibrium framework needed to move beyond
these cases.
4.1 Geographical structure and wages.
The single country on which we focus contains three locations. One is a hinterland region,
producing an ‘outside good’ using labour alone under constant returns to scale. This good isnumeraire, and labour productivity in the sector gives fixed wage w0. The other two are cities,
home to sectors z and functions A and B. Labour is perfectly mobile between cities and theoutside region.
City workers face additional urban costs of commuting and high land prices. This means thatthe cost of living may vary across locations, in which case labour mobility implies that theequilibrium wages paid by producers in each city, w1, w2, may differ from w0 and from eachother. Urban costs depend on city size as described by the simplest form of the standard
urban model (the Alonso-Mills-Muth model, Henderson and Thisse 2004). Each household
occupies one unit of land and the rent in city j at distance r from the centre is hj(r). All urban
jobs are in the city centre (CBD), and commuting costs are cj per unit distance. Workers
choose residential location within and between cities so (since final goods prices are the same
everywhere), real wages are equalised when w j  c j r  h j (r )  w0 for all j, z. In a linear cityin which there are K spokes from the CBD, along which people live and commuting takes
place, population is L j  Krj* , where rj* is the edge of the city (length of each spoke). At thecity edge land rent must be zero, so w0  w j  c j r j*  w j  c j L j / K giving the city-size
equationsL1  w1  w0 K / c1 ,L2  w2  w0 K / c 2 .

(11)

It should be noted that Lj denotes both the number of residents and the number of workers in
the city.
These equations simply say that larger cities have to pay higher wages in order to cover thecommuting costs and rents incurred by workers. Finally, we note that rent in each city can be

expressed as, h j ( z )  w j  w0  c j r  c j L j / K  r , so integrating over r and adding over all
spokes, total rent in a city of size Lj is2

H j  c j L j / 2K .

(12)

Thus, while workers’ utility is equalised across all locations, the productivity gap associatedwith w1, w2 > w0 is partly dissipated in commuting costs, with the rest going to recipients ofland rents.
13

4.2

Free entry and trade

We now turn to specification of a full general-equilibrium model in which prices, outputs
numbers of firms, and trade are all endogenous. We present the equations for the spillover case,since that is the more complicated of the two. Most of the notation needed for this is already inplace. We now need have two countries, domestic and foreign. Variablewhich will
denote the “domestic” country’s total production of sector z. In equilibrium, this will equal thesum of domestic purchases of domestic goods,, and foreign purchases of domestic
goods,.

denotes domestic purchases of foreign goods.

The model now becomes larger in terms of dimensions and features a lot of simultaneity. In
math programming language, it is a non-linear complementarity problem, in which corner
solutions (which firm types are active or inactive in which industries in which cities) is a crucial
feature of the model. Because of this, we discretise the number of sectors: in the simulations tofollow model development, there are 51 z sectors. The variables of the model are as follows(other are computed after model solution):
Non-negative variables
labor demand or employment in city iwages in city i
output of function k = (A,B) in city jlabor requirements in function k in city j
total output of sector z (all firm types)
domestic demand for foreign goods
number of firms of type l, 2, F in sector z
price of (domestic) good zWith the dimension of z equal to 51, the model has 318 non-negative variables complementary
to 318 weak inequalities. A strict inequality corresponds to a zero value for the complementary
variable.
First, the supply-demand relationships for labor demand in the two cities are given as follows,
where z denotes complementarity between the inequality and a variable.

Second, wages are given from (11)
14

z

(13)

z

(14)

z

(15)

z

(16)

Third, output levels of the two functions in the two cities are given by
z

(17)

z

(18)

z

(19)

z

(20)

Fourth, the labor input coefficients (inverse productivity) are given by
z

(21)

z

(22)

z

(23)

z

(24)

The number of active firms of each type in each sectors is complementary to a zero-profit
condition, that unit cost is greater than or equal to price. We use a simple formulation of thefragmentation cost:. Note that all inequalities are homogeneous of degree 1 inwages and prices.
z

(25)

z

(26)

z

(27)

Total output of good z is given by the sum the outputs across firm types.

