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Working Paper Series
Firm Fragmentation and Urban Patterns
WP 05-03
Esteban Rossi-Hansberg
Stanford University
Pierre-Daniel Sarte
Federal Reserve Bank of Richmond
Raymond Owens III
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Firm Fragmentation and Urban Patterns¤
Esteban Rossi-Hansberg
Stanford University
Pierre-Daniel Sarte and Raymond Owens IIIy
Federal Reserve Bank of Richmond Working Paper 05-03
June 5, 2005
Abstract
We document several empirical regularities regarding the evolution of urban structure in the largest U.S. metropolitan areas over the period 1980-1990. These regularities relate to changes in resident population, employment, occupations, as well as the
number and size of establishments in di¤erent sections of the metropolitan area. We
then propose a theory of urban structure that emphasizes the location and integration
decisions of …rms. In particular, …rms can decide to locate their headquarters and
operation plants in di¤erent regions of the city. Given that cities experienced positive
population growth throughout the 1980s, we show that our theory accounts for the
diverse facts documented in the paper.
JEL: R12, R14
Keywords: Population Growth, City Structure, Multiple Plants, Firm Integration
¤
A previous draft of the paper circulated under the title “Urban Growth and the Location of Economic
Activity in Cities”.
y
We thank Glenn Ellison and seminar participants at the NBER Urban Economics Conference and
Columbia University for useful comments. We also thank Matthew Harris and Jon Petersen for excellent
research assistance. The views expressed in this paper are solely those of the authors and do not necessarily
represent those of the Federal Reserve Bank of Richmond or the Federal Reserve System.
1
1. INTRODUCTION
The internal structure of U.S. metropolitan areas has evolved dramatically over the last
three decades. This evolution exhibits striking patterns that hold for a wide range of cities.
If one divides metropolitan areas into a center county and edge counties, employment and
residential population have increased both at the center and at the edge. However, over this
period, we also observe an important increase in the share of city residents, employment, and
establishments at the edge. This shift in economic activity to the edge of the city is more
pronounced for non-management occupations than for managers. In addition, the size of
establishments decreased in both areas throughout this period. The …rst part of this paper
is devoted to documenting these changes in U.S. cities.
What accounts for the evolution of U.S. urban structure over this period? Much of the
urban literature attributes the migration of residents to the edge to decreases in transport
costs. Explanations of this type, however, are generally not consistent with the migration of
workers and …rms to the edge.1 Furthermore, there exists a more fundamental problem with
all available explanations for subsets of these phenomena. Speci…cally, such explanations rely
on mechanisms that decrease agglomeration forces at the center thereby explaining how the
share of economic activity can increase at the periphery, but not the simultaneous increase
in the level of economic activity at the center. These theories are also silent on the issues of
functional (management versus non-management) and establishment shifts.2
This paper proposes a theory aimed at addressing the full set of facts we have just described. The key concept we emphasize relates to …rms’ ability to break down their production process into headquarters and production plants, where either can locate in di¤erent
sections of the city. Given this margin, we show that increases in population lead to changes
in organizational structure such that a larger proportion of …rms choose not to integrate their
operations. In particular, standard agglomeration forces motivate …rms to keep only those
1
See Anas, Arnott and Small (1998), and Glaeser and Kahn (2003), for a general review of this literature,
and Burch…eld et. al. 2004 for a recent empirical study of urban sprawl in the US.
2
In theories developed by Fujita and Ogawa (1982), or Lucas and Rossi-Hansberg (2002), a decline in
transport costs tends to disentangle the location of business and residential areas and can lead to employment
concentration at the center. Because these theories incorporate a richer spatial dimension, the implications
of commuting costs, or changes in externality parameters, depend on exactly how one de…nes the city center.
These theories do not incorporate occupational choices or …rms’ integration decisions.
2
workers at the center who bene…t from interactions in downtown locations (i.e. those arising
from knowledge spillovers or, more generally, production externalities). Consequently, as
city population increases, employment rises at the center but this rise is driven primarily by
increases in managerial population and, therefore, establishments. In addition, since each
manager at a headquarter supervises several workers, and the production plants of these new
…rms are located more remotely, employment growth is even more pronounced at the periphery. These changes immediately translate into a decline in the share and a simultaneous
increase in the level of employment in the central region of the city.
Because land rents are lower nearer the city’s outskirts, population growth implies that
more …rms will integrate their operations away from the center. Therefore, increases in
population also lead to an increase in managerial employment at the periphery, reenforcing
the decline in the share of center employment. Ultimately, however, the combined set of
changes resulting from a rise in city population implies a concentration of managers at
the center. Furthermore, consistent with the data, rising urban population also leads to a
decrease in the share of establishments at the center, and a decrease in establishment sizes
as more establishments become non-integrated …rms.
One interpretation of the theory we present, and the empirical evidence more broadly, is
that with …rms sending their larger and more routine operations to the periphery, city centers
are steadily becoming management or administrative hubs. Examples of …rms breaking
up their operations geographically within a given metropolitan area are ubiquitous across
industries. For instance, the Federal Reserve Bank of New York moved its cash and check
processing center to neighboring East Rutherford, NJ in 1992; the Washington Post moved
its printing operations away from its headquarters in downtown Washington to neighboring
Spring…eld, VA in 1999; the tire manufacturer Michelin, headquartered in Greenville, SC,
located its rubber production operations in nearby Anderson County in 2000; and Home
Depot is currently moving a distribution center to McDonough Georgia outside Atlanta, the
location of its corporate headquarters.
In a related paper, Chatterjee and Carlino (2001) seek to explain systems of cities and
argue that the deconcentration of U.S. metropolitan employment stems from an increase
in aggregate employment. The theory they present is one where an aggregate increase in
3
employment raises densities faster in small metropolitan areas than in larger ones, since
large metropolitan areas already have high employment densities and cannot accommodate
the new workers cheaply enough. Our paper shares with Chatterjee and Carlino (2001) the
focus on population growth as the main engine driving the structural change of U.S. cities. In
contrast to their work, however, we model a representative city, rather than the interaction
between cities, and focus on its internal structure. That is, we study the spatial allocation
of employment within a city. We also analyze the location decisions of establishments and
agents in di¤erent occupations. The new set of facts we uncover leads us to emphasize …rms’
integration decisions, which we argue can rationalize observed changes in city structure as a
result of population growth alone.
Along a di¤erent line of work, Duranton and Puga (2004) argue that cities have moved
from being sectorally specialized to becoming functionally specialized. They contend that
decreases in the cost of communication between headquarters and plants have lead to the
location of headquarters in cities and the location of production plants in smaller towns. Our
view of the changes in city structure shares many elements with their work. In particular,
Duranton and Puga (2004) also model explicitly the …rm’s decision to integrate its headquarters and production plants. In their view, this integration decision has implications across
metropolitan areas. In fact, we argue that …rms’ ability to separate headquarters and plants
is also key in explaining changes in the internal structure of cities. Our paper di¤ers from
Duranton and Puga (2004) in that we do not view changes in communication technology as
the force underlying changes in urban structure, but instead show that the latter changes
emerge simply from rising population. Further, our framework has implications for the share
of establishments located in di¤erent sections of the city that are consistent with the data.
Davis and Henderson (2004) provide evidence that complements our …ndings. They observe
that …rms take advantage of services and production externalities (which decline with distance) at the city’s business sectors by locating their headquarters at the center and their
operation plants elsewhere in the city.
The rest of the paper is organized as follows. Section 2 presents our data organized in nine
di¤erent stylized facts. Section 3 presents a simple urban framework that incorporates the
…rm’s integration decision. Section 4 shows that increases in population lead to changes in
4
city structure consistent with the diverse stylized facts we present, and Section 5 concludes.
2. SOME FACTS ON THE EVOLUTION OF CITY STRUCTURE
This section documents a set of regularities in the evolution of city structure that we care
to rationalize with the simple theory proposed in this paper. We document these facts for
the decade spanning the 1980s, although most of the empirical regularities hold from 1970
to 2000. The reason is that for some of these regularities, in particular the ones that involve
the location of agents with di¤erent occupations, we do not have data covering a longer
period. Thus, we chose to homogenize the time period and document our stylized facts over
the same decade.
Given our focus on the structure of cities, we separate the city into two locations: center
and edge. The center is the area encompassed by the central county of the city. The edge is
the set of counties that surround the central county. The central county always includes the
central business district of the city, or the downtown area, and is generally much larger than
just the downtown area. Our study relies on the 50 largest U.S. Metropolitan Areas according
to their 1999 population. In particular, for each city, we use the most extensive de…nition
of metropolitan area available, either Metropolitan Statistical Area (MSA) or Consolidated
Metropolitan Statistical Area (CMSA), as de…ned by the O¢ce of Management and Budget.
Our data originates from four sources: the Census Bureau “Commute to Work” data, the
Census County Business Patterns, the Housing and Urban Development State of the Cities
database, and the Bureau of Economic Analysis (BEA) Regional Economic Information
System. Since the theory we propose is that of a city rather than a system of cities, and
we wish to abstract from idiosyncratic city characteristics, our analysis focuses only on time
changes in city structure during the 1980s, and not on the state of city structure across U.S.
cities at a given point in time.
2.1 Changes in Absolute Population Levels
The …rst set of facts we mention are well known. As shown in Figure 1, overall population
increased throughout the 1980s in all but one city in our sample. City population growth
5
averaged 21.3% over that period, while Pittsburgh’s population contracted by 0.38%. All
averages presented in this section are weighted averages using population shares, and are
shown as horizontal lines in the bar graphs and circles in the scatter plots.
Figure 1: Population Growth, 1980-1990
Orlando
Las Vegas
Austin
Phoenix
Atlanta
Tampa
Sacram.
Jackson.
Raleigh
Norfolk
Dallas
Seattle
D.C.
LA
Nashville
Charlotte
Miami
San Ant.
San Fran.
Minneap.
Salt Lake
G. Rapids
Columbus
Portland
Richmond
Greenville
Greensboro
Memphis
Denver
Baltimore
Philadel.
Indianap.
