Optimal Spatial Growth of
Employment and Residences
Written by Ralph Braid
Journal of Urban Economics 24(1988)
Presented by Jing Zhou
1. Introduction
• 1.This paper examines the location patterns of employment and
residences in an urban area characterized by irreversible land use
commitments and smoothly growing population.
• 2. The main problem here is how to use the land efficiently. That is,
how to divide the land between employment and residential use to get
maximum economic return. Should we use land by complete
integration, complete segregation or a mixture of different uses?
• 3. Assume all output is shipped to the center and there’s commuting
costs for workers to go to employment areas.
• 4. The optimal solution should minimize the PDV of aggregate
shipping costs plus commuting costs
2. Basic Assumptions of the Model
• 1. Population increases over time, and at time t, is N(t)
• 2.Each unit of output is produced with 1 unit of labor and a
fix amount of land equal to 1/a1(C.R.S)
• 3. Output is maximized and equal to N(t). So at any given
time, output is a constant
• 4. All of the output is shipped to the center of the urban
area for export and shipping cost is K1 per mile.
• 5. Each resident has a completely inelastic demand for 1/a2
units of residential land and has commuting costs of K2
per mile. K2 should be no less than 0( no outcommuting)
3. Equilibrium for Residential and
employment land use
• Assume the land can be only used for two purposes: employment and
residential
• Let f(x)=fraction of land at distance x devoted to residential use
• 1-f(x)=fraction of land at distance x devoted to employment use
• b(t)=employment boundary(beyond boundary employment land
development has not proceeded at time t)
• B(t)=residential boundary
• So we have
•
b(t ) 2x(1 f ( x))dx  N (t ) / a

•
(1)
0
1
B(t )
•
(2)
0 2xf ( x)dx  N (t ) / a2
4.Shipping and Commuting Cost
• Total shipping costs of output to the center at time t are:
TSC (t )  b(t ) 2x(1 f ( x))a (k x)dx
•
(3)
0 
1 1
• Total commuting costs are:
•
TCC (t )   B(t ) 2xf ( x)a (k x)dx  b(t ) 2x(1 f ( x))a (k x)dx
0 
0 
2 2
1 2
(4)
• The first term on the right-hand side is total commuting
costs if all consumers had to commute to the center and the
second term corrects for the fact that employment locations
are generally not at the center
5. Optimal Condition for Land Use
• 1. Since output is a constant at any given time. The optimal
problem here is to minimize the TSC(t)+TCC(t) over time.
This is equal to find the minimum PDV of TSC(t)+TCC(t)
• 2. PDV of TSC(t)+TCC(t) is(note 1+r is approximately er):
J  0 ert (k  k )(2a )b(t ) (1 f ( x)) x 2dx  k (2a ) B(t ) f ( x) x 2 dxdt
2
1 0
2
2 0
 1
•
(5)
• 3. SO the problem is to find f(x), b(t) and B(t) that can
minimize J subject to (1) and (2)
5. Optimal Condition(2)
• After some first-condition and variance change, finally we get:
• Where Tb(x) is the employment development time at distance x and
TB(x) is the residential development time at distance x
 (b(t )  x)ert dt  k a 
0  (k  k )a Tb
(B(t )  x)ertdt
1 2 1 ( x)
2 2 TB( x)
•
(11)
• This is the optimal condition for land use. It is intuitive. Suppose one
unit of land at distance x is switched from residential to employment
use.The first term is the PDV of savings in shipping costs. The second
term gives the PDV of dissavings in commuting costs. These two must
be exactly equal to each other otherwise the first order condition is
violated.
6. The Solution for The uniform Growth Path
• Suppose that population grows uniformly at a constant percentage rate
N (t )  N ent
n, so that
(12)
0
• Assume the fraction devoted to employment use is f tentatively
• Then input(12) into (1) and (2), we get
b(t )  b ent / 2
0
B(t )  B ent / 2
0
Tb( x)  (2 / n) ln( x / b )
0
TB( x)  (2 / n) ln( x / B )
0
• Where b and B are the employment boundary and residential boundary
at time 0
0
0
6. The Solution for The uniform Growth Path(2)
• Input all the above into equation (11), finally ,we will get
n / 2r
B / b  (k  k )a / k a 
0 0  1 2 1 2 2
f  (a / a )(b / B )2 / 1 (a / a )(b / B 2 )
1 2 0 0 
1 2 0 0 
The first equation gives us the ratio of the
residential boundary to the employment boundary.
The second equation gives us the fraction of land
that is reserved for residential use at each distance
from the center. It is independent of x.
6. The Solution for The uniform Growth Path(3)
• Now consider the relationship between k2*a2 and (k1k2)*a1
• 1.If k2*a2=(k1-k2)*a1, B0=b0, f=a1/(a1+a2)
• At this situation, the residential boundary and employment
boundary are identical, so there’s complete integration of
employment and residences and no commuting occurs.
• 2. If K2*a2>(k1-k2)*a1, we get B0<b0.
• There will be outcommuting. Impossible!
6. The Solution for The uniform Growth Path(4)
• Consider when K2*a2<(k1-k2)*a1, the situation this paper
focuses on, we get
• B0>b0, which means B(t)>b(t)
• And f<a1/(a1+a2)
• At each moment, residential development occurs within
the residential boundary, but a constant fraction of the land,
1-f, is reserved for future employment.
• Employment development occurs within the employment
boundary and fills in the land that was left undeveloped
during the earlier residential development
So, what the optimal land use should be at any
given time?
• For an urban land, at any given time, it should have a
employment boundary and a residential boundary.
• The employment boundary should be inside the residential
boundary
• Land within the employment boundary is a mixture of
employment use and residential use.
• Land between the employment and residential boundaries
should be a mixture of vacant and residential use.
• Land outside the residential boundary is completely
undeveloped.
7. Some comparative static analysis
• As k1/k2 increases, B0/b0 increases, f decreases. This is intuitive
because when shipping cost is much greater than commuting cost,
expanding the employment boundary is inevitable in order to save total
shipping cost. Also people will reserve a larger part of land for future
employment development to lower the total shipping costs
• As r increases, B0/b0 decreases, f increases. As r approaches infinity,
B0/B1 approaches 1, we get completely integrated. This is true
because very high interest rate mitigates against withholding any land
from current development since current costs are weighted much more
heavily than future costs
• As n increases, B0/b0 increases and f decreases. This is true because
people need more land for future population increase.
8. Conclusion and future work
• Conclusion: The residential area expands outward
over time, but a constant fraction of the land is
always reserved for employment development
because of the existence of the shipping costs
• Future extension work:
• 1. A model of growing urban area with two
income groups. The result is similar( Braid, 1989)
• 2. Allow residential land to be converted to
employment land at some costs
Comments
• This paper provides a convincing mathematical
explanation for a urban development model. But there’s no
data evidence provided.
• This paper is based on the assumption of C.R.S. But it is
true that in real life, many industries show the I.R.S. So the
benefit of converting residential land to employment
should not only include the saving on shipping costs but
also the I.R.S. benefits
• Another thing is the social costs, more specially, the
pollution. When considering the fraction devoted to
employment use, the increase in social costs should also be
considered. In other word, even at C.R.S condition, output
may be be equal to N(t)