15

z

(28)

The final element is to specify the demand size of the model, which links outputs, prices, and theexternal foreign market. We give the demand functions here in order to complete specificationof the model and then return to the utility and budget constraint which generate these demand
functions in a short appendix.
The domestic country is assumed small as an importer, and so all foreign prices for the z sectors
are given by an exogenous value , common across all sectors. Domestic and foreign goods
within a sector are CES substitutes with an elasticity of substitution ε > 1. Sectoral composites(domestic and foreign varieties) are Cobb-Douglas substitutes. The agricultural good R is treatedas a numeraire. It is additively separable with a constant marginal utility and hence income does
not appear in the demand functions for the Q goods (though we will introduce a demand shifter
later).
The market clearing equation for the domestic good z is that supply equal the sum of domestic
and foreign demand. αd and αf are “short hand” scaling parameters for domestic and foreign, that
could depend on the relative market sizes for example (see appendix). θd and θf are the weights
on the domestic and foreign varieties in the nest for each sector z.

z

(29)

Domestic demand for foreign goods is not needed to solve the core model, but is needed forwelfare calculations after solution. These are given byz(30)

As noted above, the core model is then 318 weak inequalities complementary with 318 nonnegative unknowns.

4.3

Symmetric Ricardian and spillovers cases in general equilibrium

Figures A1 to A7 present simulation results that develop economic implications of the model.
Figure A1 presents the symmetric Ricardian case, with fragmentation costs t on the horizontal
axis. Each column of the figure is a solution to the model for that value of t, as will be the casein the following figures (the jagged line is a consequence of the discreteness of sectors). The
results naturally qualitatively resemble Figure 2 earlier in the paper. With all firms integrated,the middle sector (there is an odd number of sectors, 51) is produced in both countries.
Figure A2 shows further results for this case in four panels. The upper left panel gives
Herfindahl concentration indices for sector concentration and function concentration for eachlevel of fragmentation costs. The sector concentration is the sum over sectors of the squared
16

share of industry output located in city 1 plus the squared share in city 2 divided by the numberof sectors (a normalization such that the index equals 1 if each sector produces only in one city).
Output of the fragmented sectors is divided according to the task intensities of those sectors.
The function concentration index is the sum over tasks and cities of the share of task i producedin city j squared, divided by 2 (a normalization such that the index equals 1 if each task is
produced in only one city). The upper left panel illustrated a principal result of the paper, thatcities become more specialized in functions and less in sectors as fragmentation costs fall.
The upper right panel of Figure A2 illustrated an effect which was not discussed in previous
sections. The fall in fragmentation costs improves the competitiveness of the urban(manufacturing and services) sectors relative to the rural (agricultural) good. With trade balancein urban sectors calibrated to zero at zero fragmentation costs, the trade balance with the foreign
world is negatively related to fragmentation costs. Ease of internal transport andcommunications is a source of comparative advantage.
The bottom left panel of Figure A2 graphs the producer wage and welfare (recall all workers
earn a wage of w0 after commuting costs and land rent). Note from equations (15) and (16) thatthe producer wage is proportional to urban population or city size. Thus the very flat producer
wage shown in the bottom left of Figure A2 indicates that a lowering a fragmentation costs
doesn’t have a big effect on city size: increased outputs depress product prices some and so fromthe free-entry conditions, producer wages (city populations) don’t change much. The increase in
welfare as fragmentation costs fall is small. Part of potential welfare gains is dissipated byfalling prices (worsening terms of trade) due to the increased domestic productivity. Average Q
prices are 2.5% lower with full fragmentation than under fully integrated production. This fall in
prices also holds down urbanization (producer wages and employment) as fragmentation costs
fall.
The bottom right-hand panel of Figure A2 relates to the findings of A. Markusen and Barbour(2003) mentioned earlier. The graph shows results (arbitrarily) for city 1 and function A.
Across sectors, we use the results to get the “function intensity” of actual employment; that is,what is function A’s share of employment in sector z. Then the column of these employment
intensities are correlated with the function intensity a(z) of integrated production for each sector.
So, for example, if all sectors are fragmented, then the employment share of function A is 1.0 in
all sectors in city 1, and hence the correlation of these with the overall A-function intensity of
sector A is zero. If all sectors are integrated, the employment intensity is the same as the sector
function intensity for all sectors in which there is employment. Since there is no employment insome sectors, the correlation when all sectors are integrated is somewhat less than one.
This correlation in the bottom right-hand panel of Figure A3 is something that we can examinein the data. The theoretical results indicate that as fragmentation costs fall, the function(occupational) specialization of cities becomes less correlated with sectoral (industrial)
specialization.
Turning to the spillovers case, Figure A3a shows results confirming those in Figure 3 earlier.
17