Cincinnati
Houston
Kansas
Boston
New York
St. Louis
OK City
Hartford
Rochester
Dayton
Louisville
Chicago
Detroit
Milwaukee
Buffalo
Cleveland
New Orl.
Pittsburgh
80%
70%
60%
50%
40%
30%
20%
10%
0%
-10%
Moreover, population in most cities increased both at the center and at the edge. To
illustrate this point, Figure 2 plots population changes in these areas from 1980 to 1990.
In almost all cities, changes are positive both at the center and at the edge and, for some
cities, very large. The population of Las Vegas, for example, grew by more than 80% in
the edge counties and more than 50% at the center. A 1% increase in population at the
edge is associated with a 0.6% increase in population at the center. The correlation between
changes in population at the center and at the edge is 0.69. All 50 cities in our sample grew
in terms of edge population, and only 7 declined in terms of population at the center. In
the latter cases, this decline is always small except for New Orleans whose population fell
by 12.8% at the center, but increased by 9.7% at the edge.
Figures 1 and 2 are indicative of overall city population growth, and the fact that this
population locates both at the center and in peripheral counties. A question which then
arises is: has city population growth led to a change in the link between employment location
and residential location? As cities become larger, one might expect residential sprawl (i.e.
residents locating at the boundary of the city) and employment concentration at the center
(see Fujita and Ogawa [1982] and Lucas and Rossi-Hansberg [2002]). In general, our view of
the data is that this phenomenon did not dominate in the U.S.: changes in employment are
in general paralleled by changes in the number of residents in both areas of the city.
6
A fact consistent with this view is that net commuting between the center and the edge,
as a percentage of total population, hardly changed throughout the 1980s. Net commuting
represented 8.98% of total MSA population in 1980 and 8.38% in 1990. Average residential
growth in the 1980s amounted to 15.4% at the center and 25.7% at the edge. Similarly,
average employment growth in the 1980s was 14.8% at the center and 28.3% at the edge. This
similarity in the size of changes in employment and residents across city areas is surprising,
and suggests that the link between employment and residential location is a key component
of urban structure. In addition, this evidence suggests that commuting costs did not decline
in any signi…cant way during the 1980s.
Figure 2: Population Growth, 1980-1990
70%
Center Population
60%
50%
40%
30%
20%
10%
0%
-10%
-20%
0%
20%
40%
60%
80%
100%
Edge Population
Figure 3: Employment Growth, 1980-1990
Center Employment
80%
60%
40%
20%
0%
-20%
-10%
0%
10%
20%
30%
40%
50%
60%
70%
80%
Edge Employment
Figure 3 depicts employment growth across all cities in our sample (resident growth is
identical to population growth, which is presented in Figure 2). The …gures indicate that
7
residential population and employment grew in the majority of cities during this period.
More speci…cally, resident and employment growth is positive at the edge in all cities. At
the center, employment growth is negative in only nine cities while resident growth is negative
in only seven cities.
2.2 Changes in Population Shares Across City Areas
The facts we have just presented relate to absolute quantities of employment and resident
growth. We now address changes in the shares of residents and employment at the edge
during the 1980s. Our data show that the shares of both residents and employment have
generally increased at the edge. That is, while levels grew everywhere, the share of population
has shifted from the center to the edge of U.S. cities. The average increase in the share of
employment at the edge during the 1980s was 3.11% while the average increase in the share
of residents at the edge was 2.64%. Put simply, economic activity in the U.S. is moving
to the periphery. Figure 4 depicts changes in resident and employment shares at the edge
during this period. It is clear from the graph that in most U.S. cities, the share of individuals
who both reside and work at the edge increased.
Figure 4: Change in Resident and Employment
Share at the Edge, 1980-1990
Employment Share
16%
12%
8%
4%
0%
-4%
-8%
-6%
-4%
-2%
0%
2%
4%
6%
8%
Resident Share
Given the shift in employment shares towards the periphery, one might wonder whether
the increase in edge employment was driven by particular industries. In other words, the
facts above could have resulted from speci…c industries moving away from city centers while
8
other industries, perhaps less labor and land intensive, moved to the city center. This does
not appear to be the case. The average employment share at the center declined from 0.42
to 0.38 in manufacturing and from 0.47 to 0.43 in services. That is, average employment
shares decline by about the same percentage in both sectors.
2.3 Changes in the Location of Managers
To gain further insight into the change in employment shares at the center, we examine
the change in employment across occupations. In particular, we divide employment into two
classes: management and non-management occupations. The …rst class includes managers
and professional workers. The second includes what the Housing and Urban Development
State of the Cities database classi…es as non-management workers.3 The latter category
includes technicians, sales, administrative support, precision workers, laborers and machine
operators, and service workers.
The share of managers at the center declined in all but 3 cities in our sample. The average
change in the manager share during the 1980s was -4.71%. The share of non-managers at
the center also declined in all but a handful of cities during this period, with an average
decline of -5.63%. Although these facts are informative about changes in city structure,
comparisons related to the relative location of managers and non-managers are also telling.
Figure 5a presents the di¤erence between the fall in manager and non-manager shares across
cities. Note that at the center, manager shares fell less rapidly than non-manager shares
in approximately 75% of the cities throughout the 1980s. There are only 12 cities (out of
50) where the reverse is true. Even accounting for the fact that some of these exceptions
are among the largest U.S. cities, including New York and Chicago, the weighted average in
Figure 5a remains positive at 0:23%. Figure 5b compares the change in the ratio of managers
to non-managers at the center with the change in the same ratio at the edge: an alternative
way of looking at the same phenomenon.
3
The de…nition of center and edge in the HUD database di¤ers somewhat from our previous breakdown
in terms of counties. These data divide the city into a business center (de…ned by the geographical city
boundary) and the CMSA/MSA area outside of it. For consistency, we extend the business center area to
that of the central county, using the assumption that densities of managers and non-managers are constant
within each area.
9
Figure 5a: Change in Management Share less the Change in NonManagement Share at the Center, 1980-1990
6%
4%
2%
0%
-2%
Richmond
Norfolk
Rochester
Minneap.
Greensboro
St. Louis
Cleveland
Sacram.
Dayton
Buffalo
Kansas
Louisville
Indianap.
Austin
Seattle
Greenville
Tampa
Baltimore
Memphis
Jackson.
Salt Lake
Detroit
Columbus
Portland
Milwaukee
Cincinnati
New Orl.
D.C.
Pittsburgh
Phoenix
G. Rapids
Philadel.
Miami
Hartford
OK City
Denver
Atlanta
New York
Orlando
Dallas
Boston
San Ant.
San Fran.
Chicago
Nashville
Charlotte
Las Vegas
LA
Houston
Raleigh
-4%
Figure 5b: Change in Management : Non-Management Ratio at
the Center Relative to the Edge, 1980-1990
15%
10%
5%
0%
Minneap.
D.C.
Richmond
Greenville
Austin
Cleveland
Rochester
Hartford
Pittsburgh
Sacram.
Buffalo
St. Louis
Norfolk
Seattle
Dayton
Detroit
Atlanta
Cincinnati
Kansas
New Orl.
Indianap.
Greensbo
Louisville
Boston
Salt Lake
Jackson.
Tampa
Milwauke
Baltimore
Memphis
Columbus
Portland
Philadelp.
G. Rapids
Charlotte
Phoenix
Denver
San Fran.
Chicago
Miami
OK City
Nashville
Dallas
Orlando
New York
L. Vegas
San Ant.
Houston
Raleigh
LA
-5%
Consistent with the observation we have just made, Figure 5b shows that the ratio of
managers to non-managers rose more rapidly at the center than at the edge in 86% of the
cities in our sample.4 The average di¤erence in the change of these ratios was 2.82%. In
general, therefore, our interpretation of the data is that city centers are becoming management or administrative hubs, with managers heading operation plants at the boundary of
the city where land is cheap. In fact, we shall use this interpretation of Figures 5a and
5b in the model we develop below. Observe that although managers were generally more
concentrated in city centers throughout the 1980s, every single city over this period saw an
increase in the overall ratio of managers to population. In other words, cities as a whole are
also evolving towards administrative functions. Figure 5c illustrates the change in the ratio
of management to population, with a mean of approximately 3.89% across all cities. The
4
The ratio of managers to non-managers increased both at the center and at the edge in every city.
10
evidence presented in Figure 5c only reinforces the …ndings in Duranton and Puga (2004)
who argue for the functional specialization of cities.
Figure 5c: Change in Total Management : Population Ratio, 19801990
8%
6%
4%
2%
Houston
Boston
Baltimore
Pittsburgh
Hartford
Richmond
Dallas
Philadelp.
D.C.
St. Louis
Rochester
New York
Cincinnati
Austin
San Ant.
Raleigh
New Orl.
Atlanta
Cleveland
Indianap.
Denver
Chicago
Buffalo
Milwaukee
San Fran.
Seattle
Charlotte
Sacram.
Minneap.
Kansas
Columbus
Dayton
Norfolk
Greenville
Nashville
OK City
Portland
Detroit
Tampa
Memphis
Salt Lake
LA
Jackson.
Louisville
Orlando
Phoenix
G. Rapids
Miami
Greensboro
Las Vegas
0%
2.4 Changes in the Location and Characteristics of Establishments
Having set out some facts regarding changes in population location within U.S. cities, we
now turn to the location of establishments in di¤erent parts of the city. Consistent with overall population and employment growth throughout the 1980s, the number of establishments
also increased in all but three U.S. cities, both at the center and at the edge.
Figure 6: Establishment Growth at the Center and at
the Edge, 1980-1990
Center Establishments
100%
80%
60%
40%
20%
0%
-20%
20%
40%
60%
80%
100%
120%
Edge Establishments
The number of establishments grew on average by 30.1% at the center and by 50.5% at the
boundary. Hence, while the net entry of …rms or plants is more pronounced at the city edge,
…rm entry is also substantial at the center. The correlation between establishment net entry
11
at the center and edge is 0.45. Figure 6 illustrates these changes. Note that in some cities,
net entry of establishments at the edge exceeds 80% over our sample period and, in some
cases, reaches as high as 110%. Although central counties experienced signi…cant net …rm
entry throughout the 1980s, more establishments located at the boundary over that period.
