Equivalence between welfare and entropy approaches to residential location

Regional Science and Urban Economics 14 (1984) 147-173. North-Holland

EQUIVALENCE BETWEEN WELFARE APPROACHES TO RESIDENTIAL Lam-G&an

AND ENTROPY LOCATION*

MATTSSON

Royal Institute of Technology, S-100 44 Stockholm, Sweden Received July 1983, final version received November 1983 An optimization model for residential location in an urban area is presented. As the objective, a welfare measure is considered which is derived as aggregated expected utility based on utilitymaximizing individual behaviour. The utilities include travel costs for work trips as well as a measure of the disutility with high-density living. The problem of finding a welfare maximizing housing allocation is shown to be equivalent to an entropy maximizing problem provided that a certain condition is met. By considering a dual formulation, a computationally more expedient problem is obtained. The model approach is illustrated by a few applications to the Stockholm region.

1. Introduction Normative location models have now a long tradition in urban and regional modelling. A major theoretical achievement in recent years is the consistent integration of behavioural models of utility maximizing individuals in the evaluation criteria of locational and transport analysis through random utility theory [Williams (1977), Williams and Senior (1978), Coelho and Williams (1978)]. This research is comprehensively assembled in Wilson et al. (1981). Following this tradition we consider the problem of optimal housing allocation in an urban region with respect to welfare measures consistent with utility maximizing individuals at’ the micro level. To this end we first formulate a model of the locational choices of home and workplace made by the individuals in the region. In applying the random utility approach it is assumed that the utility a particular individual associates with a certain location of his home and his workplace can be expressed as a sum of a deterministic term and a random variable with known distribution. The deterministic term typically includes for each alternative the resulting cost of the journey to work [see e.g., Broughton and Tanner (1983)]. *A preliminary version of this paper was presented at the 23rd European Congress of the Regional Science Association in Poitiers 1983. The author wishes to thank Erik Anders Eriksson, Per Olov Lindberg, Lam Lundqvist and JSrgen W. Weibull for valuable discussions and the Swedish Council for Building Research for financial support. 01660462/84/$3.00

0 1984, Elsevier Science Publishers B.V. (North-Holland)

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Another aspect of importance for the choice of where to live is the local residential density or inversely the amount of land available per resident. This has been explicitly recognized in many analytical models within urban economics [see e.g., Mirrlees (1972)]. In the rich literature of discrete allocation models for land use and transportation analysis on the other hand, this aspect is usually ignored. Two early exceptions are Lundqvist (1973) and Holm and Lundqvist (1977). In analogy with these models it will be assumed that the deterministic term in the individuals’ utility also incorporates some measure of the local residential density. If we now let the region be divided into a number of zones and assume that each individual maximizes his utility, the expected demand for housing and employment in each zone may be derived. If the corresponding supplies are exogenously given, there is no reason to believe that the various local markets automatically will be in equilibrium, even if total demand and total supply in the region are equal. As will be shown, however, by making the standard assumptions of multinomial logit models, it is always possible tointroduce suitably defined prices related to the various markets, such that expected demand equals supply in every local market. Furthermore, by imposing a certain normalization condition, these equilibrium prices will be uniquely determined by the exogenous supplies. We may now define the welfare measure to be associated with a particular allocation of housing and workplaces over the zones. We simply take the expected maximum utility sum across the individuals at normalized equilibrium prices. Apart from the residential density term included in the utility, this is the same welfare measure as the one used by Broughton and Tanner (1983). While they are using their measure to evaluate a finite number of alternatives, however, we will use it as the objective function in an optimization problem. In this respect our approach is similar to the one suggested by Leonardi (1981) for normative facility location. More specifically, we consider the distribution of workplaces as exogenously given. We then seek the distribution of a given amount of housing subject to upper and lower bounds for each zone, such that the welfare measure is maximized. Since the welfare measure is computed at equilibrium prices, the underlying spatial interaction patterns in the optimization are consistent with cleared local markets and utility maximizing individuals. It was mentioned earlier that the individuals are assumed to choose jointly where to live and work. As will be shown, however, the resulting welfare maximization problem - if the individuals have their workplaces fixed and only choose where to live - is equivalent to the first one. Unfortunately, the discussed welfare problems cannot be optimized in a straightforward way due to the fact that the normalized equilibrium prices in the welfare measures are just implicit functions of the housing allocation.

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However, if the residential density term in the utility of the different alternatives meets a certain condition, a computationally tractable solution technique is available. The reason is that an equivalence then can be established between the welfare and a certain entropy maximization problem. In this respect this paper extends the results in Coelho and Williams (1978). The dual of this latter problem is an unconstrained convex minimization problem. Hence, a number of efficient computer algorithms are available. The number of variables in this dual problem is equal to the number of residential zones which allows a fairly fine zonal subdivision. A final theoretical result concerns how the welfare maximizing housing allocation may be obtained as a market equilibrium. It is shown that in certain cases this can be accomplished by imposing a suitable congestion tax related to the local housing supply. The use of the suggested approach is illustrated by a few applications to the Stockholm region. Finally, some extensions of the underlying choice model to include mode choice and accessibility to service in the locational decisions are outlined. 2. Two welfare maximization

problems

We consider an urban region consisting of a finite number of zones. Some non-empty subset of these, indexed by j E J = { 1,2, . . . , n}, where n 2 1, consists of zones containing workplaces. Such zones will be referred to as job zones. Let the n-dimensional vector w E R;, where R + denotes the positive reals, be the exogenously given positive supply of workplaces, i.e., Wj >O is the number of workplaces in job zone j E J. Let W=cjEJ Wj denote the number of workplaces in the region. In some (possibly other) non-empty subset of the zones, residential location is possible. This subset is indexed by i E I = (1,2,. . . , m>, where m2 1. These zones are referred to as residential zones. (Note that one and the same physical zone can be both a job zone and a residential zone and not necessarily indexed in the same manner.) The housing supplies in the residential zones, expressed in terms of residing workers, will be endogenous variables in the analysis, forming an m-dimensional vector h. We will only consider vectors corresponding to positive housing supply in each residential zone, i.e., h E R”, . Two somewhat different models for the demand in the various zones for, in the first case, housing and employment, and in the second case only housing will be formulated. These models are based on the random utility approach. It will be shown that there exist in both cases equilibrium prices in the various markets such that all markets are cleared in a stochastic sense. Based on these demand models, two closely related welfare maximization problems will be posed.

