lecture 6: urban structure and growth - princeton universityerossi/macro/lecture_6_522.pdf ·...

Lecture 6: Urban Structure and GrowthEconomics 522

Esteban Rossi-Hansberg

Princeton University

Spring 2011

ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 1 / 33

Lucas and Rossi-Hansberg (2002)

Analyzes the economic forces that determine the internal structure of cities

Circular city where a single good is produced using land and labor

Basic elements:I People consume goods and residential landI There is a production externalityI Workers face commuting costs


The Model

The total land area of the city, πS2, is divided between production use andresidential use

We describe locations within the city by their polar coordinates, (r , φ)

For any location r , letI θ(r ) be the fraction of land used for production.I n(r ) be the employment density� employment per unit of production land� atlocation r

I N(r ) be the number of workers housed at r , per unit of residential landI `(r ) units of land per personI c(r ) consumption per person


City Structure


Agglomeration

The production externality is assumed to be linear, and to decayexponentially at a rate δ with the distance between (r ,0) and (s,φ):

z(r) = δZ S0

Z 2π

0sθ(s, φ)n(s, φ)e�δx (r ,s ,φ)dφds,

where

x(r , s, φ) =hr2 � 2 cos(φ)rs + s2

i1/2.

Since allocations are symmetric, we can write

z(r) =Z S0

ϕ(r , s)sθ(s)n(s)ds,

where

ϕ(r , s) = δZ 2π

0e�δx (r ,s ,φ)dφ.


Commuting CostsA person employed at r and living at s must travel the distance j r � s jtwice daily. Assume that such a worker supplies

e�κjr�s j �= 1� κ j r � s j

hours of labor at location r

H(r) is the stock of workers that remain unhoused at r , after employmentand housing have been determined for locations s 2 [0, r)Let

y(r) = 2πr [θ(r)n(r)� (1� θ(r))N(r)],

then

dH(r)dr

= y(r) + κH(r) if H(r) > 0,

dH(r)dr

= y(r)� κH(r) if H(r) < 0,

H(0) = 0 and

H(S) � 0.


Land Use

LetS+ = clfr 2 [0,S ] : H(r) > 0g

(where clfAg denotes the closure of A),

S� = clfr 2 [0,S ] : H(r) < 0g

andS0 = fr 2 [0,S ] : H(r) = 0 and y(r) = 0g


Firm Problem and Household Problem

Firm Problem at location r

q(r) = g(z(r))f (n(r))� w(r)n(r)= max

nfg(z(r))f (n)� w(r)ng

where w(r) is the wage rate and q(r) is the business bid rent. Letn(w(r), z(r)) and q(w(r), z(r)) be the maximizing values

Household�s problem at r

w(r) = c(r) +Q(r)`(r) = minc ,`[c +Q(r)`],

subject toU(c , `) � u,

where Q(r) is the residential bid rent. Let N(w(r)) and Q(w(r)) be themaximizing values


Land use and Wages

Land use: It will be assumed that land is allocated to its highest-value use.In context, this means that

θ(r) > 0 implies q(r) � Q(r),

andθ(r) < 1 implies q(r) � Q(r).

Wage no arbitrage condition: Free mobility of labor implies a sharprestriction on equilibrium wages w(r):

e�κjr�s jw(s) � w(r) � eκjr�s jw(s)

for all r , s 2 [0,S ].


Equilibrium

An equilibrium is a pair of piecewise continuous functions θ and y , and acollection (z , n,N,w , q,Q,H) of continuous functions, all on [0,S ] such thatfor all r ,

1 w (r ) satis�es wage no arbitrage condition,2 n(r ) = n(w (r ), z(r )) and q(r ) = q(w (r ), z(r )),3 N(r ) = N(w (r )) and Q(r ) = Q(w (r )),4 θ(r ), q(r ), and Q(r ) satisfy 0 � θ(r ) � 1, land use conditions5 y (r ), n(r ),N(r ), θ(r ) and H(r ) are constructed as above6 H(S) = 0, and7 z , θ, and n satisfy the production externality equation


Assumptions

The functions n, q : R2+ ! R+ and N, Q : R+ ! R+, are continuouslydi¤erentiable. Both n and q are decreasing in w and increasing in z ; both Nand Q are increasing in w

The �rst derivative nz (w , z) satis�es

limz!∞

nz (w , z) = 0,

andlimz!0

nz (w , z) = ∞,

for all w > 0.

