The design of a location experiment: A continuous formulation

Transpn Res. Vol. 7, pp. 31-38. Pergamon Press 1973. Printed in Great Britain

THE DESIGN OF A LOCATION EXPERIMENT: A CONTINUOUS FORMULATION RICHARD V. EASTIN and PERRY SHAPIRO Department of Finance and Business Economics, University of Southern California, Los Angeles, California 90007 (Received 29 August 1972)

IN A PREVIOUSpaper (Eastin and Shapiro, 1973) it was shown how the entropy maximizing

approach used by Wilson (1967,1969, 1970) can be combined with postulates on individual behavior to form models of residential location and trip distribution. In that analysis a discrete probability function was found to be a convenient mathematical description of the distribution of a known total population among a known finite number of geographical partitions or zones. Hence, when we considered the entropy of the population distribution we used the entropy,

(1)

H=-~p,lOgpi i-1

of the corresponding discrete probability function as our model, where pi was defined to be the proportion of the total population residing in the ith zone from the central business district (CBD) (i.e. the probability that a randomly chosen individual lives in zone i). A number of results about the relationship between transportation and location were obtained for populations in which there is but one income level. Although the effects of changing the one individual income level for the city were examined, it was not possible to deal with a population with a continuous income distribution. It is true that Wilson and others have shown that it is possible to deal with a countable set of income levels; however, from a practical standpoint, the number of levels that can be handled is rather small. A continuous formulation of the entropy problem would allow the incorporation of as large and finely divided an income set as is desirable. If we wish to model a geographical population distribution which is perceived as being a continuous function of distance, x, from the CBD, we must use a continuous probability density function, f(x), rather than the discrete probability distribution, pi (i = 1, . . ., n). Clearly, we need to develop a measure, H(f), of the entropy of the continuous probability density function, f(x), analogous to the discrete measure given by expression (1). Following Goldman (1953), the most natural way of going from the discrete to the continuous measure of entropy is to define it as

where hxi = Xi+l- xi and j(xi) is the ordinate of the frequency function at distance Xi from the CBD. Equation (2) is the same as equation (1) when Ax, = 1 for all i. However, there is one problem with this formulation which can be seen by rewriting it as Hcf> = lim - C [f(Xi) logf(xJ] aXi + lim - I; Lf(xJ log AX(] hxi AZ*0

Al%+0

31

32

RICHARD V. EASTINand PERRY SHAPIRO

Consider the case in which Axi = Axj = Ax for all i and j. Then the above expression can be written H(f) = -

“f(x) logf(x) dx + lim - log Ax ,;4f (x) h (3) AZ-O J‘0 f Clearly, the second term on the right of (3) goes to infinity as x approaches zero, for all distributions of population, f(x). Thus, (3) is not a unique description of a population distribution: the measure is the same for all distributions. This problem can sensibly be corrected by calibrating the measure with respect to some distribution of minimum entropy. The measure will be calibrated by arbitrarily picking some high density population distribution which will be the highest density the measure will handle. The resulting entropy expression is such that, if the entire population were to live in a circle of some small radius 2 around the center of the city, the entropy of that distribution would be zero. Let f*(x) represent the distribution in which the entire population is maximally concentrated, and define the calibrated entropy measure as S(f) = H(f)- H(f*). Using (3), rewrite the measure as S(f> = - @Lf(x) logy(x) -f*(x) logf*(x)] dx. Since f * will be the same for all questions we ask, and, furthermore, since we will be looking for maximum values of S(f), we can conveniently write the measure as S(f) = -

-nf(x) logf(x) dx I0

(4)

Assumptions of the model

The model is presented in its most simplistic form in order to ease the derivation of the analytical results. It is assumed that the urban area considered has only one zone of economic activity, called the CBD, which is located in the center of a homogeneous plain. The model can be expanded without limit to account for the realities of multi-destination trip distributions as well as topographical irregularities. These realities must, of course, be dealt with in the application of the model, but since we are mainly interested in the properties of a continuous description of spatial behavior in an urban area with a multiincome population, we shall ignore these complications at this point. Furthermore, it is assumed that all citizens of the town take one round trip a day: they travel from their place of residence to the CBD and return. In the urban area so far postulated, the only thing that distinguishes one location from another is its distance from the CBD. Each individual (or family) in the area has the same log-linear utility function u= qaZbP where q is the quantity of space consumed by the individual, Z is a composite commodity, or a vector of commodities made up of all other goods consumed and I is the amount of leisure time taken. Leisure is chosen as a utility producing commodity on the implicit assumption that the variable encompasses all pleasurable uses of time (work and commuting are not considered pleasurable). In order to insure the concavity of the utility function and the existence of an optimum, add the restriction that 01+/3+ y < 1. During each day each individual divides his time (24 hr) between work, for which he receives wage rate W; leisure, during which he pursues pleasant activities such as eating, sleeping and recreation; and travel to and from work. It is assumed that all travel takes place at a constant velocity of 2v m.p.h., and at a constant marginal pecuniary cost of 2y dollars/mile. Thus, for simplicity, we ignore the important effects of congestion. Each family unit maximizes its utility function subject to the budget constraint w(24-Z-x/v)

