Title: An Integrated Behavioral Model of Land Use and Transport
System: a Hyper-Network Equilibrium Approach.
Authors:
Mr. Mario Bravo, Universidad de Chile
Mr. Luis Briceño, Universidad de Chile
Dr. Roberto Cominetti, Universidad de Chile
Dr. Cristián E. Cortés, Universidad de Chile
Dr. Francisco Martínez, Universidad de Chile 1
1
Corresponding Author: POBox 228-3 Santiago, Chile, Phone: 562-9784380, Email: [email protected].
1
Title: An Integrated Behavioral Model of Land Use and Transport
System: a Hyper-Network Equilibrium Approach.
Authors: Mario Bravo, Luis Briceño, Roberto Cominetti, Cristián E. Cortés,
Francisco Martínez2
Abstract:
The households’ decisions, from their residential location to their members’ trip choices
through the network, are analyzed in a long term equilibrium approach by building a model
that assumes that all consumers’ choices (location, travel and route choices) are represented
by logit or entropy models. Our approach assumes that consumers optimize their combined
residence and transport options, which are represented as a set of paths in a network defined
by extending the transport sub-network with another fictitious sub-network that represents
land use and demand options. In this context, we study the equilibrium of the land use and
transport system (LU&T) as a network equilibrium problem. We define an equivalent
objective function and the set of system constraints, which optimized on consumers
willingness to pay and transport travel times, yields the set of static equilibrium conditions.
In the land use market they reproduce the action (bid-choice) market clearance where every
household is allocated at her highest utility option and dwellings are assign to best bidders
(Martínez and Henríquez, 2003). The transport equilibrium yields the routes assignments
and travel times on all links and routes following the Markovian Traffic Equilibrium
problem (Baillon and Cominetti, 2006). At equilibrium no household is better off by
choosing a different option for residential location, no household member is better of
choosing a different set of trips’ destinations, modes and routes (frequency is fixed).
We study the static equilibrium with road congestion differentiating the cases with and
without interactions among agents; these interactions include endogenous neighborhood
quality for residential choice and trip destination choices based on endogenous land use
attractions. In the case without interactions the method defines a strictly convex, coercive
and non-restricted objective function (under usual conditions). Our method provides a
comprehensive characterization of the solution, regarding existence and uniqueness, and an
algorithm to obtain the solution with well-defined convergence properties. In the case with
agents’ interactions the equilibrium is not unique. We define the unified problem as a set of
fixed-point equations that characterizes the equilibrium. The algorithm to identify solutions
is based on fixed-point iterations. In both cases the model is applicable real size problems,
with heterogeneous population and locations, and multiple trip purposes.
2
Corresponding Author: POBox 228-3 Santiago, Chile, Phone: 562-9784380, Email: [email protected].
2
1.- INTRODUCTION
One of the major complexities in modeling big urban areas for planning purposes is to
properly represent the strong connection between the transportation system and the spatial
distribution of residential and non-residential activities. On the one hand, the activities’
spatial patterns represent the major determinant of generation and attraction of trips
from(to) each zone, while, on the other hand, the transportation system is also a relevant
input for location decisions through the resulting measures of accessibility due to the
transportation system design along with the demand conditions for such a system.
Therefore, changes in land use and activities directly affect the transportation demand
patterns, which can eventually change accessibility, and so on. This multilevel process
establishes a complex global equilibrium of the entire urban system, and a close
formulation of such a problem can be explored through the interaction between the
equilibriums of each subsystem.
In the specialized literature, we can find several articles dealing with the interaction
between land use and transport (LU&T). Some models postulate the equilibrium in land use
taking into consideration the importance of the transport in the location decisions, through a
measure of the generalized transportation costs, which are assumed known for the land use
equilibrium mechanism. The generalized costs are normally obtained from network
assignment models, computed for fixed location patterns. However, from this framework,
the authors do not solve simultaneous land use-transport equilibrium. Instead, they find the
global equilibrium by means of iterative calculations of partial equilibrium of land use on
the one hand, and transport on the other; these schemes are known as bi-level models in the
literature. The major concern with regard to these models (apart from the expensive
computation involved in solving these problems) is that we can neither ensure existence nor
uniqueness of the procedure solution. Therefore, any heuristic method designed for this
purpose does not ensure convergence to the global optimum. From all these observations,
we can say that to ensure existence as well as uniqueness of LU&T equilibrium is so far an
open research topic.
In this research, we first develop an integrated LU&T model based on a variational
inequality formulation for the equilibrium. A convex optimization problem is defined on an
extended network, representing the decisions not only taken at the transport system level
but also at the land use system, both in the same graph. We ensure existence as well as
uniqueness of the optimum under reasonable assumptions. The first order conditions of this
problem reproduce the equilibrium conditions of two previously developed models, the
“Random Bidding and Supply Model” (RB&SM) by Martínez and Henríquez (2006) for
urban location and the “Markovian Traffic Equilibrium” proposed in Baillon and Cominetti
(2006) for private urban transport networks. Thus, the proposed model yields the
equilibrium of the integrated LU&T system. One important feature of such models is that
all agent decisions can be modeled by a logit model, which also applies in the case of the
proposed model for the global equilibrium. Moreover, a solution algorithm is developed,
from which its convergence to the global LU&T equilibrium is also proved.
3
The aforementioned model does consider neither externalities among households nor
agglomeration economies among firms. In fact, including these types of interaction does
not allow writing the problem as a variational inequality. Nevertheless, we recognize that
land use externalities can constitute an important antecedent when taking medium and long
term location decisions, for both households and firms. In the case of households, location
decisions of certain socioeconomic group members are usually affected by the
socioeconomic characteristics of the neighbors, resulting in segregation, among other
phenomena. In the case of firms, we observe important interactions among firms that, for
example, generate incentives to stay together, called agglomeration economies.
Hence, also in this paper we develop a modified LU&T model, which includes externalities
in the locations of the agents. Such a model is formulated as a multidimensional fixed
point, and represents the simultaneous equilibrium of land use, generation of trips,
distribution and network assignment. The approach is based on splitting the entire system
into smaller (simpler) subsystems. The interrelation among subsystems allows us to prove
existence and uniqueness of the global equilibrium solution. With this added complexity of
including externalities in the model, there is no closed-form for the optimization problem.
However, the system still can be seen as an expanded network considering the entire
modeled process, as described in detail in Section 4. A solution algorithm for this model is
also proposed, which under very reasonable assumptions ensures convergence to the
equilibrium.
Thus, in Section 3 and 4 the two above described approaches are presented, both conceived
to represent the interaction between the land use market and the transport system, written
under an integrated formulation: a variational approach for the case without externalities,
and a fixed-point approach to add externalities in the model. Both approaches ensure not
only existence but also uniqueness of the equilibrium solution, under very reasonable
assumptions.
In the next section, the most relevant previous research efforts to model the interaction
between land use and transportation, and the LU&T equilibrium, are briefly described.
2.- BACKGROUND
In the specialized literature, we can find several examples of interactions between the land
use market and the transportation system (LU&T). According to Chang (2006), the models
can be categorized in Spatial Interaction, Mathematical Programming, Random Utility and
Bid-Rent models. However, the problem of formulating the equilibrium LU&T is still an
open problem, so far described and formulated by using some simplified models (in most
cases heuristics), from which there is no analytical way to establish conditions of existence,
uniqueness and convergence of such an equilibrium.
The first step to find an integrated formulation LU&T is to properly model the trip structure
behind the system. The spatial interaction model proposed by Lowry (1964) and later
generalized in Wilson (1970), introduces the concept of cost impedance between zones,
explicitly represented by a cost function. Wilson (1970) postulated a model based upon the
4
maximization of the system entropy. Although this case considers a fixed cost between
zones, the author introduces a relative measure of the zone attractiveness. The model is not
really able to completely explain the relation between land use and transport, mainly due to
considering constant transportation costs.
A relevant land use model to discuss is the context of this paper is the RB&SM, which
belongs to the “Bid-Rent” type modeling (Martínez and Henríquez, 2006). Real estate
transactions are commanded by an auction mechanism, under a best bid rule. In this
scheme, the resulting willingness to pay for each location describes the behavior of the
decision takers, as proposed by Alonso (1965). The RB&SM model is an extension of the
Random Bidding Model (RBM) previously developed by Martínez and Donoso (2001).
Another relevant land use model within the spatial interaction approach, is the doubly
constrained entropy model (Roy, 2004), similar to the proposal by Wilson (1970) with the
difference that in this case the location is determined by the agents´ willingness to pay
assumed to be known. From this model, the logit probabilities proposed by Ellickson
(1981) are obtained, because the entropy maximization approach and the multinomial logit
are equivalents when the parameters of the latter are estimated with the maximum
likelihood method (Anas, 1981). Then, the RB&SM model can also be derived by
formulating an optimization problem as a maximization of an entropy function (with
neither externalities nor scale economies).
The aforementioned land use models find the equilibrium considering the transport system
and its interaction with the land use market. However, in all cases the generalized costs for
transport are assumed exogenous.
The trip assignment models determine the route to be followed by each trip once both the
mode and destination have been chosen (see Ortúzar and Willumsen, 1994). The process is
modeled through the Wardrop network equilibrium conditions, for both determinist and
stochastic traffic assignment (Sheffi, 1985). Nagurney and Dong (2002) propose an
integrated model and formulates the network assignment problem as a variational
inequality, reproducing the Wardrop conditions. Baillon and Cominetti (2006) on the other
hand, developed a markovian equilibrium model for stochastic assignment. Unlike the other
network equilibrium models (route based decision), the markovian equilibrium models a
chain of decisions, where at each node the user decides the next link to get in, pursuing the
minimization of the expected travel time to reach a predefined destination, regardless of the
assignment decisions taken before.
The link between transport and land use may be identified by access measures, which can
be derived from a rigorous microeconomic framework (Martínez, 1995), and has been used
to identify the complex relationship between user benefit measures from both the land use
and the transport systems (Martínez and Araya, 2000). They explicitly compute the
transport benefits associated to origin and destination zones (accessibility and
attractiveness) in the context of a doubly constrained spatial interaction model.
One way to deal with the integrated problem is to use mathematical programming in the
modeling. That consists on writing the decision problem as an optimization problem. The
5
associated objective function must ensure existence and uniqueness of the global minimum.
Chang and Mackett (2005) formulate a bi-level problem to integrate both levels. At the
superior level, the location problem is faced under a bid-rent approach by computing the
access (accessibility and attractiveness) of the zones. At the inferior level, the network
decisions are made taking into account the access measures decided at the higher level.
This procedure, however, does not ensure the existence of equilibria. Another model of this
type is the one proposed by Boyce y Mattsson (1999) based on mathematical programming,
in which the equilibrium at the transport network level, as well as that of land use, are
solved through optimization problems. The formulation satisfies equilibrium conditions at
the transport level, however, there are no equilibrium conditions attained at the location
level.
The objective of this paper is to model the interaction between the transport and the land
use system, ensuring existence and uniqueness of equilibrium, adding also a fixed-point
approach to deal with externalities in location. Moreover, efficient algorithms to solve these
models are proposed, ensuring convergence toward the global optimum of the entire urban
system.
Before formulating the model, a glossary of the problem variables is presented:
Exogenous Data
- N : Set of nodes of the transport network.
- D : set of destinations (d) of the trips.
- C : set of nodes for household type (h) searching for a place to be located
- I: set o f zones where households can get a location.
~
- A : set of location arcs (ã).
- A : Set of links of the transport network.
- Ah : Set of links whose tails are node h.
- Ah : Set of links whose heads are node h.
- s a (.) : performance function of link a  A depending on the traffic flow.
- i a0 , j a0 : tail and head nodes of link a 0 , respectively.
- Q j  {i  H  N \  a  Ai , j a  j} : set of nodes reachable from node j through a single
link.