There is a region of multiple equilibria: one in which all sectors are integrated and one in whichsome (middle) sectors are fragmented. Results corresponding to those in Figure A2 for the
Ricardian case are qualitatively the same as for the Ricardian case, and thus we won’t show themhere.
One thing that is qualitatively different between the Ricardian and spillovers cases is the effectof increasing demand (increases in αd αf in (29 and (30)) on the equilibrium regime. In the
Ricardian case in which the λ’s are constants, a symmetric situation (w1 = w2) means that theboundaries between the integrated and fragmented sectors do not depend on demand (also true inthe partial-equilibrium case as seen in (3)).
However, in (7) and here in (21)-(24) we see that increases in total market demand will affect the
λ’s and hence will affect regime boundaries in the spillovers case. Figure A3b shows the effecton the regime boundaries following a 50 percent increase in αd and αf. For middle levels of t,
additional sectors will now fragment as shown, which implies increases function specializationand lower sectoral specialization for a given level of fragmentation costs.
Although the level of demand does not affect the integration / specialization pattern in the
Ricardian case, an increase in demand does lead to large cities in both the Ricardian andspillovers cases. These results are shown in Figure A4. Higher demand shift up the employment/ city size curves as shown. So urbanization follows from higher demand. Although we have notmodeled income elasticities or demand here, we can think of this as a parable for a world inwhich the urban sectors have a high income-elasticity of demand such that rising per-capita
incomes (for whatever reason) shift demand toward the income-elastic urban goods, thereby
increasing urbanization.

4.4

Asymmetric cases

Figures A5 and A6 consider some asymmetry between the sectors/cities. Figure A5 assumes that
. That is, city 1 has a comparative and absolute advantage in function A,
while city 2 has a comparative advantage in function B, but no absolute advantage. Forintermediate or high levels of fragmentation costs, the result in Figure A5 is that city 1 will havea larger range of integrated industries. The intuition follows from a simple argument bycontradiction. Suppose that the solution was symmetric across cities. Then if sector (1-z) (z >0.5) is just breaking even in city 2, there would be positive profits for sector z in city 1.
Figure A6 shows a similar result for the spillovers case: here only function A has spillovers, but
in both cities (in contrast to the Ricardian case where only λA is smaller in city 1 only). In
equilibrium however, the spillovers case is similar: city 1 will have an a comparative andabsolute advantage in function A, while city 2 has a comparative but not absolute advantage infunction B.
These results show up as differences in city size/employment (which in turn translate into
producer wages), shown in the right-hand panels of Figures A5 and A6. The city size difference18

is large when all industries are integrated and small when all are fragmented (though largest in
the middle for the spillovers case). Again, the intuition follows from a simple argument bycontradiction. If city sizes (employment) were the same, then producer wages would be thesame, in which case there must be positive profit opportunities in city 1 and/or losses incurred incity 2.
The convergence in city sizes as fragmentation costs become small seems to be in large part aterms-of-trade effect: as fragmentation costs fall, the relative prices of goods with low sector
indices (located in city 1) fall a lot more in general equilibrium than the prices of the high index
goods. An alternative way to think about this is that the high productivity of city 1 workers in
the A function means that less workers are required to produce those tasks at given output prices
and hence city 1's employment falls some in response to that increased productivity.

