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A Probabilistic Analysis of Shopping Center Trade AreasAuthor(s): David L. HuffSource: Land Economics, Vol. 39, No. 1 (Feb., 1963), pp. 81-90Published by: University of Wisconsin PressStable URL: http://www.jstor.org/stable/3144521 .Accessed: 11/01/2011 13:34

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A Probabilistic Analysis of Shopping Center Trade Areast

By DAVID L. HUFF*

A FUNDAMENTAL prerequisite in de- termining the potential market de-

mand for the products or services of a prospective retail firm or of agglomera- tions of prospective retail firms, within an urban area, is a geographical deline- ation of the region containing the prob- able customers for such goods. Such a region is called a retail trade area. A thorough knowledge of the characteris- tics and limits of a trade area is essen- tial. The proponents of a proposed re- tail facility must have this information in order to evaluate realistically the likely success of the venture. Not only does a knowledge of the retail trade area provide a basis for estimating potential sales but it also makes it possible to de- termine investment requirements for land, buildings, and fixtures, as well as the kinds and extent of merchandise of-

ferings, promotional activities, etc. The methods employed to delineate

intra-urban retail trade areas of pro- posed facilities generally involve either

subjective judgments based on observa- tions of similar conditions from which generalizations have been drawn or the use of empirically derived mathematical formulations. The latter are often re- ferred to as gravity models because of their close similarity to the physical as well as to the conceptual properties of the traditional Newtonian gravity model in physics. Such models are preferred by analysts over subjective evaluations

for several reasons. First, as will be

pointed out later, gravity models have a rather extensive historical background in trade area analysis. Consequently, empirical evidence is available to indi- cate that in many cases the use of such models has provided fairly good approxi- mations of the limits of a number of re- tail trade areas. Secondly, such models are easy to use and relatively inexpensive, thus adding to their appeal.

Historical Background of Gravity Models in Retailing

One of the first to formally apply the gravity concept to retail trade area analysis was William J. Reilly.' Reilly's focus, however, was directed toward de- termining the relative retail pulling power of two competing cities on an in- tervening area rather than on intra- urban retail trade area determination. His hypothesis was that two cities attract retail trade from an intermediate city or

t The author wishes to acknowledge the Real Estate Research Program of the Graduate School of Business Administration at the University of California, Los Angeles, as well as Resources for the Future, Inc., in Washington, D.C. for the fi- nancial assistance which helped to make this study possible. In addition, he expresses his appreciation to Professors Leo Grebler and Clay Sprowls for their suggestions and comments.

* Assistant Professor of Business Administration, University of California, Los Angeles.

1William J. Reilly, Methods for the Study of Retail Relationships (Austin, Texas: Bureau of Business Research Studies in Marketing, No. 4, 1929).

82 LAND ECONOMICS

town in the vicinity of the breaking point approximately in direct proportion to the populations of the two cities and in inverse proportion to the square of the distances from the two cities to the intermediate town. The mathematical expression of Reilly's hypothesis is as follows:

Ba Pa ) Db)2 Bb Pb ( Da

where Ba = the proportion of the trade from the intermediate city attracted by city A;

Bb= the proportion of the trade from the intermediate city attracted by city B;

Pa = the population of city A; Pb = the population of city B; Da = the distance from the inter-

mediate town to city A; and, Db = the distance from the inter-

mediate town to city B.

Reilly subjected his hypothesis to ex- tensive empirical examinations and was so impressed with its predictive ability that he called it "The Law of Retail Gravitation." To illustrate the mechanics involved with operating the model, con- sider the following hypothetical illus- tration: City A has a population of 100,000 and City B has a population of 200,000; and town x is 20 miles from

city A and 30 miles from city B. By placing these values into Reilly's formu- la, we can calculate the proportion of

people in town x that retail stores in City A and city B are likely to receive. That is,

2

Ba (100,000 30

Bb 200,000 )2- -= 1.125

The percentage of the population of town x attracted to city A would thus be

57 percent, and the remaining 43 per- cent would be the proportion attracted to city B.

Since Reilly's study was published in 1929, a number of empirical studies have been conducted using his model. Most noteworthy of these have been the stud- ies carried out by P. D. Converse for various communities in Illinois.2

In 1947, the Curtis Publishing Com- pany made a significant adaptation of Reilly's formula to determine the break- ing point between two cities." Such a boundary line, i.e., where Ba = Bb, or point of equilibrium, represents the point up to which one city exercises a dominant trading influence and beyond which another city dominates. Another way of defining it would be that the breaking point represents the .5 proba- bility position between two cities. The mathematical version of this adaptation is indicated below:

Dab Bb

where Bb = the breaking point between city A and city B in miles from B;

Dab= the distance separating city A from city B;

Pb = the population of city B; and

P, = the population of city A.