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2.1. Joint residential and job choice

To specify the first demand model, imagine an individual choosing where to live and work in the region. In conformity with the random utility approach, we assume that there is a random variable associated with each alternative describing his utility and that he chooses the utility-maximizing alternative. Let Uij be the (random) utility associated with living in zone iE:I and working in zone jo J. Following e.g., McFadden (1974) it is assumed / that Uij=-Cij+Zij ViEI and VjEJ, where cij, i E I and j E J, are deterministic values representing systematic differences (deterministically modelled relationships) in the evaluation of the various alternatives, while Z,,, i E I and j E J, are random variables reflecting interpersonal differences and unobserved factors influencing the locational decisions. We assume that Zij, iE1 and jE J, are statistically independent and identically distributed according to the extreme value distribution Pr[Zijsz]=exp(-e-‘)

for

z&R,

and thus [see David (1970, pp. 205206)]

ECzijl =Y, OCzijl = Klfi, where y ~0.577 is Euler’s constant. By this assumption and the principle of utility maximization, the choice probabilities will be of the multinomial logit type or gravity type [cf. McFadden (1974) and Cochrane (1975)]: qij=Pr[UijLUi,jS

Vi’El andVj’EJ]

(1) =e

VieI

and

VjEJ,

and E

=y+logCTe-“j.

(2)

(In summations and maximizations here and in the following, it will be tacitly understood that the index runs through the whole relevant index set.)

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The statistical assumptions are somewhat less restrictive than they might seem at first glance. The utilities Uij may in fact be regarded as linear transformations of underlying utilities Vi j = - c:j + 2: j with arbitrary common variance. To see that, let Z:j, i E I and j E J, have the distribution Pr [.Z: j 5 z] = exp ( - e -@)

for

ZER

where

/3>0.

It is then easy to see that pZ:j will be distributed as Zij and thus

Hence B may be conceived as a concentration measure parameterizing the variance of the underlying utilities. By setting Uij = /XJi j, and thus Cij = /3cij, we obtain for each i E I and each jeJ Pr [Uij 2 UifjS Vi’ EI and V~‘E J] =Pr[UijzUv,,j,

Vi’eI

and Vj’EJ],

and, of particular interest,

Hence, in the comparisons determining the choice probabilities it is irrelevant whether they are performed in terms of the underlying utilities Uij or in terms of the utilities Uij of standardized variance. Moreover, by maximizing the expected maximum of the transformed utilities [i.e., maximizing (2)], also any non-decreasing function of the expected maximum of the underlying utilities [i.e., any such function of the left-hand side of (3)] will be maximized. We will now make some further specification of the deterministic cost terms in the utilities. Given a housing supply over the various residential zones described by the vector h, we assume Cij = Eij(h) + pi + rj

ViEI

and

VjEJ,

where C,,(h) is a (deterministically modelled) cost (or disutility) of living in zone i and working in zone j when the housing supply is h, pi is a price related to the housing market (the rent level) in zone i, and rj is a price related to the job market (minus the wage level) in zone j. The term Eij(h) typically includes the travel cost connected with the journey to work. The

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reason for letting the cost term be h-dependent is that we will also allow for the possibility of including a residential density measure in the cost term. -We now consider a population of individuals, all behaving according to the choice model (l), and where the cost terms are in accordance with the specification above. Then their aggregated demand for housing and employment over the zones are random variables in contrast to the deterministic supplies. Accordingly, the concept of market equilibrium requires a modification: A market in a zone is said to be in equilibrium if expected aggregated demand equals supply. ’ Regarding the exogenously given supply vector of workplaces w E R;, and a given housing supply vector h E R”,, we may ask whether it is possible to find price vectors p and r such that all housing and job markets in the region will be in equilibrium. An obvious necessary condition is that the total demand and supply for housing and the total demand and supply for employment in the region all are equal, i.e., that the size of the population is W=G Wj and that Ci hi = IY However, these conditions being fulfilled, there is always a possibility of finding price vectors p and r such that all markets will be in equilibrium. These equilibrium prices will be uniquely determined up to a common additive constant for the prices in the housing markets and likewise in the labour markets. In order to attain uniqueness, we impose the condition on the prices that the total value of the housing supply as well as the total value of the supply of jobs must be zero. These normalized prices may then be interpreted as differences between the prices in the various markets and the average price level in the region. Expressed formally we have the following proposition: Proposition 1. Assume he R”,, xi hi = W and that C(h) is an m x n-matrix where the i, j-entry Eij(h) is any real-valued function of the housing supply vector h. Then there exist unique price vectors p E R” and r E R” such that, for cij=Eij(h)+pi +rj ViEZ and VjEJ, W Cqij=hi j

ViEZ,

Wcqij=wj 1

VjEJ,

and CPihi 203

Before proving the proposition

(6)

[based on a result by Evans (1970)] we

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1.53

note that (4) expresses the equilibrium condition that the expected housing demand aggregated over W individuals equals the supply in each residential zone and (5) corresponding relationship for each job zone. Proof The existence. By Theorem 2 in Evans (1970) there exist positive numbers rci and pj, i E I and j E J, such that gij

=

qpj

e-Eij(h)

satisfies Tqij=hi/K

ViEI,

T@ij=WjfW

‘djEJ.

