A dynamic spatial urban model: A generalization of Forrester's urban dynamics model

L’rhiin Swen,.\. Vol 4. pp. 93 10 I20 Pergamon Prerc Ltd 1979 Prmted m Great Britam

A DYNAMIC SPATIAL URBAN MODEL: A GENERALIZATION OF FORRESTER’S URBAN DYNAMICS MODEL? R. BURDEKIN IBM (UK) Scientific

Centre,

Peterlee,

Co.. Durham,

SR8 1BY U.K.

Abstract-The paper describes a development of the Urban Dynamics model into a spatially-disaggregated model. The resulting model is submitted to a number of tests and policy changes. An extension of the model to simulate urban spread and development without the assumption of an a priori fixed area is also discussed. The paper goes some way to providing a synthesis of the Urban Dynamics model with the static but spatially disaggregated Lowry model. the forerunner of the most widely used type of land use model. The flexibility of the system dynamics method may lead one to overlook certain problems of specification and calibration. three of which are illustrated using the Urban Dynamics model.

1. INTRODUCTION IT IS Now a decade since the publication of J. W. Forrester’s book Urban Dynamics [l]. The years since have seen a formidable growth of literature devoted to criticism, analysis and development of the original model. On the other hand, however, the model and the methodology have made little progress in becoming an accepted planning tool. This may have been because many of these developments have been centred around the original model which was unacceptable to many people for a variety of reasons. Although the present paper also takes the Urban Dynamics model as its starting point, it shows the flexibility of the method in dealing with one of the major criticisms of the model. that of its lack of a spatial dimension. The resulting model simulates the development of housing and industry over a city divided into 16 zones, It is submitted to a number of tests and policy changes. An extension of the model to simulate urban spread and development without the need to assume a priori a fixed city area is also discussed. 2. A BRIEF DYNAMICS

DESCRIPTION AND

OF LOWRY

THE

URBAN

MODELS

A brief general description of the Urban Dynamics model [l] is given in order to clarify the later developments. The city is envisaged by Forrester as having a fixed, spatially homogeneous, area. The model deals with the interplay of population, housing and industry within this area. The working population is divided into three categories or levels on the basis of skill and status-the managerial, labour (skilled) and underemployed (unemployed and unskilled). The total population is related to these groups by assuming that each worker has a family. Each population category has a corresponding housing level-premium. worker and underemployed. The model has three groups of industrial units-new, mature and declining-which employ workers from all three population groups. At any point in time the city is completely described by the value of these different stocks of population, housing and industry as well as a few additional auxiliary stocks. The main bulk of the model is concerned with the calculation of the rates or flows which bring about a change in these stocks from one interval to the next. Thus given initial values for all of the stocks one is able to calculate the change until time period

t This work II.5 4 2

A

was

mainly

done as part of a PhD thesis at the University 93

of Sheffield.

U.K.

R. BUKIXKIN

94

1. and hence the value of all the stocks at time period 1. This information may then be used to build up a picture of the city at time period 2 and so on. In the case of population these flows are births, deaths, migration and inter population group movement. The stocks of housing and industry are changed by new construction, demolition of old buildings and ageing from a higher category into a lower one. These different flows depend on conditions within the city as characterised by the values and ratios of different levels. To take a specific instance the construction of new premium housing depends on housing demand (the ratio of population to housing), the land density, the social mix (the ratio of managers to labour and underemployed). the tax situation and a confidence factor, which represents expectation as to future growth. This process may be given a more behavioural interpretation by considering it as a simulation of the decision processes used by housing developers i.e. that they balance demand, land cost (density), expected growth etc. in deciding the rate of housing construction. Certain of the ideas in this spatial generalisation of the Urban Dynamics model originated in Lowry’s model of Pittsburgh [2]. This was a large. static, spatially disaggregated model which predicted the effect. at equilibrium, of changes in the pattern of households and jobs in response, for example, to changes in the pattern of communication. Variants and developments of this model [3,4, 5,6] have become widely established as planning aids. There have been several efforts to develop this model into a dynamic one [6,7,8,9] and the work described below can also be developed into a further possible scheme [IO].

3. POSSIBLE REMEDIES DIMENSION IN THE

TO THE LACK OF A SPATIAL URBAN DYNAMICS MODEL

Many commentators have directed their strongest criticism of the Forrester model at its lack of a spatial dissaggregation, its fixed area and its omission of suburbs. On the first of these points it would seem that the spatial preoccupations of planners has perhaps led them to underestimate the many uses of such a model, as this, in laying down guidelines on, for example, job creation which hold irrespective of their precise spatial application. However, a spatial model opens up many possibilities for more detailed analysis and policy testing, although even in these cases. the opportunities for the radical alteration of spatial patterns may be limited and a simpler non-spatial model may give sufficient guidance. The criticisms of the model’s fixed area assumption as being unrealistic of city growth ignores Forrester’s (p. 2, Cl]) assertion that the area treated in the model “would be points as to how such ‘areas’ only a part of our larger cities.” This raises important are added to the existing city and about the interactions between the ‘areas’ which make up the city, such as the relationship between the core of the city and its suburbs and the impact of commuting. In this latter case, if net commuting of, say, managers, is not very important. then it will mean that fewer managers need be provided within the city and one can treat the situation as a reduction in the number of managers required per industrial unit. Such a change has no very great effect on the city’s growth pattern [l 11. Although some writers [e.g. 12, 13, 141 have used a two-zone. core-suburb model to study the effects of commuting, the extension of the model to treat the interrelated growth of a number of city zones has not proved to be as straightforward. both conceptually and practically (in the sense of simulated hypothetically, as Urban Dynamics) as it first appears. A possible method of introducing a spatial dimension would be to use the population housing and industry totals of the model as inputs to a spatial allocation model. but this does not properly integrate space with the other factors governing development and, in addition, a series of lags would have to be incorporated to deal with delays in the allocation process.

A dynamic spatial urban model

95

Another method stems from the idea of linking up several Forrester-type models, one for each zone of the city. Kadanoff and Weinblatt [ 151 have applied this idea to a two zone core-suburb model and suggested its use for a multi-zone model. To implement this scheme the flows of zonal migrants have to be divided into two: those to and from the environment and those between zones. The first of these is dealt with as before but, for the second, a rate dependent on the present populations and attractiveness (as defined by the Forrester equations) of the donor and recipient zones is used. The calculation of each zone’s attractiveness uses only its normal rates to take account of the favourability of its location and makes allowance for changes in this attractiveness caused, say, by growing employment opportunities in adjacent zones. The large number of interzonal flows even for a small number of zones could soon make such a scheme too unwieldy. Housing and industrial development reacts solely to the changes in population pressure within each zone and developers are not assumed to take any wider view of a particular zone’s suitability. The growth mechanisms of industry reveal other difficulties for a split into basic and service industries, i.e. location insensitive and location sensitive industries, becomes necessary. The scheme described below integrates spatial factors more naturally and consistently into the model. It should be noted that the main aim of the paper is to illustrate how this adaptation can be made and although it is hoped that the model does give an insight into some aspects of urban behaviour no claim is made as to the accuracy of either timings or projected totals. 4. A DYNAMIC

SPATIAL

MODEL

In the following model the city is taken as a 100,000 acre square divided into 16 equal squares of side 3.125 miles (Fig. 3). The model simulates the development of this city over time and its response to certain spatial changes. The three types of housing and industry levels described above are identified within each of the zones. In addition a further group of industry-service industryis identified in each zone. It is partly dependent upon the attractiveness of the zone’s location. The development of this housing and industry unfolds in time with the annual iterations of the model but whereas previously there was only a single calculation for each of the levels there are now 16-one for each zone. The flow equations in each zone for both housing and industry levels are dependent on a combination of city-wide and zonal characteristics. For example, in contrast to the premium housing construction in Urban Dynamics described above, the present model conceives of construction within any zone being governed by a combination of three city wide factors: overall housing demand, expectations as to future industrial growth and taxes, and four zonal factors: social mix, density, locational attractiveness and expectations as to housing growth. The implementation of these ideas is described in more detail below, The population is not disaggregated but is calculated only for the city as a whole and is modelled as in Urban Dynamics. The main reason for this is to reduce complexity and as a first approximation zonal population could be taken to be proportional to their respective housing in the zone. The generalisation to include population disaggregation is considered in section 9. As it stands the model carries the implication that the population’s locational decision is secondary to the decision as to whether to move to the city or not. In contrast to the second scheme outlined in section 3, the model, in effect, assumes that housing and industrial developers judge the potential attractiveness of particular sites for prospective clients and do not just respond to the pressure of population in a zone. The model is written in FORTRAN rather than DYNAMO, the more usual language used in this type of model. The use of FORTRAN entails manually putting the model’s equations into the correct run time order but it also offers a flexibility, particularly

R. BUKDERIN

96 Initial

Conditions

Loop once for each year *

Zone RateEquations

Loop 16 times

zone Level

Loop 16 times

City Level Equations

*

housing and industry Fig. 1. The Equation

Structure

of the model.

in the use of arrays, which certainly the earliest versions of DYNAMO did not possess. The equation structure of the model is shown in Fig. 1. Many parts of the model are identical to parts of the Urban Dynamics model and the names, definitions and values used for variables, parameters and functions in the Urban Dynamics model are retained unchanged wherever possible. Previously citywide variables which have now become zonal ones are distinguished by adding an ‘x’ to the names. Appendix 1 is a key to the variable names use in this paper and Appendix 2 gives a full listing of the model together with a note wherever changes in function values etc. have had to be made.

5. THE

PREMIUM

HOUSING

SECTOR

The modelling of the premium housing sector is now considered in some At some time period K the housing in the city is described by: PHX(I,K), the number of premium housing units in zone I at time K, WHX(I, K), the worker housing in zone I at time K.

detail.

and UHX(1,

K), the underemployed

housing

in zone I at time K.

Using a difference equation formulation the state of, say, premium I at the next time period L will be given by: PHX(I,

L) = PHX(I, K) + DT*(PHCX(I,

where PHCX(1, KL) is the rate of time interval K to L, PHOX(1. KL) is the rate of I during the time interval K and DT is the length of the time

premium filtration to L interval

housing from

KL) - PHOX(1, construction

premium

and is taken

housing

KL))J-

in zone

to worker

PHX(I)

= PHX(1)

the time suffices are retained

+ DT*(PHCX(l)

in the text for clarity.