=p,Z+h(x)q+rx

The design of a location experiment

33

where x is the distance of the family’s residence from the CBD, h(x) is the rent per unit space at distance x, pz is the price (or price vector) of the composite commodity and rx is the pecuniary cost of a round trip to the CBD. The maximization of utility subject to the budget constraint can be set as a Lagrangian function L = q~‘z~lY+ X[w24-wz-wx/u-pzZ-h(x)q-rx]

The first-order conditions for maximization yield the equations o!q”-1ZjlY-

Ah(x) = 0

#fIq’xZb-1P- Ap, = 0 yq~Z~Wl--

xw = 0

Using these conditions and combining them with the budget constraint, one can find the expression for the expenditures on housing space to be

Wq = --$-(24wConverting to k= OI/((Y +/?+r),

wx/v-rx)

a more convenient notation let y E 24~7, t 3 & [(l/u) +(r/v)] and rewrite the above expression as

and

h(x) 4(x, Y) = k 0 - o=) The variable y is called the individual’s or family’s income. In this case y is not the standard definition of income, which is the money received by an individual for the sale of his time to others. Instead, income is considered to be the implicit amount an individual pays himself for his own time. That is, he can be viewed as buying time from himself to use for leisure as well as travel. This is a convenient formulation, for it allows us to account for differential evaluations of travel time arising from different incomes. Since h(x) is the rent per unit space and q(x, y) is the residential space occupied by a person of income y at distance x, h(x)q(x, y) is the total housing expenditure of a family of income y living at distance x from the CBD. A family’s housing expenditure is only part of the total expense directly associated with living at distance x. The other expense is the cost of commuting to and from work. This commuting expense is the implicit amount the individual pays himself for that time plus the pecuniary cost of purchasing the actual transportation-a total commuting expenditure of fyx per day.t The total cost of occupying an amount of space at a given distance, then is the housing expenditure plus the commuting costs. We call this total cost incurred by an individual with income y at distance x his locational expenditure, which we label L(x, y). It is obvious from our previous discussion that L(x, y) = kO, - tyx) + tyx. Unlike most models of residential location this model does not depend on the constancy of J~(x, JJ) for all families of the same income. In the traditional treatment every family of a given income is assigned to reside at a fixed distance. We shall assume, rather, that the average expenditure on location E@) is a known function of income.f For the present, the functional form of the average is unspecified. Later, different forms are used to demonstrate certain propositions about the relationship between income and residential density. t The variable I can be conceived as a measure of the efficiency of transportation. Since t varies inversely with a and directly with r, the smaller r, the more efficient is the transport system. 2 It is not unreasonable to assume some knowledge of &). The journey to work and housing expenditure data from the 1970 Census Statistics should allow us an empirical measurement of &). 2

. 34

RICHARD

V.E~sm~and

PERRY SHAPIRO

(a) Equilibrium distribution Letf(x, JJ) be the joint probability density function of x and y. The entropy off(x, y is given, in a slightly modified version of (3), as

“=f

m 5, (x, y, 0) log f(x, y, 0) do dx dv (5 0 0 0 sss Consistent with our emphasis on empirically determinable parameters, we shall assume w know the distribution of income go in the town. In this paper go is the probabilit density function over income y. We now seek the population distribution f(x, y) for whicl Scf) will achieve a maximum subject to a known J?(JJ) and a known go. In order to formulate the maximization problem, we first look at the expression fo g0 and &Y) as S(f) = -

g(v) = $f(x, J

Y)dx

(6

and from the definition of L(x, y) &)gcV)

=

/,[ky+(1 -k)tyxl_f(x,

Adx

(7

b3Y)-kYldY) = omcl-w VdcG Y)h s

(8

Equation (7) can be rewritten in a more convenient form

For ease, define JY.f(x,

YLYI= - ,?CG d logfk Y>dx I

The maximization can be written as a problem in the calculus of variations by writing max

=JV-(x7 Y),~1dy I0 subject to (6) and (8). Form the function F”Lf(x, u), rl

= m-k J4,Yl+ w [g (Y)- jornfkYW]

It can be shown (see Elsgole, 1962, pp. 130-131) that the necessary condition

for a car

strained maximum is &-&%)

