Infrastructure Investment and Skill Distribution in a
System of Cities
Hesham M. Abdel-Rahman
University of New Orleans
Department of Economics and Finance
1
Income disparities
• Several studies have indicated that income
disparity the U.S. has been raising during the past
two decades, Murphy and Welch (1993).
• Machin (1996) showed that income inequality has
been rising in the UK within group as well as
between group.
Skills in cities
Bacolod, Blum, and Strange (2009) that large cities
have more skill concentration than small.
2
Productivity in cities
• Infrastructure provision makes the workers more
productive in cities relatively compared to home
production [Marshall (1920)]
• It has been documented empirically that worker
are more productive in cities than in ruler areas
[Baum-Snow and Pavan (2009]
• Moretti (2004) documented empirically that
productivity is higher in cities with larger
concentrations of college graduates.
3
Investment in Infrastructure
• Public good affect individual utility.
• Enhancing accessibility .
• Education and labor force training.
Urban system and skill distribution
Abdel-Rahman and Wang (1995, 1997)
Abdel-Rahman (2002)
Abdel-Rahman (2010)
4
Questions to be addressed
1. What are the factors that determine the sizes and
skill distribution?
2. What are the factors that determine income
disparity within a system of cities?
3. Would the allocation of skilled and unskilled
workers by a central government in a system of
cities bet different from allocation though self
selection by households (self-organization)?
5
The model attempts to explain four
equilibrium configurations
(i) core-periphery, or sorting equilibrium, in
which one type of city is populated with
skilled workers, S, while the other type is
populated with unskilled workers, U;
(ii) integrated, or heterogeneous equilibrium,
in which skilled and unskilled workers
collocate in the same city, M;
6
(iii) mixed system, in which both integrated
cities, M, and homogenous cities of either
type S or U coexist in the economy;
(iv) completely mixed equilibrium, in which
we have integrated cities of type M as well as
homogenous cities of S and U types.
7
Main finding
1. Formation of cities of different sizes is the result
of investment in different types of infrastructure.
2. Sorting equilibrium will arise because
workers/households have different levels of
utilization and preferences for local public goods.
3. Skilled and unskilled workers collocate in the
same city if there is economy due to joint
provision of infrastructure.
8
4. Productivity improvement due to knowledge
spillover among skilled and unskilled workers in
cities results in the collocation in an integrated
or heterogeneous city.
5. Yes the allocation of skilled and unskilled
workers by a central government will be
different from self selection.
6. The model also characterizes two types of
income disparities that arise in various
equilibrium configurations. It has been shown
that depending on the observed equilibrium
configuration, the model results in different intra
and inter-regional income disparities.
9
The Model
• one sector, spatial general-equilibrium model.
• The final good X is produced with skilled N S  (1   ) N
and unskilled NU   N workers.   (0,1)
• The economy consists of a system of cities, j=S,
U, and M.
• Cities are formed by a profit maximizing
developers (fixed set up cost).
• Households have identical preferences in the only
final good X.
• Each household is endowed with one unit of time.
10
•If local government provides infrastructure that is used
by skilled workers, FS, the city will be populated with
only skilled workers and will be called city type S. On
the other hand, if the local government provides
infrastructure that is used by unskilled workers, FU, the
city will be populated with unskilled workers and will
be called unskilled city, U. Finally if the local
government provides infrastructure that can be used by
both skilled and unskilled workers, FM, the city will be
called mixed city M.
•It is assumed that the cost of infrastructure required for
the formation of mixed city is given byFM   ( N MS / N MU )
where NMS and NMU are the quantity of skilled and
unskilled worker in the mixed city, respectively.
11
•Diversity of skilled workers in a given city results in higher
productivity. Thus, skilled worker acquire knowledge through
interaction and exchange of information with diversified group
of skilled workers in the city.
12
Households and City formation:
U i  G( Fi ) xi
xi(rj )  trj  R(rj )  Wij
f fj
N j   2rj dr  f fj
2
r0fj
ALR j   R(rj )2rj dr   N j
3/ 2
0
City size
 j  N j
3/ 2
 Fj  N *j  ( Fj /  ) 2 / 3