(5.1)

I during

housing

the

in zone

as 1 yr.

f As the old value of a variable is no longer required once the new one has been calculated, of computer storage needed for the model may be reduced by coding equation 5.1 as: However,

in zone

- PHOX(1))

the amount

A dynamic

spatial

urban

model

91

These two rates have to be built up in terms of the values of different levels at time K. The construction rate (PHCX) in each zone is seen as a combination of a desired rate constrained by the available supply of labour.

PHCDX(I,K)*LCR(K)

PHCX(I,KL)=

(5.2)

where

PHCDX(1,K) is the construction

rate desired

in zone I at time

K

and

LCR(K)is the labour construction ratio and is taken as city-wide rather than zonal, reflecting an assumption that travel distance within the city will not affect the supply of labour available for construction work in a particular zone. The desired premium housing construction rate (PHCDX) is taken as having a constant (normal) rate which is modulated by conditions within the zone and city, i.e.: PHCDX(1,

KL)= PHCN*PHMX(I,K)*PHX(I,K)

(5.3)

where

PHCN is the normal assumed

premium housing to be the same in each zone

construction

rate,

which,

for simplicity,

is

and

PHMX(1, K) is the premium The heart of the rate equation

housing

multiplier

is the expression

(PHMX) which relates the rate to the conditions

in zone I at time for the premium within the zone

K. housing multiplier and the city. It is

given by: PHMX(1,

K) = PHAM(K)

* PHEM(K)

*PHGMX(1,

* PHTM(K)

K) *PHLCX(1,

* PHPMX(1,

K) * PHLMX(1,

K)

K) (5.4)

The first three factors on the right hand side are citywide factors. They are overall housing demand (PHAM), business confidence (PHEM) and taxes (PHTM) and are determined in the same way as in Urban Dynamics (Section A4, Appendix A, Cl]). The next three factors are the zonal equivalents, in zone I at time K, to previously citywide factors used in Urban Dynamics. They are the social mix factor (PHPMX), which is related to the ratio of premium to other housing, the land density factor (PHLMX) and the housing growth expectations factor (PHGMX). The final factor, PHLCX, reflects the favourability of the zone’s location for prospective householders which is taken to depend upon the zone’s accessibility to jobs. It is calculated here as a function of the ratio of the zone’s accessibility to jobs as measured by the number of jobs and the distance to them and the average accessibility to jobs for all the zones. This ratio is known as the relative managerial job potential and is defined as: NZONES* where NZONES is the number of zones (16 in this case), MJX(1, K) is the number of managerial jobs in zone I at time K. and T(I, Q, K) is the value of the trip distribution function between zones at time K. T(I, J, K) = D3(I, J, K).

(5.5)

I and

Q

The trip distribution function incorporates the deterrent effect of distance and is taken in the model as the cube of the distance, D(1, Q, K),between zones I and Q at time K. This distance is taken as that of the straight line between the mid points of zones I and Q at time K. The distance within a zone, D(1, I, K),is put equal to half the width of the zone. Detailed descriptions of the meaning and possible forms of the trip distribution function can be found in Lowry [2], Wilson [S] and Batty [6].

98

R. BURDEKIN

c------I

Density+ . . . . . .

..a.*.......................

I

T------ Social e.......................... *

t

t

)

)

I I , , T--

Growth : 1 ) T- Loc*jicm .

t t tt

Zone1 Factors

.

.

) a..... PHX -

tttt PHCX I A b City Factors 4 4 4

t I

1

I

I

I

I I I I 1 1 ) I 1

-

PHX

WHX PHOX -w WHX

--PHX

+wHx

UHX -LMx

INDUSTRY - Zone 1 INDUSTRY - zone 2

-mix

INDUSTRY - Zone 16

WH

PH

Uli

City Total

INDUSTRY

* .

TOTAL HOUSING

1

I

t

t

POPULATION __-----___-__-_-_____Houe[*g Demand -............*.......

TV’....,.,,

__________________-_____--‘_____----__

.

T,JT&

t

,

I ) _---- Bueiness Confidence_......................,,.*........**......

Fig. 2. The Premium

Housing

Sector.

All the factors in equation 5.4 are non-linear functions of various ratios of stocks within the city at time K and thus tracing back through equations 5.4 to 5.1 it can be seen that the flow into housing which will take place between time K and time L is fully specified in terms of the city’s state at time K. The determination of the behaviour of the premium housing sector is shown diagramatically in Fig. 2. The filtration (or ageing) from premium to worker housing (PHOX) is related to the inverse of the premium housing multiplier (PHMX) (calculated using a function of the logarithm of PHMX), it could equally well be related to a further series of factors in a similar way to that for PHCX. PHOX(1,

KL) = PHON *(l/PHMX(I,

K)) * PHX(1,

K)

(5.6)

The behaviour of the other two housing categories is modelled by a similar transformation on the equations of sectors A5 and A6 of Urban Dynamics. The location multiplier for worker housing is directly related to labour job potential and for underemployed housing demolition inversely related. This latter implies that slum demolition is reduced in areas of favourable location for the underemployed, although one might argue that the underemployed are unable to exercise such an influence. Housing is initially mainly in zones 5 and 9 (Fig. 3) with small residuals in the remaining zones so that growth can begin there once conditions become favourable.

6. THE

INDUSTRY

SECTORS

In the Urban Dynamics model it is assumed the city’s industrial products which is constrained

that there is an infinite by the supply of land

+

b

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

+

Fig. 3. The Zones

+

of the city

demand for and labour.

A dynamic

spatial

urban

99

model

This rather unsatisfactory view of industrial development is retained in dealing with the zonal allocation of basic (location insensitive) industry. The growth of basic industry within each zone is essentially the overall city growth shared out to the zones in proportion to their normal rates. Basic industry has three categories, viz: new, mature and declining. The equations for new basic industry behaviour are built up in a similar way to those of premium housing. The rate of construction of new industry is again equal to a desired rate modified by construction labour availability. This desired construction of new basic industry is equal to: NECN(1) * EM (K) * BIND (K)

(6.1)

where NECN(1) is the normal rate of new enterprise construction in zone I, EM (K) is the city new enterprise construction multiplier at time K and BIND(K) is a weighting of existing industry in the city at time K. The normal rate of construction (NECN) is put equal to 1 or 0.0001 depending on whether growth is or is not allowed in a zone (an exogenous decision). The new industry construction multiplier depends on the city’s overall state and is not zone related. There is no location factor and the density factor is related to the overall city density. This mechanism implies that basic industry does not compete with housing for land within a zone but has a prior claim. The filtration and demolition of the three basic industry categories are related as in the Urban Dynamics model with the exception that declining industry demolition depends on the density of the zone rather than the city. Basic industry is initially concentrated mainly in zones 5 and 9 with residual amounts in the other zones. Throughout the run basic industry growth is only allowed in zones 1. 5, 6. 9, 10 and 13. In addition to these three categories of basic industry there is a single category of service industry which is taken to be sensitive to location and whose growth depends on the demand by households within the city. Desired construction of service industry is equal to: SZCN * SIMX(1,

K) * SIX(1, K)

(6.2)

where SICN is the normal rate of service industry construction, SIMX(1, K) is the service industry multiplier in zone I at time K. and SIX(1. K) is the stock of service industry is zone I at time K. The service industry multiplier is related to overall business confidence, taxes and labour availability and a population service demand factor which seeks to achieve a normal ratio within the city of one service unit per 900 people. Its zonal factors are land density, locational attractiveness and demand for household services which seeks to maintain a normal ratio within the zone of one service unit per 150 houses. The location factor uses relative housing potential to measure the accessibility of a zone to potential customers of the services. The calculation is similar to that of equation 5.5 but with jobs in each zone being replaced by the total housing. As before service industry is initially concentrated in zones 5 and 9, but is allowed to develop in any zone. 7. THE

BEHAVIOUR

OF THE

MODEL

The overall initial conditions are those of the original Forrester model with the zonal variables mainly distributed in zones 5 and 9. It is in these two zones that most of the early growth takes place and because housing depends on zonal density, which

R. BURIXKIN

100

is higher in these two zones than for the city as a whole, the growth is faster over the first 50 yr than for the Urban Dynamics model. These two zones become saturated, but, because basic industry is dependent on the overall city density, jobs are still being created and thus there is pressure for expansion and the growth of housing begins to spread outwards. This spreading introduces lags into the development process and subsequent overall growth is slower than for the Urban Dynamics model. The city takes longer to reach equilibrium unless, as in the present model, normal construction rates are increased. The stages of zonal growth and saturation are repeated a number of times. After zones 5 and 9 come zones 6 and 10, 1 and 13, 2 and 14, 7 and 11, 3 and 15, 8 and 12 and finally 4 and 16. Finally the city attains an equilibrium distribution in which the zone totals of worker and underemployed housing units are roughly commensurate with employment opportunities but premium housing is distributed inversely. The equilibrium totals of population and jobs for the city are higher than for the Urban Dynamics model but those for the housing levels lower. This latter is due in part to the higher rates of filtration and demolition of the housing groups. Figures 4A-D show the development of housing and service industry, over 200yr for zones l-8 (which are symmetrical to the results for zones 9916). Table 1 gives a tabulation of zone totals for housing and service industry at years 200 and 300. by which time development has reached equilibrium. The column headed WHLCX shows the locational attractiveness of each zone for worker housing. 30.0...................................................

24.0..........:.........:.........:.........:.........:

ZN 8 ZN 3,4 ZN 7

* 628aaaa55 ZN 1,2.5,6 5555556666666666666666666666666666666 0.08888888....... 200 YEARS 160 40 80 120 0

Fig. 4A. Model

Base Run-Premium

Housing

@CO’s)

A dynamic

spatial

urban

101

model

6. .

28. 7.

1: 28. 7. . 6 : 270 1 22780 . o.o8888888888888.........................i~o.......... 200 YEARS 80 120 0 40

Fig. 4B. Model

Base Run-Worker

Housing

(000’s).