4.5

Multi-function example

Figure A7 presents results for a multi-function case, the analysis of which is very preliminary asof this draft of the paper. The simulation is for a symmetric Ricardian case with 12 sectors andsix functions. The difficulty with a multi-function case is that there is no unambiguous way tothink about function intensities of sectors in a multi-function case (an old problem characterizing
the n-good, n-factor Heckscher-Ohlin analysis of the 1970s). Factor intensity is inherently abinary concept. What we have done in this simulation is shown in the matrix of function
intensities in the upper right-hand panel of Figure A7. Middle sectors 6 and 7 have relatively
even function intensities across functions. As we move up or down the list of industries, factor
intensities move toward being more uneven, with low-index sectors being intensive in functions
1-3 and high-index sectors being intensive in functions 4-6. This is obviously a very simple casewhere intensity rankings are pretty clear. The lower right-hand panel of A7 show the matrix of
λs, with city 1 having a comparative advantage in low function index functions. Combining
these two right-hand panels, city 1 will have an unambiguous comparative advantage in low
index sectors.
Another problem is how to define fragmented firms. There are potentially a great many firmtypes, where “type” is defined by the number of functions in city 1 with the rest in city 2. In
addition, some potential firm types might do each of two functions in separate cities and some inboth cities. The count of potential firm types is large. This problem is quite familiar to those of
us working in the multinationals’ literature. What we have done in the simulations for Figure A7
is simply assume a single fragmented firm type, with has functions 1-3 in city 1 and 4-6 in city 2.
With only three firm types, the top left panel of Figure A7 looks similar to earlier results. The
advantage of multiple functions is that we can now talk more meaningfully about city function
concentration across functions (with only two functions, the concentration of A and B acrosscities is the same in the symmetric case). In the middle panel of Figure A7, we show EllisonGlaeser indices for the employment concentration of each industry across cities. Similar to ourearlier results on the Herfindahl index of sector concentration, here we see results by sector andsee that sector concentration falls with falling fragmentation costs, and falls most markedly for19the middle sectors. Low fragmentation costs produce heterogeneous concentration figures acrossindustries while they are homogeneous with high fragmentation costs.
The bottom left panel of Figure A7 show Ellison-Glaeser indices for employment concentration
by function. As per our earlier results, these increase as fragmentation costs fall and increase
more for the “fringe” functions. The indices of function specialization become morehomogeneous as fragmentation costs fall.
This analysis of a multi-function case is extremely preliminary in this draft as noted earlier.
However, any more complete analysis is going to run into the difficulties here in defining
function intensities and in somehow limiting the number of firm types.

5.

Toward empirical analysis

We have just begun an empirical investigation into the issues raised in this paper as of thewriting of this draft. In order to show where we are going and to solicit comments, we haveattached some data plots to the end of the paper. We will try to work with two details data
sources.
First, data on the concentration/dispersion of employment by industry (sector) has been obtained
from the US Census Bureau’s Country Business Patterns. This gives us 965 5-digit NAICS
industry codes. For each of these, we calculate an Ellison-Glaeser employment concentrationindex for each industry across states. These are plotted in Figure B1, grouped according to thebroad 1-digit classification. The plot shows a lot of dispersion within the broad catagories.
Second, data on the concentration/dispersion of employment by occupation (function) has beenobtained from the Bureau of Labor Statistics’ Occupational Employment Statistics. This gives
568 catagories. For each of these, we calculate an Ellison-Glaeser employment concentrationindex for each occupation across states. These are plotted in Figure B2, grouped by broadclassification. The plot shows a lot of dispersion within the broad categories as in the case of theindustry employment concentration indices.
Figures B3 and B4 give a time-series picture, plotting 2010 figures again 2000 values.
Somewhat to our disappointment at this early stage, there is no strong evidence of increased
dispersion or concentration over this time period. Obviously, a lot more work needs to be done.

6.

Summary and conclusions

Under construction. Comments and suggestions most welcome.