2P.D. Converse, "Retail Trade Areas in Illinois" (Urbana, Illinois: University of Illinois Bulletin, Business Studies No. 4, 1946).

8Frank Strohkarck and Katherine Phelps, "The Mechanics of Constructing a Market Area Map," Journal of Marketing, April 1948, pp. 493-96.

SHOPPING CENTER TRADE AREAS 83

The derivation of this model from

Reilly's original formula is as follows:

(i) Since Ba = Bb

(ii) Then (Pa Db• - 1 Pb Da

(iii) And Db Pb Da -Pa

Db _ Pb (v) Therefore Dab - Db -

Dab -D I Pa (vi) And Dab

- 1 Db Pb

Dab (vii) Thus = Db

J Pa

Pb

To illustrate the operation of this

formulation, let us assume that the popu- lation of cities A and B are 400,000 and

200,000 respectively. Furthermore, let us assume that the distance separating these two cities by the most direct route is 80 miles. By substituting these data in the formula, we find that the breaking point from city B is:

80 Bb = = 33.3 miles

400,000 1 + 200,000

Correspondingly, the breaking point from city A would be 46.7 miles.

The modified version of Reilly's for- mulation as developed by the Curtis Pub- lishing Company has also been used ex- tensively to estimate trading areas of proposed shopping centers within cities.' Generally, the square footage of each retail center is substituted for popula- tion and the travel time between retail centers is substituted for physical dis- tance, i.e., miles. Again, to illustrate the use as well as the mechanics of oper- ating the model, consider the following example: Assume that a group of in- vestors are contemplating financing the construction of a proposed shopping center within a particular section of a city. The site for the facility has been chosen and the number as well as the proximity and size of competing centers has been determined. By using the mod- ified gravity model, the breaking points from the proposed cented to each of the existing centers are calculated. The hy- pothetical data used, as well as the re- sults of such computations are shown in Table I. These results are also mapped to delineate the potential trading area of the proposed facility as shown in Fig- ure I.

TABLE I-HYPOTHETICAL DATA USED IN DELINEATING TRADING AREA OF PROPOSED

SHOPPING CENTER

Sq. Footage Travel Breaking Point Shopping of Selling Time From Shopping

Center Space From A Center to A

A 200,000 0 0 B 100,000 15 6.2 C 150,000 20 9.3 D 50,000 10 3.3 E 300,000 25 13.8

' Leon W. Ellwood, "Estimating Potential Volume of Proposed Shopping Centers," The Appraisal Journal, October 1954, pp. 581-89.

84 LAND ECONOMICS

Such a delineation is usually superim- posed over a census tract map as illus- trated in Figure 1 to determine the num- ber of persons within the confines of the trading area. This information is then used as a basis for determining the purchasing power potential of such a region and, as a consequence, the po- tential sales and profitability of a shop- ping center.

FIGURE 1-HYPOTHETICAL TRADE AREA MAP OF A PROPOSED SHOPPING CENTER

c

----- TRADE AREA BOUNDARY

O COMPETING SHOPPING CENTERS

e PROPOSED SHOPPING CENTER

Limitations of Gravity Models

Despite the contribution that the grav- ity model has made to retail trade area analysis, there are a number of limita- tions associated with its use." First, the calculation of breaking points to deli- mit a retail trade area conveys an im- pression that a trading area is a fixed boundary circumscribing the market po- tential of a retail facility. Sufficient empirical evidence is avail- able, however, to indicate quite clearly that a trading area represents a series of radial zones in which the proportion of consumers patronizing a given retail

facility varies from zone to zone. For example, suppose that an analyst consid- ers using the breaking point formula to delimit the trading area of a proposed shopping center located in an area which includes a preponderance of so-called light industries and non-competing com- plementary commercial activities. Since such a trading area would be determined solely on the basis of calculating the breaking points between competing cen- ters, the resulting evidence is likely to be highly distorted. On the other hand, if the analyst had conducted a cross-sec- tion analysis of residential areas and tried to determine the proportion of households that were likely to patronize the proposed center as well as compet- ing centers, the resulting trading area would probably be considerably differ- ent and much more realistic. The situa- tion just described applies particularly to cases where the breaking point for- mula has been used to delineate the trad- ing area of the central business district.

