MULTIPLE EQUILIBRIA IN SPATIAL ECONOMIC
TRANSPORT INTERACTION MODELS
Francesco Russo, Giuseppe Musolino
Department of Computer Science, Mathematics, Electronics and Transportation
Mediterranea University of Reggio Calabria, Reggio Calabria (Italy)
phone: +39.0965.875272, fax: +39.0965.875231
email: [email protected]; [email protected]
1. INTRODUCTION
There is a two-way relationship between spatial economic and transport systems.
The former affects transport, conditioning travel demand patterns. Conversely, the
latter plays an important role in the spatial organization and economy of an area
(national, regional, urban), affecting activity location, production levels and trade
patterns. The above mutual interactions are part of what we define as a Spatial
Economic Transport Interaction (SETI) process.
Figure 1 shows schematically the components and interactions involved in the SETI
process. As regards the spatial economic system, the endogenous components are
activity generation and location (with land use at the urban scale); the exogenous
component is transport accessibility, the endogenous interactions are related to
generation and location, and the exogenous interaction is accessibility to location. As
regards the transport system, the endogenous components are transport demand
and supply, the exogenous component consists in the level and spatial distribution of
activities, the endogenous interactions are related to demand and supply, and the
exogenous interaction concerns activities on travel demand.
components
Activities
Transport system
interactions
Spatial-economic system
Travel demand
Activity generation
Transport supply
Activity location
(land use)
Accessibility
Fig. 1. Spatial economic and transport systems: components and interactions
Considering the three planning dimensions (Russo and Rindone, 2007), in the time
dimension the process of interaction between the spatial economic system and the
transport system moves to a strategic scale; in the study-in-depth dimension, it has
directional scale (in which objectives and strategies are defined); in the spatial
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dimension, it may be related to two different scales. At the national scale, attention is
devoted to estimating the competitiveness of the different activities, defining
production levels and location convenience: hence we refer to a National Economy
Transport Interaction (NETI) process. At the urban scale, the focus is on effects of
transport mobility on the spatial organization of an area (e.g. location of residential,
services and production activities) and on land use: hence we refer to a Land Use
Transport Interaction (LUTI) process.
The paper presents a general formulation of a SETI model, which has two interacting
macro-models: a transport macro-model and a spatial economic macro-model. The
SETI model simulates the transport and spatial economic systems by means of
market mechanisms, where demand and supply interact, providing prices and
quantities. In the transport macro-model, user behaviour is simulated through
demand models which estimate emission, distribution, mode and path choices.
These choices are driven by utilities, which include transport costs provided by a
network model. Demand-supply interaction is simulated through an assignment
model, which estimates transport costs (prices) and flows (quantities) on the network.
If the available supply (transport facilities and services) is limited, congestion costs
arise. The spatial economic macro-model is composed by a generation model which
estimates demand (consumption) levels and a location model which simulates where
supply (production) is located across zones. Location choices are driven by utilities,
comprising supply (production) prices plus transport costs. Subsequently to demandsupply interaction, supply (production) prices and quantities are estimated in each
zone. Due to supply constraints, a rent could be generated. Both the transport and
spatial economic models provide the mechanisms to bring the demand in line with
the available supply.
The paper investigates the above mechanisms, describing several circular
dependencies, called multiple equilibria, within the SETI modelling framework. The
transport macro-model contains a circular dependency among travel demand flows,
link flows and link costs, which is formalized through the assignment model, and
another generated by the elasticity of travel demand on other choice dimensions than
that of the path. The spatial economic macro-model has a circular dependency
involving trade coefficients, selling prices and acquisition costs, and another arising
when production capacity is limited. The generation model contains a circular
dependence among selling prices and technical coefficients. Finally, the transport
and spatial economic macro-models are mutually interacting: the transport macromodel provides transport utilities for the location model; the spatial economic macromodel provides trade flows for the demand model.
To the best of our knowledge, only the first two circular dependencies have been
formalized and solved as a fixed point problem. The other ones were implemented in
a number of operational models, but no theoretical formulations have been stated.
The paper is structured in four sections. The first section reports the motivation
behind the study and introduces the problem of systems equilibrium versus the one
of systems dynamics. Section two presents a literature review on SETI models. In the
third section the proposed SETI model is formalized and the circular dependencies
are presented and highlighted.