For premium housing (Fig. 4A) growth in zones 5 and 9 peaks at around year 20 and subsequently the growth peaks in successive pairs of zones about every 10-15 yr. Eventually it disappears completely from the inner more accessible zones, displaced by industry and lower status housing. This is particularly noticeable in zones 2 and 14 where premium housing is sustained for many years before going into decline. Worker housing (Fig. 4B) follows the same pattern but then, because of its stronger locational factor and the land released by premium housing it becomes established in all zones. However, competition from basic industry and underemployed housing forces its greatest growth not in those zones most accessible to jobs (1, 5, 6, 9, 10, 13) but in two of the zones adjacent to them (2 and 14). Underemployed housing (Fig. 4C) is less volatile than the other two housing types and its final distribution is roughly proportional to the locational attractiveness of the zones. After the initial activity in zones 5 and 9 subsides through overcrowding the main growth of service industry (Fig. 4D) skips zones 1, 13, 6 and 10 in which housing has become firmly established and land scarce, and spreads into and establishes itself in all the remaining zones. Service industry is included in the calculation of the ‘business confidence’ multiplier. The effect of this is to dampen the oscillations of industrial growth.

102

R, BLRIXKIIC

ZN 5 ZN 6 ZN 1

ZN 7

ZN 3,s ZN 4

it i .3 : .8 ..,..,.,...........‘....*......I.

:5

: 61

;8 2 * 70

278. .5 28. 61: 288 . ; 1, 0.08888888888888888~.....,~,,.,....~*........***...~~. 200 YEARS 120 160 80 40 0

Fig. 4C. Model

Base Run-

~!ndcrcmplo~ed

Housing

(Ohs},

Once basic industry growth in those zones where it is allowed (1, 6, 10, 13) has caught up with that in the core zones, 5 and 9, there is little difference in subsequent development in the different zones. Its overall growth is explained, as in the original Forrester model, by its dependence on the density of the city and the state of the labour market. Some of the properties of the model are now investigated through changing some of its assumptions. The city totals of housing and industry are fairly insensitive to changes in single tables or parameters, as was Urban Dynamics (see section 11). The chief reason for this is that housing and industry are constrained withjn a fixed area which puts a limit to their numbers and at the same time they are bound strongly together by the need of workers to have both jobs and houses. However, their distribution amongst the zones is sensitive mainly because of the interdependence between the zones which allows a shortfall in some zones to bc compensated for by increases in others while still maintaining an equivalent set of totals for the city as a whole. As an illustration of this the non-linear function for worker housing location was made more sensitive by about 10% (WHLCT-0.3, 0.4, 0.6. 0.75, 1.0, 1.3. 1.5, 1.65, 1.75, 5 * 1.8. see fisting in Appendix 2). The resulting city state at year 300 is shown in Table 2. It can be seen that although the city totals of housing and industry are Iittle changed their distribution within the city has been greatly affected, particularly premium housing

A dynamic

spatial

urban

model

103

. . . .*.....,...........,.........,....*.,..,,,

,...

..

8. . 278. 1. 734 . 2788 . 6. 2788 , 61 0.088888888888888,..,,,....**..,.*..........*.......,. 160 200 YEARS 40 120 0 80

Fig. 4D. Model

Base Run--Service

Industry

(Ws).

which has been displaced from zones 7 and 11. Worker housing shifts towards the more favourable zones. The reason for the sudden demise of premium housing from zones 7 and 11 is the influence of the social factor, PHPMX. As worker housing increases in zones 7 and 11, because of their increased attractiveness, the ratio of premium to other housing and hence PHPMX both decrease thus reducing new premium housing construction as PHPMX’s effect is not offset by the increased locational attractiveness of the zone. The reduction in premium housing construction causes the ratio of premium to other housing to decrease still further and so premium housing enters a self-sustaining decline. When the social factors are omitted from the model (Table 3) premium housing is not only retained in zones 7 and 11, but also manages to establish itself in zones 2 and 14, which it had not done even in the base run (Table 1). Service industry disappears from the left hand line of zones as a result of high density and poor location. The model takes about three times as long to reach equilibrium although the major part of the development is finished by the year 300. This is because a zone’s locational attractiveness reflects changes in surrounding zones comparatively weakly and, without the effect of the social factors which tend to reinforce particular modes of behaviour, mutual adjustment between zones is prolonged.

104

Table

1. Model

Base Run=-State

at years 200 and 300

CITY TOTALS AT YEAR 200

w

I

”

96723. 469765. 421211. PH m ml 150159. 285377. 281420. ZONES

1

P”

0.

2 3 4 5 6 7 a

I*,::: 18705. 0. 0. 17994. 19424.

1: 11 12 13 14 15 16

0. 17994. 19424. 0. 11. 18945. 18703.

!lTi 20717.

OH

22536. 20734. 12068. 11375. 24416, 2417.1. 13671. 11740. 24416. 24171. 13671. 11740. 22536. 20734, 12068. 11375.

25070. 14442. 14379. 19214. 19392. 15397. 14079. 19214. 19392. 15397. 14079. 20717. 25070. 14442. 14379.

CXTY TOTALS AT Yam w I

PH 0.

2 3 4 5 6 7 a 9 10 11 12 13 14 13 16

Table

0. 19114. 18486. 0. 0. 17854. 19635. 0. 0. 17854. 19635. 0. 0. 19114. 18486.

2. More

CITY

lb

Table

PH 0. 0.

Sensitive

!a

3. More Sensitive

UH

Worker

96026. 456353. 420159. PH WH 141247. 304549. 2a3az

14777. 22063. 0. 0. 11370. 16802. 0. 0. 11370. 16802. 0. 5611. 14777. 22063.

0. 0.

62,::.

w

“3

Ma

71266. 292222. 6163. SI 51:. 6078. 5104::. Na

856. 0. 0.

85:: 856. 0. 0. 856. 856. 0. a,:: 0. 0. 0.

Worker

CITYTcmE ATEq”ILIBRrm m L ”

3 4 5 6 7 8 9 10 11 12 13 14 15 16

454. 11743. 538. 14237. 417. 11533, 150. 62538. 164. 62872. 611. 15822. 577. 15103. 150. 62538. 164. 62872. 611. 13822, 577. 13103. 113. 62247. 454. 11743. 538. 14237. 417. 11533.

DI

144oP.

WHLCX

1.45 0.71 0.59 0.52 1.57 1.55 0.71 0.57 1.57 1.55 0.71 0.57 1.45 0.71 0.59 0.52

SI 110. 464.

540, ‘21. 151. 165. 612. 577. 151. 163.

612. 577. 110. 464. 540. 421.

Housing

LJ 61814. 11955, 14271. 11614. 62128. 62492. 15849. 15109. 62128. 62492. 15849. 15109. 61814. 11955. 14271. 11614.

Location

DI 14080,

WHLCX 1.44 0.71 0.59 0.53 1.57 1.55 0.71 0.57 1.57 1.55 0.71 0.57 1.44 0.71 0.59 0.53

Factor

300

21590. 22310. 25114. 2062‘. 22855, 12203. 10614. 27525. 8186. 8900. 0. 20100. 24186. 0. ZOOb3. 24130. 0. 24869. 21205. 24309, 1,233. 9918. 0. 20100. 24186. 0. 20063. 24130. 0. 24869. 21205. 9918. 24309. 11233. 0. 21590, 22310. 0. 25114. 20624. 22855. 12203. 10614, 8186. 27525. 8900.

ZONES PB 1 0. 2 5611.

SI 113.

0. 0. 0. 852. 852. 0. 0.

852. 0. 0. 0.

OH

22555, 20735, 11996. 11459. 24427. 24193, 13720. 11642. 24427. 24193. 13720. 11642. 22355. 20735. 11996. 11459.

u 96529. 4‘8891. 420488. PH wt! UH 149378. 288144. 282346.

ZONE! 1

X8

NE 832.

832. 852.

0

207:. 2SOB3. 14323. 14430. 19281. 19442. 154‘4. 13945. 19281. 19442. 15444. 13945. 20770. 25083. 14323. 14450.

TOTALS AT WAR HP I

UJ

300

96694. 448al3. 420943. PH m m 130178. 285478. 281455. zoN!zs 1

NJ

71313. 293552. 6134. NE SI LJ 5112. 6047. 512191.

MJ

03

MB

71173. 290035. 6162, NE SL LJ 5136. 6142. Wa51a. NE

SI

856. 0. 0. 0. 856. 856.

32. 481. 502. 402. 115. 161. 842. 536, 115. 161. 842. 536. 32. 481. 502. 402.

0.

85:: 856. 0. 0. 856. 0. 0. 0.

Housing

LJ 59624. 12332. 13578. 11493. 60798. 61811. 20260. 14362. 60798. 61811. 20260. 14362. 59624. 12332. 13578. 11493.

WLCX

1.5‘ 0.72 0.57 0.45 1.70 1.70 0.80 0.54 1.70 1.70 0.w 0.54 1.54 0.72 0.57 0.45

and No Social

"J MB HJ 71334. 286116. 6304. s* LJ 52::. 6037. 509026. LJ

Ii8

“H

NE

SI

22318. 22715. 17233. 12472. 21238. 20910. 19372. 16013. 21238. 20910. 19372. 16015. 22318. 22715. 17235. 12472.

22134. 18632. 13832. 10238. 23763. 23828. 16591. 12882. 23763. 23828. 16591. 12882. 22134. 18632. 13832. 10238.

875. 0. 0. 0. 875. 875. 0.

0. 421. 612. 492. 2.

59590. 11236. 13695. 13263. 58829.

639.

16343. 59590. 11236. 15695. 13263.

a7:: 875. 0. 87;: 0. 0. 0.

DI

13735.

0.

421, 612. 492.

Factors

DI 13562.

WHLCX 1.53 0.70 0.60 0.48 1.68 1.70 0.74 0.58 1.68 1.70 0.74 0.58 1.53 0.70 0.60 0.0

A dynamic spatial urban model

Table CITY

4. Changed

Initial

TOTALS AFTER 300 YEARS w L 0 96658. 448729. 421013. P” mi “H 150098. 285839. 281806.

ZONES P” 1 0. 2 0. 3 0. 4 16410. 5 0. 6 0. 7 0. 8 19249. 9 0. 10 0. 11 20575. 12 19685. 13 16410. 14 19249. 15 19685. 16 18835.

