Add-Ons to the Standard
Mono-Centric Urban Model
including the switch from Mono to Poly-Centric
Daniel Gat
Faculty of Architecture and Town
Planning
Center for Urban and Regional Studies
Technion – Israel’s Institute of Technology
Prepared for the Annual Meeting of
The Israeli Regional Science Association
at Haifa Universityon the 21st of June 2005
Copyright © 2003 by Daniel Gat
Purpose of this Presentation
• To show a version “x” of the urban monocentric
model that is widely used
• To compare it to a competing version “y” that I
consider is better
• To point out that “y” is better since it accounts for
important relationships ignored by “x”
• To unfold the simplest derivation of model “y” while
exposing these important relationships
• To point out the needed improvements (add ons)
• To derive some of the add-ons
• To show the switch from Mono to Poly
2
The “x” version
• Beginning with alonso, countless studies of the urban
spatial economy start with the consumer’s welfare
function; it is a function of
– the consumed housing services,
– other consumption after rent and after travel expenses
– leisure after travel time (used by Alonso but optional)
• This widely used model equates housing with land
• As the presentation unfolds you will see that this
specification spells trouble:
• It is totally incapable of handling the important relationship:
Floor-area Rent and Value / Foor-area Ratio (FAR) / and
Optimal Land Value.
3
The “y” version
• Beginning with Muth who brought in the the supplyside of housing, some authors have substituted floorarea for land as the measure of housing
• This seemingly petty improvement is in fact
a Quantum Step
• It enables derivation of the close and yet subtle
relationship between Floor-area and Land
• These relations are of key importance to those who
strive to understand the free-market as well as the
planning-constrained spatial structure of the city
• According to my experience over the stretch of my
professional and academic career – few people do
4
Structure of Presentation
1.
motivation for theoretical study of cities:
to explain stylized facts
2.
foundation: standard mono-centric model (housing)
3.
add-ons to the standard model i (housing)
4.
5
1.
time to market and the cost of capital
2.
quality of location
3.
urban re-development and optimal FAR
4.
traffic congestion
5.
people congestion
6.
non-housing landuse
extension from Mono to Poly-Centric
Copyright © 2005 by Daniel Gat
stylized facts about cities
•
The further away you live from the center of the city (other things
being equal – they never are…)
–
–
–
The less you pay for housing (for a unit of floor area)
The bigger the size of your home
The bigger the size of the plot your home stands upon
•
When consumers offer higher rent (per square foot) for a home,
builders will build at a higher floor area ratio. This is likely to occur
close to centers or to urban amenities.
•
Land value is monotonically related to floor-area rentals and prices,
and it acts as a highly leveraged option on the underlying real estate
prices.
•
Quality pays. Not just urban centers, but other attractors too
(shoreline, park-edge, etc.) will work to increase rentals & prices as
well as land values and FAR. Perceived and true low quality will
depress rentals and values, and discourage urban renewal.
6
Copyright © 2003 by Daniel Gat
stylized facts about cities (continued)
•
Large cities are often associated with high rents, prices, land values, FAR
and densities, as well as with a higher per-capita income. How are all these
related to each other?
•
Urban Re-Development, where and when it occurs through market forces, is
characterized by a major leap of FAR. This reflects a major leap in Land
Values
•
Small changes in Rentals and floor-area values are leveraged into Big
changes in Land Value. This is the beauty and the scurge of real estate
development. Therefore, smart developers will choose a bad location and
strive to boot-strap it, i.e. to create a new address.
•
Worsening traffic congestion will cause greater demand for and supply of
housing close to the city center (or to centers – in a polycentric city) and will
trigger the appearance of new outlying centers
•
Density averters will offer less rent close to the center and density seekers
will offer more. Thus, the two groups will "self-organize" their location in
urban space so as to reflect their preferences.
7
Copyright © 2003 by Daniel Gat
Part One
Modeling the
Monocentric City
La nd P ric e a nd F A R G ra die nt s
6.00
5.00
V(x)
4.00
lambda
FA R
agrilambda
3.00
spcindex
dendex
cityline
2.00
1.00
-
8
2.00
4.00
6.00
8.00
10.00
Copyright © 2003 by Daniel Gat
Monocentric Assumptions: Consumers
• one type of consumer, with
• equal income, equal tastes
• non equal locations (tell joke # 1)
•
all work at the single city center, the CBD
• consumer utility depends on:
– amount of consumed housing floor space (really a service)
– income, net of housing-rent and travel expense
– quality of location
• all housing is leased from absentee owners
• at equilibrium consumers have equal utility at any location x
u(x) = constant everywhere in the city
9
Copyright © 2003 by Daniel Gat
Monocentric Assumptions: Housing Producers
• Housing is produced by combining 2 inputs
– Land
– Capital
•
•
•
•
Producers strive to maximize rate of Profit
Maximum achievable profit is Zero
No upper limit on FAR imposed by Zoning. Therefore…
…No upper limit on value of land which depends on
‘Location’ and ‘Quality
• Rent, asset and land value are maximized at each location
• Optimization is achieved by selecting appropriate FAR, the
floor-area-ratio. (FAR = floor-area/site-area)
Consumer Welfare
utility is a Cobb Douglas function of
•
•
•
S: Housing floor-space consumed
Z: Income net of rent and travel expenses
Q: Quality of Location
u ( x)  S  Z 1 Q ( x)
 S   y  S ( x)  R( x)    m  x 