- Q j  { j a \ a  A j } : set of nodes reached from node j through a single link.
-  ijh (t )  i jh (t a   jjha (t ) ; a  Ai ) : Minimum expected time for a trip from the node i to
the node j , with the family of functions i jh belonging to the class  defined in
Baillon and Cominetti (2006), i.e., the class of functions which can be expressed in the
form  ( x)  E (min{x1  1 ,..., x n   n }) where   ( i )in1 is a random vector with
continuous distribution and E ( )  0 .
6
- H h : total amount of agents of type h .
- Si : number of real estate supply units at zone i .
- Nhd : number of trips generated from a household of type h with destination d.
- zhi : willingness to pay component corresponding to the value of location amenities,
including accessibility and neighborhood quality given by a household h for a place
within zone i.
Steady state variables
- t a : travel time of link a
- ri : rent of a real estate located at zone i .
- bh : monetary disutility index (bid) for type agent h .
- Hhi : number of households located at location i at equilibrium. H is the matrix of
households locations.
3.- INTEGRATED LAND USE – TRANSPORT MODEL WITHOUT
EXTERNALITIES: BASE MODEL (FIXED SUPPLY)
In the markovian equilibrium scheme (MTE) by Baillon and Cominetti (2006), the authors
develop a traffic flow assignment model on a transportation network, assuming known the
trip distribution patterns. They search for the equilibrium by means of the minimization of
an objective function defined on the transport network. In this first formulation we extend
the MTE model, but considering a more complex network, in which fictitious arcs and
nodes are added in order to represent the agents’ location decision. Then, the original
objective function is modified to represent the LU&T integrated problem.
In Figure 1 the extended proposed network G(N,A) is depicted. In this scheme a household
h choose the optimal path including an initial arc from a fictitious node h to the location
node i. This node also belongs to the network from which the members of household h will
start their trips. The trip generation process is represented by a second layer in the network.
Each trip chooses an optimal path through the transport network and finishes at a
destination node d. We assume that the number of trips generated is constant for each
household type, and also that the destination in the transport network is decided by the
optimal path; only one transport mode is assumed in the formulation framework.
The cost assigned to the location arcs represents the willingness to pay of the households
for the land use around the end (head) node of such an arc. The location is decided through
a bid-rent mechanism in the location decisions. Under this approach, the willingness to pay
function represents the agents’ behavior in the location decisions (Alonso, 1965). In this
model, the following willingness to pay function is postulated:
7
g Households’ population
h
w
or
k
Land use
equilibrium
d
ns
po
rt
N
et
j
i
Tr
a
d'
Transport
equilibrium
Figure 1: Urban system equilibrium: Hyper-network representation
Bhi (bh , t )  bh  zhi   N hd id (t )
(1)
d D
Note that this form for the willingness to pay is derived from assuming: a quasi-linear
subjacent utility function, in at least one of the consumption goods; an exogenous
household income; and that the consumer chooses only one location for residence. It can be
shown that under these assumptions bh  yh  uh / h , where yh is the income, uh the utility
level and  h the marginal utility of income (Martínez and Henríquez, 2006). Thus, the first
term shows the utility level reached by a household of type h, which has to be the same for
all households of the same type under equilibrium conditions. The second term is assumed
to be constant and it captures how a household of type h values the attributes of a specific
zone i, in this case assuming that location externalities are exogenous to the equilibrium
process. Finally, the third term provides a measure for the transport cost, as explained
ahead in this paper, assuming only one trip purpose and N hd trips per each destination zone
d exogenously given.
The values of state variables at the optimum are found by solving the following
optimization problem:
ta
min (t , r , b)    sa1 ( z )dz   H hbh   Si ri 
t , r ,b
aA 0
hC
iI
1

