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Modeling residential location choice in relation to housing location and road tolls on congested urban highway networks
Transportation Research Part B 33 (1999) 581±591
www.elsevier.com/locate/trb
Modeling residential location choice in relation to housing location and road tolls on congested urban highway networks David Boyce a
a,* ,
Lars-G oran Mattsson
b
Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago IL 60607, USA Department of Infrastructure and Planning, Royal Institute of Technology, S-100 44 Stockholm, Sweden
b
Received 16 June 1998; received in revised form 6 November 1998; accepted 21 January 1999
Abstract We present a reformulation of the residential location submodel of the Integrated Model of Residential and Employment Location as a network equilibrium problem, thereby making travel costs by auto endogenous. The location of housing supply is examined as a welfare maximization problem for both useroptimal and system-optimal travel costs using concepts of bilevel programming. Finally, we brie¯y discuss how the employment submodel can be reformulated, and the entire model solved as a variational inequality problem. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Residential location; Network equilibrium; System-optimal tolls
1. Introduction An Integrated Model of Residential and Employment Location (IMREL) was formulated and applied as a policy analysis tool in the Stockholm region by Anderstig and Mattsson (1991, 1998) and Mattsson (1989). Their formulation was based on random utility theory with ®xed travel costs. Using a heuristic approach, the location choice model was solved sequentially with a useroptimal route choice model (trip assignment) to determine the location and travel choices of all workers in the Stockholm region. Then, these choice patterns were used to investigate the welfare maximizing location of housing supply. In this paper we show how to reformulate the residential location submodel of IMREL to include the user-optimal route choice conditions. The reformulation of the employment location submodel follows analogously. Then, we clarify the relationships between the residential submodel and the welfare maximizing housing location model and describe a heuristic for solving the *
Corresponding author. Fax: +1 312 996 2426; e-mail: [email protected]
0191-2615/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 1 - 2 6 1 5 ( 9 9 ) 0 0 0 0 7 - 7
582
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
resulting bilevel programming problem. Finally, we investigate an upper bound on its optimal solution based on system-optimal road tolls. The objective of the paper is primarily pedagogical. We seek to show how the contributions of the random utility and network equilibrium approaches can be synthesized to yield more comprehensive and operational models in which travel times and costs for the auto mode are endogenous. The paper is organized as follows. In Section 2 we introduce the residential location submodel of IMREL in the random utility framework. Then we show how to reformulate this model in the network equilibrium framework, and state the properties of the resulting location and travel choice pattern. In Section 3 we de®ne a welfare function for this pattern and formulate a bilevel programming problem for the location of housing supply. A heuristic solution procedure is then described. By introducing road tolls, the user-optimal solution can be shifted to a system-optimal solution, and the bilevel problem reduced to a single-level problem, as shown in Section 4. Finally, we discuss how this approach can be extended to include the employment location submodel of IMREL in Section 5. 2. Formulation of the residential location submodel In the residential location submodel of IMREL, there is one worker per household; each household requires one housing unit. The employment locations of the workers are assumed to be exogenously given; however, see Section 5 for a relaxation of this assumption. Each residential zone i has a market clearing housing price qi h which depends on the given and ®xed distribution of the housing supply h h1 ; h2 ; . . . ; hI , as well as the demand; hi is the proportion of the total housing supply H located in zone i. Associated with each zone i is a ®xed housing disutility function di hi that depends on the density of housing in the zone; this function could also include ®xed components of the zone's attractiveness. Finally, cij is the generalized travel cost between residential zone i and workplace zone j, as perceived by the traveler. Based on these de®nitions, the utility of a residential location in zone i for worker n is Uijjn ÿqi h ÿ di hi ÿ cij nij ; where nij is an independently Gumbel 0; l-distributed random part of the utility, and l is inversely proportional to the standard deviation of . Given a workplace in zone j, the probability of choosing a residential location in zone i is expÿl qi h di hi cij Pijj P : i expÿl qi h di hi cij Given wj , the proportion of all of workers employed in zone j, the proportion of workers living in zone i and working in zone j, Pij , is: Pij wj Pijj and hi
X X Pij wj Pijj j
j
assuming as we do that housing demand is in balance with housing supply. Mattsson (1984, Prop. 2) proves there exists an equilibrating housing price vector q h such that all markets clear. From
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
583
the proof of this proposition it follows that the equilibrium price vector is unique up to an additive constant. Next we show how to derive the same result from the network equilibrium approach. In so doing, we provide a basis for solving for the user-optimal travel costs cij . We formulate the network equilibrium problem (NEP) in the usual way as an equivalent minimization problem for the auto mode only with one worker per auto. Extensions of the formulation to include the transit mode and auto passengers are given by Boyce and Daskin (1997). See Oppenheim (1995) for a direct utility interpretation of the following formulation, as well as the discussion in Section 3. Related formulations of residential location models were proposed by Wilson (1970), Boyce (1980), She (1985, p. 173) and Safwat and Magnanti (1988). The present formulation diers from previous ones in that both origin and destination constraints are imposed. As in the random utility model, the proportion of workers residing in zone i and working in zone j, Pij , is constrained by hi , the proportion of housing units in zone i, and wj , the proportion of jobs in zone j. T is the total number of workers residing in the region. The equivalent minimization problem is formulated as follows: Z XX 1 X fa 1 XX ca x dx Pij di hi Pij lnPij ; NEP : min Z1 Tijr ;Pij T a 0 l i j i j X Tijr TPij 8ij; s:t: r
X
Pij hi
8i;
Pij wj
8j;
j
X i
Tijr P 0
8ijr; P where the link ¯ow fa ijr Tijr daijr ; 8a and daijr 1 if link a belongs to route r, and 0 otherwise. The generalized link travel cost ca fa is a function of the link ¯ow fa . These functions are assumed to be dierentiable, positive, strictly increasing with ¯ow and separable, which guarantees that the ®rst term is strictly convex. Likewise, the housing disutility functions are assumed to be strictly increasing functions of the housing supply. In this NEP formulation of the residential location problem, the density disutility terms are constant; they are included here to clarify the relationship with the random utility formulation and to introduce the notation for the housing location problem considered in the next section. The negative of the entropy function is known to be strictly convex. Hence, the objective function is also strictly convex. The Lagrangian equation is: ! Z X XX XX 1 X fa 1 XX L ca x dx Pij di hi Pij lnPij uij TPij ÿ Tijr T a 0 l i j i j i j r ! ! X X X X qi Pij ÿ hi rj wj ÿ Pij i
j
j
where uij ; qi and rj are Lagrange multipliers.