20

Appendix 2: specification of utility and income.
The specification of utility (welfare) is quite standard for trade models. The Q goods are a twolevel ces nest. Domestic and foreign varieties for any z sector have an elasticity of substitutionof ε > 1 whereas goods from different z sectors are Cobb-Douglas substitutes. R is the
agricultural good, giving a standard quasi-linear utility function(A1)
where β is a scaling parameter. Income (Y) is given the sum of wages (net of commuting costs
and rents = ) for all rural and urban workers ( ) plus land rents H1 and H2 from (12).
(A2)
The domestic economy’s budget constraint is that Y is spend on R (used as numeraire) plus
domestic and foreign urban goods.
(A3)(A3) can be substituted into (A1) to replace R.

(A4)
Maximization of (A4) with respect to the Q’s (and equivalently for foreign) yields the demand
functions in the body of the paper, which do not depend directly on Y as is the usual result inquasi-linear preferences. Domestic demand for domestic good z for example is:(A5)
where αd is a scaling parameter that is increasing in β (βd which could differ from the foreign βf).
Suppose θd = θf = 0.5 and all. Then α = 2 in the demand functions implies β = 21/ε
and Qij = 1. Parameters αd and αf in the demand functions in section 2 are increasing in the β ofthe domestic or foreign economy, and increases in the α’s or β’s can represent increases in or
differences in market size.1
1 1

My algebra indicates that the relationship between the β in (A1) and the α in the demand functions above

are related by. Because of the concavity of the log formulation of utility, β must more than double
to double market demand (α) at constant prices.

22

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Grossman, G. M. and E. Rossi-Hansberg (2008) ‘Trading Tasks: A Simple Theory of Offshoring’, American Economic Review, 98, 1978-1997.
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Michaels, G., F. Rauch and S. Redding, (2013) ‘Task Specialization in U.S. Cities from 18802000’, Working Paper 656, Economics Dept, Oxford University.
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Patterns,’ International Economic Review, 50(1), 143-186.

24

Figure A1: Symmetric Ricardian Case
Ricardian comparative advantage, free entry, no spillovers

Sectors

Integrated in 2

fragmented
middle (integrated)industry produces inboth countries

0.125

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fragmentation cost t

Figure A2:

Symmetric Ricardian Case(fragmentation cost t on horizontal axes)

1.1

0

1

Ͳ0.01
Ͳ0.02

sectorconcentration

0.9
0.8

trade balance inmanufacturing /services

Ͳ0.03
Ͳ0.04

0.7

Ͳ0.05

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functionconcentration

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indexed to zeroat full fragmentaion

Ͳ0.08
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correlation acrosssectors: function
employment withfunction intensity of
sectors

0.4

welfare1.01

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producer
wage

0

0.99

Ͳ0.2
0

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0

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0.15

Figure A3a: Symmetric Spillovers Case
Figure A3b: Symmetric Spillovers Case
Spillovers, free entry, no Ricardian comparative advantage
Spillovers, free entry, no Ricardian comparative advantage
Base Case: region of multiple equilbria
Base case, + 50% increase in market size

Integrated in 2

Sectors

Integrated
in 2base casemultiple equilibria(1) all firms integrated(2) some sectors fragmented

increase market
size 50%

fragmented

fragmented

fragmentation cost t

fragmentation cost t

Figure A4: fragmentation cost versus market size and city employment (bothcities have same employment)

spillover case city employment (size)

Ricardian case city employment (size)

1.35

1.35

1.3

1.3

1.25

1.25

50% larger market size

1.2

50% larger
market size

1.2

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1.15

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1.05

1

1

0.95

Ricardian base case

0.95

0.9

spillovers base case

0.9

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0

0.02

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fragmentation cost t

0.08

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0

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fragmentation cost t

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Integrated in 1
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in 1

Figure A5: Asymmetric Ricardian Case
City 1: comparative and absolute advantage in function A