e 


Bhi ( bh ,t )  ri 
(2)
hC
iI
8
This problem may be explained as the dual of the following doubly constrained maximum
entropy problem (P), assuming transport costs as fixed:
min   H hi Z hi (t ) 
( P)
H hi
s.a.
iI
hC
1
hi
 Hh
 hC
H
hi
 Si
 iI
hC
hi
[ln( H hi )  1]
hC
H
iI
H
 iI
where Z hi (t )  zhi   N hd id (t ) . This problem conceptually represents the maximization of
d D
bids under stochastic bids, or the representation of the auction prices.
| A|| I ||C |
 . Moreover, the number of households
In this model, the function  :
located at each zone i , is bounded from above by the zone supply, which is assumed to be
fixed in this scheme. Additionally, equation (2) is invariant to shifts in b and r, that is
min (t , r , b)  min (t , r  c, b  c) c  . Like in the markovian equilibrium model,
the objective function in (2) is strictly convex and coercive, under the assumptions (Hyp
A), b1  0 and  Si   H h . These conditions ensure both existence and uniqueness of
iI
hC
the solution at the optimum.
The family id  E , with  dd  0 ; travel time functions sa :
(Hyp A)=


are strictly
increasing continuous, with lim sa ( x)   and sa (0)  t  0 ; and  (t 0 )  0 for
0
a
x 
d
i
all i  d
The first order conditions of problem (2) characterize the equilibrium, not only in the bidrent market (RB&SM) but also in the transport system, through the traffic markovian
equilibrium MTE. Analytically, the first optimality condition is
h  C