i
584
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
Taking partial derivatives with respect to the unknown variables, we obtain: oL 1X ca fa daijr uij ÿ1 P 0 oTijr T a
8ijr;
oL 1 ln Pij 1 di hi uij T qi 1 rj ÿ1 P 0 oPij l
8ij:
From the complementary slackness condition, we have that: ! 1X a Tijr ca fa dijr ÿ uij 0 8ijr: T a Interpreting the optimality conditions with respect to Tijr , we have: P 1. if Tijr > 0, then Pa ca fa daijr Tuij ; 2. if Tijr 0, then a ca fa daijr P Tuij . Hence cij Tuij is the user-optimal travel cost from zone i to zone j. The above conditions correspond to Wardrop's ®rst principle of user-optimal route choice. Next, consider the conditions related to Pij for Pij > 0. (Note that Pij cannot equal zero if the travel costs and other terms are ®nite since ln 0 ÿ1.) Rearranging terms, we have, ln Pij l ÿqi rj ÿ di hi ÿ cij ÿ 1 or Pij expl ÿqi rj ÿ di hi ÿ cij ÿ 1: Summing with respect to i, X X Pij wj exp lrj ÿ 1 expÿl qi di hi cij i
or
i
X expÿl qi di hi cij : exp lrj ÿ 1 wj = i
Therefore,
or
wj expÿl qi di hi cij Pij P i expÿl qi di hi cij
Pijj
Pij expÿl qi di hi cij P wj i expÿl qi di hi cij
We see that the above expression is identical to the function based on random utility theory. Moreover, the housing price variable qi can be seen to be equivalent to a balancing factor for the constraint that insures that the housing supply in zone i equals the housing demand. This equivalence indirectly proves the existence of an equilibrium price vector; the price vector is also unique up to an additive constant.
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
585
The residential location and travel choice model formulated above can be readily solved with a partial linearization algorithm (Evans, 1976; Patriksson, 1994). The parameters of the model (l; q) and other implicit parameters such as generalized cost coecients can be estimated using maximum likelihood methods (Boyce and Zhang, 1998). 3. Formulation of the housing location problem From random utility theory, and in particular the assumption of independent Gumbel (0, l) distributions, the expected utility of a worker in zone j is known to be 1 X c Vj ln expÿl qi h di hi cij h ; l l i where c is Euler's constant, which is omitted in the following. Now, qi h and cij h are written as a function of h, the housing location pattern, to remind us that the residential location and travel choice pattern depends on the location of housing. Note also that ÿVj can be equivalently interpreted as the composite cost of workers in zone j in the network equilibrium framework. We propose the following function as a welfare criterion for the location of housing: X X 1X V h wj ln expÿl qi h di hi cij h hi qi h: l j i i The ®rst term represents the consumer surplus of the employees resulting from housing location pattern h. The second term represents the producer surplus associated with pattern h. Note that the inclusion of producer surplus mitigates the problem that the equilibrium housing price vector is unique only up to an additive constant; these ambiguous prices only aect the division of the total surplus between consumer and producer surplus. To ®nd the housing supply pattern that maximizes this welfare function, we need to solve: X X 1X U : max V h wj ln expÿl qi h di hi cij h hi qi h; h l j i i s:t: li 6 hi 6 ui X hi 1;
8i;
i
where q h; c h solve Z XX 1 X fa 1 XX ca x dx Pij di hi Pij lnPij ; L: min Z1 Tijr ;Pij T a 0 l i j i j X Tijr TPij 8ij; s:t: r X
Pij hi
8i;
Pij wj
8j;
j
X i
Tijr P 0
8ijr:
586
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
In the above formulation, li and ui are respectively exogenously given lower and upper bounds on the proportion of the housing supply located in zone i. The above problem is a bilevel programming problem: the upper level problem U seeks to ®nd a housing location pattern that maximizes the welfare function V h. The lower level problem L represents the aggregate behavior of employees in their residential location and travel choices, given the location pattern of jobs w, and of housing h. In general, bilevel maximization problems are nonconcave, and therefore do not necessarily have a unique maximum. Therefore, we consider a general heuristic procedure to ®nd a good solution to this problem. The heuristic begins with a feasible solution, and then seeks to improve on it by means of a search in the vicinity of the current solution. The heuristic is described as follows: 1. Set k 0 and choose a feasible housing supply vector hk . 2. Solve NEP, given hk , yielding the UO travel costs cij hk and the zonal housing prices qi hk . 3. Evaluate the welfare function V hk and check whether V hk P V hkÿ1 . 4. Attempt to improve upon the solution hk , such as by choosing a search direction based on the gradient of V hk , holding q and c ®xed. At each trial solution, NEP should be evaluated. A dual approach such as described by Mattsson (1987) could also be used. If V hk1 > V hk , continue; otherwise, stop. The general notion of the heuristic is that the search should continue so long as improvement in V h is achieved. When no further improvement is found, the search is terminated.