Indeed, the change in the share of establishments at the center is negative in all but a few
cities, as Figure 7 illustrates. The average change in the share of establishments located
at the center is -3.65%. In 64% of the cities, the share of establishments was larger at the
center than at the edge in 1980, with an average establishment share of 54.9% at the center.
Therefore, while more than one half of the establishments were located at the center in 1980,
many new …rms chose to locate near the city’s outskirts during the subsequent decade which
lead to a signi…cant decrease in the share of …rms residing at the center.
Figure 7: Change in Establishment Share at the Center, 1980-1990
8%
4%
0%
-4%
-8%
Raleigh
Charlotte
Rochester
Austin
Columbus
Phoenix
Salt Lake
Greensb.
Houston
Buffalo
G. Rapids
Pittsburgh
San Ant.
Dayton
Tampa
Memphis
Greenville
OK City
Seattle
San Fran.
L. Vegas
Sacram.
Minneap.
Miami
Boston
Hartford
Louisville
Dallas
Indianap.
New York
Cleveland
Cincinnati
Orlando
Nashville
Jackson.
LA
Kansas
Philadel.
St. Louis
Portland
D.C.
Chicago
Milwaukee
Detroit
Richmond
Denver
Norfolk
Atlanta
New Orl.
Baltimore
-12%
How were establishment sizes, measured in number of employees, a¤ected during this
period? In general, we …nd that establishment sizes declined over the 1980s. This …nding
is consistent with other evidence in the literature regarding the average size of …rms in the
U.S. (see Garicano and Rossi-Hansberg [2003]). Establishment sizes declined on average by
4.08% at the center and 3.92% at the edge. Figure 8 shows changes in establishment sizes
both at the center and at the edge. We can see that for most cities, establishment sizes fell
in both regions. This …nding, however, does not hold for all cities.
12
Figure 8: Change in Establishment Size at the
Center and at the Edge, 1980-1990
Center Establishment Size
15%
10%
5%
0%
-5%
-10%
-15%
-20%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
Edge Establishment Size
About half of the cities in our sample experienced a decline in average establishment
size in both regions simultaneously. Establishments tend to be larger at the center than at
the boundary, with 21.4 employees per establishment at the center versus 17.1 employees
per establishment at the edge in 1980. The last characteristic of urban economic activity
that we wish to establish in this section concerns the relationship between changes in the
number of managers and the number of establishments across cities. Figures 9a and 9b
illustrate this relationship. Observe that the number of establishments and managers are
highly correlated: the correlation is 0.81 at the center and 0.62 at the edge. At the center,
a 1% increase in the number of managers is associated with a 0.89% increase in the number
of establishments, where the …t is characterized by a surprisingly large R2 statistic, 0:65.
In contrast, at the edge, a 1% increase in the number of managers is associated with only
0.57% more establishments, with the relationship having a much lower R2 . This evidence
suggests that the number of establishment per manager at the center is larger than at
the edge, and that changes in the number of establishments are more tightly related to
changes in managers at the center than at the edge. Although our measure of managers also
includes professional workers, the fact that managers and establishment are so closely linked
gives us some con…dence that many of these agents are in fact performing administrative or
managerial roles.
13
Figure 9a: Change in Managers and Change in
Establishments at the Center, 1980-1990
Change in Establishments
(CE)
100%
CE = 0.8899(CM) + 0.0499
R2 = 0.6501
80%
60%
40%
20%
0%
-20%
-20%
0%
20%
40%
60%
80%
100%
Change in Managers (CM)
Figure 9b: Change in Managers and Change in
Establishments at the Edge, 1980-1990
Change in Establishments
(CE)
120%
100%
CE = 0.5726(CM) + 0.1943
R2 = 0.3831
80%
60%
40%
20%
20%
40%
60%
80%
100%
120%
Change in Managers (CM)
2.5 Stylized Facts on Urban Structural Change
We now summarize the set of stylized facts presented for the …fty largest U.S. cities over
the 1980-1990 decade. Throughout the paper, we shall refer to these stylized facts by the
number we assign to each below.
1. Overall population growth.
2. Resident population growth at the center and at the edge of cities.
3. Employment growth at the center and at the edge of cities.
14
4. A similar reduction in resident and employment shares at the center.
(a) Present both in services and manufacturing.
5. City centers increasingly becoming management or administrative hubs:
(a) An increase in the share of managers compared to the share of non-managers at
the center,
(b) An increase in the ratio of managers to non-managers at the center compared to
the same ratio at the edge,
(c) An increase in the total number of managers relative to total employment.
6. An increase in the number of establishments at the center and at the edge of cities.
7. A decline in the establishment share at the center.
8. A decline in establishment size both at the center and at the edge of cities.
9. The number of establishments and managers are more tightly related at the center
than at the edge.
In the next section, we propose a simple model of city structure and explain how Fact 1
alone can lead to stylized Facts 2 through 9. Put another way, it is possible to think of the
changes in residents, employment, occupations, and establishments described above as the
result of urban population growth. At the heart of the theory we present lies a decision on
the part of …rms to either integrate their operations in one location or separate them into
headquarters and production plants. We argue that adding this additional margin to urban
models is crucial in explaining the diverse set of changes observed in the internal structure
of U.S. cities.
3. THE MODEL
This section presents a theory of the internal structure of cities simple enough to remain
analytically tractable yet rich enough to address the diverse set of facts outlined in the
15
previous section. Since our goal is to illustrate the main forces that lead to these empirical
regularities, we model cities as consisting of only two areas: the center of the city and what
we call the edge. We think of these two areas as the model equivalent of the central and edge
counties in the data. Given this parallel, we assume that the central area of the city is given
exogenously by Lc > 0. The edge area is endogenous, and we denote it by Lb > 0, where
b stands for the city boundary: We assume that residential land rents at the edge are given
by an agricultural land rent, R ¸ 0, that represents its opportunity cost. Land rents at the
center of the city are endogenous and determined in equilibrium. Total city population, P , is
exogenously given. In fact, we shall argue below that the theory we develop can rationalize
the set of stylized facts presented above simply as the result of population growth. Our
theory, therefore, is a partial equilibrium theory that takes as given two key elements from
national economic behavior, namely, agricultural land rents and city population sizes. Any
theory of city structure must take a stand on the variables to be determined at the aggregate
rather than the city level, and our choice is driven by the set of facts that we seek to explain.
The key insight of our model is that allowing …rms to separate their location into headquarters and production plants implies that city growth leads to a set of empirical regularities
regarding urban structure. Headquarters develop knowledge and, therefore, experience external e¤ects from other headquarters. Knowledge transactions are carried out in headquarters
which tend to agglomerate in high rent areas of the city. Production plants, in contrast,
carry out more routine tasks that do not lead to knowledge spillovers and, consequently,
tend to locate in areas where land rents are low. Production plants in our framework can be
interpreted as either manufacturing plants, retail stores, or other production facilities.
3.1 Firms
The city produces and consumes one good, the price of which we normalize to one. A …rm
is made up of a manager who hires workers to produce. The manager and her workers can
locate at either the center or the edge of the city. We refer to the location of the manager as
the …rm’s headquarters and the location of her workers as the …rm’s production plant. The
number of workers a manager can hire is determined by whether the …rm is located in only
one location (an integrated …rm), or whether the headquarters and production plants reside
16
in di¤erent locations. In the former case, the manager …nds it less costly to communicate
and interact with workers that are located close by so that she can oversee a larger set of
workers, ncc = nbb = ®± > 1, where nij denotes the number of employees of a …rm with
headquarters in area i and production plant in area j. In contrast, if the manager decides
to set up a non-integrated …rm, she needs to spend additional resources to monitor and
interact with her workers, who are physically removed, and her span of control is given by
ncb = nbc = ®± > 1, where ± < ±. This assumption is motivated by Fact 9 above. In other
words, in the data, changes in the number of managers are clearly positively correlated with
changes in the number of establishments both at the center and at the edge. In fact, the
correlation between changes in managers and changes in the number of establishments is
signi…cantly higher at the center than at the edge in the 1980s, which is consistent with a
constant number of managers per …rm (abstracting from composition e¤ects which are not
present at the center).5
In our model, the location of a …rm’s headquarters matters signi…cantly in that it determines its productivity. In particular, …rm productivity depends on the number of managers
located in the area of the city where the …rm’s headquarters are also located: a production
externality. Total output of a …rm with headquarters in area i and production plant in area
j is given by AEi nij , where Ei denotes the number of managers in i and A is a city-wide
productivity parameter. A …rm has to pay labor costs given by a city-wide wage w > 0
(since all agents in the city are assumed identical) times the number of workers it hires, nij ,
as well as land rents. We assume that the …rm needs to hire one unit of land per worker
in order to operate, so that total land rent paid by this …rm is given by Rj nij .6 Firms are
owned by managers whose earnings are given by …rms’ pro…ts. It follows that a manager
who owns the …rm we have just described earns
Fij = (AEi ¡ w ¡ Rj )nij :
(1)
The problem of a manager is then to choose the location of the …rm’s headquarter and
5
See Garicano (2000), and Garicano and Rossi-Hansberg (2003), for organizational models that yield
smaller team sizes as communication costs increase.
6
Note that managers do not require land. This assumption allows for a much simpler analysis of the
model without driving any of our central results.
17
production plant to solve
F = max fFij g for i; j 2 fc; bg and subject to
ij
8
< ®± if i = j
nij =
:
: ®± if i 6= j
Put alternatively, managers decide whether to locate at the center or at the edge of the city
and, from that location, whether to operate integrated or separate production facilities.
3.2 Individuals
A population P of identical agents lives and works in the city. Agents consume the only
good produced in the city and they live where they work. The latter assumption is justi…ed
by the fact that in the data, employment and residents in both areas of the city have moved
to the edge in similar proportions, as summarized in Fact 4. Recall also that the share of net
commuters between the center and edge stayed remarkably constant throughout our sample
period.