(t , b, r )  0
bh

H h   exp(  ( Bhi (bh , t )  ri ))
(3)
iI
Defining H hi : exp(  ( Bhi (bh , t )  ri )) that corresponds to the total number of type h
families located at zone i , the following land-use equilibrium condition is fulfilled:
H h   H hi
h  C
(4)
iI
Moreover, by replacing the expression for Bhi (bh , t ) in (3), we obtain:
9
exp( bh ) 
Hh
 exp( ( zhi   Nhd id (t )  ri ))
(5)
d D
iI
Thus, replacing (3) in the definition of H hi yields:
H hi  H h 
exp(  ( Z hi (t )  ri ))
exp(  ( Bhi (bh , t )  ri ))
 Hh 
 H h  Pi / h
exp(

(
Z
(
t
)

r
))
exp(

(
B
(
b
,
t
)

r
))


hj
j
hj
h
j
jI
(6)
jI
where Pi/h represents the probability for a household of type h to prefer a real estate at zone
i . Note that Martínez (1992b) obtains this probability (which is called “choice”), assuming
that, on the one hand, the household’ surplus, defined by hi  Bhi (bh , t )  ri , distributes
Gumbel identical and independent (iid) with dispersion parameter  , and on the other
hand, the household choose in order to maximize such a surplus.
From equation (3) we can also deduce that, for each h  C , (13) applies:
bh 

 1
ln   exp(  ( Z hi (t )  ri ))   ln( H h ),
  iI
 
1
(7)
Note that equation (7) reproduces fixed point problem for the RB&SM market clearance
equilibrium condition.
A second optimality condition is
i  I ,

(t , b, r )  0  Si   exp(  ( Bhi (bh , t )  ri ))
ri
hC
(8)
which is equivalent to S i   H hi . From this relation, we can obtain:
hH
exp(   ri ) 
Si
 exp(  Bhi (bh , t ))
i  I
(9)
iI
and
H hi  Si 
exp(   Bhi (bh , t ))
 Si  Ph / i
 exp(  Bgi (bg , t ))
(10)
gC
10
where Ph/i is called the “bid” probability in the RB&SM. This represents the probability
that the consumer of type h is the best bidder in location i competing with all bidders in C,
which is derived in Martínez (1992) by assuming bids distributed iid Gumbel with
parameter  .
Moreover, from (10) we obtain that, for each i  I :
ri 

 1
ln   exp(   Bhi (bh , t ))   ln( Si ),
  hC
 
1
(11)
Expression (11) is, again, the same as the result obtained in the RB&SM model, and can be
interpreted as the expected value of the maximum expected willingness to pay among the
households asking for a place to be located.
In addition, a third condition is provided by the equilibrium on the transport network, from
where the following first order conditions are obtained:
a  A,
B

1
(t , b, r )  sa1 (ta )   exp(  ( Bhi (bh , t )  ri ))   hi (bh , t ),
ta
 hC
ta
iI
where
Bhi
 d
(bh , t )    N hd  i (t ) .
ta
ta
d D
Then,

(t , b, r )  0
ta

  d
 sa1 (ta )      H hi N hd  i (t )
d D iI  hC
 ta
By defining gid :  H hi N hd , representing the total number of trips whose destination is d
hC
starting at zone i , we obtain:
1
a
s (t a )
d
 id
d 
 g
(t )    g ,
(t )   vad  wa ,
ta
ta
d D iI
d D
d D
d
i
(12)
from where, the equilibrium condition on the arcs belonging to the transport network
system is reproduced, t a  s a (wa ).
11
From such results, we can prove that the solution of the optimization problem in (2)
simultaneously satisfies the equilibrium conditions of both models, RB&SM and MTE.
Remark
Several extensions of the proposed model can be analytically developed. One
straightforward extension is to add the distribution of trips problem into the integrated
modeling framework. Another interesting extension is to incorporate the real estate supply
as an endogenous variable. For further details on these, see Briceño (2006)
4.- INTEGRATED LAND USE – TRANSPORT MODEL WITH EXTERNALITIES
4.1.- General framework
The objective of this part of the paper is to prove existence of the static equilibrium
associated to the location, generation, distribution and assignment as a joint global
equilibrium considering externalities. Since externalities (either positive or negative) do not
allow us to write the equilibrium as an optimization problem, we propose a four stage
approach in order to conveniently split the problem into: Generation,
Distribution/Assignment, Land Use equilibrium and Location externalities. The objective
of this scheme is to capture the properties of each subsystem, stating a multidimensional
fixed point equation, in order to prove the existence of the global equilibrium. The proof of
existence in the fixed point equation is quite hard due to the complexity of the model when
adding externalities that influence the agents’ decisions.
In simple words, the modeling approach considers people searching for a location. They
make their decision based upon (among other things) accessibility, attractiveness and the
features of other agents located at each zone (externalities). Those aspects associated with
access are provided by the transport equilibrium, i.e., the generation and distribution of
trips and the travelers’ assignment on the physical transport network. Besides, such
transport equilibrium depends on the location of households and firms in the city. Thus, as
noticed here, there are too many endogenous variables that are intrinsically connected, and
form part of the global equilibria studied in this paper.
In order to understand how the land use process enters in this modeling approach, let us
express the willingness to pay (bid) for agent type h and zone i as follows:
Bhi  bh  bhi
(13)
Expression (13) shows the interactions between transport and land use. The arguments for
existence we will develop in this paper consider any quasi-linear bid function of the form
Bhi  f ( H , O,  )  bh , with f being a smooth function (of class C 2 ). We can even consider
a bid depending explicitly on the travel time.
12
In Figure 2 we graphically show the proposed 4-stage methodology: first, we consider the
vector ( H hi ) hi belonging to a simplex |C||I | (provided that it has to fulfill the typical total
summations and no-negativity constraints). Then, the trip generation is computed by means
of the evaluation of a function. After that, the network assignment, along with the trip
distribution problem, is solved obtaining as a result the accessibility parameters. These are
used to construct the willingness to pay functions that enter the land use equilibrium to
satisfy the conditions associated to the total agents to be located. Then, the location
problem with externalities is solved. At each stage, the other stages’ parameters will be
considered fixed, defining the general fixed point (corresponding to the global equilibrium)
simply as the composition of the proper functions.
Loc. Externalities
Fixed Point(Stage 4)
Dist./Assig.
Stage 2