1.4

1.35

Integrated in 2
Sectors

1.3

employment incity 1

1.25

fragmented

1.2

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employment
in city 2

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1

Integrated in 1

integrated firms in city 1
and fragmented firms

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0.108

0.105

0.103

0.100

0.098

0.095

0.093

0.090

0.088

0.085

0.083

0.080

0.078

0.075

0.073

0.070

0.068

0.065

0.063

0.060

0.058

0.055

0.053

0.050

0.048

0.045

0.043

0.040

0.038

0.035

0.033

0.030

0.028

0.025

0.023

0.020

0.018

0.015

0.013

0.010

0.008

0.005

0.003

0.000

fragmentation cost t

fragmentation cost t

Figure A6: Asymmetric Spillovers Case
spillovers in function A only

Integrated in 2

1.2

Sectors

1.15

employment incity 1

1.1

fragmented
1.05

1employment
in city 2

Integrated in 1
0.95

fragmentation cost t

0.9

0.125

0.123

0.120

0.118

0.115

0.113

0.110

0.108

0.105

0.103

0.100

0.098

0.095

0.093

0

0.090

0.088

0.085

0.083

0.080

0.078

0.075

0.073

0.070

0.068

0.065

0.063

0.060

0.058

0.055

0.053

0.050

0.048

0.045

0.043

0.040

0.038

0.035

0.033

0.030

0.028

0.025

0.023

0.020

0.018

0.015

0.013

0.010

0.008

0.005

0.003

0.000

integrated firms in city 1
and fragmented firms

0.01

0.02

0.03

fragmentation cost t

0.04

0.05

0.06

FigureA7:MultiͲtaskRicardianModel
12sectors

6functions
Matrixoffunctionintensities
K1

K2

K3

K4

K5

K6

I12

I12

0

0

0

0.333

0.333

0.333

I11

I11

0.083

0

0

0.333

0.333

0.250

I10

0.083

0.083

0

0.333

0.250

0.250

I9

0.083

0.083

0.083

0.250

0.250

0.250

I8

0.167

0.083

0.083

0.250

0.250

0.167

I7

0.167

0.167

0.083

0.250

0.167

0.167

I6

0.167

0.167

0.250

0.083

0.167

0.167

I5

I5

0.167

0.250

0.250

0.083

0.083

0.167

I4

I4

0.250

0.250

0.250

0.083

0.083

0.083

I3

0.250

0.250

0.333

0

0.083

0.083

I2

0.250

0.333

0.333

0

0

0.083

I1

0.333

0.333

0.333

0

0

0

integrated in 2

I10
I9
I8
I7

fragmented

I6

I3

integrated in 1

I2
I1
TCOST

0.015 0.017

0.02 0.022 0.024 0.026 0.029 0.031 0.033 0.035 0.038

0.04 0.042 0.044 0.047 0.049 0.051 0.053 0.056 0.058

EllisonͲGlaesertypeindexforsectorconcentation

MatrixofLambdas
K1

K2

K3

K4

K5

K6

I12

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

LAM1

0.9

0.94

0.98

1.02

1.06

1.1

I11

0.706

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

LAM2

1.1

1.06

1.02

0.98

0.94

0.9

I10

0.454 0.454 0.454 0.454 0.454 0.454

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

I9

0.250 0.250 0.250 0.250 0.250 0.250 0.250

1

1

1

1

1

1

I8

0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116

1

1

1

1

1

I7

0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030

1I6

0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030

1I5

0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116 0.116

1I4

0.250 0.250 0.250 0.250 0.250 0.250 0.250

1

1

1

1

1

1

1

1

1

1

1

1

1

I3

0.454 0.454 0.454 0.454 0.454 0.454

1

1

1

1

1

1

1

1

1

1

1

1

1

1

I2

0.706

1

1

1

1

1

1

1

1

1

1

1

1

1

1

I1
TCOST

1
1
0.015 0.017

1

1

1

1

1

1
1
1
1
1
1
1
1
1
0.02 0.022 0.024 0.026 0.029 0.031 0.033 0.035 0.038

1

1

1

1

1
1
1
1
1
1
1
1
1
0.04 0.042 0.044 0.047 0.049 0.051 0.053 0.056 0.058

EllisonͲGlaesertypeindexforfunctionconcentation
K1

1 0.810 0.809 0.808 0.807 0.806 0.640 0.496 0.495 0.495 0.494 0.493 0.492 0.491 0.490 0.271 0.271 0.270 0.270 0.123