Consumers order consumption according to a linear utility function. Therefore, given that
the price of consumption goods is normalized to one, they solve
©
ª
U = max F ; w :
(2)
Since all agents are identical and, in equilibrium, some agents become managers while others
become workers, F = w. Furthermore, the fact that all agents have the option to set up
integrated or non-integrated …rms in any set of locations yields, in equilibrium, F = Fij for
all operating …rms with headquarters in i and production plants in j.7
3.3 Equilibrium
We denote by Eij the number of managers operating …rms with headquarters in i and
operation plants in j. Hence, the total number of managers at location i, Ei , is given by
Ei = Eii + Eij :
7
(3)
We abstract from di¤erences in agents’ human capital or ability that may lead to di¤erences in wages or
managerial rents. We could introduce these di¤erence only to reproduce our …ndings in terms of e¢ciency
units of labor.
18
Since the number of units of land at the center is exogenously given by Lc , and …rms rent
one unit of land per worker, the number of workers at the center is given by
Lc = Ecc ncc + Ebc nbc :
(4)
Analogously, the number of workers at the edge is also given by the number of units of land
used at the boundary,
Lb = Ecb ncb + Ebb nbb :
(5)
Land use at the edge, however, is endogenous and the area occupied by the city expands or
contracts as the economic environment changes. It follows that the total number of workers
in the city is given by W = Lc + Lb . Total city population is given by these workers plus
those individuals who become managers of …rms with headquarters at the center and at the
edge. Therefore, labor market equilibrium requires that
Ec + Eb + W = P:
(6)
We are now ready to de…ne a competitive equilibrium for this city:
A competitive city equilibrium is a set of scalars fEc ; Eb ; Ecc; Ebb ; Ecb ; Ebc ; Lb ; Rc ; w; F ;
Fcc; Fbb ; Fcb ; Fbc g such that:
1. Agents solve (2), managers solve (1), and w = F = Fij for all …rms of type ij that
operate in location, i; j 2 fc; bg :
2. Equilibrium conditions (3), (4), (5), and (6) are satis…ed.
3. Population size is given by P , land available at the center by Lc , and land rates at the
boundary by Rb = R.
The number of establishments at the center is given by
Sc = Ecc + Ecb + Ebc ;
| {z }
(7)
Ec
where Sc counts integrated production units at the center, Ecc , headquarters at the center
used by managers who operate plants in the periphery, Ecb , and production plants at the center run by managers residing at the boundary, Ebc . Similarly, the number of establishments
19
at the edge is de…ned as
Sb = Ebb + Ebc + Ecb :
| {z }
(8)
Eb
Within the framework of our model, average establishment sizes at the center and at the
edge are given by (Lc + Ec )=Sc and (Lb + Eb )=Sb respectively. Establishments are of only
three sizes: size one in the case of headquarters of non-integrated …rms, size ®± in the case of
the operation plant of a non-integrated …rm, and size 1 + ®± in the case of integrated …rms.
The model we have just laid out potentially yields di¤erent types of equilibria. These
types correspond to di¤erent sets of …rms (i.e. integrated or not) operating in di¤erent areas
of the city. The fact that spans of control di¤er across integrated and non-integrated …rms
rules out equilibria where all types of …rms coexist. In essence, if di¤erences in spans of
control are such that a …rm …nds worth it to locate its headquarters at the center and its
production plant at the edge, then the reverse cannot be true. We formalize this result in
the next proposition. Proofs of all propositions are included in the Appendix.
Proposition 1 There are no equilibria where both integrated and non-integrated …rms operate at all locations.
Of the remaining cases, the one corresponding to the type of city encountered in the data
has most headquarters locating at the center (which in fact de…nes what we call the center
and what is de…ned as a central county in the data). We now show that this case exists as
an equilibrium of our model under a mild parameter restriction. In this equilibrium, there
are no …rms whose headquarters are at the edge but whose production plants reside at the
center. Thus, the equilibrium we have just described is such that
F = Fcc = Fcb = Fbb = w and Fbc < F ;
(9)
so that Ecc ; Ecb ; Ebb > 0 and Ebc = 0: Because land rents are much lower at the edge than at
the center in the data, …rms that have operation plants at the center and headquarters at
the edge are indeed very rare. Perhaps the most compelling reason to focus on this type of
equilibrium is Fact 9. This fact shows that the number of managers is very tightly connected
to the number of establishments, especially at the center where the number of managers and
establishments move almost one for one in Figure 9a. In fact, in the model equilibrium with
20
Ebc = 0, the number of establishments and managers does move one for one at the center
since Sc = Ec from (7). This will not be the case at the edge, however, as Sb = Eb + Ecb
in (8), where Ecb captures establishments that are stand-alone operation plants. As in Fact
9, therefore, this equilibrium of the model implies that changes in managers are less closely
related to changes in establishments at the edge than at the center. Thus, we prove all
results below only for this case.
3.4 Equilibrium Allocation
Given the restriction Ebc = 0, we now construct an equilibrium allocation for our model.
From (3), we know that Eb = Ebb since Ebc = 0, and that Ecb = Ec ¡ Ecc where, by equation
¡ ¢
(4), Ecc = Lc = ®± . Then, the number of workers in the city is given by
Lc + Lb = W
so that
µ
¶
±
Lc 1 ¡
(10)
+ Ec ®± + Eb ®± = W:
±
Condition (9) implies that integrated and non-integrated …rms at the center earn equal
pro…ts,
(AEc ¡ w ¡ Rc )®± = (AEc ¡ w ¡ R)®±;
(11)
as do integrated …rms across locations,
(AEc ¡ w ¡ Rc )®± = (AEb ¡ w ¡ R)®±:
These relations implicitly link the number of managers working at the center and boundary
according to
Eb = Ec +
R ¡ Rc
.
A
(12)
Equality between rents and wages then implies that
(AEc ¡ w ¡ Rc )®± = w:
From equations (11) and (13), we further have that
(AEc ¡ w ¡ Rc )®± = w = (AEc ¡ w ¡ R)®±:
21
(13)
Consequently,
R ¡ Rc =
µ
1
1
¡
®± ®±
¶
w=¡
|
µ
¶
± ¡±
w;
®±±
{z }
(14)
¤<0
so that Rc > R under our maintained assumption regarding the span of control, ± > ±. That
is, land rents are larger at the center than at the edge, an implication which follows from
the assumption that Ebc = 0. The fact that most cities see land rents decrease as one moves
away from the center is well known and reinforces our focus on an equilibrium with this
feature.
From equation (13), and substituting for Rc using (14), we obtain, after some manipulations,
1
1
1 + ®±
1 + ®±
Rc
R
w + ;=
w+
(15)
Ec =
A
A
A
A
which implies Ec0 (w) > 0; the number of managers at the center increases with city wages.
Using (14) and equation (12), we can solve for the set of managers at the edge as a function
of wages,
µ ¶
1
1 + ®±
¤
R
Eb = Ec +
w+ :
w=
A
A
A
(16)
Therefore, the number of managers at the edge also increases with wages, but at a slower
rate since the rent di¤erential decreases with wages and reduces the incentives to locate at
the boundary.
Now consider the market clearing equation (6) given by,
µ
¶
±
Lc 1 ¡
+ Ec ®± + Eb ®± = W = P ¡ Ec ¡ Eb :
±
Substituting for the number of managers in both regions, the demand for workers becomes
µ
¶
±
W = Lc 1 ¡
+ Ec ®± + Eb ®±
(17)
±
µ
¶ µ
¶
±
®± + ®±
2 + ®± + ®±
= Lc 1 ¡
+
R+
w
(18)
A
A
±
which is linear and increasing in pro…ts or wages (w). Since higher pro…ts resulting from
greater externalities are associated with more managers in both areas of the city, the demand
for workers increases as pro…ts rise. The supply of workers is given by
µ
¶
1
1
2 + ®±
+ ®±
2R
P ¡ Ec ¡ Eb = P ¡
¡
w;
A
A
22
(19)
which again is linear but decreasing in pro…ts or wages (w). Because larger pro…ts motivate more agents to become managers, the supply of workers decreases with pro…ts. The
equilibrium wage can then be found by equating (17) and (19). That is
µ
¶
µ
¶
¡
¢
±
1
1
AP ¡ ALc 1 ¡
¡ 2 + ®± + ®± R = 4 + ®± + ®± +
+
w
®± ®±
±
so that city wages are given by
w=
³
´ ¡
¢
±
AP ¡ ALc 1 ¡ ± ¡ 2 + ®± + ®± R
4 + ®± + ®± +
1
®±
+
1
®±
:
(20)
With the equilibrium wage in hand, we can solve for all equilibrium variables of the model.
For the comparative statics in the next section, it is helpful to denote the denominator of
(20) as D > 0: An equilibrium of the type in which we are interested exists only if
¢
¶ ¡
µ
2 + ®± + ®± R
±
P
+
> 1¡
;
Lc
ALc
±
(21)
which ensures that w > 0. This restriction essentially requires city population densities
that are large enough to make the creation of a city pro…table given land rents at the
edge. Population densities must also be large enough so that agglomeration e¤ects guarantee
that some non-integrated …rms’ headquarters choose to locate at the center. The following
proposition obtains directly from the equilibrium wage derived above.
Proposition 2 The equilibrium wage, w, is an increasing function of population size, P ,
and city-wide productivity, A. It is a decreasing function of the supply of land at the center,
Lc , the span of control parameter, ®, and edge land rents, R.
Note that the wage increases with population size. While this …nding matches the fact
that wages are generally higher in larger cities, it nevertheless seemingly con‡icts with the
standard intuition that wages fall as the supply of workers increases. There are two characteristics of our set up that contribute to overturning this intuition. First, the wage is
the compensation of workers, but since all agents in the city are identical, it also represents
pro…ts of entrepreneurs or managers. Hence, since an increase in overall population creates
new managers, the demand for workers also increases. Second, the production externality
23
in our framework implies that the more managers operate in a given location, the higher
the productivity (output per worker) of all …rms in that location. As population grows, and
more …rms operate in the city, this e¤ect contributes to raising manager rents and worker
wages. As we have just mentioned, the prediction regarding wages and city size can be directly contrasted with data. In 2003, for instance, the average wage in the largest …ve cities
in our sample was $42,976, as compared to just $34,340 for the …ve smallest cities.8
The e¤ect of the model’s other parameters on wages are more standard. Wages increase
with city-wide productivity and decrease with the amount of land available at the center.