Oih
LU-Eq.
Stage 3
Generation
Stage 1
H hi
Bhi  f ( H , O,  )  bh
H  Simplex
Figure 2: 4-stage methodology to prove existence of urban system global equilibrium
In the next subsections, each stage is analytically presented, highlighting the interactions
with the other procedures. However, in Section 4 the global equilibrium is studied.
4.2.- Generation
For this stage, we consider a growth factor type model as follows:
 hi  thi H hi  Ohi
(14)
which differs from the model in Section 3 where thi  N h i was fixed.
The expression (14) for the total number of trips generated at zone i for agent of type h
comes from assuming a deviation from the traditional fixed rate generation procedure,
13
namely  hi small and positive. Note that this perturbation allows us to establish that

Ohi > >0 , i, h and H  K   x 
|C || I |

/
x
hi
hi

 S , xhi  0  with   Min  hi .
i ,h

Let us denote  1 the function that associates a configuration ( H hi ) hi with the generated
trips (Ohi )i ,h , that is 1 (( H hi ) hi )  (Ohi )i ,h
, where 1 :
|C || I |

|C || I |
is of class C 1 .
4.3.- Distribution/Network Assignment
At this stage, we jointly find the equilibria associated to both, the trip distribution and
network assignment problems. For the distribution problem, and in order to add generality
to the embedded attraction process, we focus our analysis on the simply constrained case, in
which the total number of trip generated at each zone is assumed to be known. The joint
equilibrium is reached by minimizing the following objective function:
ta
Min (t ,  )    sa1 ( z )dz   Ohi ( H ) hi  
t ,
aA 0
iN hC
1
hC h
  exp( (
d N iN
h
dh
i
(t )   hi   d ( H ))) (15)
This model is an extension of the markovian traffic equilibrium (MTE), which models the
network and the distribution of trips simultaneously. We assume that any node of the
transport network is destination of some trip, i.e D=N. Hereafter we assume also that
sa : R   R are the strictly increasing travel time functions with lim sa (wa )   . As in
wa 
the previous model, we assume the hypothesis (Hyp A) only asking the functions in family
E to be of class C2 .
Now, we can say that the optimal solution of problem (19) satisfies the following first order
conditions:

1
(t ,  )  Ohi 
 hi
h
obtaining
 exp( (
d N
h
exp( h hi ) 
dh
i
(t )   hi   d ( H )))  ( h )  0

Ohi   exp(  h ( idh (t )   hi   d ( H ))),
d N
Ohi
dh
exp(





h
i (t )  h   d ( H ))
d N
and

 1
 hi  ln   exp( h  idh (t )  h   d ( H ))   ln Ohi
h  dN
 h
1
This last equation shows the relationship between the Lagrange multipliers vector  and
travel times, which makes bids in equation (13) dependent on the travel times in the traffic
network.
Let us define
14
gidh : exp(h ( idh (t )   hi   d ( H ))) 
Ohi  exp( h ( idh (t )   d ( H )))
 Ohi  Pd / ih
 exp(h ( ikh (t )   k ( H ))
(16)
kN
where
Pd / ih
exp( h ( idh (t )   d ( H ))

 exp(h ( ikh (t )   k ( H ))
kN
This expression corresponds to the logit probability of choosing as trip destination zone d,
conditional on the trip origin zone i and the individual of type h. Thus, gidh measures the
total number of trips generated at zone i choosing destination d for individual h. Besides,