Since CC 4”ij = 1 we have i

j

Gij

=nipj

e-““(h)

“J’C I

i

nipj

e-‘ij(h).

j

By letting pi=-logrr,+(~hilog~i)/U:

iEI

‘;=-logPj+(~WjlOgPj);~

jEJ,

and

the eqs. (6) and (7) will be satisfied. Moreover, 4”ij may be rewritten as

4”ij=e~p{-(cij(h)+pi+r~))

i

C~exp{-(Eij(h)+pi+ri)}.

Hence, iij = 4ij for cij = Cij(h) +pi + rj Vi E I and Vj E J, which proves the existence. The uniqueness. Suppose there are two sets of price vectors p’, r1 and p2, r2 fulfilling the requirements of the proposition and denote the corresponding choice probabilities by qfj and qFj respectively. Due to Theorem 1 in Evans (1970) qfj = qFj VieI and Vje J. Thus

exp - (p,l + rf) = k exp -(pF + r$

VieI

and

VjeJ,

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where k is some constant. By taking the logarithm we obtain p;-pF=r;--r;-logk

VieZ

and

VjeJ.

Hence p1 =p2 +(& 5,. . . , [) and r1 = r2 +(a q, . . . , r~) for some constants t and q. Since both sets of price vectors satisfy eqs. (6) and (7) these constants must be zero, which proves the uniqueness. Q.E.D. Before stating our first welfare maximization following definitions.

problem,

we make the

1. Suppose hE R”, and xi hi = W Let p*(h) E R” and r*(h) E R” be, according to Proposition 1, the unique normalized equilibrium price vectors in the housing and labour markets as functions of the housing supply h. Moreover, let the equilibrium choice probabilities qi j(h), i E Z and j E J, and the aggregated expected equilibrium utility V(h) be defined as the corresponding values of qij and W(E[maxi,j Uij] -y-log W) for cij =Eij(h)+pf(h)+rj*(h).

Dejinition

As defined, the aggregated expected equilibrium utility, V(h), for a given housing supply h is the sum of W individuals’ expected maximum utility at the equilibrium prices p*(h) and r*(h) modified by the constant term W(y +log W) (to facilitate the notation). It corresponds to what Coelho and Williams (1978) termed locational surplus. By (2) we have V(h)= W logCCexp{-(F;j(h)+p,*(h)+r,*(h)))log i

j

W

>

(8)

We are now in the position to state the first welfare maximization problem. Regarding the distribution of workplaces, w, as given, we want to find that housing supply vector h which - subject to certain constraints maximizes the social welfare function V(h). The constraints are in the form of lower, li, and upper, ui, bounds on the housing supply, hi, in each residential zone i EZ. These lower bounds may represent existing housing stocks while the upper bounds may come from various restrictions concerning housing construction and land availability in the different zones. As a regularity condition, we assume that 0< li 5 ui Vi E Z and xi li 5 WsCi Ui which guarantees that the set of housing supply vectors h E R”,, Ci hi = q fulfilling the constraints is non-empty. Stated formally we define The welfare maximization problem 1 (PI):

max V(h)

subject to

h

li~hi~ui

VieZ

and

Chi=W

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2.2. Purely residential choice In the microeconomic founding of this second housing demand model, the individuals have already their workplaces fixed and are choosing just where to live in the region. For a utility maximizing individual working in zone jE.J, the choice probabilities for the different residential zones iel then is Vi’EI].

L$j=Pr[UijLUifj

In analogy with the derivation of (1) and (2) we obtain qFj =edcij

Ce-cij

QieI

and

QjeJ,

I

(9)

and

[i 1 I

E maxUij

=y+logze-cij

QjEJ.

(10)

Observe that while for the previous model Ci cj qij = 1, in this case xi qFj = 1 for each j E J. Consider now the following specification of the cost term ~~~=~~~(h)+p,

QiEI

and

VjeJ

with the same notation as earlier. Assume there are W individuals having their workplaces distributed over the job zones according to the exogenously given vector w. The labour market will then be cleared by definition. As before, the demand for housing in a residential zone will be a random variable and we consider equilibrium in the same stochastic sense as for the earlier model. Hence we may ask whether for a given housing supply vector h E R”, there is a price vector p such that the expected demand for housing equals the supply in each zone. The answer is yes provided that Ci hi = W Interestingly enough, these equilibrium prices are the same as those associated with the housing markets in the earlier model. By imposing the same normalization condition - the total value of the housing supply vector must be zero - the normalized equilibrium price vector will be p*(h). We summarize in the following proposition where (11) expresses the equilibrium condition: expected housing demand equals supply in each residential zone. Proposition 2. Assume he R”,, Ci hi = W and that c(h) is an m x n-matrix where the &j-entry c,,(h) is any real-valued function of the housing supply vector h. Then p=p*(h) is the unique m-dimensional vector such that, for

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ViEI and VjGJ,

TWjqf’j=hi

ViEI,

and

CPihiCO*

(12)

Proof: The existence. Let p=p*(h). Then 42 =qij(h)/ci qij(h) by (1) and (9), and (11) follows immediately from (4) and (5). Clearly (12) is satisfied due to (6). The uniqueness. For any PER” the quantities Wjq~j will be of the form ‘Tcipje-“jth) where rci and pj are positive constants, i E I and j E J. If the eqs. (11) are satisfied, qFj will be uniquely determined due to Theorem 1 in Evans (1970), since Ci Wjq~j = Wj by definition for all jE J. Suppose (11) holds for two price vectors p1 and p2. Then by the uniqueness argument

exp{ -(Cij(h) +P,‘>> _ exp{ -(ctj(h) +d)> Ce~p{-(i;ij(h)+p!)}-Cexp{-(cij(h)+P~)}

viE1

and

“jjEJ’

Thus by fixing j= 1, say, and by taking the logarithm, it follows that

for some constant <. By (12), however, < =O.