4. System-optimal travel choices with road tolls Three urban areas in Norway have introduced road pricing, in the limited form of toll rings, as a means of generating additional revenues for road improvements (Ramjerdi, 1995). A similar road pricing system was also considered in Stockholm (Anderstig and Mattsson, 1998). Although these systems are not link-based or ¯ow-dependent, it is increasingly reasonable to consider the system-optimal variant of the residential location and travel choice problem in which travel costs are assumed to consist of the generalized cost of travel incurred by the user plus a road toll designed to yield system-optimal route choices. To explore this case, a system-optimal variant of NEP is stated as follows: XX 1X 1 XX fa ca fa Pij di hi Pij lnPij ; T a l i j i j X Tijr TPij 8ij;
min Z2
Tijr ;Pij
s:t:
r X
Pij hi
8i;
Pij wj
8j;
j
X i
Tijr P 0
8ijr;
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
587
P
where fa ijr Tijr daijr ; 8a. As with NEP, this formulation is an equivalent optimization problem designed to yield certain desired optimality conditions. Whether the objective function has a direct interpretation has not been investigated; again, cf. Oppenheim (1995). The optimality conditions for route ¯ows Tijr are as follows: oL 1X ca fa fa dca fa =dfa daijr ÿ vij P 0 oTijr T a
8ijr;
where vij is now the Lagrange multiplier related to the ¯ow conservation constraint. The interpretation of the optimality conditions for route ¯ows is that for each origin±destination pair, the marginal travel costs on all used routes are equal, and no unused route has a lower marginal travel cost. The marginal travel cost on link a is given by: ma fa ca fa fa dca fa =dfa : In keeping with the above motivation for this problem, let the toll on link a be: sa fa fa dca fa =dfa : We write the toll as a function of the ¯ow on link a to make explicit the dependence of these congestion tolls on the link ¯ows. As with the user-optimal case, the proportion of workers with jobs in zone j and residences in zone i is given by wj expÿl qi h di hi mij h ; Pij h P i expÿl qi h di hi mij h P where mij h a ca fa sa fa daijr for all used routes from zone i to zone j. Next, we evaluate the objective function of the system-optimal formulation at the optimal solution f h; P h by substituting the natural logarithm of Pij h into the objective function, and simplifying the resulting expression by use of the origin and destination constraints: ( ! X X 1X 1 XX fa hca fa h Pij h di hi Pij h lnwj ÿ lqi h Z2 f h; P h T a l i j i j ) X exp ÿ l qi h di hi mij h di hi mij h ÿ ln i
X X 1X 1X fa hca fa h hi di hi wj lnwj ÿ hi qi h di hi T a l j i i ÿ
XX i
j
Pij hmij h ÿ
X 1X wj ln expÿl qi h di hi mij h: l j i
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D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
Using the identity relating origin±destination-based costs to link-based costs, we have: XX 1X 1X Pij hmij h fa hma fa h fa hca fa h sa fa h T a T a i j Therefore, XX 1X 1X fa hca fa h ÿ Pij hmij h ÿ fa hsa fa h T a T a i j and 1X 1X fa hsa fa h wj lnwj T a l j " # X X 1X ÿ hi qi h wj ln exp ÿ l qi h di hi mij h : l j i i
Z2 f h; P h ÿ
Thus the two terms in the large brackets equal V^ h, the producer surplus plus the consumer surplus, as de®ned in Section 3 for the welfare maximization problem, now based on the marginal travel costs including the system-optimal tolls paid by consumers. Hence, we have: ÿZ2 f h; P h V^ h
1X 1X fa hsa fa h ÿ wj lnwj : T a l j
In the residential location and travel choice problem, the last term is a constant. From a random utility theory viewpoint, ÿZ2 f h; P h is the aggregate utility associated with the residential location and travel choices, subject to system-optimal tolls, plus the aggregate producer revenue from housing, the aggregate toll revenue and the constant term related to employment location. Hence, we now rede®ne V^ to include also the toll revenue in producer surplus, and designate it V~ : The bilevel welfare maximization problem with system-optimal location and travel choices is 1 maxV~ h V^ h h T s:t: li 6 hi 6 ui X hi 1;
X
fa hsa fa h;
a
8i;
i
where q h; m h; f h solve the system-optimal network problem. Since at optimality ÿZ2 f h; P h was shown to equal the welfare function V~ h constant, this bilevel programming problem reduces to the following single level maximization problem.