(t ,  )  0
ta

sa-1 (ta )     gidh 
iN d N hC
 idh (t )
,
ta
obtaining the transport network equilibrium condition as follows
sa ( wa )  ta
(17)
dh
since wa     gidh   i (t ) corresponds to the total vehicle flow on link a (Baillon and
iN d N hC
ta
Cominetti, 2006). Note that equation (17) guaranties that the network attains the MTE.
The model summarized in (15) corresponds to the dual of the simply constrained entropy
model, where the parameter  hi is the Lagrange multiplier associated to the origin
constraints. Moreover, note that the number of zones and nodes do not have to match
necessarily. However, the formulation above can be easily adapted (by defining proper
fictitious links with cost zero) to such a case.
As before, let us denote  2 the function that assign the Lagrange multipliers associated
with the solution of (15) to a set of parameters ((Ohi )i ,h , ( H hi ) hi ) . That is,
2 :
2| H || I |

| H || I |
Lemma 1. If (Hyp A) holds, then  2 is smooth ( C 1 ).
Proof: First, provided that for any set of parameters ((Ohi )i ,h , ( H hi ) hi ) the problem (15) is
feasible and the involved function is convex as well as coercive, there exists a unique
solution of the problem. That is, the function  2 is well defined. Then, by using the
Implicit Function Theorem we conclude that the function is smooth in the
parameters ((Ohi )i ,h , ( H hi ) hi ) .
15
4.4.- Equilibrium in Land Use
At this point of the construction, the term bh in the agents’ bid is studied. Such a term
corresponds to an adjustment of the perceived utility (or cost) for the individual h to get
located somewhere. The bh that corresponds to the equilibrium can be obtained from the
equation that ensures to locate all the agents. Analytically,
H
iI
hi
 Hh
h  C
(18)
When the H hi terms correspond also to those obtained for the location equilibrium (see
section 4.5), equation (18) becomes:

iI
Si exp(  Bhi )
 Hh
 exp( Bgi )
h  C
(19)
gC
Note that equation (19) summarizes the first order condition of the following problem:
Min
(b) 
|C|
b
b
hC
The normalization
b
hH
h
h
1
 S ln( exp( B
 iN
i
hi
hC
))   H hbh
hC
(20)
0
 0 is needed since (b  1c)  (b) , c 
. Note that the
function  in problem (20) can take values in |H | and can be visualized as a function in
| H |1
(to avoid the mentioned normalization) by fixing the value b0  0 . Thus, we obtain
an unrestrictive problem of the form:
Min
(b) 
|C|1
b
1
 S ln(  exp(  B
 iN
i
jN
hi
))   H r br
(21)
rC
The function  is strictly convex and coercive. Then, there exists a unique minimum bh of
the previous problem that fulfills the first order conditions. Note that, at this stage all terms,
except bh , are assumed fixed in equation (13), and also note that for any location structure
in the city (Hhi) there exists a unique vector bh that adjusts each type of agent utility to
make sure that all users are located.
Let us call
the function that associates a set of parameters
3
((Ohi )i ,h , ( H hi ) hi , ( ) hi ) with (bh )hH the unique solution of problem (21). Since the
solution is unique, the function  3 is well defined.
16
Lemma 2. If
H
h
h
 S  Si , the function  3 is of class C 1 .
i
Proof: Analogously to Proof of Lemma 1, one can conclude that  3 is C 1 in the
parameters ((Ohi )i,h , ( H hi ) hi , ( ) hi ) .
4.5.- Location externalities
In this section we study the fixed point associated with the location model. In order to
include the different behaviors as a result of the idiosyncrasy differences among the agents
involved in a market cluster, the bids are assumed to distribute iid Gumbel. From that, the
probability that one of the H h agents gets located at zone i as a result of an auction, is
given by the following the probability that an agent type h is the highest bidder (Martínez,
1992). That is,
Pi / h 
exp(  Bhi )
 exp( Bgi )
gC
By assuming fixed the real estate supply, we get
H hi 
Si exp( Bhi )
 exp( Bgi )
gC
Since Bhi is a function of vector ( H hi ) hi through the function f and the through the vectors
 and O , then we propose to analyze the problem from the following fixed point structure
which will be very useful later to define the whole equilibria:
H hi 
Si exp(  hi ( H ))
 exp( gi ( H ))
(22)
gC
where the functions  hi ( H ), h  C, i  I are differentiable, with continuous derivative (of
 K with:
class C1). Then, let us consider the function  : K 
hi ( H ) 
Si exp(  hi ( H ))
 exp( gi ( H ))
(23)
gH
17


Here, K   x  |C||I | /  xhi  S , xhi  0  is compact and  H h  S  Si . Note that it is
hi
h
i


possible to define H in a more restrictive way, by imposing  xhi  Si , i  I and
x
hi
h
 H h , h  C .
i
Let us now formulate the following lemma:


Lemma 3. If  H h  S  Si , K   x  |C||I | /  xhi  S , xhi  0  , and the functions
hi
h
i


1
 hi ( H ), h  C, i  I are of class C , then exists c  0 such that    0, c  there exists
one and only one H  K such that ( H )  H with  defined in (27).
Proof. Let M rlhi  max |
H K
 hi
( H ) | ; we have that M glhi   because functions  hi are of
H rl
class C1. Then, after some analytical work we get: |


 hi
( H ) |  Si  M rlhi   M rlgi  .
H rl
gC


|C|| I ||C|| I |
Applying any norm in
, and using the finite increments theorem, we get:
|| ( H )  ( H ) ||  M || H  H || H , H  K


