Add-Ons to the Standard Mono-Centric Urban Model including the switch from Mono to Poly-Centric...

Add-Ons to the StandardMono-Centric Urban Model

including the switch from Mono to Poly-Centric

Daniel GatFaculty of Architecture and Town

PlanningCenter for Urban and Regional Studies

Technion – Israel’s Institute of Technology

Prepared for the Annual Meeting of The Israeli Regional Science Associationat Haifa Universityon the 21st of June 2005

Copyright © 2003 by Daniel Gat

2

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

3

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.

4

The “y” version• Beginning with Muth who brought in the the supply-

side of housing, some authors have substituted floor-area 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

5

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)

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

4. extension from Mono to Poly-Centric


6

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.


7

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.


8

Part One

Modeling the Monocentric City

Land Price and FAR Gradients

-

1.00

2.00

3.00

4.00

5.00

6.00

- 2.00 4.00 6.00 8.00 10.00

V(x)

lambda

FAR

agrilambda

spcindex

dendex

cityline


9

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


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)

1

1

( ) ( )

( ) ( ) ( )

u x S Z Q x

S y S x R x m x Q x

Consumer Welfareutility is a Cobb Douglas function of • S: Housing floor-space consumed• Z: Income net of rent and travel expenses• Q: Quality of Location

where• y: income ($ per month per household)• x: distance from CBD (km)• R(x): Rent ($ per sqm)• Z Income net of housing and travel • tau: unit cost of travel ($ per km)• m: mobility, number of monthly trips per hhld• alpha: weight of housing in consumer utility

12

Bid Rent Curve

• bid rent declines assimptotically away from CBD

• when quality is uniform (assumed unity) then Q(x) can be ignored

• wherever rent goes, value follows

• But what is R0? We assume a minimal per-household space standard S0, that is socially acceptable at the city center

1

00

( )( ) 1

m Q xR x R x

y Q

1

0( ) 1m

R x R xy

Bid Rent, R(x), is what the typical consumer is willing to pay for one unit of space at Location x

1

0

( )( ) 1

m R xV x V x

y k

0 0 00

yR S y R

S

Recall: U(x) = constant utility throughout the city


Housing Production• Housing floor-space F(x) is produced at location x

using a Cobb Douglas production function in Land and in Capital

• Producers intend to maximize profits Π, end up earning zero (excess profit)

• 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.

1CALF

( ) ( )V x F x L C


14

Optimizing Housing Production• Producers maximize profit by selecting optimal floor area ratio

FAR

• Causing rent and land value maximization

• 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.

1 1

( ) (1 ) ( )FAR x A V x

1 1

1 1

1

0

( ) (1 ) ( )

(1 ) 1

x A V x

mA V x

y

Land Value is a highly leveraged derivative of Rent and Floor-space Value


15

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

6 1 1

0

101 1

k mDensity AV x

y y

• Integrating the circular volume defined by rotating the density gradient 360o and multiplying by the household size η we get the urban population:

1

6 1 1

0

10 21 1

cbd

x

x

k mPop AV x x dx

y y


16

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:

1 1

0

1

(1 ) ( )

agyx

mAV

• By plugging this x wiggle value into other functions, we can get their values at the fringe. (FAR, R and V; Lambda we know…)


Monocentric Urban Gradients

Land Price and FAR Gradients

-

1.00

2.00

3.00

4.00

5.00

6.00

- 2.00 4.00 6.00 8.00 10.00

V(x)

lambda

FAR

agrilambda

spcindex

dendex

cityline

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)

18

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

19

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

* *

* *

0

*1 0

Tk T k t

k T k T

CNPV e V F L e dt

TC

e V F L ek T

Revenue is expected only after completion

site stays idle during construction

Construction capital is payed out in equal installments

20

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:

11*

*

1*

( ) (1 ) ( )1

k T

k T

k Tx e A V x

e

• 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)

*

*

1 11

*0( ) (1 ) 1

1

k T

k T

e mFAR x k T V A x

ye

21

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

1

1

( ) ( )

( ) ( ) ( )

u x S Z Q x

S y S x R x m x Q x

• This is reflected in the rent function R(x) and in all the other functions

dependent on rent:1

0( ) 1 ( )m

R x R x Q xy

22

3-D Models

• a clown’s hat : monocentric city with uniform quality

• a crater :

quality at the center has fallen – so have prices and FAR

-20

-10

0

10

20

-20

-10

0

10

20

-

200

400

600

800

1,000

1,200

1,400

1,600

1,800

-20

-10

0

10

20 -20

-10

0

10

20

-

200

400

600

800

1,000

1,200

1,400

1,600

1,800

23

3-D Models (contd)

• a crater with a spike:

the core is viable, but the surrounding ring is blighted

• uneven quality surface with its peaks and troughs

• non-symmetric gradient: the super-imposition of uneven quality onto the symmetric clown’s hat

-20

-10

0

10

20

-20

-10

0

10

20

$0

$200

$400

$600

$800

$1,000

$1,200

$1,400

$1,600

$1,800

$2,000

-20

-10

0

10

20

-20

-10

0

10

20

-

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

-20-10

010

20

-20

-10

0

10

20

-

200

400

600

800

1,000

1,200

1,400

1,600

1,800

-20

-10

0

10

20

-20

-10

0

10

20

-

200

400

600

800

1,000

1,200

1,400

1,600

1,800

+

=

24

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?

