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APPENDIX B
A LOGITBASED LAND ALLOCATION MODEL
WITH ENDOGENOUS PRICE SIGNALS
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This appendix describes a prototype landuse transport interaction modeling system that serves
as a target for the shorttem~ mode! development work envisaged in Chapter 5. The intention is to
specify a system that can make use of an existing transport model with appropriate modification, so
the larger system is to some extent just a "bolt on" to the existing transport model. A further
intention is Mat this system could be implemented In a United States context without going beyond
reasonable levels of time and resource requirements. A principal design objective was to include
explicit representation of the overall system response to policy variables concerning both (a)
transport service provision and (b) zoning regulations and tax/subsidy structures. At this point it is
a very preliminary specification.
This prototype system has three models. as follows:
I. transport model;
2. landuse model; and
3. developer model.
The transport model takes activities distributed among the model zones and determines the
resulting patterns of transport flows and costs. or (dis~utilities. It loads the transport demands that
arise from these distributions to network representations of the available transport supple, taking
account of congestion effects and demand elasticities. The landuse model takes exogenously
determined activity totals for the entire model area arid allocates them among the model zones. The
allocation process is influenced by transport costs and by the price for space in each zone. among
other things. The price for space is adjusted according to the demand for space relative to the supply
for space in the zone, and We rate at which space is used by an activity is elastic with respect to the
price for the space. The trar~sport model and the lar~duse model interact in that the activity
distributions determined by the lar~duse model are used by the transport model and the tray el costs
determined by the transport model are used by the landuse model. The developer model takes the
prices for space and the quantity of space and determines Me extent to which the supply of space we
change through demolition and redevelopment and nest construction consistent with zoning
regulations and tax/subsid` structures. The resulting new space supply is farther input to the land
use model.
The structure of Me Interaction among these models is shown in Figure B.l, which distinguishes
among the "models", "inputs" arid "information" components of the system.
Temporal dynamics are simulated by making "time steps" in general from time t to time t+ ~ as
shown in Figure B.1. The landuse model provides activity distributions as inputs to the transport
model in the same time period. The activity distributions and the accessibilities from time t form
inputs to the mode] for time t+! in order to simulate lags in the system response to changes in
transport conditions and inertia in the system generally. The developer model determines the
changes in space from time t to time tot.
Ideally, each time step would cover a period of 2 years. It is expected this would provide
adequate representation of Apical boombust cycles in space development arising because of
inertia in developer activities. Lear time steps are seen to be too aggregate, in spite of the
advantage that they can be matched to census years. Some transport model runs might be skipped.
and the accessibil~ties from previous time period used as proxies. in order to save computer run
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Figure B.1
Structure of Interactions in Prototype System, Showing "Models", "Inputs"
and "Information" Components
.... .
,. . . . .
. .
.
.. ~ ~. . ..# ~. . .
.: Me ad: i.. .:..
.. . .. .
~. .
I
... .
~. .. ~ . ~ .
~ : Time t+l:
.. . .
: : :
.
space totals >:
:t and )
~ distributions >:
~ .
. Am.
.
= land use . :: activity  
: :: :: serviced ~~~ model 1:; ~dlstnbutlon ~model I::
~ : ~: ~~ of: IT ~ . :: my I t
 ~ ~ ~~; i ;~ ,~ :;,  Am_ i;   , \ ~
~ if:::   ::. a: ( bevel ~ / \ ~ space oh:
:  ~::::'~'~':.'~.'~ Em. I :: \ W:
~ : :~::: :::: : : ;:: ~ : \: ~  I \  T
.  .....
i;,. , ., ~ ,:
... ~ .......
: .': ~ ~':~:~.~
... ~ 
K;~ I \
': ~  I ::~ \
~ ~ ~ ~\
I: : ::~: :( accessibilities)
Lopers space zoning:
 L model I :; taxes/su~
space totals ~
and )
rli.ctnh~tic~n.~ a/
..
  ~ ~: .
i::...
, i.::
transport ~ transport <: activity ale lane use ~ ~ activity
curviness _   model ~ ~ d~stubut~onsy model ~: ~ totals
.
::

. ~ .  . . 
.. ~.  . . .
.. . .
.
. .
....~ ..~. ~ ~ . ... i..
... .
.
..
., <~ ~ ~
i\
( accessibilities) \
 B2
~ :f~
I moues 1 of I taxes/sudsidies ~
~1
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times; but it is expected that this wall become less and less of an issue as computer run speeds
. .
continue to Increase.