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
max
hi ;Tijr ;Pij s:t: C1 and C2
589
XX 1X 1 XX fa ca fa ÿ Pij di hi ÿ Pij lnPij T a l i j i j 0 1 XX 1X 1 XX B C max @ max ÿ Z2 ÿ fa ca fa ÿ Pij di hi ÿ Pij lnPij A hi Tijr ;Pij T a l i j i j
ÿ Z2 ÿ
s:t: C1
s:t: C2
max ÿ Z2 f h; P h hi s:t: C1
max V~ h ÿ hi s:t: C1
1X wj lnwj ; l j
where C1 denotes the set of constraints: li 6 hi 6 ui ; X hi 1;
8i;
i
and C2 the set of constraints: X Tijr TPij 8ij; r
X Pij hi
8i;
j
X Pij wj
8j;
i
Tijr P 0 8ijr; P where fa ijr Tijr daijr ; 8a. Note that ca fa and di hi are assumed to be strictly increasing, and hence the ®rst two terms of ÿZ2 are convex; therefore, their negative values are concave. We believe this problem can be solved with a partial linearization algorithm with side constraints, as suggested by Larsson and Patriksson (1994). Investigation of this idea remains a topic for future research. In addition to its interest from the viewpoint of congestion pricing, the single-level formulation of the welfare maximization problem provides an upper bound on the solution of the bilevel welfare maximization problem with user-optimal residential location and travel choices. This information provides an improved termination criterion for the heuristic stated at the end of Section 3. 5. Extensions In IMREL an employment location submodel determines the location of jobs from the viewpoint of employers. The objective of the employment location submodel is to maximize the sum of accessibility to the labor force, which is dependent on household location, a measure of local conditions in the employment location, and a positive or negative rent (shadow price) associated
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D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
with the employment zone. This model can also be formulated as a network equilibrium problem with user-optimal route choice, given the location of households. The model can also be solved with the Evans algorithm. The residential and employment location submodels can each be formulated as a variational inequality problem. Then, these two problems can be summed to obtain a single variational inequality. This formulation suggests a solution algorithm by means of the diagonalization technique. In eect, each submodel would be solved, holding ®xed the location variable in the other model. This procedure would be repeated in sequence until convergence is observed. In general, if such a sequence converges, it converges to the equilibrium solution. Using this model of joint residential and employment location as a lower level problem, the upper level welfare maximizing location of housing supply can be investigated, as originally envisaged by Anderstig and Mattsson. Extensions of the framework to include transit modes and multiple time periods during the workday are also possible. A more dicult extension of the modeling framework would consider the location or relocation of the small fraction of households or workplaces which move in each time period of N years, holding constant most of the locators. This extension would enable the model to be applied in a long-run dynamic manner. An additional extension concerns the relaxation of the unrealistic and untenable assumption of one worker per household. Acknowledgements The ideas for this paper were conceived while the ®rst author was a Guest Professor at the Department of Infrastructure and Planning, Royal Institute of Technology, Stockholm, Sweden. The ®rst author gratefully acknowledges the support of the (US) National Science Foundation through the National Institute of Statistical Sciences, and the second author that of the Swedish Transport and Communications Research Board. References Anderstig, C., Mattsson, L.-G., 1991. An integrated model of residential and employment location in a metropolitan region. Papers in Regional Science 70, 167±184. Anderstig, C., Mattsson, L.-G., 1998. Modelling land-use and transport interaction: policy analyses using the IMREL model. In: Lundqvist, L., Mattsson, L.-G., Kim, T.J. (Eds.), Network Infrastructure and the Urban Environment, Springer, Berlin, pp. 308±328. Boyce, D.E., 1980. A framework for constructing network equilibrium models of urban location. Transportation Science 14, 77±96. Boyce, D.E., Daskin, M.S., 1997. Urban transportation. In: Revelle, C., and Mc Garity, A. (Eds.), Design and Operation of Civil and Environmental Engineering Systems, Wiley, New York, pp. 277±341. Boyce, D.E., Zhang, Y., 1998. Parameter estimation for combined travel choice models. In: Lundqvist, L., Mattsson L.-G., Kim, T.J. (Eds.), Network Infrastructure and the Urban Environment, Springer, Berlin, pp. 177±193. Evans, S.P., 1976. Derivation and analysis of some models for combining trip distribution and assignment. Transportation Research 10, 37±57. Larsson, T., Patriksson, M., 1994. Equilibrium characterizations of solutions to side constrained asymmetric trac assignment models. Le Matematiche XLIX, 249±280.
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Mattsson, L.-G., 1984. Equivalence between welfare and entropy approaches to residential location. Regional Science and Urban Economics 14, 147±173. Mattsson, L.-G., 1987. Urban welfare maximization and housing market equilibrium in a random utility setting. Environment and Planning A 19, 247±261. Mattsson, L.-G., 1989. Normative housing allocation and work trip energy use. In: Lundqvist, L., Mattsson, L-G., Eriksson, E.A. (Eds.), Spatial Energy Analysis, Avebury, Aldershot, pp. 335±358. Oppenheim, N., 1995. Urban Travel Demand Modeling. Wiley, New York. Patriksson, M., 1994. The Trac Assignment Problem. VSP, Utrecht, The Netherlands. Ramjerdi, F., 1995. Road pricing and toll ®nancing. Institute of Transport Economics. Oslo, Norway. Safwat, K.N.A., Magnanti, T.L., 1988. A combined trip generation, trip distribution, modal split, and trip assignment model. Transportation Science 22, 77±96. She, Y., 1985. Urban Transportation Networks. Prentice-Hall, Englewood Clis, NJ. Wilson, A.G., 1970. Entropy in Urban and Regional Modelling. Pion, London.