with M  max Si  M rlhi   M rlgi  .
hilk
gC


That is, provided that M  0 holds, there exists some c  0 such that, for all    0, c )
there exists a unique H  K such that ( H )  H . The previous result is obtained by
choosing c such that c M  1 along with Banach’s fixed point theorem.
Note that, because the dispersion parameter of the Gumbel  distribution is proportional to
the inverse of the variance of the distribution  , then the condition for Lemma 3 to hold
implies a minimum dispersion of bids.
4.6.- Existence and Uniqueness of Global Equilibrium
In this section, we will focus on identifying the conditions to be fulfilled to ensure the
existence and uniqueness of equilibrium in the integrated model. Finding the equilibrium of
the global system is equivalent to find a fixed point of the equations analyzed separately in
Section 3. In summary,
H
iI
hi
 Hh
sa (wa )  ta
h  C
a  A
(24)
(25)
18
H hi 
Si exp( Bhi )
 exp( Bgi )
(26)
gC
g
d N
dh
i
 Oih
 hi  thi H hi  Ohi
(27)
(28)
where the bids Bhi are of the general form presented in equation (13). In short, the
integrated problem as in (24)-(28) can be written as follows:
 hi ( H ) 
Si exp(  Bhi ( H ))
 exp( Bgi ( H ))
(29)
gH
where Ohi  (1 ( H ))hi ,  hi  ( 2 (O, H ))hi and bh  (3 ( H , O,  ))h . That is, implicitly
and through the equations defined in the previous section, we can define the global
equilibrium in just one equation, where  : K 
 K . Note that, if we find any H  K
such that  ( H )  H , then equations (24)-(28) are satisfied. It is important to note that the
whole construction provided before collapses in this fixed point equation.
Theorem 1 [Existence of Global Equilibria].


 S   Si , K   x  |C||I | /  xhi  S , xhi  0  , and (Hyp A) holds, then
hi
h
i


  0 there exists H  K such that  ( H )  H with  defined in (29).
If
H
h
Proof . By using lemmas 1 and 2, we obtain that functions Bhi are of class C 1 because,
they are compositions of functions C 1 . Besides, provided that K is compact and the
function  is, in particular, continuous from K to K , then, by using the fixed point
theorem by Brouwer, there exists a solution of equation  ( H )  H , which defines the
equilibrium of the integrated model we have proposed.
In the same context, let us analyze the uniqueness of the same equilibrium.
Theorem 2 [Uniqueness of Global Equilibria].


 S  Si , K   x  |C||I | /  xhi  S , xhi  0  , and (Hyp A) then, if  is small
hi
h
i


enough, there exists a unique H  K such that  ( H )  H with  defined in (29).
If
H
h
19
Proof. The existence is ensured from Theorem 1. For proving uniqueness, from lemmas 1
and 2 we obtain, as before, that functions Bhi are of class C 1 , because they are
compositions of functions C 1 . Then, by using Lemma 3 with Bhi   hi we show
uniqueness of the equilibrium.
Remarks
1) The proof of existence of a global equilibrium for the integrated model including
externalities in location does not depend on the value of  . For the case of uniqueness, we
ask the parameter  to be small, or the variance of bids to be large enough.
2) Link travel times as well as the trip distribution between zones are output of the model.
From the global equilibrium it is straightforward to compute these variables at the
equilibrium.
5.- SIMULATIONS
The main purpose of these simulations is to compare the performance of the model without
externalities (MWE) with that of the model with externalities (ME) developed in this paper.
In the first model we used a global MSA (successive averages method), which is a gradienttype method to find the equilibrium on traffic flows, with an internal fixed point algorithm
to find the location equilibrium at each iteration. In the second model, we utilize a global
fixed point method to solve the location equilibrium with an internal MSA procedure to
obtain the transport network equilibrium at each iteration.
Two basic scenarios are generated and the equilibria obtained from the two models are
obtained. Locations externalities were represented as segregation effects between
households of different types, i.e. category dislikes living next (in the same zone) to some
others (representing the rich class), while the second one likes living close to the first one
(representing the poor class). The function that represents these externalities is linear.
Notice that these experiments are all feasible, since our ME model allow us to consider any
kind of smooth externalities.
The algorithms were implemented in MATLAB over the known Siouxfalls network, which
has 24 nodes and 76 arcs, with a travel time function of the form
sa ( wa )  ta0 [1  ba ( wa / ca ) pa ] . This function satisfies the conditions given before in order to
assure the existence and uniqueness of the equilibrium. For arc choice in the transport
network we considered a logit model with a scale parameter  independent of the node,
household cluster, destination or trip purpose. That is:
 idph (t )  
1

ln
 exp( (t
aAi
a
  dph
ja (t )))
20
The basic scenario (named 0) considers: 5 categories of households divided into two
groups: the poor class includes categories 1 to 3 and the rich class includes categories 4 and
5; only 1 trip purpose available in the whole set of nodes; high attraction factors on 5
special nodes of the network which makes that almost all network trips are attracted to this
neighborhood, called neighborhood A; inelastic population of consumers (Hh) and supply
of locations (Si), initially spread homogeneously over the network’s nodes. Besides,
through constants zhi we simulated the preference of the poor class for living in
neighborhood A, while the rich class prefer neighborhood B represented by four neighbor
nodes. Thus, neighborhood A represents a poor residential area but also an employment
district, while neighborhood B is a high class residential neighborhood.
We designed simulation that shows subsequent scenarios divided into two sets. The first
set, denominated k--scenarios, shows the influence of the differentiated amenities zhi , then
the second set, denominated k+-scenarios, shows the influence of increasing congestion
created by increasing demand Hh in all categories. In the first set, k--scenarios, zhi increases
linearly according to zhik  zhi0 (1  k / 4) , thus, for example, the influence of zhi in the -4
scenario is null. In the second set, k+-scenarios, demand and supply increases in successive
k+-scenarios as follows: H hk  H h0  2k and Sik  Si0  2k .
In the case with externalities, any bid derived form a quasi-linear utility can be used to
define a proper bid function. In the experiments, this function will be
 1

(30)
Bhi  bh  zhi  h   log(Oih )  ih   i ( H )   h ( H i )

 h

1

The term  ln(Oih )   ih   i ( H )  in (30) is the consumer’s valuation of access advantage