WH 20367. 19358. 20525. 15931. 19358. 19261. 25127. 14249. 20525. 25127. 13745. 13921. 15931. 14249. 13921. 14242.

OH 23254. 2436‘. 22828. 12798. 24364. 24455. 21770. 11914. 22828. 21770. 12267. 11600. 12798. 11914. 11600. 11283.

Pattern w 71242.

105

of Basic Industry

UJ 291830.

95;.

62. 151. 154. 362. 151. 164. 618. 533. 154. 618. 708. 566. 362. 533. 566. 415.

852. 852.

0. 852. 852. 0. 0. 852. 0. 0. 0. 0. 0. 0. 0.

60331. 61924. 62448. 10263. 61924. 62193. 15359. 14132. 62448. 15359. 18042. 14872. 10263. 14132. 14872. 11499.

WaLcx 1.49 1.56 1.47 0.57 1.56 1.57 0.82 0.58 1.47 0.82 0.67 0.57 0.57 0.58 0.57 0.52

An alternative pattern of basic industry was tried in the original model. Growth was allowed in the triangle composed of zones I, 2, 3, 5, 6 and 9 and, except for residuals in all zones, all initial housing and industry were concentrated in zone 1. The city’s development is symmetrical about the diagonal through zones 1, 6, 11 and 16 and its equilibrium state is shown in Table 4. Although the totals for the city as a whole are very close to the original (Table 1) a very different distribution of housing and industry emerges. Worker and underemployed housing are strongly represented in the industrial triangle and the central zones adjoining it, zones 7 and 10. Premium housing is in the remaining zones. Service industry establishes itself well in all the zones outside the industrial triangle.

8. AN

EXPLANATION

OF

THE

MODEL’S

BEHAVIOUR

The behaviour of models of this complexity cannot easily be analysed except by using simulation. However, some insight can be gained through examination of the rate equations for the conditions necessary for growth and equilibrium. A previous paper [l l] gave such an analysis for the Urban Dynamics model. For growth of, say, premium housing to take place in any zone its construction must be greater than its filtration into worker housing for at least some of the time. For the present model, if one assumes that the transition from inactivity to growth takes place during time interval KL then, from equation 5.1: PHCX(I, From LCR(K)

equations

5.2, 5.3 and 5.6 this implies

* PHCN * PHMX(1,

and provided

KL) > PHOX(I,

that PHX(1,

K) * PHX(I.

KL)

(8.1)

that:

K) > PHON * PHX(1. K)/PHMX(I,

K) (8.2)

K) f 0,

PHMX(L W >

/z

(8.3)

Breaking down the left hand side of equation 8.3 using equation 5.4 and noting that, if no growth has as yet taken place in the zone, the growth factor PHGMX will be equal to 1 and PHPMX and PHLMX will remain at their initial values, gives: I PHLCX(1, K) > J Cl /F(K) (8.4) where ci

= PHON/(PHCN

* PHLMX(1,

0) * PHPMX(1,

0)), a constant

106

R. BLRDWIN

and

F(K)= LCR(K)*PHAM(K)*PHTM(K)*PHEM(K) a function of the total city state. Thus growth can only take place if either the location of the zone becomes sufficiently favourable or the pressure from overall city growth becomes sufficiently strong (i.e. the right hand factor decreases). As the growth of basic industry increases the pressure for urban growth also builds up (i.e. F(K)increases) and the inequalities 8.4 and 8.1 are satisfied for zones increasingly distant from the main centre of jobs. At the same time service industry is attracted into zones outside the industrial core and thus spreads job opportunities and increases the locational attractiveness of more distant zones. This process continues until the pressure for additional growth is insufficient to overcome the penalty of distance. The present model does not attain this situation as the process is cut off by the limit set upon the number of zones. Spreading may also occur without any increase in housing or industry if locational attractiveness becomes less sensitive due to cheaper, faster travel or transport costs being considered relatively less important than previously vis-a-vis. say, housing rents. In this case those zones with below average relative job accessibility will increase their attractiveness by possibly sufficient for the inequality of equation 8.4 to be satisfied and growth to begin. On the other hand the more accessible zones will have their attractiveness reduced and this may result in demolition (PHOX) becoming greater than construction (PHCX) and in consequence premium housing (in this example) would decline. It can be seen from the above and from equation 5.1 that equilibrium in any zone I can only occur if premium housing construction there is equal to its demolition i.e.:

PHCX(1,KL)= PHOX(I,KL) This equation

is satisfied

if either

PHX(1)= 0 or (equations

(8.5) 5.2, 5.6, 8.2):

PHMX(I,K) = JC,/LCR(K) where C2 = PHONIPHCN. From equations 5.4 and 8.4 and as

(8.6)

PHGMX(1, K) = 1, at equilibrium:

PHPMX(I,K)*PHLMX(I,K)*PHLCX(I,K) = &/F(K)

(8.7)

As the right hand term L/ Cl/F(K) is the same for all zones for any particular run then, at equilibrium, provided they are not zero, the amounts of premium housing in the zones can be explained by the balance between the three factors on the left hand side of equation 8.7, viz. social, density and location. Thus there are two threshold values. The first (equation 8.4) must be passed for growth to begin at all. The second (equation 8.6) must be reached and maintained if any particular housing or industry group is going to reside permanently in a zone. Thus in those zones which regain premium housing when social factors are omitted, (Table 4) the social factors served to drive the combined value on the left hand side of equation 8.7 below the second threshold so that growth could not be sustained. A similar analysis may be carried out for the other levels in the model.

9. TWO

POLICY

CHANGES

The effects of two policy changes are now described. In the first policy new enterprise construction is permitted in zones 2 and 3 and suppressed in zones 10 and 13. The pattern of basic industry will thus change from a linear to a triangular one. Figure 5 shows the effects on housing and industry in zone 3, the zone experiencing the greatest changes, over 150 yr. Table 5 gives the totals for housing and service industry after 150 yr of the policy.

A dynamic

Table

5. Policy

spatial

urban

Run (Industry

Zone

107

model

ChangeFafter

150 yr

WHLCX 1.02 1.48 1.35 0.62 1.49 1.50 0.98 0.63 1.37 1.00 0.75 0.60

23283. 11816. 9984. 19862. 20021.

0.66 0.70 0.60 0.54

It can be seen that it needs about 40 yr for new basic industry to fully accommodate the change. During this time the locational attractiveness for housing in each of the zones will begin to change in response to the changing distribution of jobs. As a result of these changes further housing construction will be encouraged in some zones and discouraged m others which will in turn lead to a different pattern of housing. These 000’S

25.0............. ..

... ........ ... .. ..............

w . w

*

..I..

uuulJulJuuuuuulJ

uuuu

.

UHX WHX

.w.Ll.

20.0............. . . ........W......UU.................. u .PPP * .w P .w u. PP * PP . .W u. W u. P. w. uu . P. WW.u . P .Pkw.U . . Pwd .uu mwP UU 15.o..-....P..uu................................

ww

. uuu uuuu P

UUU. . uu . .uu . UUII

. .

P. P. P. P

lO.O.....................P............................. .P . NNNNNNNNNFNNNNNNNNNNNNN~MM.JNNNNNNNN NEX NNN .P NN. N. .P . N

.

. N . . P . SSN . 5.O..SS......................P........................ . NSS . s . P * .N SS. ss P. .ssss . .N . sssssssssssssssss P sssssssssssssssssss SIX .PP .PP . PHX O.ON..................................PPPPPPPPPPPPPPPP 0 30 60 90 120 150 YEARS

Fig. 5. Policy

Run (Industry

Zone

Change)-

Effects of Zone

3.

108

R. BUIUXKIN

changes will affect the attractiveness of the zones for service industry which will lead to further shifts in the distribution of jobs and hence of housing. Eventually these repercussions will die away leaving the city in a new equilibrium state. As might be expected the increase of jobs in zone 3 and the increased job accessibility of zone 7 force premium housing to decline there and to increase its numbers in the remaining premium housing zones. Premium housing is unable to re-establish itself in those zones (10 and 13) vacated by basic industry because of the large amount of worker and underemployed housing which already exist there and which force down the zones’ social attractiveness. The removal of basic industry from zones 10 and 13 leads to a decrease in density there and a growth of worker housing even though zone 13 in particular is not too well sited. Worker housing in zone 2 is reduced because of the demand for land for basic industry and competition from the underemployed. In general there is a shift of worker housing towards the industrial triangle. The movement of underemployed housing is in line with the changed distribution of employment opportunities and in fact falls in zone 13 even though worker housing has increased there. Service ipdustry is forced away from zones 2, 3 and to some extent 4 by the lack of land and increases in zones 10. 11, 13 and 14.

000’S 25.0...................................................

ZO.O..PPPPPP........................................... .P PPPPPPPPPPPPPPPPPPPPPPPPPPPPPP P

. PPPPPPPPPPPPP

PM

15.o:..wwww.............:.........:.........:.........: www wwwwwwwwwwwwwwwwwwwwwHx

. uLnJu UuLm

Lmu;“LmJulJ” . .

uuLfuuuuuuuuuuLJuuuLnllJuuuLrLlwuuuuuuuu ,

mix

lO.O...................................................

sssssssssssssssssssssssss SIX

. ssss ssss

. ss .ss

ssssssss .

.

ssss.

SS 5.0....................

.

.

. . . . . . . . ..*.........*..*.....*.

O.O................................................... 0 20 40

Fig. 6. Policy

Run (Distance

60

80

Reduction)--EfTects

100

YFARS

of Zone 3

A dynamic

Table

6. Policy

ZONES P” 1 0. * 0. 3 18988.

4 5 6 7

20927. 0. 0. 17894.

a

19086. 0. 0. 17790. 18916. 0. 0. 1a112. 18505.

9

10 11 12 13 14 15 16

spatial

Run (Distance

m 20636. 25486. 14719. 13352. 19785. 19780. 15412. 14231. 19858. 20018. 15390. 14253. 21235. 25038. 14798. 14295.

“H 22578.

24022. 13624. 11712. 23743. 23719.

13560. 11666.

23654. 23416. 13261. 11524.

21944. 20167.

urban

ReductionFafter

NE

856. 0. 0. 0.

856. 856. 0. 0. 856. 856. 0.