Vij  G( Fj ) Wij  3 2 / 3 Nj1/ 3

13
Production sector:
i
j
X j =  ij F Nij
Assumption 1.
 S  U  0 and  S  U 0
i
Wij =  ijFj
Household Utility:

i
j
V ( F )  f ( F j )  j F  3 j F j
*
ij
Assumption 2.
2/3

F j  3

2/3
1/ 3

j  S ,U , M

1 3 / 3 i 1
i
14
V
F*U
F*S
F
15
Lemma 1
i) If
and given Assumption 1, the
function Vij (F) is increasing in F. No public
good but strong positive productivity due to
infrastructure.
i) If
and given Assumption 1, the
function Vij (F) is strictly concave in F. No
public good but weak strong positive
productivity due to infrastructure.
16
Lemma 1. continued
iii)If
and given Assumption 1, the
function Vij (F) is decreasing in F. No public
good and no productivity due to
infrastructure.
iv)If
and given Assumption 1,
the function Vij (F) is strictly concave in F.
No public good and no productivity due to
infrastructure.
17
Equilibrium
Equilibrium system of cities is defined by
{Ni* , m*j , X * ,Vij*  0 j  {S , U , M }, i  {U , S }}
{R* (r ), W * ; r  rfj , j  {S ,U , M }, i  {U , S}}
j
ij
1.
Profit maximization for firms in the final good;
2.
Zero profit for all firms in all sectors;
3.
Zero profit for developers;
4.
Workers will reside in the city that provides the highest
utility.
18
Result 1: The equilibrium city size, Nj*, is larger the smaller the
transportation rate, t, and the greater the infrastructure
investment, Fj .
Result 2: The equilibrium utility level, Vij*, is larger the smaller
the transportation rate, t, and the greater the value of δi .
Result 3: The equilibrium relative number of cities is larger the
smaller the values of FS and β and the grater the value of FU.
Intra-regional income disparity is increasing in the productivity of
the skilled worker and decreasing in the productivity of the
unskilled workers. Furthermore, increasing in the provision of
infrastructure for the skilled workers and decreasing in the
infrastructure for the unskilled workers.
19
Theorem 1. Given a set of parameters that satisfy
Assumptions 1 and 2 and condition (13), there exists a unique
core-periphery equilibrium system of cities where the core is
populated with skilled workers while the periphery is
populated with unskilled workers.
FM  FU andFM  FS
Theorem 2. Given a set of parameters that satisfy
Assumptions 1 and 2 and condition (15), there exists a unique
integrated equilibrium system of cities, where all cities are
populated with skilled and unskilled workers if FM   ( / 1  .)
FM  FU andFM  FS
20
FS /FM
45O
B
f
I
g
e
D
C
C-P
FU /FM
A
Figure 2. Parameter space for Lemma 1 (iii) Parameter Space
21
Theorem 3. Given a set of parameters that satisfy Assumptions 1
and 2 and conditions (17), (18), there exist a unique equilibrium
of mixed-U system of cities, where some cities are populated with
skilled and unskilled workers while other cities are populated with
only unskilled workers, only if the indirect utility satisfy Lemma
1(ii) and (iv) for at least the unskilled workers.
THEOREM 4. Given a set of parameters that satisfy Assumptions
1 and 2 and conditions (19), FM  FS, and δM > δU there exist a
unique equilibrium of mixed-U system of cities where some cities
are populated with skilled and unskilled workers while other cities
are populated with only unskilled workers
22
FS /FM
45O
B
f
C-P
g
e
C
D
I
FU /FM
A
Figure 3. Parameter space for Lemma 1 (ii)
Parameter Space
23
Result 3: The equilibrium relative number of cities is
larger the smaller the value of Ф, FU and the
greater the value of β and FM .
Result 4: The equilibrium income disparity under the
mixed-U is larger the smaller the value of δU , FU
and the greater the value of δM , δS , and FM .
24
THEOREM 5. Given a set of parameters that satisfy
Assumptions 1 and 2 and conditions (21), (22), there exists a
unique equilibrium of mixed-S system of cities, where some
cities are populated with skilled and unskilled workers while
other cities are populated with only skilled workers, only if
the indirect utility satisfies Lemma 1(ii) and (iv) for at least
the unskilled workers.
THEOREM 6. Given a set of parameters that satisfy
Assumptions 1 and 2 and conditions (23), (24), there exists
multiple equilibrium of completely mixed system of cities,
where some cities are populated with skilled and unskilled
workers while other cities are populated with only unskilled
workers and other are populated with skilled, only if the
indirect utility satisfy Lemma 1(ii) for at least the unskilled 25
workers.
EFFICIENT CONFIGURATIONS
A central government whose objective is to choose the
distribution of workers in cities to maximize the social
welfare function, given the investments on
infrastructures.
Social welfare function
  N SVS  NUVU
26
FS /FM
C
A=[1/(1-β)]3
G
I
C-P
K
FU /FM
D
B=[1/β]3
Figure 4. Parameter space for Lemma 1(i)
(iii)
27