2. MOTIVATION AND PROBLEM STATEMENT
The study was motivated by the awareness of the growing importance of
understanding and modelling the interactions between spatial economic and
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transport systems. It is widely accepted that in economically poor areas, transport
infrastructures are considered a prerequisite for economic development. Even in
industrialized areas, since the existing transport infrastructures progressively reduce
their level-of-service due to the increasing travel demand to be satisfied, there is
interest in the effects of new or improved transport infrastructures. Nevertheless,
there is still the need of reliable and operational SETI models to estimate spatial and
economic effects arising from improvements in transport infrastructures as well as
from transport policies. On the other hand, as governments are concerned with the
cost of providing and maintaining transport infrastructures under continuous pressure
to reduce public expenditure, they require rigorous justification of the need for
transport infrastructures which, in turn, means a need for accurate assessment of the
incidence of their wider economic benefit (Miyagi, 1996). In Italy, an increasing
number of local and regional authorities rely on SETI models to support strategic
transport planning, even if there is no specific legislation that drives this process.
Several modelling frameworks have been proposed in the past decades to model
interactions between spatial economic and transport systems. Recently, efforts were
focused to integrate the two systems in order to simulate a hierarchical user decision
process both in spatial economic including generation and location choices and in
transport, including making a trip, destination, mode and route choices. As an
example, figure 2 depicts a hierarchical user decision process in which the
performances of transport supply elements (disutilities and accessibility) are
modelled through a transport network model, which may be congested or noncongested. The performances of supply elements affect both transport user choices
(such as path, mode, destination and making a trip choices), modelled through a
travel demand model; and choices associated to generation and location of socioeconomic activities, modelled through a spatial economic model. The hierarchy
structure is represented in two directions: each choice is made conditionally upon the
higher level choices; the higher level choice is influenced by the expected maximum
(dis)utility, generally represented by a logsum term, of lower level choices.
Many of the existing SETI models rely on the concept of equilibrium, which describes
a stable state in which spatial distribution of socio-economic activities is consistent
with transportation costs (travel times), which depend on congestion, that is a result
of spatial distribution of socio-economic activities. The above equilibrium implies
multiple local equilibria involving some internal processes in both spatial economic
and transport systems.
The equilibrium concept is common in economic studies. Its kernel, in a Walrasian
sense (Walras, 1874), is the idea that agents belonging to markets in the economy
make mutually consistent plans, such that no agent has incentives to revise his/her
plan for a subsequent period except as a response to exogenous influences
(shocks). It is a situation of market clearing implying that agents are achieving the
maximum benefit under the existing constraints.
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… SEex ...
… SE | SEex ...
Generation
Disutilityg
… SEi, | SE...
Location
Disutilityl
.… . Oi, | SEi...
Emission
Accessibility
… . Dj,| (Oi, SEi)
Destination
Disutilitym
..Mm | (Oi, Di, SEi)
Mode
Disutilityp
Pk | (Mm, Oi, Di, SEi)
Path
Disutilityk
Supply
Legend:
model,
data,
endogenous interactions,
congested network.
i, zone i; j, zone j; ex, exogenous; SEi, population and employment in i; Oi, origin in i; Dj, destination in j; Mm,
mode m; Pk, path k.
Fig. 2 – Structure of a hierarchical decision process
However, due to the numerous processes involved in the spatial economic and
transport systems and to the fact that they take place in different time periods,
equilibrium is considered a conventional concept due to two main reasons. First,
continuous perturbations of supply and demand components within each system
make equilibrium impossible to reach in some cases. Second, the processes both
within spatial economic and transport systems are asynchronous in the sense that,
even if they interact with one another, they have their own speed of change. In
transport systems, mobility processes have different speeds of change, in the sense
that they have different response times due to variations in transport supply (figure
2). Path and mode choice dimensions may have rapid variations; so-called day-today and within-day variations, especially for the path choice dimension, have been
described and simulated in literature (Cantarella and Cascetta, 1995; Nuzzolo et al.,
2001). The dimensions of destination and making a trip have a lower speed of
change than the previous ones. Mobility processes are the most rapid and may be
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termed very fast compared with those within the spatial economic system. Three
types of processes belonging to the spatial economic system may be identified
according to their speed of change (Wegener et al.; 1986). Fast processes involve
spatial location of employment and population (represented in figure 2 by the location
process); economic activities may change location and workers decide to apply for
jobs closer to their place of residence, or households move their residence closer to
where workplaces are located. Medium-speed processes concern economic,
demographic and technological aspects which affect the use of the physical
structures (represented in figure 2 by the generation process). Economic changes
involve the sectoral composition of employment caused by technological innovation
and changing consumption patterns. Demographic changes affect life spans (birth,
aging, and death) and household formation and composition. Technological change
plays an important role in urban transportation, involving innovations in private and
transit vehicles and in transit services. Slow processes are connected to industrial,
residential and transport construction and affect the physical structure (not
represented in figure 2). Industrial, residential and transport infrastructures are the
most permanent elements and they have only incremental changes (not considering
calamitous events and natural decay).