,

( ) , ( )

( ) ( )

F L

V x F x L

F

x L V x F F

Let the site and floor areas of the existing building be

Therefore its existing values of building and site are

Demolition cost is also related to the floor area

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

25

Conditions for market driven Urban Redevelopment

1. Make sure that vacated site’s value is greater than that of old building and dem. costs.

2. Recall that at optimality

3. Substitute

4. Obtain optimal FAR*

*

1

1( ) _

L V F F

L F V

F V V F F

F F F V V F

L L L

VFAR 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)

26

Part TwoModeling the Central Business District

33

Part Three

Modeling the Poly-Centric City

34

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. What is the mechanism by which non-central activities – mainly housing – spatially self-organize (self locate-allocate) around a given set of centers?

36

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.

37

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:

• Total Expected (monthly) Travel Expenses of that household, (assuming each household generates m trips per month)

( , )

( , )

1

( , )x i

iN

x kk

k

F eP x i

F e

1

( ) ( , ) ( , )N

k

T x m P x k x k

38

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):

• Recall the quality factor Q. It is really a function of several quality attributes:

• So re-combining multi-center Accessibility and Quality, and we get a typical estimable Hedonic Regression equation:

• It is called “Hedonic” because through it, value reflects the things we enjoy due to the attributes.

31 2

1 2

1/

/0

1/

/ / /0 1 2

( )( ) 1 ( )

( )( ) 1

M

MM

T xV x V Q x

y

Q q q q q

T xV x V q q q

y

43

References (to part 2: SPACE, in the book)

Alonso, William (1964), Location and Land Use, Harvard University Press

Anas, Alex (1978), “Dynamics of Urban Residential Growth,” Journal of Urban Economics, Vol. 5, pp. 66-87.

Anas, Alex and Ikki Kim (1996), “General Equilibrium Models of Polycentric Urban Land Use with Endogenous Congestion and Job Agglomeration,” Journal of Urban Economics, Vol. 40, pp. 232-256.

Anas, Alex and Rong Xu (1999), “Congestion, Land Use, and Job Dispersion: A General Equilibrium Model,” Journal of Urban Economics, Vol. 45, pp. 451-473.

Ben Akiva, M. and S. Lerman. 1985. Discrete Choice Analysis: Theory and Application to Travel Demand. Cambridge: MIT Press

44

Benson, Earl D., Hansen, Julia L., Schwartz, Arthur, L, and Smersh, Greg T. (1998), “Pricing Residential Amenities: The Value of a View,” Journal of Real Estate Finance and Economics, Vol. 16, Issue 1, pp. 55-73.

Brueckner, Jan K., Jacques-Francois Thisse, Yves Zenou (1999), “Why is Central Paris Rich and Downtown Detroit Poor? An Amenity based Theory,” European Economic Review, Vol. 43, pp. 91-107.

Capoza, Dennis R. and Robert W. Helsley (1989), “Fundamentals of Land Prices and Urban Growth,” Journal of Urban Economics, Vol. 26, pp. 295-306.

Fujita, Masahisa, Paul Krugman and Anthony Venables (2000), The Spatial Economy: Cities, Regions and International Trade

Hatta, Tatsuo and Toru Ohkawara (1993), “Population, Employment, and Land Price Distributions in the Tokyo Metropolitan Area,” Journal of Real Estate Finance and Economics, Volume 6, No. 1, pp. 103-128.

Henderson, j. Vernon (1977), Economic Theory and the Cities, Academic Press

45

Hotchkiss, David and Michelle J. White (1993), “A Simulation Model of a Decentralized Metropolitan Area with Two-Worker, Traditional and Female-Headed Households,” Journal of Urban Economics, Vol. 34, pp. 159-185.

Huff, David L. (1964), “Defining and Estimating a Trading Area,” Journal of Marketing, Vol. 28, pp. 34-38.

Krugman, Paul (1997), Development, Geography and Economic Theory, MIT Press

McFadden (1977), “Conditional Logit Analysis of Qualitative Choice Behavior,” Chapter 4 in Paul Zarembka

Mills, Edwin S. (1972), Urban Economics, Scott Foresman

Muth, R. F. (1969), Cities and Housing, The University of Chicago Press

Oppenheim, N. 1995. Urban Travel Demand Modelimg: From Individual Choices to General Equilibrium. NY: Wiley

46

Papageorgiou, Yorgos Y. and David Pines, An Essay on Urban Economic Theory, Kluwer Academic Publishers (1999)

Richardson, H. W. (1977), The New Urban Economics: and Alternatives, Pion

Rosen, Sherwin (1977), “Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition,” Journal of Political Economy

Romanos, M. C. (1976), Residential Spatial Structure, Lexington Books

Romanos, M. C. (1977), “Housing Location in a Linear Multi-Center Metropolitan Area,” Regional Science and Urban Economics, Vol. 7, pp. 233-250.

Tabuchi, Takatochi (1998), “Urban Agglomeration and Dispersion: A Synthesis of Alonso and Krugman,” Journal of Urban Economics, Vol. 44, pp. 333-351

Wheaton, William C. (1982), “Urban Spatial Development with Durable but Replaceable Capital,” Journal of Urban Economics, Vol. 12, pp. 53-67.

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