The transport mode} can be a standard aggregate 4step model. provided it has (a) some form of
Togitbased mode split mode} that includes transit as an alternative and (b) art assignment process that
includes capacity restraint. An ex~shn~ transport mode} can be used. with modifications to include
Togit mode split if necessary as depicted in Fissure 6.~. Activitybased and sample enumeration forms
of transport model with behavioral mode choice and network assignment components can also be
used.
The activity distributions output by the model are population (or household) and employment
totals by zone in categories consistent With the categories used as inputs to the transport mode} and
thus related to the segmentation of mp purposes in the transport model.
The travel costs are the zone to zone mode choice composite (dis~utilities (logit mode split
logsum terms) determined in the Passport mode]  with congestion on the transport networks taken
into account. These costs are fed back into the transport model to ensure mode! consistency.
Ideally, the accessibilities are zonal destination choice composite utilities (Iogit destination
allocation logsum teens) detennined in the transport model. including representation of activity
totals in zones and Gavel costs between zones. The accessibilities would then be consistent with the
spatial behavior as expressed in the estimated parameters in the destination choice models.
Alternatively, We accessibilities could be standard 'Wilsontype' expressions, using the sum of the
products of activity mates and Bagel costs raised to the exponent as follows:
Be = Ej M`jexp(pc,j)
where:
Bki = accessibility to activity of tYne k in zone
M,;j
= magnitude of activity of tvDe k (relevant coDulation and/or emnIovment
totals) in zone j
ci; = travel cost from zone ~ to zone j (a mode choice composite (dis~utility)
,0 = parameter.
The transport services inputs are the standard descriptions of networks of roads. rails and
transport services going into a transport model. including travel times, fares and tolls, service levels
and connectivities generally. These inputs provide representation of transport policy including
pricing, regulation and infrastructure development at each point in time considered.
Jr a ~rr
r A
The model takes exogenousl`dete~mined activity totals for the entire model area arid allocates
these totals among the model zones. The activity categories in the model are various population
(household) types arid employment categories based on some form of SIC/SOC classification
consistent with the categories of space considered in the model. The space categories should
include land and/or floorspace Apes related to relevant planning regulation and the SIC/SOC
classification used, along with 'ur~developed land available for development as a separate category.
BE
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Figure B.2 shows what is envisaged regarding a typical classification of activities and space types
in the model and corresponding trip purposes in the transport model.
The mode] establishes prices (actual or imputed rents) for space in each categoric in each zone.
and it uses these prices in order to take account of the supply of space in its allocation of activities.
An iterative process is used to establish prices that either (a) respect relevant space capacities or (b)
match space demand to space supply. A prototype functional form for the mode] is as follows:
T. Begin a new iteration
n=n~
where:
n = iteration counter
and if this is the first iteration, then initialize demands for space to appropriate nonzero ~ alues
(some proportion of the corresponding supply values):
Dn° = X~Sp
where:
Dn° = demand for space of type 'A in zone 'I' in iteration O
So = supply of space of type 'L in zone 'T.
X1 = parameter.
2. Adjust space unit rents in response to relative demand and supply for space:
for each type of space "f in the set of space types F
for each zone ;I' in the set of zones 'l:
Run = X ~ Remind ~ +X2¢D6n~/S6yX3] ~ X4~ ~ ~ +X5in~x6D5n~~/S6~]
where:
Rfin = rent per unit of space of type 'fit in zone 'I' in iteration n'
Rfimin = minimum rent per unit of space of type 'A in zone 'T.
Dfin~~ = demand for space oftype f in zone 'I' in iteration ;nI'
Rfin' = rent per unit of space of type ;f in zone 'I' in iteration in ~ '
X1 to x6= parameters.
Note that two alternative functional forms are included in order to facilitate two forms of
solution as indicated above. that is to either (a) respect relevant space capacities or (b) match
space demand to space supply.
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Figure B.2
Anticipated Typical Classification of Activities and Space Types
in the LandUse Model and Corresponding Trip Purposes
in the Transport Mode}
Transport Flows
Homebased
Activity Categories
Space Types
/ population/households residential
to work R /allemployment
to school ~ // Resources whilecollar
resources bluecollar ~s~
to shopping \//; \:
Am/ // m anufactunng white co liar \:
Remanufacturing bluecollar
to soc~al/recreat~onal I\ `\
in course ofwork /~ retail
to personal business
goods movement
wholesale ~
If services
education
\ govemment/health
 BS
industrial
warehousing
~ / \\~ commercial
office
government
institutional
school
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Determine space use rates consistent with space unit rents:
for each type of space 'L in the set of space types F
for each type of activity 'a' in the set of activity types 'Aft that use space type f
for each zone 'I' in the set of zones 'T:
Ufajn = xTx2exp~x3Rfin)x4~x5exp~x6Rfin)]
where:
Ufain = space of type 'A used per unit of activity of type 'a' in zone 'It in iteration
no
Note that two alternative forms of decay in the rate of use of space are included in order to
provide flexibility in calibration.