Q.E.D.

Parallel with the earlier demand model, we make the following definitions. Definition 2. Suppose hE R”, and xi hi = W Let the equilibrium choice probabilities q:‘(h) and the aggregated expected equilibrium utility V’(h) be

defined as the corresponding ~Wj(“[-:.“ij]-?-“sWj)

values of q~j and for

cij =Eij(h) +pT(h).

Then q~j(h), for the housing supply h, will be the probability that an individual working in zone j,E J will choose residential zone ic I at the equilibrium prices. Similarly, the aggregated expected equilibrium utility (or locational surplus) V”(h) is the sum of the expected maximum utilities, modified by a constant, over W individuals, each employed at one of the exogenously giyen workplaces in the region. By (10) we obtain

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VO(h)=C

Welfare

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1OgCexp {-(Cij(h)+pT(h))}-log

Wj j

location

Wj

.

151

(13)

>

VO(h) will be the social welfare function in the second welfare maximization problem to be considered. As in the first one, the problem is to find a welfare maximizing housing supply vector h, subject to the same constraints. Stated formally, we define The welfare maximization problem lo (PI’):

max V’(h)

subject to

h

l.Ih.lu. I- z-

and

I VieI

~hi=W

2.3. Equivalence

The next proposition shows how closely related the behavioural assumptions are which underly the two welfare maximization formulations Pl and Pl’. Proposition 3.

Suppose h E R”, and xi hi = W Then

and

VieI

(4 Wqij(h) =Wjd’j(h) (b) V(h) = V’(h).

VjeJ,

P~ooJ: (a) AS in the proof of Proposition follows from (5). (b) As a simple consequence of (a)

2, q~j(h)=qij(h)/Ci

C~expj-(c,j(h)+p:(h)+r~(h))j = Wev { -rj*(h)lCexp{-(c,j(h)+p*(h))}, 1 wj for any jE J. Inserted in (8) this yields V(h)=

(j

Cwj

>

log Cxexp{-(cij(h)+pT(h)+rj*(h))) (i

j

which by (13) and (7) equals V’(h).

Q.E.D.

qij(h)* Thus (a)

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The (a)-part of the proposition says that ‘the expected work trip flows at equilibrium prices are the same in the two choice settings. The (b)-part tells us that the seemingly different social welfare functi!ns in the welfare maximization problems Pl and Pl” also coincide. Since the constraints also are the same in these two problems, we obtain Corollary 1. equivalent.

The social welfare maximization

problems PI

and PI0 are

Recall that Pl and Pl” are founded on different perceptions ‘of the individual’s choice situation. In Pl, the individual chooses where to live as well as where to work while in Pl” he only chooses where to live conditional on where he works. According to Corollary 1, however, the same housing supply will maximize the aggregated expected equilibrium utility (or locational surplus) in both cases. 3. Equivalence between welfare and entropy maximization Next we turn to the question of solving the two posed (equivalent) welfare maximization problems. Since V(h) and V’(h) are dependent on h in an implicit way via the normalized equilibrium price vectors p*(h) and r*(h), the maximization is not straightforward. Fortunately, as will be shown, under a certain condition on the cost matrix C(h), the (in fact) unique solution to these problems could be identified with part of the solution to a certain entropy maximization problem. This latter problem is in its turn most easily solved by considering an unconstrained dual formulation. We start by stating the relevant entropy maximization problem. Here X is defined to be a real-valued non-negative m x n-matrix with the i&entry Xij having the interpretation of work trip flow, i.e., the number of individuals living in residential zone iEl and working in job zone j E J. As before we assume h E Ry . The entropy maximization problem (PZ):

max -CC i h,X

Xij(lOg

xij + C,j(h)),

subject to

j

l.lh.lui I- I-

VieI,

~xij=hi

CXij=Wj

VjeJ,

xii20

ViEI,

(14)

ViEI and jeJ.

This entropy maximization problem1 differs from the simplest ones of this kind by the housing-supply-dependent cost functions ‘Observe that the objective can be rewritten as -~i~jxijlog(xij/&j(h)) e-c,j(h). Thus P2 can also be perceived as an information minimization and Weibull (1977)].

problem

where Xij(h)= [cf. Snickars

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Before stating what will turn out to be a dual formulation, following condition to be imposed on the cost matrix C(h):

location

159

we consider the

Cl. For each jE J and each ieZ such that li
cij(h)=tij

where tij is some real constant and di some real-valued function on [li, Ui] such that the related real-valued function gi on [Zi, Ui] defined by gi(hi) =hidi(hi)

Vhi E [li, Ui]

is convex, non-decreasing and continuously

differentiable.

For example, any g,-function twice continuously differentiable with nonnegative first and second derivative will meet this condition [see Rockafellar (1970, Theorem 4.4)]. Hence di may be any power function with nonnegative exponent. In the applications to be considered, tij will be the travel cost associated with living in zone i and working in zone j, while di(hi) will be some indicator of the disutility as perceived by an individual to be associated with living in zone i when the local housing supply in the zone is hi. For example, di could be the number of residents per unit of space available for residential purposes [cf. Holm and Lundqvist (1977), and Lundqvist and Mattsson (1983)]. Assuming condition Cl to be met, the following unconstrained optimization problem is well-defined and actually a dual formulation of P2. The vector cp~R”’ is the vector of dual variables related to the latter part of the constraints (14) in P2. The dual problem (P3): +cPi)}-logwj