0. 856; 0. 0. 0.

109

model

a1 159,

402. 923. 560. 149. 162. 551. 520. 146. 159. 524. 495.

WHLCX

629%. 10612. 22702. 14807. 62351. 62652. 14‘90. 13816. 62327.

62663. 13899. 13269.

58. 60988. 406. 461. 362.

100yr

10682.

12480. 10317.

1.45

1.08 0.79 0.61 1.53 1.52 0.70 0.56 1.52 1.51 0.66 0.55 1.40 0.67 0.56 0.50

One interesting point is that the distribution of housing and industry resulting from this change is different from that which evolves if the same pattern of basic industry is used in the initial state (Table 4). This is due to the social factors which prevent growth restarting in those zones where a large imbalance in housing types has occurred. The other policy change was to assume that the ‘travel distance’ between zones 1, 2, 3 and 4 had been reduced by 307: through improvements in communication. This affects, in some degree, any travel distance between any of these above four zones and any of the remaining twelve zones. Figure 6 shows the effects on housing and industry in zone 3 and Table 6 shows the totals in all zones at 100 yr. The greatest effect is on underemployed housing (zones 2 and 3) and service industry (1. 3 and 4) which is a result of their stronger location factors. 10. POSSIBLE

FURTHER

DEVELOPMENTS

OF

THE

MODEL

The number of zones could be increased which would enable a more detailed examination of policy options and the model is limited in this respect only by the availability of computer time and storage. However, the comprehensibility of the resulting output, as well as data considerations, are likely to be a more effective limit. The basic industry growth mechanism in the model is unsatisfactory depending as it does on the area of the city rather than on demand for its goods from the outside environment. The inclusion of a more realistic industrial growth theory would not only strengthen the structure of the model but would allow the assumption of a fixed city area to be dropped and lead to a truer simulation of the spread of urban development. Drawing on the ideas presented in section 8, urban growth would spread outwards over successive bands of zones until such time as the distance of the zones from the centre of jobs became prohibitive or pressure of demand for jobs and housing, deriving from the outside demand for the city’s products, no longer increased. Several service industry categories could be included (Lowry [2]. has 3) each with different customer and location factors. Suburbs could easily be differentiated from the rest of the city by making changes in the structure of their equations or in the values of their parameters. For example, their taxes would be assessed for the suburb zones only rather than for the whole city. The impact of the suburbs and the effects of commuting-deterioration of housing, a shrinking tax base, falling job prospects etc. within the inner city--could be simulated without necessarily modelling zonal populations and worker movements. The simplest way to model the zonal populations would be to make no distinction between inter-zonal movements and those to and from the city. Immigration to a particular zone could then be made to depend, for example, on city-wide job availability and public spending and zonal housing and accessibility to jobs and services. The population equations would be adapted in the same way as that of housing and industry.

110

R. BURUEKIN

To separate the two types of migration one would require a similar scheme to that of section 3. Out of town migration into or out of zones would depend on a combination of city and zonal factors as above. Inter-zonal movements would be made to depend on some component of attractiveness in donor and recipient zones. However, as was pointed out previously, this would become very unwieldy, although a possible simplification would be to assume that all inter-zonal movement is channelled through a common pool. Each zone would thus have only two migration flows: those to and from the city altogether and those to and from the common intra-city migration pool.

11. THREE

POINTS

ON THE

IN THE

URBAN

GENERATION DYNAMICS

OF

BEHAVIOUR

MODEL

The flexibility of the system dynamics methodology may lead one to overlook some of the problems associated with their specification and calibration. As a comment on these issues, three points are described which arose out of tests carried out on Forrester’s Urban Dynamics model. The amount of detail in the Urban Dynamics model makes calibration infeasible and in any case, Forrester has argued that model insensitivity is sufficient for the purpose of comparing qualitative changes in behaviour, a view which has aroused much criticism and misunderstanding. Single parameter sensitivity tests (i.e. noting the effects of a change in a particular parameter, [16]) have been criticised because they do not cover the more likely occurrence of changes in sets of parameters. To examine such objections the model was run a number of times, each time with a different parameter set, Pi, P, . etc with pi

=

(PO

+

ei(PJ3 + . . Pj + ei(Pj).

. Pn + ei(Pn))

where Pa = (p,, ph.. . p,) was the original parameter set and ei(pj) was some random error on the value of pj For 200 of the runs e(p) was chosen randomly from a normal distribution about p, with standard deviations of 5% of p and for a further 200 runs the standard deviation was taken to be 10% of p. The table functions were not changed. Figure 7 shows the behaviour of the underemployed in four of these runs which were those which had the maximum and minimum values of the underemployed at year 250 for each of the two error distributions. It shows that the profiles retain their base run form and thus gives some confidence that alternative sets of model parameters would not give rise to a radical change in the basic pattern of model behaviour. However the spread of the results, particularly if compounded with possible errors in the table functions, is rather less reassuring and underlines the need to try and find supporting data whenever possible. Several simplified versions of the Urban Dynamics model have been published [ 171 and thus parts of the model appear to be redundant. It is found [lS] that if all the tax and social factors are held constant throughout the policy runs, then in general, the model reactions are similar to those of the original. Such simplifications are useful in reducing complexity and clarifying the chief forces at work in the model. The first need, however, is to check that tax and social influences are in fact small and not that they have been modelled improperly. There has been comment [19,20] on the use of multiplication for combining the different factors in the rate equation (as in equation 5.4 above). It appears that alternative ways of combining them can produce different policy outcomes. For example, Senge [20] used addition for the underemployed migration equation and found that the underemployed increased as a result of an underemployed housing construction policy whereas their numbers fell when this policy was applied to the original model. He asserted that the model was generally insensitive because the behaviour of certain key ratios

A dynamic

spatial

urban

111

model

Number of Underemployed (000's) 600.0...................................................

S.D. OF 10%

480.0.

. . . . *. . . . S.D. OF 5%

BASE RUN 360.0...

,....

S.D. OF 5x

S.D. OF 10 x

:I”:DC

.

.

.

. 120.0:.........:.....A I...........*....*.........*..*.*. . DA& , . CE . .DA . aDABE . .DBE . DCE DDEE . DDDEEE. .BEEEEE . O.OEE ,.,,.............,..,,.,....,....,............... 0 50 100 150 200 250 YEARS

Fig. 7. Runs with Maximum

and Minimum

Equilibrium

values.

had the same trend in both cases but it obviously depends on what aspects are of interest to the user. The author [lo] replaced all the multiplicative rate formulations by the average of their factors multiplied by the density factor, if there was one and also found a reversal of some of the previous behaviour. This aspect of sensitivity is not usually considered and is a difficult one to investigate. The discipline of data is needed here, perhaps even more than in the choice of parameters, if one is to have confidence in the validity of the generation of model behaviour.

CONCLUSIONS The majority of this paper has been concerned with showing the flexibility of the system dynamics methodology in dealing with some of the criticisms made of the Urban Dynamics model. In particular it is shown how this model can be generalised into a spatial model of the city and that such a model can be used to simulate a range of key urban phenomena. If, in addition, a more realistic theory of industrial growth was to be incorporated then the assumption of a fixed urban area could be dispensed with and a more satisfactory simulation of urban spread and development would result. The model is also used to test two policies of spatial change. The paper shows that Forrester’s approach is not necessarily as antipathetic to the

R. BL’KIIFKIN

112

more established land-use modelling methods, as typified by the Lowry model, as is sometimes portrayed. The flexibility of the method. however, may lead one to overlook certain problems of specification and calibration and three of these are illustrated using the Urban Dynamics model. The importance of data is crucial in this regard and sensitivity analysis alone is unlikely to be fully sufficient. A~kno~/&~~~~nra This work was done as part of a Ph.D. thesis at the University author

would

like to thank

Professor

H. Nicholson

for supervising

of Shefield it and the S.R.C. for providing

and the a grant.

REFERENCES 1. Forrester J. W. urban D~numics. M.I.T. Press, Cambridge, MA (1969). 2. Lowry 1. S. A Model uf Metropolis. RM-4305-RC, RAND Corporation, Santa Monica, CA (1964). 3. Garin R. A. A matrix formulation of the Lowry model for intra-metropolitan activity location, J. Am. Inst. Planners, 32, 361-364 (November, 1966). 4. Wilson A. Enr,op~, i/l L’rhan urld Reyionol .tfod~‘I/i~~g.Pion Press. London (1970). 5. Wilson A. C’rhu/l crm/ Rqion~ll illodrls in Gcogruphy und Planning. Wiley, London (1974). 6. Batty M. C’rhafl Modclli~~g. Cambridge University Press (1976). Santa Monica, CA 7. Crecine J. D. A D~wmic~ Model of Crhtm Srr~~t~rre. P3803. RAND Corporation, (1968). model of urban dynamics. Tovvn Planning Rw. 43, 166-186 (April 1972). 8. Batty M. An experimental University of Warwick, 9. Sayer A. Dwumir Spufirrl Modrl.~ oj’ Lrhun S~rf~~nrs,P.T.R.C. Seminar Proceedings, (July 1974). R. The Simulation and Control of Urban Development. PhD Thesis, University of Sheffield, 10. Burdekin (September 1977). R. & Marshall S. A. The use of Forrester’s systems dynamics approach in urban modelling. 11. Burdekin Envir. Planning 4, 471-485 (December 1972). MIT Press, 12. Hester J. Systems Models of Urban Growth und Development, Urban Systems Laboratory, Cambridge, MA (1969). of a Dynamic Urban Model, Ph.D. Thesis, University of 13. Babcock D. L. Analysis and Improvement California (1970). W. W. Urban Dynumics and the Suburbs in Readings in Urban Dynamics, W. W. Schroeder, 14. Schroeder R. E. Sweeney & L. E. Alfeld (Eds) Vol. 2. Wright-Allen Press, Cambridge, MA (1975). H. (1972) Public policy conclusions from urban growth models. IEEE 15. KadanotT L. P. and Weinblatt Truns. Svsr. Mtm C‘&~r. SMC-2. 159-165 (April 1972). The above issue was devoted to a series of articles on aspects of the ljrban Dynamics Model. Scnsitivitv Issue in Urban Dynamics, (op. cit. (14)). 16. Brittine K. R. and Trump J. G. The Parameter M. A simpli’fication of Forrestcr’L model of ai urban area. IEEE ?rans. Sq’st. -Man Cybrr. 17. Stonebyeaker SMC-2. 468472 (I 972). S. A. The Calibration, Potential Development and Limitations of the Forrestej 18. Burdekin R. & Marshall Model, P.T.R.C. Seminar proceedings, University of Sussex, (June 1973). 1970). 19. Berlinski D. J. Systems analysis. C’rb. dff: Q. 6, 104126 (September Formulations in Urban Dynamics, (op. cit. (14)). 20. Senge P. M. Multiplicative APPENDIX D E.M LCR MJ NECN NEX PHAM PHCDX PHCN PHCX PIIEM PHG.k4X PHLCX PHLMX PH.tIX PHON PHOX PHPMX PHTM PHX SI C‘N SIAlX SIX UHX WHX