The latter result re‡ects the fact that more …rms at the center become integrated as Lc
rises. This e¤ect reduces the number of managers per worker at the center and, therefore,
externalities in that location and manager pro…ts (wages). As rents at the boundary increase,
the advantage of setting up a non-integrated …rm falls, which again reduces externalities at
the center and wages. Finally, as …rms’ span of control, ®, increases, …rms become larger,
less agents become managers, and externalities fall along with wages and managerial rents.
4. ADDRESSING THE STYLIZED FACTS
This section shows that the model we have developed naturally leads to the changes in city
structure laid out in Section 2. From Fact 1, we know that population growth was positive
in virtually all cities in our sample. Recall that the …rst set of facts referred to population
size both at the center and at the edge. Since, in our model, agents live and work in the
same location, the model’s predictions concerning the growth in residents and employment
are identical. Therefore, if we can show that as population grows, employment increases
both at the center and at the edge (Fact 3), then the model will immediately satisfy Fact 2.
To address Fact 3, observe that two e¤ects emerge as population grows. First, the number
of managers at the center increases, as a result of the rise in agglomeration e¤ects, which
raises population at the center (recall that the worker population at the center is pinned
down at level Lc by assumption). Second, since managerial population also increases at the
boundary with urban growth, again as a result of larger externalities, so does the number
of workers given the …xed span of control. Therefore, total employment must also increase
8
See Lee (2005) for recent evidence on the urban wage premium.
24
at the edge. A portion of the additional workers at the boundary will work for managers
that head …rms with headquarters at the center. Ultimately, this reasoning implies that the
model is consistent with Fact 2.
Total employment at the center and at the edge is given by Ec +Lc and Eb +Lb respectively.
Simple di¤erentiation then leads to the following proposition, consistent with Facts 2 and 3.
All proofs in this section are relegated to the Appendix.
Proposition 3 An increase in population implies an increase in total employment at the
center, Ec + Lc , and at the edge, Eb + Lb .
The proposition above provides conclusions regarding the level of employment in both
areas of the city. However, we are equally interested in the share of employment in each
area. We have already argued that as population grows, managerial population increases at
the center. All new center managers, however, lead non-integrated …rms. The reason is that
the number of workers at the center cannot expand given the …xed amount of land and the
technology that requires one unit of land per worker. Managers choose non-integrated rather
than integrated …rms because rents at the boundary do not grow with population, since they
are pinned down by the opportunity cost of land, R. At the same time, since the center land
area of the city is constant, the price of land increases at the center which again motivates
some managers to send their operation plants to the edge. In fact, rents at the center rise
until the number of center managers who choose to operate integrated …rms reverts back to
its initial equilibrium. We can show that given a large enough population share at the center,
the increase in center employment (caused solely by the increase in managerial population)
is always smaller than the increase in managers and workers at the edge. This result implies
that the employment share at the center falls with population growth, consistent with Fact
4. We formalize this reasoning in the next proposition.
Proposition 4 An increase in population implies a decrease in the employment share at the
center,
Ec +Lc
,
P
if and only if
Ec + Lc
>³
P
1+
1
®±
´
1+
1
®±
+ 3 + ®± + ®± +
25
1
®±
:
The lower bound on the share of population working at the center amounts to restrictions
on ® and ±=± that turn out to be very mild (the proof of the proposition in the Appendix
includes a parallel restriction expressed only in terms of the exogenous parameters). To
illustrate this point, observe that in our framework, ®± determines the size of integrated
…rms which are the largest in the economy. As pointed out earlier, the average employment
size of establishments in 1980 was 21:4 at the center and 17:1 at the edge. Therefore, we
can conjecture that 1 + ®± > 21:4 and, furthermore, that ±=± > 20:4=17:1 = 1:19 since
±=± represents the ratio of the largest to the smallest operation plant. Note also that the
restriction set out in Proposition 4 becomes more severe as ®± and ±=± fall. Thus, suppose
that we very conservatively set ®± = 10 and ®± = 12: Then Proposition 4 indicates that the
share of employment at the center decreases with overall population whenever the share of
center employment exceeds 4:2%, a condition that is easily met by all cities in our sample.9
To summarize thus far, an increase in overall city population leads to …ndings for the levels
and shares of employment in di¤erent areas of the city that are consistent with the data. In
particular, population growth leads to an increase in employment levels everywhere, but also
to a shift in employment from the center to the edge in shares. These results follow directly
from …rms having the opportunity to break up their operations geographically.
It is di¢cult to understand how observed changes in levels and shares of population in
di¤erent sections of the city could be the result of forces that are not related to an overall
expansion in size (i.e. population growth). If other forces were responsible for these changes,
and since one needs to introduce scale e¤ects in order to generate cities, reductions in the
share of employment at the center will generally lead to reductions in the level of employment
as well. Hence, it seems that two elements are needed to obtain models that can reproduce
these dimensions of the data. First, one needs models where these changes are the result of
city growth. Second, these models should also allow for endogenous employment densities.
Evidently, if densities were not endogenous, given that the center county area has not changed
in the data, employment at the center would necessarily be predicted to remain constant.
The advantage of writing down a model in which agents’ occupations are explicitly consid9
If one actually uses ®±= 17:1 and ®± = 21:4, the lower bound required in Proposition 4 becomes even
less stringent at 2:5 percent.
26
ered is that it has predictions for the e¤ect of changes in exogenous variables on the location
of di¤erent occupations within cities. In Section 2, we presented a set of facts that are related
to the locations of agents working in di¤erent occupations. Speci…cally, we showed that the
changes in manager shares at the center were generally larger than those in non-manager
shares (Fact 5a). We also showed that the change in the manager to non-manager ratio was
larger at the center than at the edge (Fact 5b). Both these facts imply that managers are
increasingly concentrated at the center relative to the boundary. These facts were expressed
in terms of di¤erences in shares and ratios between the center and the edge, and not in term
of levels, shares, or ratios directly.10 Fact 5c shows that the fraction of urban population in
management positions increased during the 1980s. While our model is consistent with this
fact, other forces working at a more aggregate level – such as changing transport and communication costs across cities, or between cities and rural areas – may have helped increase
the total number of managers in the city beyond that implied by population growth alone
(as in Duranton and Puga [2004]). Hence, we concentrate primarily on the predictions of
our model for changes in the di¤erence between the center and edge management shares, as
well as manager to non-manager ratios, driven by city population growth (Facts 5a and 5b).
Consider …rst the di¤erence in the share of managers and non-managers at the center.
The analysis we just carried out suggests that population growth leads to an increase in the
number of managers at the center. In fact, it also leads to an increase in the number of
managers at the boundary. Some of the new managers establish themselves at the center
because of the production externalities generated by managers in that section of the city.
Others take advantage of the larger spans of control, as well as lower rents, at the boundary.
However, the share of managers at the center increases unambiguously since managers do
not use land and, therefore, do not drive up land rents. As we argued earlier, all employees
under the supervision of new entrepreneurs at the center work in operation plants located
at the periphery. In addition, all new managers at the boundary head integrated …rms. As
a result, the share of workers at the center must decrease and, given the rise in the share
of managers at the center, the di¤erence between manager and non-manager shares must
10
Since our model focuses on only one city and not a system of cities, it is silent on level di¤erences across
cities. In our framework, these cross-sectional di¤erences would in principle stem from di¤erent values of Lc,
R, ®, or A across cities.
27
increase following population growth.
It should be remarked that the increase in the share of managers at the center is actually
a counter-factual implication of our model. Speci…cally, the theory over-emphasizes the
concentration of managers at the center. This implication can be attenuated by requiring
that managers rent land at the headquarter’s location. This extension, however, would come
at the cost of a much more complicated setup. In addition, we view the decrease in manager
share at the center as resulting partly from the overall increase in the number of agents in
management occupations, as in Fact 5c (i.e. Ec = (Ec + Eb ) tends to fall as Ec +Eb increases).
Our model also predicts that the ratio of managers to non-managers will increase more
rapidly at the center than at the edge with population growth. Since the number of managers
at the center increases and the number of workers is …xed, it is clear that this ratio increases at
the center. At the edge, since the number of workers increases in part because of the increase
in managers of non-integrated …rms at the center, the ratio always decreases. Together these
results directly lead to Fact 5b. In addition, as population increases, all new managers at
the center run non-integrated …rms. Given that non-integrated …rms have fewer workers per
manager (smaller spans of control), this leads to an increase in the share of city residents that
become managers, as in Fact 5c. These results are stated formally in the next proposition,
consistent with Fact 5.
Proposition 5 An increase in population implies an increase in:
² The di¤erence between manager and non-manager shares at the center,
Ec
Ec +Eb
c
¡ LcL+L
:
b
² The di¤erence in the ratio of managers to non-managers between the center and the
edge,
Ec
Lc
¡
Eb
:
Lb
² The number of managers per resident,
Ec +Eb
.
P
Our model evidently has predictions for the number of establishments at the center and
at the edge of the city. First, recall that at the center, the number of establishments is equal
to the number of managers since establishments at that location are either headquarters or
non-integrated …rms, but under our assumptions never just operation plants. It follows immediately that the number of establishments at the center increases with population growth.
28
Furthermore, since new managers at the center operate only non-integrated …rms, the boundary also sees an increase in operation plants. The latter two …ndings are consistent with Fact
6. Because every new manager at the center is associated with an additional operation plant
at the edge, and the edge also experiences entry of new integrated …rms following population growth, the number of establishments at the edge must increase and, in fact, must
increase by more than the increase in center establishments. This implies that the share of
establishment at the center must fall, as in Fact 7.
Proposition 6 An increase in population implies:
² An increase in the number of establishments located at the center and at the edge, Sc
and Sb ; with the number of establishments increasing more rapidly at the edge than at
the center.
² A decrease in the share of establishments at the center,
Sc
.