Although the original classification of Wegener et al. (1986) referred to the urban
context, we feel it may be generalized to regional and national contexts. Moreover,
the classification does not introduce segmentation among the mobility processes, as
reported above. But, it includes all among the fast ones: they “have an ambiguous
temporal structure. Seen as a short-term phenomenon, they are planned and
completed within hours. Seen in a longer time frame, they form habitual patterns that
do not change much faster than workplace and household locations” (Wegener et al.,
1986).
Nevertheless, modelling equilibrium presents some benefits. The first is that
equilibrium analysis is static (time-independent); while non-equilibrium analysis has a
dynamic nature and requires the treatment of time, as emerged from the above
considerations. An intermediate approach is called quasi-(or pseudo) dynamic, where
the systems are described as a succession of equilibrium configurations over
discrete time periods. Secondly, several equilibrium approaches to simulate both
spatial economic and transport systems may be found in the literature which allow
the formulation of mathematical models and provide access to operational
algorithms. For example, the user equilibrium approach (Wardrop, 1952; Dafermos,
1980; Daganzo, 1983; Cantarella, 1997, Cascetta, 2006) is a common practice in
transport models. User equilibrium models are solved by means of specific iterative
algorithms, but sometimes these algorithms are improperly used to simulate nonequilibrium conditions. In our opinion, they require specific analysis through dynamic
process models.
3. LITERATURE REVIEW ON SETI MODELS
The paper investigates the multiple equilibria present inside one popular category of
SETI models based on a Multi-Regional Input-Output (MRIO) framework. MRIO was
originally developed to represent national economies, subdivided into sectors and
zones (regions). In the national context, attention is focused on production location
and on travel (freight and passenger) demand estimation, neglecting land use
aspects. The basic concept lay in Keynes’s theory (Keynes, 1936), who introduced
the principle of effective demand, whereby production is determined by consumption.
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In the sphere of Keynes’s theory, Leontief (1941) first proposed an IO model to
simulate inter-dependencies between economic sectors through fixed technical
coefficients. Further theoretical developments from the original IO framework, able to
reproduce the spatial representation of the economy, were later proposed (Isard,
1951; Chenery, 1953; Moses, 1955). They introduced trade coefficients to calculate
exogenously interregional trade patterns and locate production across zones,
although they did not specify any model to estimate them. Several NETI models were
proposed, which incorporates a location model into the IO framework in order to
obtain an endogenous estimation of trade coefficients. Initially, trade coefficients
were estimated through entropic-gravitational location models (Leontief and Strout,
1963; Wilson, 1970). However, after the proposition of random utility theory
(Domencich and McFadden, 1975), they were estimated through discrete location
models (de la Barra, 1989; Echenique and Hunt, 1993; Cascetta et al., 1996). The
economy and freight travel demand have been extensively simulated at a national
scale (Cascetta et al., 1996; Russo, 2001; Marzano and Papola, 2004; Kochelman et
al., 2005).
In the urban context, a distinction must be made between models with exogenous
transport costs and models with endogenous transport costs (LUTI models). A model
belonging to the former category was proposed by Lowry (1964), which simulates
location patterns of residential and service activities, given the level and location of
basic (exogenous) employment. Lowry’s contribution was of capital importance and
several later attempts were made to overcome the original limitations of the model
and extend its applicability. Garin (1966) improved the Lowry model by casting the
entire model in matrix notation. Macgill (1977) presented the Lowry model as an
input-output model. Macgill's work marks the first attempt to build a metropolitan
Input-Output (IO) model that captures inter-sectoral linkages. Lowry’s model was
further developed by Putnam (1973, 1983, 1991), who proposed two models to
respectively locate residents (DRAM, Disaggregate Residential Allocation Model) and
employment (EMPAL, Employment Allocation Model) across zones. Several LUTI
models have been proposed like TRANUS (de la Barra, 1989), MEPLAN (Echenique
and Hunt, 1993; Echenique, 2004), IRPUD (Wegener, 1998) and DELTA (Simmonds,
2000). A detailed classification of SETI models is reported in Russo and Musolino
(2007, 2008).