4. Determine space use prices consistent with space trait rates:
for each type of space 'f indite set of space types F
for each type of activity 'a in the set of activity types 'Af. that use space type f
for each zone 'I' in the set of zones 'l:
Pai Hi Ufai
where:
Pain = price per unit of activity of type ;a' in zone 'I in iteration ;n'
5. Allocate the modelwide total activities among zones taking into account the influences of
relative prices, accessibilities. other attributes. the distribution in the previous time period and
. · . ~
an a priors welg. :ltlng :
for each type of activity 'a in the set of activity types 'Al
for each zone 'I' in the set of zones I^:
expf X1P~in + x2B~,~~ + x30 t x4G~j~~~ ~ xsW~;~ x6Z~j)
. . . . . .
Hain = Ha [ . ]
Ei exp( x1Pa,n ~ x9Bai'l ~ x30ai x4Gait3 ~ xsWaj + x6Zai)
where:
Ha = modelwide total activity of type a (determined exogenous to modeling system)
Hain = activity of type "a allocated to zone ~ in iteration 'n'
Baja = accessibility for activity of type a. in zone 'I' in time period 'tl'
0aj = term representing vector of other attributes of zone 'I' relevant to activity
of type 'a'
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Gait = proportion Ha,lHa in solution in time period tT' (provides an inertia teen)
Waj = size term indicating a pnori value for proportion HailHa
Zai = zone specific constant tenn for activity a' in zone 'I'
X1 to x6= parameters.
Note that the superscript ;t, representing time period 'I', has been omitted from terms for clarity.
6. Determine the total demand for space of each type by each activity consistent with space use
rates:
for each type of space 'A in the set of space types F
for each type of activity ~a' in the set of activity types Oaf' that use space type f
for each zone I' In the set of zones 'I :
Dfai = Ufai Hai
where:
Dfajn = demand for space of type 'L by activity ;a' in zone ~I' in iteration ;n'
7. Determine the total demand for each troupe of space:
for each type of space ;f in the set of space types F
for each zone 'I. in the set of zones 'I :
Dain = ~ a Dfain
8. Check for convergence:
for each type of space `f in the set of space types F
for each type of activity' 'a' in the set of activity types ;Af'
for each zone 'I' in He set of zones 'I :
if Hain = Hairy within a specified tolerar~ce
and
either Rf,n = Rfi~} within a specified tolerance (for solution (a) above, where prices respect
relevant space capacities)
or Den = sfin within a specified tolerance (for solution (b) above, where
Remarry is matched to space supply
then stop mode] for year it'
otherwise return to step ~ 4. ~ above.
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The developer model simulates the actions of those preparing space for use for each category of
space and activity considered. Space is developed in response to space rents determined in the
previous time period. The effects of dilapidation, demolition arid redevelopment, new den elopment
arid some degree of 'dampening' ~ ticipation of aggregate market conditions are all represented.
A prototype functional form for the developer model is as follows:
T. Determine Toss of space through dilapidation:
for each type of space f in the set of space types F
for each zone 'Ii in the set of zones 'l:
~n = XiS6t
where:
= TOSS of space of type 'L in zone 'T. though dilapidation
S6t = supply of space of type 'f in zone 'T in time period t'
X1 = parameter.
Note that this assumes a straightforward constant rate of loss within a given type of space. A
reasonable farther enhancement might be to add a term to represent the influence of the age of
structures on the rate of loss due to dilapidation.
2.
Determine 'zonespecific' loss of space through demolition for redevelopment, taking account
of pressure for redevelopment to the most profitable allowable space type:
for each type of space 'A in the set of space types F
for each zone 'I' in the set of zones 'I':
fi = X ~ Sfi ~ ~ (Rki Cam; I Cfi ~ ~ /~ Cfi ))
where:
~n = TOSS of space of type 'A in zone 'I' though demolition
Rfit = rent per unit of space of type 'L in zone 'I' in time period ;t'
Cfit = cost to develop unit of space of type ~ plus taxes less subsidies in zone 'I'
in time period 't.
Maxi {x}= maximum value of x over the set of k types of space that zoning rules will
permit space of type ~ to become (could also be an expected maximum
consistent with logit formulation)
X1 to x3= parameters.
The representations in this component as well as the next two components of the developer
mode! imply that developers consider conditions and make decisions regarding each zone
independently. A representation that developers also consider the aggregate result of decisions
in all zones is included further below.