Second, the vast majority of market analysts who have used the gravity model to delineate intra-urban retail trade areas have naively assumed that the exponent 2 which Reilly had originally estimated for intra-urban trade movements would possess the same value within urban areas. Such an assumption appears ques- tionable. Studies have shown that the exponent has ranged from 1.5 to over 3, depending on the trip type being ana- lyzed as well as the geographical setting involved."

6 David L. Huff, "A Note on the Limitations of Intra-Urban Gravity Models," Land Economics, February 1961, pp. 64-66.

6 See, for example, Gerald A. P. Carrothers, "An Historical Review of Gravity and Potential Models of Human Interaction," Journal of American In- stitutes of Planners, vol. 22, No. 2, 1956.

SHOPPING CENTER TRADE AREAS 85

Finally, the gravity model possesses very little theoretical content. That is, it does not reveal why observed regulari- ties occur as they do. Throughout its thirty-three year history, the model has been used primarily as a tool and, as a consequence, little has been done in the way of examining the behavioral impli- cations associated with its use.

Purpose and Scope of this Study

In view of these limitations this study attempts to develop an improved formu- lation for delineating trade areas of pro- posed shopping centers. The analysis will utilize the conceptual properties of the gravity model but its focus will be on the consumer rather than on the re- tail firm per se. Since the consumer is really the primary object of any trade area analysis, an explicit understanding is needed not only of the factors affect- ing his choice of a shopping center but also of the choice process itself which

gives rise to observable spatial behavior.' It is felt that once these conditions are known, considerably greater insights will have been achieved concerning the dy- namic characteristics of retail trade areas. Furthermore, knowledge of such condi- tions should make it possible to prepare more reliable estimates of intra-urban retail trade areas.

Measuring a Shopping Center's Utility The utility of a shopping center to

a consumer is based upon a host of dif- ferent factors. Any attempt to measure the relative intensity or weight of all of these factors would be doomed to fail- ure. Furthermore, the difficulty would become compounded if an attempt was made to ascertain the variations among such weights that would inevitably exist from consumer to consumer. Conse- quently, what is desired in this study is

to discover and specify only a few rele- vant variables that will enable predic- tions to be made reasonably well and consistently.

Empirical evidence suggests that two variables exert such an influence on a consumer's choice of a shopping center that they may be the only variables need- ed to predict such behavior according to the criterion cited above." These two variables include: (1) the number of items of the kind a consumer desires that are carried by various shopping centers; and, (2) the travel time that is involved in getting from a consumer's travel base to alternative shopping centers.

Impact of Merchandise Offerings. A large number of consumer shopping de- cisions are made under conditions of uncertainty, i.e., a consumer does not know in advance whether a particular shopping center will definitely fulfill a specified purchase desire. However, a consumer does have an a priori knowl- edge of the probability that various shop- ping centers might satisfy his shopping demands. Such a probability is based, for the most part, on the number of items of the kind that he desires and feels are carried by various shopping centers. Presumably, the greater the number of items carried by such centers, the greater is the consumer's expecta- tion that his shopping trip will be suc- cessful.' Therefore, consumers will show

7Spatial Behavior will be used throughout this study to refer to the observable courses of action that consumers take with respect to their choice of a shopping center. A shopping center will refer to complementary and competing agglomera- tions of retail firms which are geographically con- tained. A shopping center, according to this defi- nition, may be either planned or unplanned.

SGerald A. P. Carrothers, op. cit. 9 William J. Baumol and Edward A. Ide, "Variety

in Retailing," Management Science, October 1956.

86 LAND ECONOMICS

a willingness to travel further distances for various goods and services as the number of such items available at various shopping centers increases. In the pres- ent study, the preceding condition is assumed to be solely a linear function. Since it would be extremely time con- suming and laborious to ascertain the number of different types of goods of- fered by various shopping centers in any empirical analysis, this factor can be ap- proximated by using the square footage of selling space devoted to the sale of such items.

Impact of Travel Time. The utility of a shopping center to a consumer is also influenced significantly by the ef- fort and expense that is perceived to be involved in traveling to various shopping centers. The anticipated costs of trans- portation, the effort involved in prepar- ing for as well as making the trip, and other opportunities that must be fore- gone, tend to detract from the utility of a shopping center the further it is from the consumer's travel base.