1. KEY TO

SYMBOLS

Distance Between Zones Enterprise multiplier Labour construction ratio Manager jobs New enterprise construction normal New basic enterprise (zone) Premium housing adequacy multiplier Premium housing construction desired (zone) Premium housing construction normal Premium housing construction (zone) Premium housmg enterprise multiplier Premium housing growth multiplier (zone) Premium housing location multiplier (zone) Premium housing land multiplier (zone) Premium housing multiplier (zone) Premium housing obsolescence normal Premium housing obsolescence (zone) Premium housing population multiplier (zone) Premium housing tax multiplier Premium housing (zone) Service industry construction normal Service industry multiplier (zone) Service industry (zone) Underemployed housing (zone) Worker housing (zone)

A dynamic

APPENDIX

C

C

C

C

C

C

spatial

2. A LISTING

urban

OF

113

model

THE

MODEL

IMFLICIT REAL*8(A-2) INTEGER DT,FINTIM,I,J,IJ,LCNTRL,TIME ZONAL VARIABLES DIMENSION BDMX (lG),DIDMX(16),DIDX (16),DIEMX(l6),DILMX(l6), DIX (16),EDMX (16),SIOMX(l@,ELCX (16),ELMX (16), 1 SILX (16),SIMX (16),EMX (16),HUTX (16),LDCX (16), 2 LDIX (16),LFOX (16),LJX (16),LURX (16),MBX (16)s MBDX (16),MJX (16),MPRX (lG),NECDX(l6), 2 NECX (16),NEDX (16),SICX (16),SIDX (16),NEX (16)~ 5 PHAX (16),PHCX (16),PHCDX(16),PHDIS(l6),pHGMX(l6), PHLCX(~~),PHLMX(~~),PHMX (16),pHOMX(l6),pHOX (16), ! PHPMX(16),PHX (16),PuTX (16>,SFHDR(l6),SFHDX(16), 8 SHDCX(~~),SHDIS(~~),SHDMX(~~),SHDX(16),SHLMX(l6) 9 DIMENSION SICDX(~~),SIDIS(~~),SILCX(~~),SIX (16),UHX (16), ~HAX (16),NHc~x(16),WHCx (16),WHDIS(l6),~GMX(l6)s 1 WHLCX(16),WHLMX(16),WHMX (16),wHOMX(l6),~OX (16), 2 3 WHUMX(16),WHX (16) ARRAYS FOR NON-LINEAR TABLE FUNCTIONS DIMENSION BDMT (14),DIEMC(l4>,DILMT(l4),ED~ (14),EGMT (14)n ELJMT(14),ELMT (14),SILEIT(l4),EMbfI (14), 1 ETMT (14>,LAHMT(l4),LAJMT(l4), SFPDT(14), 2 LATMT(l4),LAIJMI(14),LCRT(14),LDMT (14),LEMT (14)) 3 LLFT (14),LSMT (14),LUMT (14),MAHMT(l4),MAJMT(l4), 4 MAPMT(~~),MATMT(~~),MDMT (lO),MLMT (14),MSMT (14)s PEMT (~~),PHAMT(~~),PHEMT(~~),PHG~(~~),PHLCT(~~), : PHLMT(~~),PHOMT(~~),PHP~(~~),PHTMT(~~),SFHDT(~~)B 7 SHAMT(l4),SHDCT(14),SHLMT(l4),SILCT(l4),TCMT(14), 8 TRT (14),UAMMT(14),UDMT (14),UEMT (14),UFm (14) 9 DIMENSION UHMT (14),u~~Mr(l4),uJMT (14),ULJW14>,-(14)s WHEMT(14),WHGMT(l4),WHLCT(14),WHLMT(l4),~OMT(l4), 1 WHTMT(14),WHUMT(14) 2 LAG TIME CONSTANTS DATA AMMPT,LAMPT,LMMPT,LRPT,MAMPT/20.,15.,15.,5.,10./ DATA ~AT,P~T,T~PT,VMMPT,UTLPT,WHAT/10,,10.,30.,10.,10.,10./ ORIGINAL URBAN DYNAMICS PARAMETERS DATA AMF,DIAV,DICF,DIDF,DIDN,DIL,DIM/l.,lOO.,.3,l.,.O3,1O.,l./ DATA EF,LAF,LAN,LBR,LDN,LFS,LHCL/1.,1.,.03,.01,.02,6.,.6/ DATA LMF,LMN,LPH,LPP,W,MAN,MBAV/1.,.02,.1,.2,1.,.03,300./ DATA MBCF,MBDN,MBL,MBM,MDN,MPBR,MPFS/.5,.05,15.,2.,.02,.0075,5., DATA NEAV,NECF,NECL,NECN,NEDN,NEGRI/500.,1.,20.,.05,.08,.03/ DATA NEL,NEM,PHAV,PHCL,PHCN,PHF,PHGRI/20.,4.,30.,2.,.03,1.,~03/ DATA PH0N,PHPD,SHDF,SHDN,TAN,TLP,TMP/.03,3.,1.,.02,50.,200.,150 DATA TPCN,TUP,UAN,UBR,UDN,UFS,UHAV/250.,300.,.05,.015,.02,8~,5., DATA UHPD,UMF,UMN,WHAV,WHCL,WHCN,WHF/12.,1.,.1,15.,1.,.03,1./ DATA WHGRI,WHON,WHPD/.03,.02,6,/ ADDITIONAL SERVICE INDUSTRY PARAMETERS DATA SIAV,SICL/400,,20./ DATA SICN,SIDN,SIL,SIM,SPRH,SPSR/.lO,.O8,2O.,4.,l5O.,9OO./ CHANGED URBAN DYNAMICS PARAMETERS DATA PHCN,WHCN,NECN,SHDN/0.06,0.06,0~06,0.06/ DATA PHON,WHON,NEDN/O.O6,O,O6,O.O6/

114

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ORIGINAL URBAN DYNAMICS TABLE FUNCTIONS DATA BDMT/2.0,1.8,1.5,1,0,0,7,0.5,8*0,4/ DATA D1EMT/0.4,0.5,0.7,1.0,1.6,2.4,8*4,0/ DATA DILMT/1.0,1.2,1.6,2.2,lO*6.0/ DATA EDMT/2.0,1.8,1.5,1.0,0.7,9*0.5/ DATA EGMT/0.2,0.6,1.0,1.4,1,8,2.2,8*2.2/ DATA ELJMT/0.001,0.05,0.15,0.4,1,0,1.5,1.7,7*1.8/ DATA EL~/1.0,1.15,1.3,1.4,1.45,1.4,1.3,1,0,0,7,0.4,0.0,3*C DATA EMMT/0.1,0.15,0.3,0.5,1.0,1.'4,1.7,1.9,2.0,5*2.0/ DATA ETMT/1.3,1.2,1.0,0.8,0,5,0.25,8*0.1/ DATA LAHMT/1.3,1,2,1.0,0.5,0.2,0.1,8*0.05/ DATA LAJMT/2.6,2.6,2.4,1,8,1.0,0.4,0.2,0.1,6*0.05/ DATA LATMT/1.2,1.0,0.7,0.3,10*0.3/ DATA LAUMT/0.4,0.8,1.0,1.2,1.3,9*1.3/ DATA LCRT /0.001,0.5,0.9,1.1,1.15,9*1.2/ DATA LDMT/8.0,4.0,2.0,1.0,0.5,0.25,0.125,7*0.1/ DATA LEMT/0.2,0.7,1.0,1.3,1.5,1.6,8*1.7/ DATA LLFT/0.0,0.01,0.03,0.1,10*0,3/ DATA LSMT/2.4,2.0,1.0,0.4,10*0.2/ DATA LUMT/0.2,0.7,1.0,1.2,1.3,9*1.4/ DATA MAHMT/1.3,1.2,1.0,0.5,0.2,0.1,8*0.05/ DATA MAJMT/2.7,2.6,2.4,2.0,1.0,0.4,0.2,0.1,6*0.05/ DATA MAPMT/O.3,0.7,1.0,1.2,1.3,9*1.3/ DATA MATMT/1.4,1.0,0.7,0.3,10*0.3/ DATA MDMT/8.0,4.0,2.0,1.0,0.5,0,25,0.125,0.1,6*0.1/ DATA MLMT/0.2,0.7,1.0,1.2,1.3,9*1.3/ DATA MSMT/2.3,2,2,2.0,1.6,1.0,0.5,0.2,0.1,6*0.05/ DATA PEMT/0.2,0.6,1.0,1.6,2.4,3.2,8*4.0/ DATA PHAMT/0.001,0.001,0,01,0.2,1.0,3.0,4.6,5.6,6*6.0/ DATA PHEMT/O.2,0.6,1,0,1.4,1.8,2.2,8*2.2/ DATA PH0MT/2.8,2.6,2,0,1.0,0.5,0,3,8*0.2/ DATA PHTMT/1.2,1.0,0.7,11*0.3/ DATA SHAMT/3.6,2.0,1.0,0.6,10*0.4/ DATA SHLMT/1.0,1.2,1.6,2.2,6.0,9*6.0/ DATA TCMT/2.0,1.6,1.3,1.1,1.0,0.9,8*0.8/ DATA TRT/0.3,0.5,1.0,1.8,2.8,3.6,8*4.0/ DATA UAMMT/0.3,0.7,1.0,1.2,1.3,1.4,8*1.5/ DATA UDMT/8.0,4.0,2.0,1.0,0.5,0.25,0.125,7*0.1/ DATA UEMT/0.2,0.7,1.0,1.3,1.5,1.6,8*1.7/ DATA UFWT/0.9,0,8,0.5,0.33,0.25,9*0.2 / DATA UHMT/2.5,2.4,2.2,1.7,1.0,0.4,0.2,0.1,6*0.05/ DATA UHPMT/1.0,1.2,1.5,1.9,2.4,9*3.0/ DATA u~/2.0,2.0,1.9,1.6,1.0,0.6,0.4,0.3,0.2~0~15,4*0~1/ DATA ULJRT/1.15,0.8,0.5,0.25,10*0.L/ DATA w~~~~/0.001,0.05,0.1,0.3,1.0,1.8,2.4,2.8,6*3.0/ DATA WHEMT/0.3,0.7,1.0,1.2,1.3,1.4,8*1.4/ DATA WH0MT/2.2,2.0,1,6,1.0,0.7,0.5,8*0.4/ DATA WHTMT/1.2,1.0,0.7,0.3,10*1.3/ CHANGED URBAN DYNAMICS TABLE FUNCTIONS DATA EGMT/0.6,0.8,1.0,1.2,1.4,1.8,8*2.0/ DATA P~~~T/0.6,0.8,1.0,1.2,1.4,1.6,8*1.8/ DATA PHGMT/0.8,0.9,1.0,1.1,1.2,1.3,8*1.4/ DATA PH~/2.5,2.4,2.3,2.15,2.0,1.8,1.5,1.1,0~6,0~~,4*0~~/ DATA PHPMT/0.5,0.8,1.0,1.1,1.2,9*1.3/