www.elsevier.com/locate/trb
Modeling residential location choice in relation to housing location and road tolls on congested urban highway networks David Boyce a
a,* ,
Lars-G oran Mattsson
b
Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago IL 60607, USA Department of Infrastructure and Planning, Royal Institute of Technology, S-100 44 Stockholm, Sweden
b
Received 16 June 1998; received in revised form 6 November 1998; accepted 21 January 1999
Abstract We present a reformulation of the residential location submodel of the Integrated Model of Residential and Employment Location as a network equilibrium problem, thereby making travel costs by auto endogenous. The location of housing supply is examined as a welfare maximization problem for both useroptimal and system-optimal travel costs using concepts of bilevel programming. Finally, we brie¯y discuss how the employment submodel can be reformulated, and the entire model solved as a variational inequality problem. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Residential location; Network equilibrium; System-optimal tolls
1. Introduction An Integrated Model of Residential and Employment Location (IMREL) was formulated and applied as a policy analysis tool in the Stockholm region by Anderstig and Mattsson (1991, 1998) and Mattsson (1989). Their formulation was based on random utility theory with ®xed travel costs. Using a heuristic approach, the location choice model was solved sequentially with a useroptimal route choice model (trip assignment) to determine the location and travel choices of all workers in the Stockholm region. Then, these choice patterns were used to investigate the welfare maximizing location of housing supply. In this paper we show how to reformulate the residential location submodel of IMREL to include the user-optimal route choice conditions. The reformulation of the employment location submodel follows analogously. Then, we clarify the relationships between the residential submodel and the welfare maximizing housing location model and describe a heuristic for solving the *
Corresponding author. Fax: +1 312 996 2426; e-mail: [email protected]
0191-2615/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 1 - 2 6 1 5 ( 9 9 ) 0 0 0 0 7 - 7
582
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
resulting bilevel programming problem. Finally, we investigate an upper bound on its optimal solution based on system-optimal road tolls. The objective of the paper is primarily pedagogical. We seek to show how the contributions of the random utility and network equilibrium approaches can be synthesized to yield more comprehensive and operational models in which travel times and costs for the auto mode are endogenous. The paper is organized as follows. In Section 2 we introduce the residential location submodel of IMREL in the random utility framework. Then we show how to reformulate this model in the network equilibrium framework, and state the properties of the resulting location and travel choice pattern. In Section 3 we de®ne a welfare function for this pattern and formulate a bilevel programming problem for the location of housing supply. A heuristic solution procedure is then described. By introducing road tolls, the user-optimal solution can be shifted to a system-optimal solution, and the bilevel problem reduced to a single-level problem, as shown in Section 4. Finally, we discuss how this approach can be extended to include the employment location submodel of IMREL in Section 5. 2. Formulation of the residential location submodel In the residential location submodel of IMREL, there is one worker per household; each household requires one housing unit. The employment locations of the workers are assumed to be exogenously given; however, see Section 5 for a relaxation of this assumption. Each residential zone i has a market clearing housing price qi h which depends on the given and ®xed distribution of the housing supply h h1 ; h2 ; . . . ; hI , as well as the demand; hi is the proportion of the total housing supply H located in zone i. Associated with each zone i is a ®xed housing disutility function di hi that depends on the density of housing in the zone; this function could also include ®xed components of the zone's attractiveness. Finally, cij is the generalized travel cost between residential zone i and workplace zone j, as perceived by the traveler. Based on these de®nitions, the utility of a residential location in zone i for worker n is Uijjn ÿqi h ÿ di hi ÿ cij nij ; where nij is an independently Gumbel 0; l-distributed random part of the utility, and l is inversely proportional to the standard deviation of . Given a workplace in zone j, the probability of choosing a residential location in zone i is expÿl qi h di hi cij Pijj P : i expÿl qi h di hi cij Given wj , the proportion of all of workers employed in zone j, the proportion of workers living in zone i and working in zone j, Pij , is: Pij wj Pijj and hi
X X Pij wj Pijj j
j
assuming as we do that housing demand is in balance with housing supply. Mattsson (1984, Prop. 2) proves there exists an equilibrating housing price vector q h such that all markets clear. From
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
583
the proof of this proposition it follows that the equilibrium price vector is unique up to an additive constant. Next we show how to derive the same result from the network equilibrium approach. In so doing, we provide a basis for solving for the user-optimal travel costs cij . We formulate the network equilibrium problem (NEP) in the usual way as an equivalent minimization problem for the auto mode only with one worker per auto. Extensions of the formulation to include the transit mode and auto passengers are given by Boyce and Daskin (1997). See Oppenheim (1995) for a direct utility interpretation of the following formulation, as well as the discussion in Section 3. Related formulations of residential location models were proposed by Wilson (1970), Boyce (1980), She (1985, p. 173) and Safwat and Magnanti (1988). The present formulation diers from previous ones in that both origin and destination constraints are imposed. As in the random utility model, the proportion of workers residing in zone i and working in zone j, Pij , is constrained by hi , the proportion of housing units in zone i, and wj , the proportion of jobs in zone j. T is the total number of workers residing in the region. The equivalent minimization problem is formulated as follows: Z XX 1 X fa 1 XX ca x dx Pij di hi Pij lnPij ; NEP : min Z1 Tijr ;Pij T a 0 l i j i j X Tijr TPij 8ij; s:t: r
X
Pij hi
8i;
Pij wj
8j;
j
X i
Tijr P 0
8ijr; P where the link ¯ow fa ijr Tijr daijr ; 8a and daijr 1 if link a belongs to route r, and 0 otherwise. The generalized link travel cost ca fa is a function of the link ¯ow fa . These functions are assumed to be dierentiable, positive, strictly increasing with ¯ow and separable, which guarantees that the ®rst term is strictly convex. Likewise, the housing disutility functions are assumed to be strictly increasing functions of the housing supply. In this NEP formulation of the residential location problem, the density disutility terms are constant; they are included here to clarify the relationship with the random utility formulation and to introduce the notation for the housing location problem considered in the next section. The negative of the entropy function is known to be strictly convex. Hence, the objective function is also strictly convex. The Lagrangian equation is: ! Z X XX XX 1 X fa 1 XX L ca x dx Pij di hi Pij lnPij uij TPij ÿ Tijr T a 0 l i j i j i j r ! ! X X X X qi Pij ÿ hi rj wj ÿ Pij i
j
j
where uij ; qi and rj are Lagrange multipliers.