As a consumer perceives his journey, the disutilities or dissatisfactions associat- ed with a given shopping expedition rise rather markedly with incremental in- creases in distance, taking the form of a

parabolic function. This condition sim-

ply reflects the notion of opportunity costs. That is, the consumer has only so much time that he can devote to various activities in any given time pe- riod if he is going to accomplish any one of them. Consequently, if the consumer devotes more time to one activity at the expense of others, the costs of these fore- gone opportunities rise sufficiently to act as a check against further time losses. Therefore, a shopping center's utility or value to consumer is inversely related to the effort and expense involved in getting from the consumer's point of

origin to a given shopping center. Since the effort and expense involved with a

shopping trip are inextricably linked with time, a suitable measure for these factors would be the travel time involved in making a given journey.

Formal Expression of Utility Ele- ments. Now that the two variables af-

fecting a shopping center's utility have been described, they can be formally expressed as follows:

SJ

Tu P (Cu) =

n S3

j=1 \T1/

where P (Cu) = the probability of a con- sumer at a given point of origin i traveling to a giv- en shopping center j;

SJ = the square footage of sell- ing space devoted to the sale of a particular class of goods by shopping cen- ter j;

Ti = the travel time involved in getting from a consum- er's travel base i to shop- ing center j; and,

X = a parameter which is to be estimated empirically to reflect the effect of travel time on various kinds of shopping trips.

The Use of the Model in Determining a Shopping Center's Trading Area

Once having estimated X for both

clothing and furniture shopping trips, it is possible to determine the trading area of a shopping center for any given

SHOPPING CENTER TRADE AREAS 87

product class." The steps involved in

making such a delineation are as follows:

1. Divide the area surrounding any exist- ing or proposed shopping center into small statistical units. These units could be Census enumeration districts, if they are not too large, or they could be simply squares within a constructed grid. Such units represent the i's in the model.

2. Determine the square footage of retail selling space of all shopping centers (the j's) included within the area of

analysis. 3. Ascertain the travel time involved in

getting from a particular statistical unit i to each of the specified shopping cen- ters j.

4. Calculate the probability of consumers in each of the statistical units going to the particular shopping center under investigation for a given product pur- chase. That is,

Si

Tij P (Cij) = (For all i's)

n Sj

J=1 Tij )

where P (CiU) = the probability of con- sumers from each of the ith statistical units going to a specific shopping center j; and,

A = the parameter estimate appropriate to the type of product class, e.g., 3.191 for clothing and 2.723 for furniture.

5. Map the trading area of the shopping center in question by drawing lines con- necting all statistical units having like probabilities.

An illustration of a partial retail trade area map pertaining to a shopping center for clothing purchases is shown in Figure 2. It can be seen that a retail trade area is not a fixed line circumscribing a shopping center

FIGURE 2-EQUIPROBABILITY CONTOURS CIRCUMSCRIBING SHOPPING CENTER J1

J14

.01

*.70

.90

.99

so

but rather a series of zonal probability con- tours radiating away from a shopping cen- ter.

However, a retail trade area map por- trayed solely in terms of probability con- tours is insufficient in making a market analysis. In addition to the likelihood of consumers from various statistical units patronizing a proposed shopping center, it is necessary to know the expected number of such consumers from each of the units. For example, it might be that a given con- tour possesses a high probability value but the consumers within its confines may be few in number. As a consequence, the ex- pected number of consumers from contours possessing lower probability values may be much higher and thus should be given more significance. The following step reflects this point.

10 The results of a recent empirical study estimated X for clothing and furniture shopping trips. These estimates were 3.191 and 2.723 respectively. David L. Huff, Determination of Intra-Urban Retail Trade Areas (Los Angeles, California: Real Estate Re- search Program, Graduate School of Business Ad- ministration, University of California, 1962).

88 LAND ECONOMICS

6. Calculate the number of households within each of the statistical units. Then, multiply each of these figures by their

appropriate probability values to deter- mine the expected number of consumers

(expressed in households) who will pa- tronize the shopping center in question for a particular product purchase. That is,

Si

Tu E (Cl) =

? Ci (For all i's)

n Sj( j=I Tij

where E (CU)= the expected number of consumers from each of the ith statistical units go- ing to a particular shop- ping center j; and

Ci = the number of consum- ers in the ith statistical unit.

As was true in the case of the probability contour map, calculating the expected num- ber of consumers from each of the statis- tical units is also inadequate per se in as- certaining the market potential of a pro- posed shopping center. It could be, for example, that a given contour possesses the largest expectation in terms of the number of consumers who are likely to patronize the proposed center. However, in terms of purchasing potential, another contour possessing a much smaller expected number of consumers may have a greater disposable income level and thus greater purchasing potential. Since the level of consumer in- come is a primary determinant in affect- ing the kinds as well as the volume of prod- uct sales, it is obvious that such data must be included if a realistic appraisal of the market is to be made. Step 7 demonstrates how such information is used in arriving at a sales expectation for a proposed shop- ping center.