A dynamic

DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA

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wHEMT/0.6,0.8,1.0,1.2,1.4,1.6,8*1.8/ wHGMT/0.8,0.9,1,0,1.1,1.2,9*1.3/ ~L~/2.5,2.4,2.3,2.15,2.0,1.8,1.5,1.1,0.6,0.1,4*0.0/ WHUMT/0.8,0.9,1.0,1.1,10*1.2/ ADDITIONAL TABLE FUNCTIONS ELCX/1.0,3*0.0001,2*1,,2*0.0001,2*1.0,2*0.0001,1.0,3*0.0~1 PHLCT/.6,.6,.7,.85,1.0,1.15,1.3,1.4,1.5,5*1.6/ SFPDT/.6,.8,1.,1.2,1.5,9*1.8/ SFHDT/.8,.8,.85,.9,1.0,1.15,1.3,1.5,5*1.!3/ sHDCT/1.3,1.3,1,2,1.1,1.0,0.9,0.8,0.7,0.6,5*0.5/ S1LCT/.5,.5,.6,.75,1.0,1.3,1.5,1.65,1.7,5*1.7/ sIL~/2.2,2.2,2.1,2.05,2.0,1,9,1.7,1.4,0.8,0.2,4*0.0/ WHLCT/.5,.5,.6,.75,1.0,1.25,1.4,1.5,1.6,5*1.6/ ZONAL INITIAL CONDITIONS NEX/4*0.01,100.,3*0.01,100.,3*0.01,4*0.01/ MBX/4*0.001,500.,3*0.001,500.,3*0.001,4*0.001/ D1X/4*0.001,50.,3*0.001,50.,3*0.001,4*0.001/ PHX/4*50.,2150.,3*50.,2150.,3*50.,4*50./ WHX/4*50.,10150.,3*50.,10150.,3*50.,4*50./ UHX/4*10.,480,,3*10.,480.,3*10.,4*10./ S1X/4*1.,400.,3*1.,400.,3*1.,4*1./

INITIAL CONDITIONS IN THE CITY AREA = 100000 AREAX = AREA/16 DATA AMMP,LAMP,LMMl?,LRP,MAMP,UMMP,UTLP/1.,1.,1.,1.,1.,1.,75./ PH = 0 wH=o UH = 0 NE=0 MB =o DI = 0 SI =o DO 100 J = 1,16 PH = PH+PHX(J) WH= WH+WHX(J) UH = UH+UHX(J) NE = NE+NEX(J) MB = MB+MBX(J) DI = DI+DIX(J) SI = SI+SIX(J) PHAX(J) = (l.O-PHGRI*PHAT)*PHX(J) WHAX(J) = (1.0~WHGRI*WHAT)*WHX(J) MP = 4000. L= 25000. U= 1000, NEA = NE+SI-NEGRI*NEAT*(NE+SI)

C HAV BAV AV LUR TCM TAI

= = = = = =

PHAV*PH+WHAV*WH+UHAV*UH NEAV*NE+MBAV*MB+DIAV*DI+SIAV*SI HAV+BAV TABLE(TCMT,LUR,O.O,3.0,0.5) (TMP*MPFS*MP+TLP*LFS*L+TUP*UFS*U)*TCM/AV

116

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TRNP = TAIlTAN FINTIM IS THE LENGTH OF THE RUN, DT THE TIME INTERVAL FINTIM = 301 DT =1 LCNTRL = FINTIM/DT+l C THE START OF THE LOOP FOR THE ANNUAL ITERATIONS TIME = 0 DO 200 I = 1,LCNTRL C THESE ARE THE CITY LEVEL EQUATIONS IF (I.EQ.~) ~0 To 210 C OMIT THE LEVEL CALCULATIONS THE FIRST TIME THROUGH u= U+DT*(UA+UB+LTU-UD-UTL) L= L+DT*(UTL+LB-LTM+LA-LD-LTU) MF= MP+DT*(LTM+MPB+MA-MD) AMMP = AMMF+(DT/AMMPT)*(AMM-AMMF) UTLP = UTLP+(DT/UTLPT)*(UTL-UTLP) UMMP = UMMP+(DT/UMMPT)*(UMM-UMMP) LMMF = LMMP+(DT/LMMPT)*(LMM-LMMP) LAMP = LAMP+(DT/LAMPT>*(LAM-LAMP) MAMP = MAMP+(DT/MAMPT)*(MAM-MAMP) NEA = NEA+(DT/NEAT)*(NE+SI-NEA) TRNP = TRNP+(DT/TRNPT)*(TRN-TRNP) LRP = LR~+(DT/LRPT)*(LR-LRF) C ZONE LEVEL CALCULATIONS DO 220 J = 1,16 = PHAX(J)+(DT/PHAT)*(PHX(J)-PHAX(J)) Pm(J) = WHAX(J)+(DT/WHAT)*(WHX(J)-WHAX(J)) W(J) PHx(J) = PHX(J)+DT*(PHCX(J>-PHox(J)) tJHX(J) = WHX(J)+DT*(WHCX(J)-WHOX(J)+PHOX(J)) Urn(J) = UHX(J)+DT*(WHOX(J)-SHDX(J)) N=(J) = NEX(J)+DT*(NECX(J)-NEDX(J)) SIX(J) = SIX(J)+DT*(SICX(J)-SIDX(J)) MBX(J) = MBX(J)+DT*(NEDX(J)-MBDX(J)) DIX(J) = DIx(J)+DT*(MBDX(J>-DIDx(J)) CONTINUE 220 CALCULATE THE CITY TOTALS OF THE ZONAL VARIABLES C PH =o DI =o WH=o UH = 0 NE =o MB = 0 ST =o DO 230 J = 1,16 PH = PH+PHX(J) WH= WH+WHX(J) UH = UH+UHX(J) NE = NE+NEX(J) MB = MB+MBX(J) DI = DI+DIX(J) 230 SI = SI+SIX(J) 210 CONTINUE C AUXILIARY EQUATIONS WHICH HELP TO DETERMINE CITY FLOW RATES C C

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TAX SECTOR TR = TABLE(TRT&.44*DLOG(TRNP),-2.0,4.0,1.0) HAV = PHAV*PH+WHAV*WH+UHAV*UH BAV = NEAV*NE+MBAV*MB+DIAV*DI+SIAV*SI AV = HAV+BAV TC = AV*TAN*TR P = MP*MPFS+L*LFS+U*UFS TPCR = (T~/P)/TPcN LUR = L/U TCM = TABLE(TCMT,LUR,O.O,3.0,0.5) TN = (T~*M~F~*M~+TLP*LF~*L+T~~*uF~*~)*TcM TAI = TN/AV TRN = TAIlTAN OVERALL CITY DENSITY HUT = PH+WH+UH PUT = NE+MB+DI+SI LFO = (LPH*HUT+LPP*PUT)/AREA NEW BASIC INDUSTRY SECTOR MJ = NE*NEM+MB*MBM+DI*DIM+SIM*SI MR =MP/MJ EMM = TABLE(EMMT,MR,0.0,2.0,0.25) ELJM = TABLE(ELJMT,LRP,O.O,2.0,0.25) ETM = TABLE(ETMT,l.44*DLOG(TR),-2,0,4.0,1.0) NEGR = (NE+sI-NEA)/((NE+SI)*NEAT) EGM = TABLE(EGMT,NEGR,-0.1,0,15,0.05) ELM = TABLE(ELMT,LFO,O.O,l.O,O.l) CITY SERVICE FACTOR (NORMALLY 1 UNIT PER 900 HOUSES) SR = P/(SI*SPSR) SFPD = TABLE(SFPDT,SR,.5,1,5,.25) PREMIUM HOUSING - CITYWIDE FACTORS MHR = Mp*MpFs/(pH*p~p~) PHAM= TABLE(PHAMT,MHR,O.O,2.0,0.25) PHEM = TABLE(PHEMT,NEGR,-0,1,0.15,0.05) PHTM = TABLE(PHTMT,1,44*DLOG(TR),-2.0,4.0,2.0) WORKER HOUSING - CITYWIDE FACTORS LHR = L*LFS/(WH*WHPD) WHAM= TABLE(WHAMT,LHR,O.O,2.0,0.25) WHTM = TABLE(WHTMT,l.44*DLOG(TR),-2.0,4,0,2.0) NHEM- TABLE(WHEMT,NEGR,-0.2,0.3,0.1) UNDEREMPLOYED HOUSING - CITYWIDE FACTORS UHR = (~*uF~)/(uH*uHPD) SHAM= TABLE(SHAMT,UHR,O.O,2.0,0.5)