i
584
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
Taking partial derivatives with respect to the unknown variables, we obtain: oL 1X ca fa daijr uij ÿ1 P 0 oTijr T a
8ijr;
oL 1 ln Pij 1 di hi uij T qi 1 rj ÿ1 P 0 oPij l
8ij:
From the complementary slackness condition, we have that: ! 1X a Tijr ca fa dijr ÿ uij 0 8ijr: T a Interpreting the optimality conditions with respect to Tijr , we have: P 1. if Tijr > 0, then Pa ca fa daijr Tuij ; 2. if Tijr 0, then a ca fa daijr P Tuij . Hence cij Tuij is the user-optimal travel cost from zone i to zone j. The above conditions correspond to Wardrop's ®rst principle of user-optimal route choice. Next, consider the conditions related to Pij for Pij > 0. (Note that Pij cannot equal zero if the travel costs and other terms are ®nite since ln 0 ÿ1.) Rearranging terms, we have, ln Pij l ÿqi rj ÿ di hi ÿ cij ÿ 1 or Pij expl ÿqi rj ÿ di hi ÿ cij ÿ 1: Summing with respect to i, X X Pij wj exp lrj ÿ 1 expÿl qi di hi cij i
or
i
X expÿl qi di hi cij : exp lrj ÿ 1 wj = i
Therefore,
or
wj expÿl qi di hi cij Pij P i expÿl qi di hi cij
Pijj
Pij expÿl qi di hi cij P wj i expÿl qi di hi cij
We see that the above expression is identical to the function based on random utility theory. Moreover, the housing price variable qi can be seen to be equivalent to a balancing factor for the constraint that insures that the housing supply in zone i equals the housing demand. This equivalence indirectly proves the existence of an equilibrium price vector; the price vector is also unique up to an additive constant.
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585
The residential location and travel choice model formulated above can be readily solved with a partial linearization algorithm (Evans, 1976; Patriksson, 1994). The parameters of the model (l; q) and other implicit parameters such as generalized cost coecients can be estimated using maximum likelihood methods (Boyce and Zhang, 1998). 3. Formulation of the housing location problem From random utility theory, and in particular the assumption of independent Gumbel (0, l) distributions, the expected utility of a worker in zone j is known to be 1 X c Vj ln expÿl qi h di hi cij h ; l l i where c is Euler's constant, which is omitted in the following. Now, qi h and cij h are written as a function of h, the housing location pattern, to remind us that the residential location and travel choice pattern depends on the location of housing. Note also that ÿVj can be equivalently interpreted as the composite cost of workers in zone j in the network equilibrium framework. We propose the following function as a welfare criterion for the location of housing: X X 1X V h wj ln expÿl qi h di hi cij h hi qi h: l j i i The ®rst term represents the consumer surplus of the employees resulting from housing location pattern h. The second term represents the producer surplus associated with pattern h. Note that the inclusion of producer surplus mitigates the problem that the equilibrium housing price vector is unique only up to an additive constant; these ambiguous prices only aect the division of the total surplus between consumer and producer surplus. To ®nd the housing supply pattern that maximizes this welfare function, we need to solve: X X 1X U : max V h wj ln expÿl qi h di hi cij h hi qi h; h l j i i s:t: li 6 hi 6 ui X hi 1;
8i;
i
where q h; c h solve Z XX 1 X fa 1 XX ca x dx Pij di hi Pij lnPij ; L: min Z1 Tijr ;Pij T a 0 l i j i j X Tijr TPij 8ij; s:t: r X
Pij hi
8i;
Pij wj
8j;
j
X i
Tijr P 0
8ijr:
586
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In the above formulation, li and ui are respectively exogenously given lower and upper bounds on the proportion of the housing supply located in zone i. The above problem is a bilevel programming problem: the upper level problem U seeks to ®nd a housing location pattern that maximizes the welfare function V h. The lower level problem L represents the aggregate behavior of employees in their residential location and travel choices, given the location pattern of jobs w, and of housing h. In general, bilevel maximization problems are nonconcave, and therefore do not necessarily have a unique maximum. Therefore, we consider a general heuristic procedure to ®nd a good solution to this problem. The heuristic begins with a feasible solution, and then seeks to improve on it by means of a search in the vicinity of the current solution. The heuristic is described as follows: 1. Set k 0 and choose a feasible housing supply vector hk . 2. Solve NEP, given hk , yielding the UO travel costs cij hk and the zonal housing prices qi hk . 3. Evaluate the welfare function V hk and check whether V hk P V hkÿ1 . 4. Attempt to improve upon the solution hk , such as by choosing a search direction based on the gradient of V hk , holding q and c ®xed. At each trial solution, NEP should be evaluated. A dual approach such as described by Mattsson (1987) could also be used. If V hk1 > V hk , continue; otherwise, stop. The general notion of the heuristic is that the search should continue so long as improvement in V h is achieved. When no further improvement is found, the search is terminated.