+C(~ihP(rpi)-gi(hP(~i))), >

where ho, i E I, is any function from R to [Zi, ui] satisfying

hP(qi)=Ui

if

li =Ui

Ek:)-l(~i)

if

CpiEE(k)7.dC”i)17

= li

if

Cpi
or

Observe that if & is strictly increasing,

‘pi >d(ui),

then Z$ is uniquely

(15)

defined and

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continuous by (15). If this holds for every ieZ, the objectives of P3 is obviously well-defined and continuous. If gj is constant on some interval (15) does not determine h: uniquely. As can be shown, however, the objective of P3 is well-defined anyhow and also continuous. We are now ready to formulate a duality relationship between the two optimization problems P2 and P3. Proposition 4. Suppose CZ holds. Then there exist solutions to P2 and P3 and their optimal values are equal. Let the solution to- P2, which is unique, be (&a. Then, for any solution 4 to P3, &=-&tij

i

)2ij

=X~j(~)

ViEI,

and

ViEZ

and

VjEJ,

where X~j(~)=Wj

~expi-(tij+qoi)l

exp{-(tij+cpi)} I

(

VieZ

>

and jeJ.

Moreover, @ being a solution to P3 is equivalent to, for each i E I, ~x,“j(@)=hf’(4i)

(16)

for some function h:, satisfying condition (15).