1

 Q ( x)
where
•
•
•
•
•
•
•
y:
x:
R(x):
Z
tau:
m:
alpha:
income ($ per month per household)
distance from CBD (km)
Rent ($ per sqm)
Income net of housing and travel
unit cost of travel ($ per km)
mobility, number of monthly trips per hhld
weight of housing in consumer utility
Bid Rent Curve
Recall: U(x) = constant utility throughout the city
Bid Rent, R(x), is what the typical consumer is willing to pay for one
unit of space at Location x
1
• bid rent declines assimptotically
away from CBD
    m  Q( x) 
R( x)  R0  1 
 x

y
Q
 0 

• when quality is uniform
(assumed unity) then Q(x) can
be ignored
  m 
R( x)  R0 1 
 x
y


• wherever rent goes, value
follows
  m 
R( x)
V ( x)  V0 1 
 x 
y
k


• But what is R0? We assume a
minimal per-household space
standard S0, that is socially
acceptable at the city center
12

1

1

R0  S0    y  R0 
y
S0
Copyright © 2003 by Daniel Gat
Housing Production
• Housing floor-space F(x) is produced at location x
using a Cobb Douglas production function in Land
and in Capital
F  AL C

1
• Producers intend to maximize profits Π, end up
earning zero (excess profit)
  V ( x) F ( x)    L  C 
• Notice that floor-space creation is “instant” so that
there is no financing fee (all captal is invested at once
and all revenue is received at that time). This is the
short version; the long version including construction
and payout delays and cost of capital will appear
later.
Copyright © 2003 by Daniel Gat
Optimizing Housing Production
• Producers maximize profit by selecting optimal floor area ratio
FAR
1 
FAR( x)  A  (1   ) V ( x)
1

• Causing rent and land value maximization
 ( x)   (1   )
  (1   )
1 

1 

 A  V ( x) 
 A V0 
1

1

  m 
 x
1 
y


1

Land Value is a
highly leveraged
derivative of Rent
and Floor-space
Value
• Note the alpha times beta in the distance decay exponent,
reflecting 2 tradeoffs: housing vs income in consumption and
Land vs Capital in production.
• City outer boundary is easy to get by setting the urban land value
equal to the exogenous rural land value.
14
Copyright © 2003 by Daniel Gat
Density of Population and City Size
• Dividing FAR by per capita space S(x) and multiplying by a million,
we get the number of households per square kilometer of site
area. Covert to hhlds per sqkm of city land by multiplying by :
1 
1
106    k
Density 
1      AV0  
y
  m 
 x
1 
y