K2

1

1

1

1

1

1 0.824 0.668 0.667 0.667 0.666 0.665 0.665 0.664 0.664

K3

1

1

1

1

1

1

1 0.838 0.838 0.837 0.837 0.837 0.836 0.836 0.836 0.693 0.693 0.692 0.692 0.567

K4

1

1

1

1

1

1

1 0.838 0.838 0.837 0.837 0.837 0.836 0.836 0.836 0.693 0.693 0.692 0.692 0.567

K5

1

1

1

1

1

1 0.824 0.668 0.667 0.667 0.666 0.665 0.665 0.664 0.664

K6
TCOST

0.53 0.529 0.529 0.529 0.315

0.53 0.529 0.529 0.529 0.315

1 0.810 0.809 0.808 0.807 0.806 0.640 0.496 0.495 0.495 0.494 0.493 0.492 0.491 0.490 0.271 0.271 0.270 0.270 0.123
0.015 0.017 0.02 0.022 0.024 0.026 0.029 0.031 0.033 0.035 0.038 0.04 0.042 0.044 0.047 0.049 0.051 0.053 0.056 0.058

Memoryjog:lambdasarelaborinputrequirements
inverseofproductivity
City1hascomparativeadvantageinlowKfunctions

FigureB1:EllisonͲGlaeser indicesofemploymentconcentrationbyindustry(sector)
965industries,groupedbybroadclassification

Naics Sector

mean

1
Farming
0.110
2Mining
0.110
3Manufacturing 0.073
4WholesaleRetail 0.030
5BusinessServices 0.039
6ͲͲ8 PersonalServices 0.016
Total|0.053

sd

numberof
industries

0.088
0.180
0.086
0.063
0.071
0.030

15
72
365
189
180
144

0.090

965

Sourcedata:USCensusBureau
CountyBusinessPatterns
ResultsarederivedfromStatelevelinformationonproductionbyindustry
WhenEGindexishighthereisgeographicconcentration
Someindustrieswithineachbroadsectorareconcentratedandothersdispersed
Manufacturingismoreconcentratedonaverage

FigureB2:EllisonͲGlaeser indicesofemploymentconcentrationbyoccupation(function)
586occupations,groupedbybroadclassification

OCC1

mean
1 Management,Business,andFinancialOccupa0.009
2 Computer,Engineering,andScienceOccupat 0.049
3 Education,Legal,CommunityService,Arts,an0.022
4 HealthcarePractitionersandTechnicalOccup0.006
5 ServiceOccupations
0.019
6 SalesandRelatedOccupations
0.015
7 OfficeandAdministrativeSupportOccupation0.007
8 Farming,Fishing,andForestryOccupations 0.068
9 ConstructionandExtractionOccupations 0.057
10 Installation,Maintenance,andRepairOccupa0.025
11 ProductionOccupations
0.034

No.ofsdfunctions
0.016
48
0.090
70
0.044 102
0.006
38
0.032
81
0.027
20
0.012
52
0.071
11
0.130
53
0.048
48
0.048
63

12 TransportationandMaterialMovingOccupations
13 MilitarySpecificOccupations
Total 0.027 0.062 586

ThesearederivedfromStatelevelinformationonemploymentbyfunction
WhenEGindexishighthereisgeographicconcentration
Health,salesandadministrationfunctionsarelessconcentratedgeogrpahicallySomefunctionswithineachbroadcategoryareconcentratedandothersdispersed

Sourcedata:BureauofLaborStatistics,
OccuptionalEmploymentStatisitics

FigureB3:Change inEllisonͲGlaeser indicesofemployment
concentrationbyindustry(sector),2000Ͳ2010

FigureB4:Change inEllisonͲGlaeser indicesofemployment
concentrationbyoccupation(function),2000Ͳ2010