4. System-optimal travel choices with road tolls Three urban areas in Norway have introduced road pricing, in the limited form of toll rings, as a means of generating additional revenues for road improvements (Ramjerdi, 1995). A similar road pricing system was also considered in Stockholm (Anderstig and Mattsson, 1998). Although these systems are not link-based or ¯ow-dependent, it is increasingly reasonable to consider the system-optimal variant of the residential location and travel choice problem in which travel costs are assumed to consist of the generalized cost of travel incurred by the user plus a road toll designed to yield system-optimal route choices. To explore this case, a system-optimal variant of NEP is stated as follows: XX 1X 1 XX fa ca fa Pij di hi Pij lnPij ; T a l i j i j X Tijr TPij 8ij;
min Z2
Tijr ;Pij
s:t:
r X
Pij hi
8i;
Pij wj
8j;
j
X i
Tijr P 0
8ijr;
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
587
P
where fa ijr Tijr daijr ; 8a. As with NEP, this formulation is an equivalent optimization problem designed to yield certain desired optimality conditions. Whether the objective function has a direct interpretation has not been investigated; again, cf. Oppenheim (1995). The optimality conditions for route ¯ows Tijr are as follows: oL 1X ca fa fa dca fa =dfa daijr ÿ vij P 0 oTijr T a
8ijr;
where vij is now the Lagrange multiplier related to the ¯ow conservation constraint. The interpretation of the optimality conditions for route ¯ows is that for each origin±destination pair, the marginal travel costs on all used routes are equal, and no unused route has a lower marginal travel cost. The marginal travel cost on link a is given by: ma fa ca fa fa dca fa =dfa : In keeping with the above motivation for this problem, let the toll on link a be: sa fa fa dca fa =dfa : We write the toll as a function of the ¯ow on link a to make explicit the dependence of these congestion tolls on the link ¯ows. As with the user-optimal case, the proportion of workers with jobs in zone j and residences in zone i is given by wj expÿl qi h di hi mij h ; Pij h P i expÿl qi h di hi mij h P where mij h a ca fa sa fa daijr for all used routes from zone i to zone j. Next, we evaluate the objective function of the system-optimal formulation at the optimal solution f h; P h by substituting the natural logarithm of Pij h into the objective function, and simplifying the resulting expression by use of the origin and destination constraints: ( ! X X 1X 1 XX fa hca fa h Pij h di hi Pij h lnwj ÿ lqi h Z2 f h; P h T a l i j i j ) X exp ÿ l qi h di hi mij h di hi mij h ÿ ln i
X X 1X 1X fa hca fa h hi di hi wj lnwj ÿ hi qi h di hi T a l j i i ÿ
XX i
j
Pij hmij h ÿ
X 1X wj ln expÿl qi h di hi mij h: l j i
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Using the identity relating origin±destination-based costs to link-based costs, we have: XX 1X 1X Pij hmij h fa hma fa h fa hca fa h sa fa h T a T a i j Therefore, XX 1X 1X fa hca fa h ÿ Pij hmij h ÿ fa hsa fa h T a T a i j and 1X 1X fa hsa fa h wj lnwj T a l j " # X X 1X ÿ hi qi h wj ln exp ÿ l qi h di hi mij h : l j i i
Z2 f h; P h ÿ
Thus the two terms in the large brackets equal V^ h, the producer surplus plus the consumer surplus, as de®ned in Section 3 for the welfare maximization problem, now based on the marginal travel costs including the system-optimal tolls paid by consumers. Hence, we have: ÿZ2 f h; P h V^ h
1X 1X fa hsa fa h ÿ wj lnwj : T a l j
In the residential location and travel choice problem, the last term is a constant. From a random utility theory viewpoint, ÿZ2 f h; P h is the aggregate utility associated with the residential location and travel choices, subject to system-optimal tolls, plus the aggregate producer revenue from housing, the aggregate toll revenue and the constant term related to employment location. Hence, we now rede®ne V^ to include also the toll revenue in producer surplus, and designate it V~ : The bilevel welfare maximization problem with system-optimal location and travel choices is 1 maxV~ h V^ h h T s:t: li 6 hi 6 ui X hi 1;
X
fa hsa fa h;
a
8i;
i
where q h; m h; f h solve the system-optimal network problem. Since at optimality ÿZ2 f h; P h was shown to equal the welfare function V~ h constant, this bilevel programming problem reduces to the following single level maximization problem.