1

• Integrating the circular volume defined by rotating the density
gradient 360o and multiplying by the household size η we get the
urban population:
1 
1
106  2 k
Pop 
1      AV0  
y

x
xcbd
  m 
x 1 
 x
y


1

dx
15
Copyright © 2003 by Daniel Gat
City Outer Boundary
• Urban development at the fringe (greenfield development) will
take place only where the value of urban land (x) is expected
to be greater than that of agricultural land ag:
 
ag
y  
x
1  
1 
1
 m  



(1


)
(
AV
)
0
 











• By plugging this x wiggle value into other functions, we can get
their values at the fringe. (FAR, R and V; Lambda we know…)
16
Copyright © 2003 by Daniel Gat
Monocentric Urban Gradients
since asset value V(x) equals R(x)/k, land value and FAR
gradients are all expressible as functions of distance x and are
graphed that way (normalized for viewing)
La nd P ric e a nd F A R G ra die nt s
6.00
5.00
V(x)
4.00
lambda
FA R
agrilambda
3.00
spcindex
dendex
cityline
2.00
1.00
-
2.00
4.00
6.00
8.00
10.00
Add-Ons to the Standard Model
• Time to Market
• Quality of Location
• Urban Re-Development and
Optimal FAR
• Traffic Congestion
• People Congestion
• Land value discount due to FAR
restrictions
18
Time to Market
• Time to market T is highly significant in real estate. Therefore it is
more appropriate to model the maximization of NPV instead of
profit. Let k* be the cost of capital, then, for any project at any
location x, monopolystic competition drives NPV to zero

T
C
k T
 k *t
NPV  e V  F    L 
e dt
T 0
C
 k *T
 k *T
 e V  F    L  * 1 e
0
kT
*

Revenue is expected
only after completion
19
site stays idle
during construction

Construction capital is
payed out in equal
installments
Time to Market (continued)
• Results are very similar to the “short Version” but reflect time and
financing. Rent and value functions are un-affected but depressing
effects on land value are important:
1
 ( x)   (1   ) 
 k* T 

 k *T 
1  e 
1 

 e  k T A  V ( x) 


*
1

• As the product k*T grows (scarce capital and/or inferior building
technology) land value becomes less valuble, more land is
substituted for capital. This means a lower FAR (also a smaller city)
 *

e k T
FAR( x)   k  T 
(1   ) V0 
 k *T
1 e


*
20
1 

  m 
A 1 
x
y 

1

1 
 
Quality
• Recall that utility depends on:
– amount of built housing space consumed (really a service)
– income, net of housing rent and travel expense
– quality of location…ignored until now
u ( x)  S  Z 1 Q ( x)
 S   y  S ( x)  R( x)    m  x 
1

 Q ( x)
• This is reflected in the rent function R(x) and in all the other functions
dependent on rent:
1
   m   
R( x)  R0  1 
 x   Q ( x)
y


21
3-D Models
1,800
1,600
1,400
1,200
• a clown’s hat :
monocentric city with
uniform quality
1,000
-20
800
-10
600
400
0
200
-
10
20
10
0
-10
20
-20
1,800
• a crater :
quality at the center
has fallen – so have
prices and FAR
1,600
1,400
1,200
1,000
800
600
400
20
200
-2010
-10
0
0
-10
10
20
22
-20
$2,000
$1,800
3-D Models (contd)
$1,600
$1,400
$1,200
$1,000
$800
$600
• a crater with a spike:
the core is viable, but the
surrounding ring is
blighted
$400
$200
-20
$0
-10
20
0
10
0
10
-10
20
-20
2.00
1,800
1.80
1.60
• uneven quality surface
with its peaks and
troughs
1,600
1.40
1.20
1.00
20
0.80
10
0.60
+
1,400
1,200
1,000
-20
800
-10
0.40
600
0
0.20
400
-10
-20
-
0
• non-symmetric
gradient: the superimposition of uneven
quality onto the
symmetric clown’s hat
0
200
-10
10
10
20
-20
10
20
0
-10
1,800
=
1,600
1,400
1,200
20
1,000
10
800
600
0
400
23
20
-20
200
-10
-20
-10
0
-20
10
20
Urban Redevelopment and Optimal FAR
•
•
If a building loses most of its value (functional obsolesence, poor
maintenance, etc.) it makes sense to demolish it to make room for a new
one. That’s logical…
But we sometimes see a functional building demolished so as to create a
vacant buildable site. For that to happen, and if we assume that V(x) is the
value of one sqm, of the old and the new building, the motivation for the
new building has to be a higher FAR. But how much higher?
Let the site and floor areas of the existing building be
F, L
Therefore its existing values of building and site are
V ( x)  F ,  ( x)  L
Demolition cost is also related to the floor area
 F
For market driven renewal to take place, the value of
the about-to-be created vacant site must be more than
the foregone value of the demolished building, plus the
cost of demolition
24
 ( x)  L  V ( x)  F    F
Conditions for market driven Urban Redevelopment
1.
2.
3.
Make sure that vacated
site’s value is greater than
that of old building and
dem. costs.
Recall that at optimality
Substitute
4.
Obtain optimal FAR*
 L V F  F
  L    F V
  F V  V  F    F
F F    F V  1  V   F 