= 0

where f ‘(x, y) = df(x, y)/dy. In this problem aF*/af ‘(x, JJ) = 0, therefore, the necessary condition for a maximum -logf(x,.Y)-1-4Y)-/&)(l-k)tYx=O Thus, the equilibrium population distribution is f(x, Y) = exp [ - 1 - fW - PW Cl- 4

WI

c

35

The design of a location experiment

The value of the multipliers h(y) and p(y) can be found by appealing to the constraint equations (6) and (8). First from (6) exp

I- 1- X.Y)I =exp[- ,W (1-V WI dx = g(y) s0

By integrating it can be shown that exp I- 1-

X.dl = AY)(~- W odd

Substitute this value of exp [- I- h(y)] into (9) and examine the expression for average locational expenditures (8) gW(l-WV

s

ompCv)(l-~)fyxexp[-~(y)(l-k)tyxldx = b%9 - bl g(v)

The integration yields

exp[ - P(Y)( 1- W 04 dv~(l--W~Y [-PQ(1 -wvxI-44 Cl- wv

11Cc 0

From L’Hospital’s rule it can be shown that x exp [-p(y) (1 - k) tyx] goes to zero as x goes to infinity. The value of the parameter p(v) can be found 1 P(Y) = L(y)-ky

(10)

From the necessary non-negativity of ,u(y), it must follow that L(y) > ky. The full expression forf(x, y) is

(1-k)w

f(xtY)=&,,)_kygti)exp

(11)

which is the joint probability density function over families with income y at distance x from the CBD. It is clear from (11) that every income class is distributed negative exponentially from the CBD. It is impossible to tell from (11) whether or not the entire population, as Clark (1951) found, is so distributed; however, it seems a likely consequence of some functional form of L(y). Although we are not yet able to answer the question of the distribution of total population [to do so would require integrating (11) over y], we are able to determine the effect of an improvement in transportation technology and the relationship between income and location. (b) The e$ect of transportation improvements on equilibrium location It can easily be shown that the partial derivative of (11) with respect to t is

(12) Becausef(x,

y)/r is clearly positive, the sign of (12) will be the sign of (13)

Expression (13) will be strictly positive if and only if L(y)>ky+(l-k)ryx

(14)

RICHARDV. EAST-INand PERRYSHAPIRO

36

But the right-hand side of expression (14) is the expenditure on location by an individual of income y living at distance x, i.e. L(x, y). Because aL/ax > 0, it is clear that for locations closer to the CBD than the average location of residents of a given income class, L(x, Y)
Equation (11) gives the joint probability density function over x and y. That is, J”, J$f(x, y) dy dx is the probability of finding a person in the population who has income between a’ and b’ and lives at a distance between a and b. In order to analyze the effects of income on the distribution it will be easier to deal with the conditional density function over distance x, given income y

c(xly)=f(X’Y)=(_l-k)ty i?(Y)

L(y) - ky

-(l-k)tyx

expE(y)- ky

The log of (15) is logc(xly)

1

(1%

= log(l-k)ty-lo&(y)-ky] - (1 - k) tyx/L(y) - ky

(16)

The partial derivative of (16) with respect to income is alogc(xly) 8Y

1 (1 -k)tx = ;-E(Y)-ky+

[r(y)-k]

= [L(Y)-ky-Cl-k)tYx]

[ky+(l -k) tyx--E(Y)] K(Y) -

ky12

Y[~(y;_ky]-[;$~;;]~~

.

It follows that alwc(xIY)

aY

=

[&Y)- b - (1-k)

L(Y) -YL'(Y) 'Y"ly~~o _ ky]2

(17)

Since c(xly) is the conditional probability density of the population with a given income y living at distance x, equation (17) tells us how that density changes with income. For instance, if (17) is positive it means that the conditional density function at distance x varies directly with income. Thus, one can interpret this result as saying that x is more attractive as income grows. The first term of (17) [l(y) - ky - (1 -k) tyx] is the difference between the average expenditure on location for a given income class, and the amount that an individual in that class who lives at distance x spends on location. As before, k(y - tyx) is the amount spent on housing and tyx is the amount spent on transportation. Since k is less than one, for a given income class, individual expenditures increase with distance. Thus average locational expenditures must be greater than individual expenditures for areas close to the CBD and the left-hand term is positive. Furthermore, the term must be negative for some areas far from the center. Since the denominator, y&(y) - ky12, is always positive, the sign of the second term of (17) is the same as the sign of z(y)-y,?‘(y). If L(y) is linear and homogeneous in y, then the second term is always zero; if it is concave it is always positive, and it is always negative if L(y) is convex in y.