Sc +Sb
As pointed out earlier, establishments are of three di¤erent sizes in our set up: Headquarters of size one, operation plants of size ®±, and integrated …rms of size 1 + ®±: Hence, the
speci…c combination of …rms of each type in a given region of the city determines average
establishment size in that region. Since there are no operation plants at the center, …rms
can only be of size one or 1 + ®± in that section of the city. As population increases, the
number of managers at the center increases and so does the share of establishments of size
one. Therefore, population growth contributes to a decrease in average …rm size in the center
region, as in Fact 8. In contrast, the boundary comprises only establishments of size ®± and
1 + ®±. First, this implies that the average size of establishments is larger at the edge than
at the center, unless most …rms are production plants and many integrated …rms reside at
the center. The latter case would make our model consistent with the larger establishment
sizes at the center observed in 1980. As population grows, the set of establishments that
are production plants increases at the edge, and so does the set of integrated …rms. In the
next proposition, we show that the increase in the number of production plants dominates,
thereby leading to a decrease in …rm size at the edge, consistent with Fact 8.
29
Proposition 7 An increase in population implies a decrease in the average size of establishments at the city center and edge,
Ec +Lc
Sc
and
Eb +Lb
,
Sb
respectively.
Thus, our model predicts that population growth reduces establishment sizes in both
regions following simple composition e¤ects. The data, however, shows that changes in
establishment size are negative in both regions for slightly more than half of the cities in our
sample. What factors may explain the behavior of establishment size in the remaining cities?
Given the theory we have just laid out, a possible answer is that lower communication costs
have led to larger spans of control (an increase in ®) and, therefore, larger …rms throughout
the city. The evidence suggests that this phenomenon did not dominate changes in average
…rm size in most cities in the 1980s, but may nevertheless be signi…cant in more recent
time periods for several cities (as argued by Garicano and Rossi-Hansberg [2003] for the late
1990s).
5. ROBUSTNESS OF THE THEORY: LABOR MOBILITY
Our analysis to this point has focused on a theory where land rents at the boundary
are given by some alternative country-wide non-urban land use. Wages and pro…ts in the
city were therefore endogenously determined. In a model of a system of cities, this would
be equivalent to assuming a perfectly elastic supply of urban land at R and high moving
costs that impede the mobility of workers between cities. One might instead imagine an
alternative construct where wages are …xed at some economy wide level, w, exogenous to the
city. Heterogeneity in the quality of land in di¤erent regions would then attract a certain
population which in turn would determine all land prices in the city. This section establishes
that all propositions derived in the previous sections continue to hold using this alternative
interpretation, provided a mild restriction on parameters.
The model remains as in Section 4, except that condition (20) now determines land rents
at the edge instead of wages,
R=
AP ¡ ALc (1 ¡ ±± ) ¡ w(4 + ®± + ®± +
2 + ®± + ®±
1
®±
+
1
)
®±
:
It is straightforward to check that @R=@P > 0; @R=@Lc < 0; and @R=@w < 0, consistent
30
with Proposition 2. We can then prove that results replicating Proposition 3 through 7
continue to hold. The derivations for a subset of these results requires that the number of
managers per establishment at the boundary be greater than one half, Eb =Sb > 1=2.11 Figure
9b suggests that this is likely the case in the data, where a given percentage change in the
number of establishments is associated with approximately twice that percentage change in
the number of managers. These results are formalized in the next proposition.
¹ an
Proposition 8 With exogenous wages, w, and endogenous land rents at the edge, R,
increase in population implies:
² i) An increase in total employment at the center, Ec + Lc , and at the edge, Eb + Lb :
² ii) A decrease in the employment share at the center,
Ec +Lc
:
P
² iii) An increase in: a) the di¤erence between manager and non-manager shares at
the center,
Ec
Ec +Eb
¡
Lc
,
Lc +Lb
b) the di¤erence in the ratio of managers to non-managers
between the center and the edge,
resident,
Ec
Lc
Ec +Eb
:
P
¡ ELbb if
Eb
Sb
> 12 , and c) the number of managers per
² iv) An increase in the number of establishments located at the center and at the edge, Sc
and Sb respectively, and a decrease in the share of establishments at the center,
Sc
.
Sc +Sb
² v) A decrease in the average size of establishments located at the center and at the edge,
Ec +Lc
Sc
and
Eb +Lb
Sb
respectively, if
Eb
Sb
> 12 .
6. CHANGES IN CITY STRUCTURE AND POPULATION GROWTH
In Section 2, we presented a set of stylized facts on the evolution of city structure. We
then argued in the three subsequent sections that observed increases in population alone
could help rationalize those facts. At this stage, therefore, it is natural to ask whether
one could establish the implications of our model more directly in the data? As a …rst
pass, we can use the data to assess whether the changes in city structure presented in
11
However, since a di¤erent price is now taken as given from the city’s standpoint, the restriction on the
share of center population in Section 3, Proposition 4, is no longer necessary.
31
Facts 2 through 9 are in fact correlated with population growth. However, one needs to be
cautious in the interpretation of such an empirical exercise. First, the theory predicts that
the e¤ect of changes in population size should lead to Facts 2 through 9 only if cross-sectional
characteristics of cities are properly controlled for. In particular, the theory has predictions
for the sign of these correlations after controlling for center county land sizes (Lc ), land rents
at the boundary (R), spans of control (®, ± and ±), and productivity (A), as well as any
changes in these variables during the 1980s. At this point, we do not have residential land
rents at the boundary for 1980, or a suitable proxy. However, we can control for land area
Lc and, to some degree, for productivity as well as spans of control using the 1980 level of
per capita income and ratio of managers to population respectively. Since our theory also
abstracts from available city infrastructure and other idiosyncratic city characteristics, we
take city age into consideration by using the decade in which the city became one of the
largest 50 cities in the U.S. This last variable helps but cannot obviously capture all crosssectional characteristics omitted from the model. Because of the size of our sample, we do
not control for changes in any of these variables. Finally, a key problem with calculating
simple correlations is that our theory does not predict a linear response of city structure to
population changes. Despite these caveats regarding the mapping between these correlations
and our theoretical results, Table 1 presents encouraging results that are consistent with the
framework introduced in this paper.
Table 1 presents correlations between population growth and the residuals obtained from
running an OLS regression of the various changes in city structure laid out in Section 2
against the controls discussed above. Observe that all but the last two correlations, the
ones related to establishment size, have the sign predicted by our theory. Some of these
correlations are admittedly low. Nevertheless, our framework does surprisingly well given
that increases in only one independent variable, namely population growth, are shown to
be consistent with eleven diverse changes in the internal structure of cities.12 The incorrect
sign on the correlation between center/edge establishment sizes and population growth is
somewhat disappointing, and indeed the model does not contain a force that would lead to
12
If we eliminate from the sample the four fastest growing cities (Orlando, Las Vegas, Austin, and Phoenix)
correlations related to occupations increase substantially (both become larger than 0.22). This result is
consistent with the non-linear response of these changes to population growth in our model.
32
larger …rms in larger cities. Here, …rms are larger only if they decide to integrate but there
are no di¤erences across integrated …rms. In practice, larger cities have larger …rms partly
because demand for …rms’ varieties is larger, a dimension from which we have abstracted.
Table 1
Correlations with Population Growth
Population Growth
Center Population Growth
0.511
Edge Population Growth
0.558
Center Employment Growth
0.496
Edge Employment Growth
0.538
Change in Edge Population Share
0.127
Change in Edge Employment Share
0.155
Change in Management - Non-Management Shares
0.133
Change in Management over Non-Management Ratio
0.128
Change in Center Establishments
0.381
Change in Edge Establishments
0.572
Change in Center Establishment Share
-0.188
Change in Center Establishment Size
0.467
Change in Edge Establishment Size
0.129
6. CONCLUSIONS
This paper makes three distinct contributions. First we document a set of facts regarding
changes in urban structure experienced by U.S. cities in the 1980s. These facts include
overall population growth; an increase in residents and employment at the center and city
boundaries; a reduction in the share of employment and residents in the center region of
cities; a concentration of managers relative to non-managers at the center; an increase in
establishments in both areas of the city but a decrease in establishment shares at the center;
and a decline in establishment size both at the center and at the edge of cities. Second, we
propose a theory that incorporates …rms’ location and integration decisions and characterize
33
the implications of such a theory for urban structure. Third, we show that population growth
alone is consistent with the set of changes observed in the 1980s, thereby highlighting the
e¤ects of population growth on urban structure.
The theory we present has urban policy implications that di¤er from more standard models
of urban structure. In particular, we provide a framework that could potentially be used to
analyze the kinds of policies aimed at “reviving city centers” that have been put in practice
in many cities across the U.S. Given that our framework includes agglomeration forces in the
form of externalities, some urban policies may improve equilibrium allocations in our setup.
However, the speci…c type of policy in place is critical. For example, Au and Henderson
(2004) show that restrictions on urban migration have had important e¢ciency costs in
China. For now, the question of whether zoning restrictions or location subsidies, as in
Rossi-Hansberg (2004), are optimal in our setup, and in general the design of these policies,
is left to future research.
In order to underscore the importance of …rms’ location decisions, as well as their decision
regarding whether or not to integrate their operations, our model abstracts from important
elements of cities typically emphasized in the urban literature. One such element is a spatial
setup in which multiple sub-centers may arise (see Fujita and Ogawa [1982], and Lucas and
Rossi-Hansberg [2002]). Other dimensions, such as the e¤ect of durable housing structures,
as in Glaeser and Gyourko (2004), and urbanization patterns in a system of cities, as in
Henderson (2003), and Henderson and Wang (2004), are undoubtedly important. In addition, our theory focuses on one particular type of agglomeration force. However, as argued
by Rosenthal and Strange (2003), di¤erent agglomeration forces interact in metropolitan
areas. One could, in principle, study any of these forces along with the …rm’s location and
integration decisions we emphasize. The explanatory power gained by incorporating these
…rms’ decisions with respect to the facts we document in this paper will, hopefully, push the
urban literature to add these margins to the rich set of frameworks available.
34
REFERENCES
[1] Anas, A, R. Arnott, and K. Small, 1998, “Urban Spatial Structure,” Journal of Economic Literature, 36:1426-1464.
