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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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 2 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 3 / 33
City Structure
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 4 / 33
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.
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 5 / 33
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.
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 6 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 7 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 8 / 33
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 ].
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 9 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 10 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 11 / 33
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))
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 12 / 33
Wage Paths
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 13 / 33
Wage Paths
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 14 / 33
Wage Paths
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 15 / 33
Equilibrium Given ProductivityUnique wage path satis�es
H(S) = 0
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 16 / 33
Example
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 17 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 18 / 33
Equilibrium
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 19 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 20 / 33
Numerical Examples
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 21 / 33
Numerical Examples
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 22 / 33
Numerical Examples
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 23 / 33
Eeckhout (2004)
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 24 / 33
Gibrat�s Law
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 25 / 33
Another Look
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 26 / 33
Another Look
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 27 / 33
Right Tail Close to Pareto
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 28 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 29 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 30 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 31 / 33
City SizeThis implies that
Si ,t�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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 32 / 33
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
ERH (Princeton University ) Lecture 6: Urban Structure and Growth Spring 2011 33 / 33