1

Infrastructure Investment and Skill Distribution in a System of Cities

Hesham M. Abdel-Rahman

University of New Orleans

Department of Economics and Finance

2

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.

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

4

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)

5

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)?

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

7

(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.

8

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.

9

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.

10

The Model

• one sector, spatial general-equilibrium model. • The final good X is produced with skilled and unskilled workers.• 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.

NNU

NN S )1(

)1,0(

11

•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 by where NMS and NMU are the quantity of skilled and unskilled worker in the mixed city, respectively.

)/(M MUMS NNF

12

•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.

13

iii xFGU )(ijjjji WrRtrrx )()(

2

0

2 fj

f

jj fdrrNfj

2/3

0

2)( j

r

jjj NdrrrRALRfj

3/2*2/3 )/( jjjjj FNFN

Households and City formation:

City size

3/13/23)( jijjij NWFGV

14

Production sector:

Assumption 1.

ijjijj NF =X i

ijijij F=W

MUSjFFFfFV jjjjjiji ,,3)()( 3/13/2*

Household Utility:

00 USUS and

Assumption 2.

13/313/23 i

ijF

15

F*SF*U

V

F

Lemma 1i) 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

18

Equilibrium

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.

Equilibrium system of cities is defined by

0,,,{ **** ijji VXmN }},{},,,{ SUiMUSj ** ),({ ijj WrR }},{},,,{,; SUiMUSjrr fj

19

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.

.

20

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.

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 .

SMUM FandFFF

SMUM FandFFF )1/(M F

21

Figure 2. Parameter space for Lemma 1 (iii) Parameter Space

FU /FM

FS /FM

C-P

I

A

B

C De

f

g

45O

22

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), , 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

SM FF

23

Figure 3. Parameter space for Lemma 1 (ii) Parameter Space

FU /FM

FS /FM

I

C-P

A

B

C De

f

g

45O

24

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 .

25

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 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.

26

UUSS VNVN Social welfare function

27

FU /FM

Figure 4. Parameter space for Lemma 1(i) (iii)

FS /FM

I

C-P

C

D

A=[1/(1-β)]3

B=[1/β]3

G

K