ZONAL JOBS AND HOUSING DO 240 J = 1,16 LDIX(J) = NEL*NEX(J)+MBL*MBX(J)+DIL*DIX(J)+SIL*SIX(J) = NEM*NEX(J)+MBM*MBX(J)+DIM*DIX(J)+SIM*SIX(J) M=(J) HUTX(J) = PHX(J)+WHX(J>+UHX(J) CONTINUE 240 CALCULATE RELATIVE POTENTIALS C CALL DISWT(HUTX,SIDIS,16,3,125) CALL DISWT(MJX,PHDIS,16,3.125) CALL DISWT(LDIX,WHDIS,16,3.125) CALL DISWT(LDIX,SHDIS,16,3.125)

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LJ =o C

ZONAL AUXILIARY EQUATIONS 250 J = 1,16 ZONAL DENSITY PUTX(J) = NEX(J)+MBX(J)+DIX(J)+SIX(J) LFOX(J) = (HUTX(J)*LPH+PUTX(J)*LPP)/AREAX BASIC INDUSTRY EM = EMM*ELM*ELJM*ETM*EGM = (NECF*NE+MBCF*MB+DICF*DI)/6.0 BIND NECDX(J) = NECN*BIND*EM*ELCX(J) EDMX(J) = TABLE(EDMT,l,44*DLOG(EM),-3.0,3.0,1.0) BDMX(J) = TABLE(BDMT,1,44*DLOG(EM),-3.0,3.0,1.0) DIEMX(J) = TABLE(DIEMT,1.44*DLOG(EM),-3.0,3.0,1.0) DILMX(J) = TABLE(DILMT,LFOX(J),O,8,l.O,O.O5) DIDMX(J) = DIEMX(J)*DILMX(J) PREMIUM HOUSING PHLCX(J) = TABLE(PHLCT,PHDIS(J),O,O,2,0,0.25) = 0.2*PHX(J)/(UHX(J)+WHX(J)) M==(J) PHPMX(J) = TABLE(PHPMT,MF'RX(J),O,O,O.1,0.02) PHLMX(J) = TABLE(PHLMT,LFOX(J),O,O,l.O,O.l) PHGRX = (PHX(J)-PHAX(J))/(PHXx(J)*PHAT) PHGMX(J) = TABLE(PHGMT,PHGRX,-O.L,0.15,0.05) PHMX(J) = PHAM*PHLMX(J)*PHPMX(J)*PHTM*PHEM*PHGMX(J)*PHLCx(J) PHCDX(J) = PHCN*PHX(J)*PHMX(J) PHOMX(J) = TABLE(PHOMT,1,44*DLOG(PHMX(J)),-3.0,3.0,1.0) WORKER HOUSING WHLMX(J) = TABLE(WHLMT,LFOX(J),O,O,l,O,O.l) WHLCX(J) = TABLE(WHLCT,WHDIS(J),O,O,2.0,0.25) LURX(J) = 0,67*WHX(J)/UHX(J) WHUMX(J) = TABLE(WHUMT,LURX(J),O,O,5.0,1.0) = (WHX(J)-wHAx(J))/(WHX(J)*WHAT) WHGRX WHGMX(J) = TABLE(WHGMT,WHGRX,-0.1,0.15,0.05) (J) WHMX(J) = WHAM*WHLMX(J)*WHuMx(J)*WHTM*WHEM*WHGMX(J)*WHLCx WHCDX(J) = WHCN*WHx(J)*WHMX(J)-PHON*PHX(J)*PHOMX(J) WHOMX(J) = TABLE(WHOMT,l,44*DLOG(WHMX(J)),-3.0,3.0,1.0) SLUM HOUSING SHLMX(J) = TABLE(SHLMT,LFOX(J),O.8,1.0,0.05) SHDCX(J) = TABLE(SHDCT,SHDIS(J),O.O,2.0,0.25) SHDMX(J) = SHAM*SHLMX(J)*SHDCX(J) SERVICE INDUSTRY SECTOR SILCX(J) = TABLE(SILCT,SIDIS(J),O,O,2.O,O.25) SFHDR(J) = HUTX(J)/(SPRH*SIX(J)) SFHDX(J) = TABLE(SFHDT,SFHDR(J),O.O,2.0,0.25) SILX(J) = TABLE(SILMT,LFOX(J),O,O,l.O,O.l) SIMX(J) = EMI..f*SILx(J)*ELJM*ETM*EGM*SILCX(J)*SFPD*SFHDx(J) SICDx(J) = SICN*SIX(J)*SIMX(J) SIOMX(J) = TABLE(EDMT,l.44*DLOG(SIMX(J)),-3,0,3.0,1.0) CONSTRUCTION JOBS LDCX(J) = PHCDX(J)*PHCL+WHCDX(J)*WHCL +NECDX(J)*NECL+SICL*SICDx(J) = LDIX(J)+LDCX(J) LJX(J) TOTAL CITY JOBS = LJ+LJX(J) LJ CONTINUE

DO C

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MORE CITY AUXILIARY EQUATIONS WHICH DEPEND PARTLY ON ZONAL ONES CITY WORKER AND UNDEREMPLOYED JOB FACTORS LR = L/LJ ULJR = TABLE(ULJRT,LR,0.0,2.0,0.5) UJ = LJ*(uLJR) UR = U/UJ LCR = TABLE(LCRT,LR,0.0,2.0,0.5) UNDEREMPLOYED POPULATION SECTOR UJM = TABLE(UJMT,UR,0.0,3.0,0.25) UHPR = 0. UHPM = TABLE(UHPMT,UHPR,O.O,O.O5,0.01) UM = UTLP/U UAMM= TABLE(UAMMT,UM,0.0,0,15,0.025) UHM = TABLE(UHMT,UHR,0.0,2.0,0.25) PEM = TABLE(PEMT,TPCR,O.O,3.0,0,5) AMM = UAMM*UHM*PEM*UJM*UHPM*AMF UDM = TABLE(UDMT,1,44*DLOG(AMM),-3.0,3.0,1.0) UFW = TABLE(UFWT,UR,0.0,4.0,1,0) uw = U*UFW LSM = TABLE(LSMT,LR,0.0,2.0,0.5) UEM = TABLE(UEMT,TPCR,O.O,3,0,0.5) LUM = TABLE(LUMT,LUR,0.0,5.0,1.0) UMM = LSM*LUM*UEM*UMF LABOUR POPULATION SECTOR LAUM = TABLE(LAUMT,LUR,O.O,5.0,1.0) LATM = TABLE(LATMT,l,44*DLOG(TR),-2.0,4.0,2.0) LAHM= TABLE(LAHMT,LHR,O.O,3.0,0.5) LAJM= TABLE(LAJMT,LR,O.O,2.0,0.25) LAM = LAJM*LAUM*LATM*LAHM*LAF LLF = TABLE(LLFT,LR,0.0,2.0,0.5) MSM = TABLE(MSMT,MR,0.0,2,0,0.25) MLR = MP/L MLM = TABLE(MLMT,MLR,O.O,O.2,0.05) LEM = TABLE(LEMT,TPCR,O.O,3.0,0.5) LMM = MSM*MLM*LEM*LMF LDM = TABLE(LDMT,1.44*DLOG(LAM),-3.0,3.0,1.0) MANAGERIAL-PROFESSIONAL SECTOR MAJM= TABLE(MAJMT,MR,O.O,2,0,0.25) MFR = MP/(L+u) MAPM= TABLE(MAPMT,MPR,O.O,O.l,O.O2) MATM = TABLE(MATMT,1.44*DLOG(TR),-2.0,4.0,2.0) MAHM= TABLE(MAHMT,MHR,O.O,3.0,0,5) MAM = MAJM*MAPM*MATM*MAHM*MAF MDM = TABLE(MDMT,1.44*DLOG(MAM),-3.0,3.0,1.0) RATE EQUATIONS FOR THE CITY = (u+L)*uAN*AMm UA UD = UDN*U*UDM = U*UBR UB UTL = uMN*uw*uMMP LB = L*LBR LTU = L*LLF LTM = LMN*L*LMMP = LAN*L*LAMP LA LD = LDN*L*LDM

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= MF*MFBR =MAN*MP*MAMP = MDN*MP*MDM THE CALCULATION OF ZONAL RATES C DO 260 J = 1,16 PHCX(J) = PHCDX(J)*LCR PHOX(J) = PHON*PHX(J)*PHOMX(J) WHCX(J) = WHCDX(J)*LCR WHOX(J) = WHON*WHX(J)*WHOMX(J) SHDX(J) = SHDN*UHX(J)*SHDMX(J) NECX(J) = NECDX(J)*LCR NEDX(J) = NEDN*NEX(J)*EDMX(J) MBDX(J) = MBDN*MBX(J)*BDMX(J) DIDX(J) = DIDN*DIX(J)*DIDMX(J) NEDX(J) = NEDN*NEX(J)*EDMX(J) SICX(J) = SICDX(J)*LCR SIDX(J) = SIDN*SIX(J)*SIOMX(J) CONTINUE 260 TIME = TIME + DT 200 STOP END MPB MA MD

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SUBROUTINE DISWT(X,RP,N,D) ROUTINE TO CALCULATE RELATIVE POTENTIAL IMPLICIT REAL*8(A-H,O-Z),INTEGER(I-N) DIMENSION X(N),Y(16),R(l6,16>,T(l6),RF(N) Z =0 NN= SQRT(N+l.) DO 2 I = l,N Y(1) = 0 = (I-l)/NN+l 11 = I-((I-l)/NN)*NN 12 DO 1 J = 1,N CALCULATE DISTANCE BETWEEN ZONES Jl = (J-l)/NN+l 52 = J-((J-l)/NN)*NN x2 = (J2-12) Xl = (Jl-11) XX= x1*x1+x2*x2 = D*DSQRT(XX) R(J,I) R(J,J) = D/2 CALCULATE TRIP FUNCTION = R(J,I)*R(J,I)*R(J,I) T(J) CALCULATE ZONE POTENTIAL POT = X(J) /T(J) = Y(I)+POT Y(I) Z = Z+Y(I) CONTINUE AVERAGE ZONE POTENTIAL = Z/N Z DO 3 I = l,N RP(I) = Y(I)/Z CONTINUE RETURN END