The function U is homogeneous of degree one, and U(c , 1) is strictlyincreasing in c


Existence of Equilibrium

TheoremThere exist a unique allocation that satis�es (1)-(4)

The wage pathw(r) = Ke�κr if r 2 S+,w(r) = Keκr if r 2 S�,wm(r) = wm(z(r)) if r 2 S0.

where wm(z(r)) solves

q(w(r), z(r)) = Q(w(r))


Wage Paths


Equilibrium Given ProductivityUnique wage path satis�es

H(S) = 0


Example


Equilibrium

An equilibrium has to satisfy

(Tz) (r) =Z S0

ϕ(r , s)sθ(s; z)n(s; z)ds

T maps nonnegative valued functions into nonnegative valued functions

T is continuous in the sup norm as w (r ; z) is continuous in z

w (r ; z) is increasing in z (point by point)

If z (r) < z all r then (Tz) (r) < z

Then by Shauder�s �xed point theorem equilibrium there exist a function z�

such that z� = Tz�

TheoremAn equilibrium exists


Equilibrium


Numerical Examples

Functional forms:

U(c , `) = cβ`1�β

g(z) = zγ

f (n) = nα

Calibration: α = .95, β = .9, γ = .04, A = u = 1, S = 10. In order forassumption A to hold we need

0 < γ < 1� α.

Results are very sensitive to κ

Increasing δ concentrates production areas


Numerical Examples


Eeckhout (2004)


Gibrat�s Law


Another Look


Right Tail Close to Pareto


A Simple Theory

Let there be a set of locations (cities) i 2 I = f1, ..., IgEach city has a continuum population of size Si ,tTotal country-wide population is S = ∑I Si ,tAll individuals are in�nitely lived and can perform exactly one job

Ai ,t is the productivity of city i at time t with

Ai ,t = Ai ,t�1 (1+ σi ,t )

where σi ,t is an exogneous productivity shock

Denote by σt the vector of shock by all cities

Shock is symmetric, iid, mean zero and 1+ σi ,t > 0

No aggregate growth in productivity


A Simple Theory

The marginal product of a worker is given by

yi ,t = Ai ,ta+ (Si ,t )

a0+ (Si ,t ) > 0 is the positive external e¤ect

Denote the wage by wi ,t , then wi ,t = yi ,t as �rms are competitive

Large cities have higher wages

Workers have one unit of time and work li ,t 2 [0, 1]Some work is lost because of commuting, so productive labor is

Li ,t = a� (Si ,t ) li ,t

where a� (Si ,t ) 2 [0, 1] and a0� (Si ,t ) < 0 is a negative external e¤ect


Consumer Maximization

Land in a city is �xed at H

Price of land given by pi ,t and an individuals consumption of land by hi ,tConsumers and �rms are perfecty mobile

Consumer solve

max u (ci ,t , hi ,t , li ,t ;Si ) = cαi ,t , h

βi ,t (1� li ,t )

1�α�β

s.t. ci ,t + pi ,thi ,t � wi ,tLi ,t

Perfect mobility implies that

u� (Si ,t ) = U all i , t

and so

Ai ,ta+ (Si ,t ) a� (Si ,t ) S� β

αi ,t � Ai ,tΛ (Si ,t )

is constant across cities


City SizeThis implies that

Si ,tΛ�1 (Ai ,t ) = K

Si ,tΛ�1 (Ai ,t�1 (1+ σi ,t )) = K

So Λ0 < 0 implies thatdSi ,tdσi ,t

> 0

If Λ is a power function

Λ�1 (Ai ,t ) = Λ�1 (Ai ,t�1)Λ�1 (1+ σi ,t )

So

Si ,t =K

Λ�1 (Ai ,t�1)Λ�1 (1+ σi ,t )

=K

Λ�1 (1+ σi ,t )Si ,t�1

� (1+ εi ,t ) Si ,t�1


Gibrat�s Law and the Size Distribution

Taking natural logarithms and letting, ln (1+ εi ,t ) � εi ,t for εi ,t small,

lnSi ,t � lnSi ,t�1 + εi ,t

and so

lnSi ,T � lnSi ,0 +T

∑t=1

εi ,t

But then, since shocks, are iid the Central Limit Theorem implies that

lnSi ,T � N


lecture 6: urban structure and growth - princeton universityerossi/macro/lecture_6_522.pdf ·...

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