Thus the proposition suggests that P2 can be solved by solving the unconstrained optimization problem P3 or equivalently by solving the nonlinear system of equations (16). To prove Proposition 4, we need the following almost trivial duality result [see Lindberg (1981)]. Lemma 1. Let f and fR be real-valued functions on some sets Y and YR respectively such that f(y) 5 fR(y) on Yc YR. Suppose f achieves its maximum 7 at j? and fR its maximum fR at jiR. Zf yR E Y and f (yR) =fRk, then f (JR) =J Proof

~~~f~(9~)~f~(~))ff(~))ff(9~).

Butf&)=3k

HencefW=f(N=$

Q.E.D. (To avoid some technical difliculties, we will prove under a somewhat stronger condition than Cl. We assume

Proof of Proposition 4.

the proposition

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that for each i E I such that 2i
.i

where It/j = log (xi exp ( -(tij R,:max L(qp, h, X), h,X Zishisui

+ ~i))/wj) - 1. Consider the relaxation of P2

subject to ~~~2-0

ViEI

and

ViEI

VjEJ.

Due to C2 h’(cp)=(hy(q,,), h$(cp,), . . ., hz(cp,)) will be uniquely defined by condition (15). Since L is strictly concave in (h, X), ho(q) and the matrix X”(q) with the positive i, j-entry xFj(cp), iE1 and jo J, is the unique solution to R, because

T& (cp, k X0(d)=O

‘diEI

and

VjEJ,

iJ

and for every iEZ such that li
(hi-hO(

& (Cp, ho(V),X) 50

Vht

E [li,

ut].

I

Let u(q) be the optimal value of R,. Then u(q) =I(cp, easily verified to be equal to the objective of P3. For any vector cp obviously

and CxFj(Vlcwj

VjE J.

h’(q),

X”(q)) which is

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Suppose there is a vector 4 such that also (16) holds. Then (h”(@),Xo($)), being the solution to R,, also satisfies the constraints of P2 and v(e)= L(+, ho(@), X”(e)) equals the value of the objective of P2 at h=h’(@) and X=X’(@). Thus by Lemma 1 (h’(@),X’(@)) is a solution to and u(@) the optimal value of P2. Moreover, since for any vector cp, v(q) =L,(cp, ho(~), X”(cp))~L(cp,ho(@),Xo(@))=L(@,ho(iJ), XO(@))=u(@), @ is a solution to P3 and, furthermore, the optimal values of P2 and P3 are equal. On the other hand, if 4 is a solution to P3, then, since z) is continuously differentiable due to C2,

which is easily verified to be equivalent to (16). It remains to prove that P3 always has a solution. The proof is rather lengthy, although it employs only elementary calculus. First we consider the case Ci Ui = IY By Proposition 2 applied on hi =ui, i E I, there exists a vector cp” E R” such that

by taking ~~j(~) = ti j and ‘pp = p:(u) Vi E I and Vj E J. Clearly, for a sufficiently large 5, hP((pP+ 5) =ui for all ie:I. Since = for 4 = q” + (t,c, . . . , <), $ clearly satisfies (16) and hence is a solution to P3. As the case xi li = W can be treated analogously, we proceed with the final case Ci li < W < ‘& ui. We shall prove the existence of a compact set @c R”’ such that inf v(R”) =inf v(Q). Since v is continuous (because g’i is strictly increasing for every iE:I due to C2) the inlinum is achieved by some @iE @ and thus P3 has a solution. Let f=maxi, j(tij), u=min, (ui) and 2’ =maxi (gi(Ui)) if this maximum exists and g’ =0 otherwise. Then u>O and g’ 20. Consider a vector cpE R” and let @= maxi (cp;) and 9 = mini (cp,). Then X~j(~)

X~j(cp”)

(17) (18)

(19)

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for some q-independent constant JC. The relation (18) follows from the observation that @((pi) = li for ‘pi 5 0 since gi(l,) > 0 due to C2. Now define the compact set ~={(cpERrn;(cpil~i;MViEI} for some M >g’ 2 0. Suppose q~$ @. Then q < - M and/or q > M. Consider first (p < -M. Then by (19) v(q)2

(

W-c

i>

Zi M+Ic,

which can be made greater than v(0) for say 8 = (0, 0, . . . , 0) E @ by choosing M sufficiently large. Next assume (p> M and 9 &’ 2 0. Then by (17)

LCW4-~W+(@i~)Q+K. i -

Let v=min(Ciui-W&>O.

Thus

v(cp)&p+(@-q$v+k.~Mv+tc. Thus again u(q) can be made greater than v(0) by choosing M sufficiently large. Finally assume (p> M and cp
Also in this final case v(q) can be made greater than v(6) by choosing M sufficiently large. By choosing M greater than all these three lower bounds, it is guaranteed that inf v(R”) = inf v(Q). Q.E.D. To Proposition 4 we have the following corollary which relates the optimal solution and the optimal value of P2 to the choice settings underlying Pl and Pl’: Corollary 2. Suppose Cl holds and let (6,$) the optimal value of P2. Then

v^=V(c)= V”(h),,

’

be the unique solution to and 6

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Ri j = Wqi j(r;>

Welfare

and entropy

= Wj&j(@

VieI

approaches

and

to residential

location

VjeJ.

Moreover, there is some real-valued function 6 on [lip Uilrn such that for which & E (li, Ui)

Proof:

Let @ be a solution to P3. By Proposition

for

all i E I

4

for any hy, i E I, satisfying (15). By (16) there is for each i E I some hP satisfying (15) such that

Hence v”=cj

Wj(lOgCi

exp { -(Cij(@ +p”i)) -log Wj), where (20)

we obtain by Proposition 2 p”i=p:(&) ViEZ. Hence by (13) fi= V”(@ In passing we have also shown ‘Rij =w~&(~ Vie I and VjE J. Assume r;i E&, ui) for some iE1. Since & = hP(~i) for some ho satisfying (15) we have pi =g:(~i)=di(i;,)+~i~(~i). Hence by (20)

p,*(fi)=p”i =t%iL&(&) +8(L), where 6 is some real-valued function on [li, Uilrn. The remaining parts of the corollary follows from Proposition 3. Q.E.D. The latter part of the corollary could be given an economic interpretation.

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For zones where the constraints on the housing supply are not binding, the normalized equilibrium price, p:(h), at the optimal supply vector, can be seen as a kind of congestion price or tax which - imposed on the individuals makes their private cost equal to the social cost. This is the case because an increase of the housing supply in zone i by one unit from the optimal supply will raise the (social) cost for those already living there by

which equals the privately perceived cost pT(fi,) up to a constant negligible in the ranking of the zones. We may now formulate the main theoretical result concerning the equivalence between welfare and entropy maximization, Assuming condition Cl, the unique common solution to the equivalent welfare maximization problems Pl and Pl” coincides with the housing supply vector I% of the unique solution to the entropy maximization problem P2. Moreover, by Corollary 2, the optimal matrix 8 to P2 gives the expected flows at the optimal housing supplies in the choice settings underlying Pl and Pl’. Proposition 5. Suppose Cl holds. Then Pl and PI0 have a unique common solution equal to the housing supply vector of the unique solution to P2. ProoJ: Let h^ be the housing supply vector of the unique solution to P2 by Proposition 4 and let v” be the optimal value. Let &#fi be any other vector satisfying the constraints of Pl. Consider the restriction of P2 where for each iE1 the bounds Zi= Ui =&. Then the maximum of this restricted problem v”
Hence ff is the unique PlQ. Q.E.D.

solution

to Pl

and by Corollary

1 also to

In sum: the common welfare maximum of the originally stated welfare problems could be attained by solving P2 which in turn is most readily solved by means of the unconstrained optimization problem P3. This results in a drastic reduction of the problem size. In Pl (and Pl’) there are m unknown variables and 2m+ 1 constraints together with an objective involving implicitly defined functions. P2 avoids the latter difficulty at the expense of an increased number of variables, (n+ l)m, as well as constraints,