[2] Au C. and V. Henderson, 2004, “How Migration Restrictions Limit Agglomeration and Productivity
in China,” Working paper Brown University.
[3] Burch…eld, M., H. Overman, D. Puga, M. Turner, 2004, “The determinants of urban sprawl: A
portrait from space,” Working paper University of Toronto.
[4] Chatterjee S. and G. Carlino, 2001, “Aggregate Metropolitan Employment Growth and the Deconcentration of Metropolitan Employment,” Journal of Monetary Economics, 48:549-583.
[5] Davis J. and V. Henderson, 2004, “The Agglomeration of Headquarters,” Working paper Brown
University.
[6] Duranton G. and D. Puga, 2004, “From sectoral to functional urban specialization,” forthcoming
in Journal of Urban Economics.
[7] Fujita, M. and H. Ogawa, 1982, “Multiple Equilibria and Structural Transition of Nonmonocentric
Urban Con…gurations,” Regional Science and Urban Economics, 12:161-196.
[8] Garicano, Luis, 2000, “Hierarchies and the Organization of Knowledge in Production,” Journal of
Political Economy, 108: 874-904.
[9] Garicano, L. and E. Rossi-Hansberg, 2003, “Organization and Inequality in a Knowledge Economy,”
Working paper Stanford University.
[10] Glaeser E. and J. Gyourko, 2004, “Urban Decline and Durable Housing,” forthcoming in Journal
of Political Economy.
[11] Glaeser, E. and M. Kahn, 2003, “Sprawl and Urban Growth,” in Handbook of Regional and Urban
Economics, Volume 4, edited by J. V. Henderson and J.-F. Thisse.
[12] Henderson, V., 2003, “Urbanization and Growth,” in Handbook of Regional and Urban Economics,
Volume 4, edited by J. V. Henderson and J.-F. Thisse.
[13] Henderson, V., H. Wang, 2004, “Urbanization and City Growth,” Working paper Brown University.
35
[14] Lee, S., 2005,“Ability Sorting and Consumer City,” Working paper University of Minnesota.
[15] Lucas, R. and E. Rossi-Hansberg, 2002, “On the Internal structure of Cities,” Econometrica,
70:1445-1476.
[16] Rosenthal, S. and W. Strange, 2003, “Evidence on the Nature and Sources of Agglomeration
Economies,” in Handbook of Regional and Urban Economics, Volume 4, edited by J. V. Henderson and J.-F. Thisse.
[17] Rossi-Hansberg, E., 2004, “Optimal Urban Land Use and Zoning,” Review of Economic Dynamics,
7:69-106.
36
APPENDIX
Proof of Proposition 1
The proof proceeds by contradiction. Suppose that both integrated and non-integrated …rms
exist in both areas of the city. Then by (2), it must be the case that
(22)
Fbb = Fcc = Fcb = Fbc :
The fact that Fcc = Fbb implies that
(AEc ¡ w ¡ Rc )®± = (AEb ¡ w ¡ R)®± or
(23)
AEc ¡ AEb = Rc ¡ R:
Similarly, the fact that Fcb = Fbc implies that
(AEc ¡ w ¡ R)®± = (AEb ¡ w ¡ Rc )®± or
(24)
AEc ¡ AEb = R ¡ Rc :
Equations (23) and (24) can only hold if Rc = R = R, in which case Ec = Eb = E. It follows
that pro…ts for an integrated …rm are (AE ¡ w ¡ R)®± while those of a non-integrated …rm are
(AE ¡ w ¡ R)®±. That is, Fcc = Fbb > Fcb = Fbc which contradicts (22).
Proof of Proposition 2
Simple partial derivatives imply that
@w
@P
@w
@Lc
and
=
A
= ¡D
A
D
³
> 0;
´
1 ¡ ±± < 0;
@w
@A
=
@w
@R
=
´
³
±
P ¡Lc 1¡
±
D
¡(2+®±+®± )
D
¡
¢
± + ± ¡ ®12 ± ¡
±+± R
@w
=¡
¡
w
@®
D2
D2
1
®2 ±
> 0;
< 0;
<0
The sign of the last term is guaranteed since the span of control of integrated and non-integrated
…rms is greater than ®± > ®± > 1:
Proof of Proposition 3
Employment in the center of the city is given by
Ec + Lc =
1+
1
®±
A
37
w+
R
+ Lc :
A
Di¤erentiating with respect to P , we obtain
1
1 + ®±
@ (Ec + Lc )
@w
=
> 0;
@P
D @P
which captures the increase in population at the center.
Total employment at the edge is given by
µ
¶
¡
¢
Lc
Eb + Lb =
Ec ¡
®± + Eb 1 + ®±
®±
¢
R¡
± 3 + ®± + ®± +
1 + ®± + ®± ¡ Lc +
=
A
A
±
so that
3 + ®± + ®± +
@ (Eb + Lb )
=
@P
D
1
®±
1
®±
w
> 0;
which yields the increase in population at the boundary.
Proof of Proposition 4
The employment share at the center is given by
Ã
!
1
1 + ®±
R
1
Ec + Lc
=
w + + Lc
:
P
A
A
P
Hence, the derivative with respect to population is
@ EcP+Lc
P @Ec ¡ (Ec + Lc )
= @P
;
@P
P2
where
It follows that
1
1 + ®±
@Ec
=
:
@P
D
+Lc
@ Ec P
@P
< 0 if and only if
@Ec
Ec + Lc
<
;
@P
P
or alternatively,
Ec + Lc
>³
P
1+
1
®±
´
1+
1
®±
+ 3 + ®± + ®± +
1
®±
:
This last condition can be re-written in terms of exogenous parameters only,
µ
¶
µ
¶
2
±
±
1
1
3 + ®± + ®± +
+
Lc > R
¡1+
¡
;
±
®± ®±
®± ±
in which case the share of employment at the center decreases with population growth.
38
Proof of Proposition 5
i)
Ec
Ec +Eb
¡
Lc
Lc +Lb
increases with P .
The share of managers at the center is given by,
1
(1 + ®±
)w + R
Ec
Ec
¢
¡
=
=
:
¤
1
1
Ec + Eb
w
)w + 2R
2Ec + A
(2 + ®±
+ ®±
Hence,
c
@ EcE+E
b
@P
R
=h
(2 +
h
1
®±
1
®±
¡
+
1
®±
i
1
)w
®±
@w
@P
+ 2R
i2 > 0:
which captures the increase in manager share at the center. The share of workers at the center is
given by
Lc
Lc
Lc
´
³
´ ³
=
=
±
®±+®±
Lc + Lb
W
Lc 1 ¡ ± +
R+
A
Therefore,
c
@ LcL+L
b
@P
=¡
µ
Lc 2 + ®± + ®±
W2
A
¶
2+®±+®±
w
A
:
@w
<0
@P
which captures the fall in worker share at the center. Thus,
c
@ EcE+E
b
@P
ii)
Ec
Lc
¡
Eb
Lb
¡
c
@ LcL+L
b
@P
> 0:
increases with P .
We analyze each term in turn. Since Lc is …xed and Ec increases with P ,
with P ,
c
@E
Lc
@P
=
1+
1
®±
Lc D
Ec
Lc
clearly increases
> 0:
To analyze the second term, note that Eb =Lb is given by
Eb
Lb
=
>
Hence,
b
@E
Lb
@P
=
<
E
´b
Lc
®± + Eb ®±
®±
³
´
1
1 + ®±
w+R
¡
¢
¡
¢
2 + ®± + ®± w + ® ± + ± R
³
Ec ¡
µ
¶
¢
1
Eb ¡
1+
¡
2 + ®± + ®±
®± Lb
Ã
!
2
1
±(2 + ®± + ®±)w + R(± ¡ ± + ®±± + ®± )
1¡
< 0;
DLb
±[(2 + ®± + ®±)w + ®(± + ±)R]
1
DLb
39
@
from which it immediately follows that that
iii)
(Ec +Eb )
P
³
E
Ec
¡ Lb
Lc
b
@P
´
> 0.
increases with P .
To see this, observe that
Ec
=
P
Ã
1
®±
1+
A
R
w+
A
!
1
:
P
Hence, the derivative with respect to population is
!
µ
¶¶ Ã
1 µ
1
1 + ®±
1 + ®±
¡
¢ R
Ec
Lc
±
=
2 + ®± + ®±
1¡
1¡
+ 1¡
P
D
P
D
AP
±
Ã
Ã
!
!
µ
¶
1
1
1 + ®±
1 + ®±
¡
¢ R
@ EPc
1
±
=
L
1
¡
2
+
®±
¡
1
¡
+
®±
c
@P
P2
D
D
A
±
0
0
1 1
µ
¶
1
2
3 + ®± + ®± + ®±
+ ±±
1 @ 1 + ®±
±
R
@
=
Lc 1 ¡
+
¡ 1A A > 0
1
1
2
P
D
A
±
4 + ®± + ®± + ®± +
®±
where the inequality follows from ± > ±. Moreover, note that
Eb
P
=
=
Hence,
1+
R
w+
PA
PA
µ
¶
¶
1 µ
1 + ®±
¡
¢ R
Lc
±
R
1¡
1¡
¡ 2 + ®± + ®±
+
:
D
P
PA
PA
±
0³
1+
1
= 2@
@P
P
@ EPb
so that
1
®±
b
@ Ec +E
P
@P
which is positive if
=
1
®±
´³
D
³
2+
0
+@
1¡
1
®±
±
±
´
Lc +
´³
1
®±
P 2D
+
Ã
1¡
3 + ®± + ®± +
4 + ®± + ®± +
±
±
6 + 2®± + 2®± +
´
Lc
2
®±
4 + ®± + ®± +
³
2
+ ±±
®±
1
1
®± + ®±
+
± ±
+ > 2;
± ±
or if
which holds trivially.
¡
¢2
2
± ¡ 2±± + ± 2 = ± ¡ ± > 0
40
2
+ ±±
®±
1
1
®± + ®±
+
±
±
´
!