 
L
 L
   L
 1  V 
FAR* ( x)  
  Existing _ FAR
  
Optimal FAR has to be much greater than the Existing FAR. Beta is between
1/4 and 1/3 and add demolition costs plus financing. These economic facts
and citizens’ resistance to higher FAR delay urban redevelopment. (joke #2)
25
Part Two
Modeling the Central
Business District
26
Part Three
Modeling the
Poly-Centric City
33
Modeling the Poly-Centric City
•
•
•
So far, so good. The monocentric model has yielded results that
reflect many “stylize facts” about cities.
However, most cities are polycentric, and we need to deal with
that “most stylized of all facts” – polycentricity - head on.
I see three highly interesting and important research questions
regarding polycentric cities:
1. Why do cities become polycentric as they grow? Population
pressures could just as well generate new mono-centric
cities…
2. What is the mechanism that divides the non-housing land-uses
into the polycentric structure (divides as in cell division?)
3.
34
What is the mechanism by which non-central activities –
mainly housing – spatially self-organize (self locateallocate) around a given set of centers?
Polycentric Extension of the Mono Model
• I side-step questions 1 and 2 and go directly to Q.3, which is in-fact
the polycentric extension of the mono-centric model.
• Romanos (1976, 1977) suggested that the “only” difference between
the poly and the mono is that the consumer now has a choice of
destinations, instead of the single CBD. Romanos could have used
McFaddens discrete-choice model (1973) to do that—but he didn’t.
• So I continue where he left off. Once we know how to compute the
consumer’s new travel budget, we are home free! The rest is exactly
as in the theory and models developed above.
36
The Poly-centric City Model
•
•
•
•
In the Mono model, trip end is identical for all commuters
In the Poly model, commuters visit the various centers probabilistically,
depending on the expected benefit from the trip and the expected cost.
It is customary to assume that cost is a function of distance, , and benefit
a function of center floorspace F.
Accordingly, the model for probability of consumer locate at x choosing to
visit center i is:
P ( x, i ) 
Fi e ( x ,i )

N
Fk e ( x ,k )
k 1
• Total Expected (monthly) Travel Expenses of that household,
(assuming each household generates m trips per month)
 P(x, k )  (x, k )
N
T ( x )  m 
k 1
37
Gradients are Back + Hedonics
• So the T(x) function is now part of all of the gradient functions, e.g.
floor area value V(x):
1/ 
 T ( x) 
V ( x)  V0 1 
Q  /  ( x)

y 

• Recall the quality factor Q. It is really a function of several quality
attributes:
Q  q1 q 2 q 3 q M
• So re-combining multi-center Accessibility and Quality, and we get
a typical estimable Hedonic Regression equation:
1/ 
 T ( x) 
V ( x)  V0 1 

y


q11 /  q2 2 / 
qM M / 
• It is called “Hedonic” because through it, value reflects the things
we enjoy due to the attributes.
38
39
40
41
42
References
(to part 2: SPACE, in the book)
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44
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45
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46