4. SETI GENERAL FORMULATION
The SETI model has two interacting macro-models: the transport macro-model and
the spatial economic macro-model. The transport macro-model has three main
components, namely the supply model, demand model and assignment model.
The supply model is represented by a network model, with a primary graph and
aggregate link cost functions (time-flow relationship). The general formulation of the
congested network model consists of a path costs vs link costs consistency equation
(1.a) and of a path flows vs link flows consistency equation (1.b):
g = ∆T c (f)
f=∆h
(1.a)
(1.b)
with
g, path costs vector;
∆, link-path incidence matrix;
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c, link cost functions vector;
f, link flows vector;
h, path flows vector.
The demand model is behavioural (random utility based) and elastic to transport
costs on the emission and mode dimensions and to trade flows, with a stochastic
path choice model
h = P (∆
∆T c (f)) d (n, v)
(2)
with
P, probability path choice functions matrix;
d, demand functions vector;
n, trade flows vector (obtained rearranging the matrix N defined below);
v=v(g)=v(∆
∆Tc(f)), transport utilities vector (obtained rearranging the transport utilities
matrix V).
The assignment model is a user equilibrium model with stochastic path choice model
and elastic travel demand:
∆T c (f*))
f* = ∆ P (∆
∆T c (f*)) d (n, v(∆
f* ∈ Sf
(3)
with
f*, link flows vector at equilibrium;
Sf, set of feasible link flows.
The spatial economic macro-model is composed by:
• a generation model with technical coefficients depending on selling prices
y = A(p) y + ye
•
with
y, activity demand vector (internal end external);
A(p), technical coefficients functions matrix;
p, selling prices vector;
ye, exogenous activity demand vector;
a location model for estimating trade coefficient matrix, T, which depends on
transport utilities, selling prices and production
T = T(V, p, x)
•
(4)
(5)
with
T, trade coefficient functions matrix;
x, production vector;
a generation-location interaction model:
N = T(V, p, x) A Dg(y) + T(V, p, x) Dg(ye)
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Spatial economic macro-model
Transport macro-model
Location model
Demand model
Supply model
Generation model
p
c=c(f)
g=∆
∆Tc(f)
g
V=V(g)
V
T=T(V,p,x)
T
p=p(q,A)
q
q=q(T, p)
A
x
ye
x =1T N
y=Ay+ye
P=P(g)
∆
d=d(N,V)
A=A(p)
f
f=∆
∆h
h
h=P d
d
N
model
y
Generation-location model
Assignment model
Legend:
N =T Dg(y)
exogenous input
endogenous input/output
Fig. 3. - SETI model: components and interactions
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with
Dg(y), matrix obtained by arranging the elements of vector y along the main
diagonal.
Finally, production vector, x, is obtained from:
x = 1T N
(7)
In both the transport and spatial economic macro-models, the presence of
physical or regulatory constraints gives rise to rents, which reflect the
congestion in the transport and spatial economic systems and provide the
mechanisms to bring the demand in line with the available supply.
5. CIRCULAR DEPENDENCIES IN THE SETI MODEL
The general framework, depicted in figure 3, shows each modelling component
and mutual interactions inside the SETI model. The framework shows several
circular dependencies that are described separately below.
5.1. Transport macro-model
The transport model presents two circular dependencies. The first involves
travel demand flows, d, link flows, f, and link costs, c, which is formalized
through the assignment model (eq. 8):
f* = ∆ P (∆
∆T c (f*)) d
f* ∈ Sf
(8)
with d=const, which means that the travel demand is assumed to be rigid to
transport costs on dimensions other than that of path choice. The above circular
dependency affects the path choice dimension of transport users, which
presents the shortest response time in relation to changes in transport supply.