D. Boyce, L-G. Mattsson / Transportation Research Part B 33 (1999) 581±591
max
hi ;Tijr ;Pij s:t: C1 and C2
589
XX 1X 1 XX fa ca fa ÿ Pij di hi ÿ Pij lnPij T a l i j i j 0 1 XX 1X 1 XX B C max @ max ÿ Z2 ÿ fa ca fa ÿ Pij di hi ÿ Pij lnPij A hi Tijr ;Pij T a l i j i j
ÿ Z2 ÿ
s:t: C1
s:t: C2
max ÿ Z2 f h; P h hi s:t: C1
max V~ h ÿ hi s:t: C1
1X wj lnwj ; l j
where C1 denotes the set of constraints: li 6 hi 6 ui ; X hi 1;
8i;
i
and C2 the set of constraints: X Tijr TPij 8ij; r
X Pij hi
8i;
j
X Pij wj
8j;
i
Tijr P 0 8ijr; P where fa ijr Tijr daijr ; 8a. Note that ca fa and di hi are assumed to be strictly increasing, and hence the ®rst two terms of ÿZ2 are convex; therefore, their negative values are concave. We believe this problem can be solved with a partial linearization algorithm with side constraints, as suggested by Larsson and Patriksson (1994). Investigation of this idea remains a topic for future research. In addition to its interest from the viewpoint of congestion pricing, the single-level formulation of the welfare maximization problem provides an upper bound on the solution of the bilevel welfare maximization problem with user-optimal residential location and travel choices. This information provides an improved termination criterion for the heuristic stated at the end of Section 3. 5. Extensions In IMREL an employment location submodel determines the location of jobs from the viewpoint of employers. The objective of the employment location submodel is to maximize the sum of accessibility to the labor force, which is dependent on household location, a measure of local conditions in the employment location, and a positive or negative rent (shadow price) associated
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with the employment zone. This model can also be formulated as a network equilibrium problem with user-optimal route choice, given the location of households. The model can also be solved with the Evans algorithm. The residential and employment location submodels can each be formulated as a variational inequality problem. Then, these two problems can be summed to obtain a single variational inequality. This formulation suggests a solution algorithm by means of the diagonalization technique. In eect, each submodel would be solved, holding ®xed the location variable in the other model. This procedure would be repeated in sequence until convergence is observed. In general, if such a sequence converges, it converges to the equilibrium solution. Using this model of joint residential and employment location as a lower level problem, the upper level welfare maximizing location of housing supply can be investigated, as originally envisaged by Anderstig and Mattsson. Extensions of the framework to include transit modes and multiple time periods during the workday are also possible. A more dicult extension of the modeling framework would consider the location or relocation of the small fraction of households or workplaces which move in each time period of N years, holding constant most of the locators. This extension would enable the model to be applied in a long-run dynamic manner. An additional extension concerns the relaxation of the unrealistic and untenable assumption of one worker per household. Acknowledgements The ideas for this paper were conceived while the ®rst author was a Guest Professor at the Department of Infrastructure and Planning, Royal Institute of Technology, Stockholm, Sweden. The ®rst author gratefully acknowledges the support of the (US) National Science Foundation through the National Institute of Statistical Sciences, and the second author that of the Swedish Transport and Communications Research Board. References Anderstig, C., Mattsson, L.-G., 1991. An integrated model of residential and employment location in a metropolitan region. Papers in Regional Science 70, 167±184. Anderstig, C., Mattsson, L.-G., 1998. Modelling land-use and transport interaction: policy analyses using the IMREL model. In: Lundqvist, L., Mattsson, L.-G., Kim, T.J. (Eds.), Network Infrastructure and the Urban Environment, Springer, Berlin, pp. 308±328. Boyce, D.E., 1980. A framework for constructing network equilibrium models of urban location. Transportation Science 14, 77±96. Boyce, D.E., Daskin, M.S., 1997. Urban transportation. In: Revelle, C., and Mc Garity, A. (Eds.), Design and Operation of Civil and Environmental Engineering Systems, Wiley, New York, pp. 277±341. Boyce, D.E., Zhang, Y., 1998. Parameter estimation for combined travel choice models. In: Lundqvist, L., Mattsson L.-G., Kim, T.J. (Eds.), Network Infrastructure and the Urban Environment, Springer, Berlin, pp. 177±193. Evans, S.P., 1976. Derivation and analysis of some models for combining trip distribution and assignment. Transportation Research 10, 37±57. Larsson, T., Patriksson, M., 1994. Equilibrium characterizations of solutions to side constrained asymmetric trac assignment models. Le Matematiche XLIX, 249±280.
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Mattsson, L.-G., 1984. Equivalence between welfare and entropy approaches to residential location. Regional Science and Urban Economics 14, 147±173. Mattsson, L.-G., 1987. Urban welfare maximization and housing market equilibrium in a random utility setting. Environment and Planning A 19, 247±261. Mattsson, L.-G., 1989. Normative housing allocation and work trip energy use. In: Lundqvist, L., Mattsson, L-G., Eriksson, E.A. (Eds.), Spatial Energy Analysis, Avebury, Aldershot, pp. 335±358. Oppenheim, N., 1995. Urban Travel Demand Modeling. Wiley, New York. Patriksson, M., 1994. The Trac Assignment Problem. VSP, Utrecht, The Netherlands. Ramjerdi, F., 1995. Road pricing and toll ®nancing. Institute of Transport Economics. Oslo, Norway. Safwat, K.N.A., Magnanti, T.L., 1988. A combined trip generation, trip distribution, modal split, and trip assignment model. Transportation Science 22, 77±96. She, Y., 1985. Urban Transportation Networks. Prentice-Hall, Englewood Clis, NJ. Wilson, A.G., 1970. Entropy in Urban and Regional Modelling. Pion, London.