¡1
1
1
RA
;
A
R
¡ 2A 2
P A
Proof of Proposition 6
The total number of establishments at the center is given by
It follows that
1
®±
1+
Sc = Ec + Ebc = Ec =
A
w+
R
:
A
1
1 + ®±
@Sc
=
> 0;
@P
D
which depicts the rise in establishments at the center. The number of establishments at the edge
is given by
Sb = Eb + Ecb = Eb + Ec ¡ Ecc
1
1
2 + ®±
+ ®±
R Lc
=
w+2 ¡
:
A
A ®±
Therefore
@Sb
@P
1
®±
2+
=
+
1
®±
+
1
®±
@w
@P
A
1
®±
2+
=
>0
D
which establishes the increase in establishments at the edge. It is also the case that
1
1
1 + ®±
2 + ®±
+
@Sc
=
<
@P
D
D
1
®±
=
@Sb
;
@P
so that the increase in the number of establishments is greater at the edge than at the center.
Finally, the change in the share of establishments with respect to population is given by
c
@ ScS+S
b
@P
(Sc + Sb )
=
=
1 @w
A @P
µ
¶
1
1+ ®±
A
@w
@P
h
¡ Sc A1 3 +
1
®±
+
(Sc +
1
)+
®±
Sb )2
Sb (1 +
The above expression is negative whenever
(1 +
Ã
1
®±
+1+
1
)
®±
1
®± )
!
1
+ ®±
1
+ ®±
(1 +
= Sc 1 +
1
1
41
< 2Sc :
2
®±
+
(Sc + Sb )2
h
¡Sc (2 +
Sb < Sc
1
®±
1
®± )
i
:
i
@w
@P
Thus, Sc > 12 Sb whenever
1
Sc
> :
Sc + Sb
3
This restriction, however, is always satis…ed since
µ 1 ¶
Sc
=
Sc + Sb
1
A
holds when
h
1+ ®±
A
3+
1
®±
w
Lc
>
A
®±
which is always the case since
1
®±
¡
1
®±
w+R
A
i
2
+ ®±
w + 3R
A ¡
µ
1
1
¡
®± ®±
¶
Lc
®±
1
3
>
;
< 0.
Proof of Proposition 7
Average establishment size at the center is given by
1+
1
Lc + A®± w +
Lc + Ec
Lc + Ec
=
=
1
1+ ®±
Sc
Ec
R
A w+ A
so that
R
A
;
1
c
1 + ®±
@ LcE+E
@Ec Lc
Lc
c
=¡
=¡
< 0:
2
@P
@P (Ec )
D (Ec )2
Hence, average …rm size at the center decreases with population growth. Average establishment
size at the edge is analogously given by
³
¡
¢
±
R
1
+
®±
+
®±
¡
AL
+
3 + ®± + ®± +
c±
Eb + Lb
³
´
=
1
1
Lc
Sb
2 + ®±
+ ®±
w + 2R ¡ A ®±
It follows that
b
@ EbS+L
b
@P
1 A
=
Sb D
1
®±
´
w
:
¶
¶¶
µµ
µ
1
Eb + Lb
1
1
3 + ®± + ®± +
¡
2+
+
<0
Sb
®± ®±
®±
when
3 + ®± + ®± +
2+
or
ALc
>¡
±®
2
Ã
1
1
®±
+
1
®±
1
®±
<
Eb + Lb
;
Sb
¡
¢2
±¡±+® ±¡±
2
+ ± ®2
¡ ±±®2 + 2±® ¡ ±®
!
R
;
±
which always holds since 1 + ± ®2 ¡ ±±®2 + 2±® ¡ ±® > 0: Consequently, average …rm size at the
edge also falls.
42
Proofs of Proposition 8
i) At the center, we have
1 @R
1
@(Ec + Lc )
=
= 0 > 0;
@P
A @P
D
where D0 = 2 + ®± + ®±: With respect to the edge, we have
¡
¢
Eb + Lb = 1 + ®± Eb + ®±Ec ¡ ®±Ecc :
Therefore,
¢ @Eb
¢
@(Eb + Lb ) ¡
@Ec
1 ¡
= 1 + ®±
+ ®±
= 0 1 + ®± + ®± > 0
@P
@P
@P
D
ii) Taking the derivative of
@ EcP+Lc
@P
Ec +Lc
P
with respect to P yields
µµ
¶
¶
1
1
1
= ¡
1+
w + R + ALc +
2
AP
®±
P D0
´
´
3
2 ³³
1
0
1 4 1 + ®± w + R + ALc D ¡ AP 5
= ¡
;
P
AP D0
which is strictly negative when
µµ
1
1+
®±
¶
¶
w + R + ALc D0 ¡ AP > 0:
The latter condition holds if and only if
·µ
¶
µ
¶¸
·
µ
¶¸
1
1
1
±
0
0
w 1+
+
D ¡ 4 + ®± + ®± +
+ ALc D ¡ 1 ¡
> 0;
®±
®± ®±
±
or alternatively if
µ
¶
1
1
1
1+
D0 > 4 + ®± + ®± +
+
®±
®± ®±
and
µ
¶
±
D > 1¡
;
±
0
which both hold since ± > ±:
iii)
a)
Ec
Ec +Eb
¡
Lc
Lc +Lb
increases with P .
Simple derivation gives
c
@ EcE+E
b
@P
³
1+
= h³
2+
1
®±
1
®±
´
w @R
@P
´
i2 > 0
1
+ ®±
w + 2R
43
and
c
@ LcL+L
b
@P
Therefore,
Lc
=¡
W
c
@ EcE+E
b
µ
¡
@P
®± + ®±
A
c
@ LcL+L
b
@P
¶
@R
< 0:
@P
> 0:
b) Ec =Lc ¡ Eb =Lb increases in P .
We have that
@
³
Ec
Lc
¡
Eb
Lb
@P
´
1 @Ec
1
=
¡ 2
Lc @P
Lb
But since @Ec =@P > 0, it is su¢cient to show that
µ
¶
@Lb
@Eb
Lb ¡
Eb :
@P
@P
@Eb
@Lb
Lb ¡
Eb < 0:
@P
@P
(25)
Note that Lb = (Ec ¡ Ecc ) ®± + Eb ®± so that @Lb =@P = ®±@Ec =@P + ®±@Eb =@P . Then the LHS
of (25) becomes
µ
¶
¢
@Eb
@Eb ¡
@Ec
®±Ec ¡ ®±Ecc + ®±Eb ¡ Eb ®±
+ ®±
@P
@P
@P
µ
¶
@Eb
@Eb
@Ec
= ®±
Ec ¡
Ecc ¡
E
@P
@P
@P b
®±
(Ec ¡ Ecc ¡ Eb )
=
D0
since @Ec =@P = @Eb =@P = 1=D0 > 0. It then follows that (25) holds if Ec ¡ Ecc ¡ Eb < 0, or
equivalently, Ecb < Eb , which holds whenever
Eb
1
> ;
Sb
2
since Sb ¡ Eb = Ecb .
c)
Ec +Eb
P
increases with P .
Ec + Eb
1
=
P
AP
µµ
¶
¶
1
1
2+
+
w + 2R :
®± ®±
Thus,
b
@ Ec +E
P
@P
=
=
µµ
¶
¶
2
1
1
1
¡
2+
+
w + 2R
P D0 AP 2
®± ®±
µ
¶
1
2
Ec + Eb
¡
:
P 2 + ®± + ®±
P
44
If more than half of the …rms in the economy are integrated, (Ec + Eb ) =P < 1=(1 + ®±=2 + ®±=2).
Hence the above expression is positive if
2
1
¸
2 + ®± + ®±
1 + ®±=2 + ®±=2
which is trivially satis…ed with equality. Observe also that when Eb =Sb > 1=2, and since Ec = Sc ,
then
1
1
1
Eb > Sb , Eb + Ec > (Sb + Sc ) + Sc
2
2
2
so that
Eb + Ec
1 1 Sc
1
> +
> :
Sb + Sc
2 2 Sb + Sc
2
iv). The total number of establishments at the center is given by Sc = Ec + Ebc = Ec : Thus, we
have
Sc =
1
®±
1+
A
w+
R
:
A
Similarly, the total number of establishments at the edge is
·
¸
1
1
Lc
1
2
Sb = Eb + Ecb =
2+
+
w+ R¡
:
A
A
®± ®±
®±
(26)
(27)
It follows that
µ ¶
@
1
R
= 0 > 0;
@P A
D
and
µ
¶
@
2
2
=
R = 0 > 0;
@P A
D
@Sc
@P
=
@Sb
@P
which also implies that @Sb =@P > @Sc =@P: To show that the share of establishments in the center
declines with population, observe that
c
@ ScS+S
b
@P
=
@R
@P
@R
(Sc + Sb ) ¡ 3 @P
Sc
A (Sc + Sb )
2
=
Sc + Sb ¡ 3Sc
<0
D0 (Sc + Sb )2
when
Sc
1
> ;
Sc + Sb
3
which is always the case as shown in Proposition 6.
v) The average size of establishment at the center is given by
45
Lc
Ec + Lc
=1+
:
Sc
Ec
Therefore,
³
@ 1+
@P
Lc
Ec
´
µ
Lc =D0
=¡
Ec2
The average size of establishment at the edge is given by
¶
< 0:
Eb + Lb (1 + ®±)Eb + ®±Ecb
=
:
Sb
Eb + Ecb
Di¤erentiating with respect to P yields
b
@ EbS+L
b
@P
=
³
´
@Ecb
b
(1 + ®±) @E
(Eb + Ecb )
@P + ®± @P
¡
=
³
@Eb
@P
@Eb
@P
S2
´¡b
¢
cb
+ @E
(1 + ®±)Eb + ®±Ecb
@P
Sb2
£
¤
(1 + ®±)Ecb ¡ ®±Ecb ¡
Sb2
@Ecb
@P
£
¤
(1 + ®±)Eb ¡ ®±Eb
which is negative when
@Ecb
@Eb
Ecb <
Eb ;
@P
@P
where
@Eb
@Ecb
1
=
= 0:
@P
@P
D
Therefore (28) reduces to Ecb < Eb , which holds when Eb =Sb > 1=2:
46
(28)
@FedFRASER