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3. Determine 'zonespecific' gain of space b T redevelopment of the tom amount of space made
available through demolition in zone, allocating among allowable types of space according to
relative profitability:
for each type of space 'A in the set of space types F
for each zone 'I' in the set of zones 'I':
exp~x~(Rf,2Cfi~) Max~gRI;jiC~)
~3fi = (~2ki) ~
Ik exp~x~(Rf,~Cfi~)  Max~(R~itC~i )~)
where:
L3n = gain of space of type ~ in zone ;I through
X1 = parameter.
4. Determine 'zonespecific' gain of space through new development, talking account of the total
developed space of all types and of the specific type the profitability of development and the
available capacity for new development:
for each type of space 'f' in the set of space tripes F
for each zone 'I' in the set of zones 'l:
~4n = X]~Skit)~(Snt)~X4+X5eXp(X6~C6~+X7~i ~ X8eXp(X9(RntC6~(Q6~+~Snt)XO
where:
Qfi~+] = total amount of space of type ;f allowable in zone 'I' in time period 't+l'
according to zoning
X1 to xo= parameters.
Note that a multiplicative form is used, which means that if the available capacity (as indicated
by the last term) is zero, then the amount of new development will be zero. Also note that two
alternative forms of function depicting increase in the development of space in response to
profitability are included in order to provide flexibility in calibration
5. Determine total net development dampening aggregate adjustment, which is the amount by
which the total net development of space is reduced in order to represent that developers will not
respond fully to all the opportunities in all zones
for each type of space 'f' ir1 the set of space types F:
~5 = ISft)~iX] I /St~)]]
where:
x~ = parameter.
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A = total net development of space of type f in all zones before adjustment
=  Llf L'f~ L3f + L4f
L]f = modelarea wide total loss of space of type f through dilapidation = ~ iron
L'f = model~ea wide tom loss of space of type 'L through demolition for
redevelopment
= ~ jL2n
L3f = modelarea wide total gain of space of type 'fir through redevelopment
= ~,L3fi
L4f = modelarea wide total gain of space through new development = ~ i~46
Sit = total space of type 'A in time period t' in all zones
= Kiwi
Note that this assumes a straightforward proportional reduction in the total net increase (or total
net decrease) of space for the entire model area without taking into account rents. profitability
or capacities. Further enhancements might consider adding terms to represent the influences of
such factors.
6. Allocate total net development dampening adjustment among zones, taking account of relative
amounts of existing space, profitability and capacity:
for each type of space 'L in the set of space types F
for each zone 'T. in the set of zones 'I':
~5 = ~5
where:
(S t)Xl(Rfit C t)X2 (Q t1 S t)X3
j (Sfit)~l(RfitCfit)~ (Qua ]Sfi~)X2
X1 to x3= parameters.
Combine the various losses and gains to establish new space total for next time period:
for each type of space 'A in the set of space types F
for each zone 'l. in the set of zones 'I':
Sp = Sfi ~ fi  ~ fi ~ ~ 6, ~ fi ~ fi .
Zoning regulations enter the mode] via the developer mode} as both (a) constraints on the
allowable amounts of space in each zone and (b) pe~itted conversions for each type of space in
each zone. Tax/subsidy structures enter the developer model as adjustments to the cost to develop
units of space in zones. God. Tax/subsidy structures can also be included in the model as adjustments
to the zone specific constants, Zai
The mode] in time penod 't+l allocates the exogenouslydeterrnined modelarea wide activity
totals for time period ;t+l' according to the new space totals, Sfi2i, along with the accessibilities
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determined for time period 't'. Bait , and the activity distributions for time period 't ~ 0' ~ thus
simulating the lagged response to chimes In travel costs and the inert respectively, in the system.
It is anticipated Mat the and developer models in this system can be calibrated given reasonably
complete observations of the distributions of activities and space in relevant categories for two or
more points in time, along with some additional indications of the corresponding patterns of
dilapidation, demolition and development of space between these points in time. Some observed
rents for space would also be helpful, but are not essential. This mentors the situation with transport
models generally" where observed network travel times (analogous to rents in land markets) are
helpful but not essential in the calibration of a capacityrestrained traffic assignment process.
It should be noted Mat the simulation of space development through time is one of the primary
areas where improvements are needed in both the state of the art and the state of the practice in and
transport interaction modeling. Accordingly, a detailed treatment and a set of fairly flexible potential
functions have been included for the developer model in this descnption of a prototype. Specific
applicatior~s in the short term might use simple forms of the functions or even combine some of the
treatment. Consideration of the potential functions indicated here as well as other potential
functions and approaches for this component, should be included in the research activities for both
the shortterm and the longterm.
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