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3m+n (yet disregarding the never binding non-negativity constraints). In contrast, P3 involves just m variables and no constraints. Finally, we want to point out the possibility of attaining the welfare maximum as a market equilibrium by means of an appropriate local housing supply tax. The result will be formulated in terms of the second, i.e., purely residential, choice model. Considering a housing supply vector h E R”, and a price vector PER”’ related to the housing markets, the expected aggregated housing demand in zone i will be (cf. (9)) cexp{-@j(h)+pi)}

hf(h,p)=CWj

j

(

J$ew(+ij(h)+Pi)l

I

Then the following proposition

>

holds.

Proposition

6. Suppose Cl holds and that for Pl” the unique solution i.e., belongs to the interior of the constraint set defined .by the lower lE(li, %lrn, and upper bounds. Define the supply related price vector, or housing tax vector, p(h) E R”, such that the ith component is pi(h)=hid’i(hi).

(21)

Then h is the unique solution of the equations hf(h, p(h)) = hi

where

hi E (li, ui),

i E 1.

(22)

In view of Proposition 5 and the third relation in Corollary 2, E clearly satisfies (22). Conversely, if h” satisfies (22), then (16) is satisfied by iE1. Then h” is the unique solution to P2. Hence @i =di(&) +&dj(&)=g’i(Ei), K= 6. Q.E.D.

ProoJ:

In other words: by imposing a local housing supply tax according to (21), the welfare maximizing housing supply h could be attained as a market equilibrium. More precisely, under the hypotheses of Cl, t%belonging to the open set (li, y)” and this housing tax, I%will be the unique housing supply for which supply and expected aggregated demand will be equal in every residential zone, Before closing this section, a remark concerning the variance of the involved random utilities is in place. All results reported refer to the case of standard deviation equal to x/$. As noted in the opening discussion, however, the results can easily be transformed to utilities with arbitrary common standard deviation, say z&,@, where p>O. What is needed is a linear transformation of the utility scale, i.e., all utilities, costs and prices must be multiplied by the concentration parameter /3. Clearly, in terms of

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this transformed scale all utilities will have the required variance to make the derived results applicable. 4. Applications to the Stockholm region

To illustrate the use of the suggested approach to residential location, we make some applications to the Stockholm county for the year 1990. For a more detailed account, see Mattsson (1984). 4.1. Model specification and data

Before applying the equivalent welfare maximization problems Pl and Pl” to the Stockholm county, a few words about the data for the exogenous parameters are in place. The county is divided into 105 zones regarded both as residential and job zones. Broadly speaking, the data employed agree with those in Lundqvist and Mattsson (1983). Thus the distribution, w, of the projected 800000 workplaces in total is taken from the adopted regional plan for the county 1990. Likewise, the lower bound on the housing supply in each zone, lj, is based on projections for 1990 of remaining parts of the 1975 housing stock. The upper bound, ui, on the other hand, is for each zone based on a recent survey of greatest possible housing construction. The lower bounds are quite restrictive. They leave only 22 percent of the total housing supply to be distributed. The upper bounds are looser. The slackness amounts to 76 percent of the total supply. Next we have to specify the cost functions in the individuals’ utilities. In accordance with condition Cl we consider cost functions of the form cij(h) = tij + di(hi),

iEI

and jEJ.

The term tij is interpreted as the travel time for work trips between zone i and j. As data for this parameter we take average car and transit travel time for 1990 with regard to expected mode use. The residential density indicator di(hi) is perceived as the disutility associated with living in zone i when the housing supply in the zone is hi. For its specification we need some further notation. Let o be the number of residents per individual in the labour force (chosen equal to 2.03 in accordance with the regional plan). Let si, iE1, be the supplies of space for housing purposes in the zones. Values for these variables have been determined according to a calibration procedure, see Lundqvist and Mattsson (1983). The applied residential indicators are of the form di(hi) =

l-awhi a si ’

(23)

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for some trade-off parameter a ~(0,l). Since hi is in terms of individuals in the labour force, ohi/si is a measure of residential density. The a-parameter transforms residential density into a travel-time-comparable scale and will be varied in the applications. Clearly this density indicator meets condition Cl. As noted in the closing discussions of the previous section, there must be a common standard deviation for the random parts of the utilities. Let this common value be rc/(&/@ for some fi> 0. Then the P-parameter describes the degree of concentration in the distribution of the utilities. 4.2. Parameter variations By changing the parameters a and /I in our specification of the equivalent welfare maximization problems Pl and Pl’, very different housing distributions may be generated. As two aggregated measures describing the resulting distribution, we consider average residential density and expected travel time for work trips. In both cases the appropriate value for each zone is weighted with the housing supply in the zone. As seen from (23) a reflects a trade-off between residential density and the other terms in the utilities. A larger a-value corresponds to less weight on the density term. Consequently, fig. 1 shows a consistent pattern of increased average density and decreased expected travel time as a is increased. The p-parameter, being inversely proportional to the standard deviation of the involved random variables, measures the concentration in their distributions. Hence, for large p the random variations are small and travel times and densities capture most of the factors which determine an

Average density [residents/ha] A 43 42

39

---we.___

38

Expected travel time [min] Fig.

1. Variations

in c( and a. Effects on average residential travel time for work trips, (xi xi

density, w(ci fi j tij)/W

@/si)/K

and expected

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individual’s locational decision. As seen from fig. 1, the effects of variations in j? are most obvious on the travel behaviour. When j3 is increased, there will be a decrease in the probability that the utility is maximized for workplaces and homes far away from each other. This explains the reduction in expected travel time. The effects on residential density are less clear. However, for large a and /I (say a>= 0.48 and /32 O.l), an increase in j? leads to higher average density. By such a choice of parameters, much weight is put on travel times, and housing supplies in zones with good access to workplaces will be favoured in the optimization. Since the workplaces in the Stockholm region are concentrated to central parts which also are characterized by high residential densities, this leads to an increase in average density. Fig. 1 also indicates that while the changes in expected travel time might be considerable (35 to 10 min), the changes in residential density are relatively less drastic (from 42 to 38 residents/ha). It should be remembered, however, that our choice of data for the lower bounds on the housing supplies only permit 22 percent of the total housing supply to be reallocated. Fig. 2 gives a detailed spatial description of the optimal housing supply for two examples of parameter choice. The height of the columns refers to the number of residents above the lower bound for each area. The two supplies could be considered as extreme cases. The housing pattern ranges from a very dense one where the supplies are concentrated to the central areas (lefthand columns) to a more spread-out pattern favouring low-density living in suburban and peripheral areas (right-hand columns). This shows how various preferences might be explicitly modelled within this approach to residential location.