A second circular dependency arises when travel demand, d, is assumed to be
elastic to transport costs in the dimensions of mode, destination and making a
trip choices:
d=d (v(∆
∆T c (f))
(9)
In this case, the travel demand model becomes:
h = P (∆
∆T c (f)) d (v(∆
∆T c (f))
(10)
and the assignment model is formalized (Cascetta, 2006) combining eqs. (9)
and (10) with eqs. (1.a) and (1.b):
f* = ∆ P (∆
∆T c (f*)) d (v(∆
∆T c (f*))
f* ∈ Sf
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The above circular dependency affects the dimensions of mode, destination and
making a trip choices of transport users, which belong to what we call mobility
processes.
5.2. Spatial economic macro-model
The spatial economic macro-model presents circular dependencies both within
the location and generation models. The location model shows a first circular
dependency among trade coefficients of matrix, T, selling prices, p, and
acquisition costs, q. Given that trade coefficients of matrix, T, may be
dependent on transport utilities, V, and on selling prices, p:
T=T(V, p)
(12)
that selling prices, p, may be dependent on acquisition costs, q, and technical
coefficients, A:
p = p(A, q)
(13)
and that acquisition costs, q, are expressed through selling prices, p, and trade
coefficients, T:
q = q(p, T)
(14)
it is possible to formalize a circular dependency among T, p and q, combining
eqs. (12), (13) and (14):
p* = p(A, q(p*, T(v, p*, x)))
(15)
The circular dependency expressed through eq.(15) holds assuming constant
values of transport utilities, V=const, of technical coefficients (ex. constant
technology), A=const, and non-limited production capacity, x ∈ [0, ∞]. It was
introduced by de la Barra (1989) and solved iteratively in several SETI models
(de la Barra, 1989; Echenique, 2004). Zhao and Kochelman (2004) formalized
eq. (15) as a fixed point problem and defined the solution existence and
uniqueness conditions.
A second circular dependency occurs among trade coefficients, T, trade flows,
N, and productions, x. It arises due to limited production capacity in each zone:
trade coefficients and productions emerge from an interaction between demand
and (limited) production of inputs in each zone. Given eqs. (6) and (7), it is
possible to formalize a circular dependency among T, N and x:
x* = 1T (T(V, p, x*) A Dg(y) + T(V, p, x*) Dg(ye))
(16)
The circular dependency expressed through (16) holds assuming constant
values of transport utilities, V=const, of selling prices, p=const, and limited
production capacity, x ∈ [0, x ] with x , vector of maximum values of production.
The above two circular dependencies are presented separately. But, they are
strictly linked (see figure 3), as the process connected to price adjustment, p, is
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mutually dependent with the process of production location, x. They are slower
than those concerning passenger and freight mobility simulated by the transport
macro-model.
The generation model contains a circular dependency among selling prices, p,
and technical coefficients, A. Given the changes in selling prices in different
zones due either to constraints in production or to transport cost modifications,
the technical coefficients are adjusted to take such changes into account:
p* = p(A(p*), q)
(17)
assuming constant values of q.
The above circular dependency is related to economic, demographic and
technological aspects (values of technical coefficients in matrix A) that we said
have a medium rate of change.
5.3. Spatial economic and transport interaction
Finally, the spatial economic macro-model and the transport macro-model are
mutually interacting. The transport macro-model provides transport utilities, V,
for the spatial economic one
N=N(V)
(18)
and the spatial economic macro-model provides trade flows, N, for transport:
V=V(N)
(19)
Combining eqs. (18) and (19), we obtain:
N*=N(V(N*))
(20)
Circular dependency between transport utilities, V, and trade flows, N, was
implemented in a number of operational SETI models (de la Barra, 1989;
Echenique, 2004), implicitly accepting that there is a stable state in which
spatial distribution of socio-economic activities is consistent with transportation
costs, which depend on congestion, that results from the spatial distribution of
socio-economic activities. The two interacting macro-models (transport and
spatial economic) simulate two systems operating asynchronously, in the sense
that, even if they interact, they have their own rate of change. No existence and
uniqueness of the solution of the circular dependency (20) is mathematically
stated. However, some considerations about the possibility of formalizing a
fixed-point problem are reported in Cascetta (2006). Preliminary analysis of the
existence and uniqueness of the solution of eq. (20) is reported in Bifulco
(2000).
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