The design of a location experiment

3

In the case in which L(v) is proportional to y, equation (17) is clearly zero for all y It therefore follows that the conditional probability density functions will be identical fo all income levels, For a given income class, if L(v) is concave in y, (17) is positive fo distances in which the locational expenditures are less than the average, i.e. when the lef expression is positive. This implies that a greater proportion of rich people than poo people live close to the CBD. Thus, the concavity of L(Y) implies that the higher a person’ income the higher is the probability that he lives close to town. Conversely, if &) i convex in y, an increasing proportion of rich people live in the areas far from the CBD. It is impossible to list all possible shapes of &). However, there is one shape which i particularly interesting for it leads to a description of residential patterns similar to those of modem cities. Consider &J) function similar to that in Fig. 1: one that is convex ove

FIG. 1

part of its range (O,v*) and concave thereafter. Such a shape would lead to the conclusio: that up to income y* people have a tendency to live farther from the CBD the higher thei income. For people with incomes higher than y* there is a tendency to move towards th CBD. This shape &v) is not unreasonable on a priori grounds and to be strongly expect& if the model is correct, of any empirical estimates. It is to be strongly expected for i predicts as we observe in the world, that the poor and the super rich (incomes above y* will dominate the central city. CONCLUSIONS

This paper has adopted the entropy maximizing methodology to handle tuba. population distributions which are continuous in both distance and income levels. On th assumptions (1) that all residents have identical log-linear utility functions over residentia space, leisure and a composite of all other goods; (2) that travel time and commuting cost are proportional to distance; and (3) that the relationship between income and averag expenditure on location is stable and observable, the entropy maximizing methodolog allows us to make the following conclusions about the most probable distribution c population : (i) The Lagrange multipliers can be solved explicitly by the use of calculus of variation to yield an analytical solution for the distribution of population. (ii) Each income class will be distributed negative exponentially with distance from th CBD.

38

RICHARD V. EASTINand PERRYSHAPIRO

(iii) An improvement in urban transportation technology will not alter the form but will lower the distance-density gradient causing population to be less densely clustered near the CBD. (iv) If average expenditure on location is proportional to income, the same proportion of the population of each income class will live in each annulus around the CBD. (v) If average locational expenditure is a concave function of income, a greater proportion of the higher income classes will live close to the CBD while the bulk of the lower income population will live farther from the CBD. (vi) If the relationship between income and average locational expenditure is convex, there will be a tendency for the major proportion of the wealthier population to live near the outskirts of the city while the poorer classes cluster near the central city. REFERENCES CLARK C. (1951). Urban population densities. J. R. Stat. Sm., Series A, 114, 490496. EAST~N,R. V. and SHAPIROP. (1973). The design of a location experiment. Transpn Res. 7, 17-29. EL~G~LE L. E. (1962). Cufculus of Variarions. Pergamon Press, Oxford. GOLDMANS. (1953). Information Theory. Prentice-Hall, New York. WILSON A. G. (1967). A statistical theory of spatial distribution models. Transpn Res. 1, 253-269. Wnso~ A. G. (1969). Developments of some elementary residential location models. J. Regional Sci. 9, 377-385. WILSON A. G. (1970). Entropy in Urban and Regional Modelling. Pion Press, London.

Abstract In an earlier article the authors combined the entropy maximizing approach to urban location theory with postulates on individual behavior to develop a discrete model of residential land use. This paper extends the earlier model so that it can deal with populations which are continuous in distance and income levels. The new formulation provides the planner with a model to predict the spatial behaviour of income classes and their response to changes in the transportation system.

R&ttmC Dans un article anterieur, les auteurs combinaient l’approche dune theorie du developpement urbain par la maximisation de l’entropie avec des hypotheses de comportement individuel pour mettre au point un modele discontinu d’utilisation du sol en zone d’habitat. Cet article developpe le modele precedent de man&e a pouvoir prendre en compte des populations caracterisdes par des crittres de distance et de niveaux de revenus continus. Cette nouvelle formulation foumit a l’amtnageur un modele de prevision du comportement spatial des individus des differentes categories de revenus et de leurs reactions aux modifications du systeme de transport.

Zusammenfassung

In einer frtiheren Abhandlung haben die Verfasser einen entropischen Maximierungsansatz in Richtung auf eine stadtbezogene Standorttheorie mit Annahmen iiber individuelle Verhaltensweisen verkniipft, urn daraus ein diskretes Model1 der Wohnflachennutzung zu entwickeln. Der vorliegende Bericht erweitert dieses Model1 in der Weise, dass es such fur Bevolkerungsgruppen gtiltig ist, die hinsichtlich Entfemung und Einkommensniveau kontinuierlich sind. Damit wird dem Planer ein Model an die Hand gegeben, mit dem er das ramnliche Verhalten von Einkommensklassen und deren Reaktion auf Veranderungen im Verkehrssystem vorherbestimmen kann.