5. Extensions and directions for future research The equivalence between random utility based welfare maximization and entropy maximization (Proposition 5) has been established under fairly restrictive assumptions. One may ask to what extent this equivalence might be generalized. To this end we will point out two possibilities of enriching the underlying choice model by including mode of transport and service accessibility in the choice of where to live and work. 5.1. A joint choice model of location and mode

Let k~K={l,..., v} denote the mode of transport and let Uijk be the random utility associated with living in zone i, working in zone j, and using mode k. We assume that the utility can be written in the separable form

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Fig. 2. Optimal housing supply by municipality in terms of residents above lower bound. Lefthand columns refer to a=032 and /?=0.2 and right-hand columns to cc=O.20 and /I=O.OS. Stockholm municipality is subdivided into three parts with Stockholm city specially indicated.

where cijk is deterministic and the random vector (2, I r, . . . , Z,,,) multivariate extreme value distributed with the distribution function

PrCZllt5zll19..., [ ( ~mnvSbJ=ew

-TT

TexP{--Zijk/(l-4)

is

1-I >1 ,

for some parameter ill [0, 1). By this assumption Zi,j,kl and Zi2j2k2 will be statistically independent as long as ir # i2 and/or jr #j, while if A#O, Zijkl and Zijk, will be correlated. The parameter A can be interpreted as an index of similarity [see McFadden (1978)]. Let qijk=Pr[UijkzUitjfk*

Vi’EI, Vj’jlEJ, VKEK], iEI,

jEJ,

kEK,

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location

and define the inclusive value 1-1

Texp{-Cijk/(l-l)}

It can then be shown that [cf. McFadden

and

iEI,

2

jEJ.

(1978)]

@j=Tqijk=e-Eij C4f+j, I

igI,

jsJ,

(24)

(25)

If we assume that iEI,

Cijk=tijk+di(hi)+pi+rj,

jEJ,

kEK,

and define l-1

>

Texp{-tijk/(l-,?))

)

and jEJ,

iEI

(26)

we obtain ~ij=~ij+di(hi)+pi+rj,

ieI

and jgJ.

(27)

By observing that (24) and (25) are of the same form as (1) and (2) it follows that normalized equilibrium prices in (27) can be defined in analogy with Definition 1. Moreover, the corresponding welfare maximization problem analogous to Pl can be solved by solving P2 for Fij(h) = fij + d,(&), i E I and jE J, provided that L&(/Q), i E I, meets condiction Cl. Hence, the inclusion of mode choice does not really affect the mathematical structure of the welfare maximization. What has to be considered is that the optimization is carried out with respect to appropriately defined travel costs, i.e., according to (26). 5.2. Choice of location with regard to service accessibility

By letting k E K denote different model might be reinterpreted as a service accessibility. We are then choosing where to live, work and might assume that Cijk=t~+~k+di(hi)+pi+rj,

available service alternatives, the previous model of locational choice with regard to considering individuals who are jointly get their service. To be more specific, we iGI,

jsJ,

kEK,

/

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location

where t; is the travel cost for work trips between zone i and j and f& is the travel cost for service trips between residential zone i and service alternative k, while the remaining symbols are defined as before. Then, by defining the service indicator

t;= ( -log

Texp{--t&/(1--A)}

1-I

>

, ifsI,

iEI,

jEJ.

we obtain Eij=t;+tf+d,(hi)+pi+rj,

Thus, by solving P2 with respect to E,j(h) = t; + tf +d,(Q iel and jE J, welfare maximum of the problem analogous to Pl will be-found provided that di(hi), iE1, meets condition Cl. 5.3. Future research The suggested approach to residential location has several important limitations. As one example, condition Cl does not allow the cost functions to reflect possible congestion in the traffic system in a realistic way. Is it possible to establish equivalence between welfare maximization of type Pl (or Pl’) and some computationally tractable problem also for such more realistic cost functions? This is an open question certainly worth future research. As another example we consider the assumption of statistical independence among the random utilities (cf. subsection 2.1).2 Some preliminary studies indicate that certain equivalence results might be obtainable by only assuming the vector of random utilities to be multivariate extreme value distributed, thus permitting a fairly general correlation structure among the utilities. ‘Note that this assumption is partially subsections.

weakened in the choice models of the last two

References Broughton, J. and J.C. Tanner, 1983, Distribution models for the journey to work, Environment and Planning A 15, 31-53. Cochrane, R.A., 1975, A possible economic base for the gravity model, Journal of Transport Economics and Policy 9, 34-49. Coelho, J.D. and H.C.W.L. Williams, 1978, On the design of land use plans through locational surplus maximisation, Papers of the Regional Science Association 40, 71-85. David, H.A., 1970, Order statistics (Wiley, New York). Evans, A.W., 1970, Some properties of trip distribution methods, Transportation Research 4, 1936.

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Holm, M. and L. Lundqvist, 1977, Spatial allocation of housing programmes: a model of accessibility and space utilization, Papers of the Regional Science Association 38, 149-168. Leonardi, G., 1981, The use of random-utility theory in building location-allocation models, Working paper WP-81-28 (International Institute for Applied Systems Analysis, Laxenburg), in: J.F. Thisse and H.G. Zoller, eds., Locational analysis of public facilities (North-Holland, Amsterdam) forthcoming. Lindberg, P.O., 1981, A relaxational framework for duality, TRITA-MAT-1981-12, Mimeo. (Department of Mathematics, Royal Institute of Technology, Stockholm). Lundqvist, L., 1973, Integrated location-transport analysis; a decomposition approach, Regional 1 and Urban Economics 3, 233-262. Lundqvist, L. and L.-G. Mattsson, 1983, Transportation systems and residential location, European Journal of Operational Research 12, 279-294. Mattsson, L.-G., 1984, Some applications of welfare maximization approaches to residential location, Papers of the Regional Science Association 54, forthcoming. McFadden, D., 1974, Conditional logit analysis of qualitative choice behavior, in: P. Zarembka, ed., Frontiers in econometrics (Academic Press, New York) 105-142. McFadden, D., 1978, Modelling the choice of residential location, in: A. Karlqvist et al., eds., Spatial interaction theory and planning models (North-Holland, Amsterdam) 75-96. Mirrless, J.A., 1972, The optimum town, Swedish Journal of Economics 74, 114-135. Rockafellar, R.T., 1970, Convex analysis (Princeton University Press, Princeton, NJ). Snickars, F. and J.W. Weibull, 1977, A minimum information principle, theory and practice, Regional Science and Urban Economics 7, 137-168. Williams, H.C.W.L., 1977, On the formation of travel demand models and economic evaluation measures of user benefit, Environment and Planning A 9, 285-344. Williams, H.C.W.L. and M.L. Senior, 1978, Accessibility, spatial interaction and the spatial benefit analysis of land use - transportation plans, in: A. Karlqvist et al., eds., Spatial interaction theory and planning models (North-Holland, Amsterdam) 253-287. Wilson, A.G., J.D. Coelho, S.M. Macgill and H.C.W.L. Williams, 1981, Optimization in locational and transport analysis (Wiley, Chichester).