7. Determine the annual average per house- hold incomes of each of the statistical units. Compare such figures to corres- ponding annual household budget ex- penditure figures in order to determine the average expected amounts spend by such families on various classes of prod- ucts, e.g., clothing and furniture. Esti- mate annual sales for the shopping cen- ter under investigation by multiplying each of the product budget figures by expected number of consumers from each statistical unit who are expected to pa- tronize the shopping center in question. Then, sum these individual estimates to arrive at a total annual sales potential by product class for the selected shop- ping center. Such a sales potential can be expressed in terms of the basic model as follows:

Si

Tij E (Au) -- Ci Bik (For all i's)

n Sj

j=I \ Tij S

where E (Au) = the expected annual sales potential for shopping center j with respect to a given product class from each of the ith statistical units; and,

B =k the annual amount bud- geted by consumers in the ith statistical unit for prod- uct class k.

The kinds of predictions afforded by the basic model can be extended even further by incorporating additional vari- ables. For example, if the average num- ber of shopping trips that consumers make with respect to various types of

product purchases within particular time

periods is known and included in the model, it is possible to estimate the ex-

pected number of consumer shopping

SHOPPING CENTER TRADE AREAS 89

trips that are likely to be made to a pro- posed shopping center in a given time

period. The basic model would be modi- fied to provide such an estimate as fol- lows:

Si

Tij E (Sam) = Ci Sm (For all i's)

j=I ($Ti)

where E (SiQm) = the expected number of shopping trips from neighborhood i to shop- ping center j for a given product purchase in a given time period m; and,

Sm= the average number of shopping trips S that consumers make with respect to a given pro- duct purchase within a given time period m.

As an illustration, suppose the devel-

opers of a proposed shopping center are interested in determining the expected number of clothing shopping trips that consumers from a particular neighbor- hood are likely to make to such a center

per month. Let us assume that the prob- ability of consumers from this neighbor-

hood patronizing the proposed center has been calculated from the model to be .80. In addition, let us imagine that there are 1000 households within this

particular neighborhood. Finally, let us assume that statistics indicate that, on the

average, consumers make two shopping trips per month for clothing purchases. The expected number of shopping trips from neighborhood i to the proposed center j per month would thus be:

E (StUm) = (.80) (1000) (2)

E (Sijm) = 1600

It is possible to carry the preceding operation a step further to determine the

expected sales of a given product class from a particular neighborhood within a specified period of time. Again, if we know the annual amount budgeted by consumers of a particular income stra- tum (Bik) for a given product class and, if AS indicates the annual number of

shopping trips made for such goods; then,

Ba,/AS would indicate the amount

spent per shopping trip; and, (Ba,/AS)

(Sm) would indicate the amount spent within a given time period of less than one year. The model would now appear as:

Sj

TO Bi,

E (Dim) = n Ci c* Sm (For all i's) jn I AS

j \-I TiJ

90 LAND ECONOMICS

where E (Dijm) = the expected dollar sales D made at shopping cen- ter j from neighborhood i for a particular prod- uct class in a given time period m; and,

AS = the average annual num- ber of shopping trips made by consumers to shopping centers for a particular product class.

Again, using the values cited in the last illustration, let us assume that each consumer from neighborhood i spends, on the average, $1,200 annually for cloth-

ing items. Therefore, the expected amount spent monthly for clothing pur- chases by all consumers from neighbor- hood i at shopping center j would be:

E(Dijm) = (.80) (1000) ( $1,200

E (Dijm) = $80,000

Conclusions

The model provides a sound concep- tual and operational basis for under- standing and determining the retail trade area of a shopping center. The retail trade area of an existing or proposed shopping center can be ascertained by: (1) dividing the surrounding area into small statistical units; (2) calculating the probability of consumers from each of these units going to a particular shop- ping center; and (3) drawing lines around all statistical units having like probabilities. A retail trade area is not a fixed line circumscribing a shopping center but rather a series of radial zonal probability contours.

Because of the model's simplicity and versatility it should be capable of being extended still further in providing a number of different kinds of valuable estimates associated with market analyses.

In spite of the model's attributes, ex- tensive empirical tests are necessary to ascertain reliable estimates of the para- meters. In an effort to meet this objec- tive, a large scale empirical study is pres- ently being undertaken by the author.