–called accessibility- from zones i. It represents the traveler’s surplus -or transport user’s
benefit- obtained from the transport system considering a single trip between zones i and d,
assuming that the travel demand is described by a singly constrained trip distribution
entropy model; see Martínez and Araya (2000) for the derivation of accessibility in the
doubly constrained model. The traveler’s surplus is obtained from the consumer’s travel
demand curve and weights the benefits of visiting activities and the corresponding trip
generalized cost. For the sake of simplicity, we assume a single trip purpose and a single
type of housing in the location model.  ih is, like in the development, the Lagrange
multiplier of the trip’s destination choice problem constrained at the origin; they explicitly
depend on travel times in the traffic network, as we shown. Here the function
 d ( H ) corresponds to an attraction function of the trip destination zone d (assumed to be
constant for simplicity). The parameter  h describes how a user of type h values the
access. zhi describes the consumer’s value of exogenous features of the zone.
Additionally, the term h ( Hi ) represents the consumer’s value of the set of location
attributes, which describe location externalities at zone i, where  h is linear. Intuitively,
21
 h represents the function of like (or dislike) of the agents of cluster of type h for other
agents located at zone i . Unlike the model in Section 3 without externalities, here two
types of externalities adjust in the equilibrium: location attributes h ( Hi ) and destination
attractions  d ( H ) ; also in the previous model  h =1 for all users.
Figure 3 assumes no externalities effects, i.e. hi ( H )  0 , which makes the two models
comparable (MWE and ME). It depicts the share of households’ classes (rich in red and
poor in blue) in neighborhood A at equilibrium, and its evolution along different scenarios.
This figure shows that the behavior of both models is similar despite their theoretical
differences. It can be seen that as the preference for the neighborhood A increases,
represented by the increase in the respective zhi in successive scenarios, the poor class tends
to concentrate in this neighborhood while the rich class concentrates in neighborhood B.
This shows the different preference for neighborhoods between socioeconomic classes. In
the second set of scenarios, the increase in congestion make more attractive for the rich
class to live next to their jobs, which explains the tendency of this class to outbid the poor
in neighborhood B. In this case, the rich class faces two opposite preference: the preference
for neighborhood B and the reduced travel costs by living in A. The figure also shows that
the ME models adjust changes faster than the MWE model.
Shares
Poor-Rich Shares in Neighborhood A (ME Model)
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
-4
-3
-2
-1
0
1
2
3
4
5
Scenario
Figure 3: The case without externalities
Figure 4 depicts the case with externalities hi ( H )  0 , considering the evolution of the
population as in the previous figure, but keeping the attraction factors fixed at the levels of
scenario 0 in Figure 3. Thus this figure only represents the k+_scenarios modelled with the
ME model. It shows that the outbidding process is lees intensive than in Figure 3, because
the segregation effects plays the role of inducing a higher preference for neighbourhood B.
Thus, in this case the rich class face more intensive opposite preferences.
Another comparison between the two models is regarding their computing performance.
The MWE model attains equilibrium in 24 seconds and the ME in 113 seconds, both for the
same precision running on a 3.2GHz PC. The difference is explained principally by the
number of times each algorithm solves the MTE problem.
22
Figure 4: The case with externalities
6.- CONCLUSIONS AND FURTHER RESEARCH
The models developed here allow the full integration of the land use and the transportation
systems, for two cases of interest.
The case without externalities builds the bases for an integrated model based on hypernetworks, i.e. an extension of the classical transport network to represent the land use
market equilibrium. The model follows the classical Beckmans’ approach of defining an
equivalent optimization problem that reproduces Wardrop’s traffic equilibrium conditions.
The contribution of our model is to extend the approach to simultaneously reproduce the
transport and land use equilibria under a unique optimization problem. The main theoretical
result is the proof of a unique solution for the LU&T system. The model can be used in real
contexts by assuming that externalities are lagged in one or more periods, i.e. consumers
make choices using information of the land use system that takes time to be acquired. In
this sense, this model is regarded as a partial equilibrium model.
The second model includes externalities, which makes no longer valid the optimization
method. In this case the LU&T problem is decomposed into four fixed point sub-problems:
Trip Generation, Distribution/Network Assignment, Land Use and Location. Despite the
increase in complexity induces by externalities, our method proved to yield a unique
solution for the global equilibrium, under the mild condition of minimum level of
dispersion on consumers’ bids. The model can also be applied in real contexts, assuming in
this case a longer term where all externalities adjust to a LU&T’s equilibrium. Unlike the
previous model, this one attains a general equilibrium in the LU&T system. Although the
analogy with the transport network classical optimization model is lost in this case, the
analogy of a hyper- network representation remains.
One limitation of both models is the treatment of the transport system as only private
transport modes. However, the extension to public transport assignment can be readily
included by using a shortest path approach with congestions in bus stops, following for
example, De Cea and Fernández, 1993), which is completely compatible with the proposed
model.
23
The hyper-network model of the urban system can be seen as a platform for modeling other
dimensions of the urban system. Further developments may include the information and the
goods markets, as additional layers in the hyper-network.
Finally, the hyper-network approach can be used to specify dynamic urban processes on the
hyper-network, including equilibrium stages along time on each submarket, in line with
Martínez and Hurtubia (2006). This would allow considering delays in infrastructure
development and the introduction of lack of information on key variables of decision
makers, like on expected future prices. The equilibrium problem so far developed provides
the basic structure for such further extensions.
ACKNOWLEDGMENTS
This research was partially financed by project Fondecyt 1060788 and the Milennium
Institute in Complex Engineering Systems.
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