A CONSUMER UTILITY MODEL FOR THE ALLOCATION OF
SALES AMONG MAJOR RETAIL CENTERS
EDGAR JAMES MCDOUGALL, JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
Edgar James McDougall, Jr.
This study represents the contributions of many people. Dr. Halbert
C. Smith, Dissertation Committee chairperson contributed encouragement,
guidance and support in the writing and critiquing of this study and through-
out this doctoral program. Dr. Jerome Milliman provided invaluable advice
during the critical stage of developing a topic and during the doctoral pro-
gram. Special thanks also go to Dr. Wayne Archer and Dr. Andrew
McCollough for their suggestions and support.
Drs. Stephen Messner, William Kinnard and Byrl Boyce, all of the
University of Connecticut, provided professional advice and criticism and
friendly support and encouragement during the writing of this study. Judy
Paesani provided expert editorial advice and did much to transform a multi-
tude of facts into a logical coherent work. Sandy Mazzola and Nancy Easton
provided typing and editing and showed a great deal of patience. Special
thanks go to Alan Sidransky who worked untiringly researching data and
programming the computer.
The Center for Real Estate and Urban Economic Studies at the University
of Connecticut provided financial support so that I could devote the neces-
sary time to this study.
Last and most important, I would like to thank my parents who provided
guidance, support and love throughout the last 30 years. I owe them all of
the opportunities I have had and the many good things which have come to
TABLE OF CONTENTS
ACKNOWLEDGMENTS ................... ........ ................ iii
ABSTRACT ..................................................... vii
ONE INTRODUCTION ...................................... 1
Statement of the Problem .............................. 2
Objectives of the Study .... ........................... 6
Significance of the Study ............ ... .. .......... 7
Setting of the Study ................................... 7
Organization of the Study ............................. 8
TWO REVIEW OF THE LITERATURE ......................... 10
Central Place Theory ................................ 11
The Assumption of Zero Economic Profits ........... 16
The Assumption that Shoppers Patronize the
Nearest Center ................................. 18
The Single Trip--Single Order Assumption ......... 24
The Homogeneous Demand Curve Assumption ....... 26
The Gravity Models ................................... 27
Reilly's "Law" of Retail Gravitation ................. 28
The Gravity Model within an Urban Area ........... 32
The Huff Probability Model ........................ .. 36
Theoretical Approaches ........................... 39
Empirical Evidence ....... ..................... 44
The Zone of Indifference Hypothesis ..................... 46
Hotelling--Center Clustering ....................... 47
Devletoglou's Dissenting View ..................... 48
Psychological Distance ............................. 51
The Outshopper ........... .............. .......... 52
Summary .......................... ................. 54
THREE TESTING THE MODELS ............. ........... .......... 56
The Method of Testing ................................. 56
The Development of the Models ......................... 57
The Central Place Model ........................... 57
The Gravity Model .................... ............ 58
The Zone of Indifference Model ..................... 60
T he D ata .............................................
MRC Characteristics ...............................
Linkage Characteristics ...........................
Buyer Characteristics .............................
The Predictions of the Models .........................
Analysis of the Predictions .......................
Analysis of the Distance Exponent, A ...............
Examination of the Statistical Measures .............
Sum m ary .........................................
THE WEIGHTED GRAVITY MODEL .......................
Additional MRC Attributes .............................
Analysis of the Regression .............................
Weighting Factor .....................................
Analysis of Results ...................................
A FRAMEWORK FOR UTILITY MAXIMIZATION AND
TIME ALLOCATION ...................................
Review of the Utility of Time Literature .......
A Framework for Decision Making .............
A Model for Decision Making .................
Utility of the Alternative Bundle of Goods .
Utility of a Good .........................
Utility of a Condition .....................
Utility of an Action .......................
Utility of an Alternative Activity .........
A Model of Consumer Shopping Behavior .....
Convenience Goods .....................
Comparison Goods ......................
Style Goods .............................
Sum mary ...................................
DEVELOPMENT OF A PREDICTIVE MODEL ...............
Convenience Goods Sales ...............................
The Central Place Model for Convenience
Goods Sales .....................................
The Gravity Model for Convenience Goods ...........
Comparison of the Predictions of the Two Models .....
Shopping Goods Sales .................................
Analysis of the Predictions of the Gravity
Model for Shopping Goods .......................
Shopping Goods Sales and Convenience Goods
Sales Combined .....................................
Local Dominance .......................................
The Interceptor Effect .................................
Analysis of the Prediction of the Final Model .............
SEVEN SUMMARY AND CONCLUSIONS .......................... 156
A INTRODUCTION TO THE CENSUS OF RETAIL TRADE ............ 163
B MAPS OF HARTFORD SMSA .................................... 165
BIBLIOGRAPHY .................................................. 169
BIOGRAPHICAL SKETCH .................................... ...... 180
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
A CONSUMER UTILITY MODEL FOR THE ALLOCATION OF
SALES AMONG MAJOR RETAIL CENTERS
Edgar James McDougall, Jr.
Chairman: Halbert C. Smith
Major Department: Real Estate
The purpose of this study is to examine the shopping process as it
relates to centers of retail agglomeration (shopping centers) by testing exist-
ing models of shopper behavior and developing an improved model of shopper
behavior. The data used are sales of the 19 Major Retail Centers (MRC's) in
the Hartford, Connecticut, Standard Metropolitan Statistical Area (SMSA).
The models tested are the central place model associated with Losch
and Christaller (the shopper is assumed to patronize the nearest center offer-
ing the desired goods); the gravity model associated with Reilly and Huff
(the shopper is assumed to be attracted by the size of the center, in square
feet of gross leasable area, and the square of the distance to the center acts as
an opposing force); and the zone of indifference model associated with
Devletoglou (the shopper is assumed to be indifferent to distances within a
certain limit). In all cases, "distance" is measured as the driving time from
the place of residence to the MRC. Sales for each MRC are obtained by using
the models to allocate total spendable income among the MRC's. The 1972
Census of Retail Trade is the data source for actual sales. The gravity model
produces estimates with an R2 of .80, significant at the .001 level; the central
place model produces estimates with an R2 of .42, significant at the .003
level; and the zone of indifference model does not produce significant results.
The gravity model produces the best estimates, but those estimates show
bias and unequal variance.
A theoretical framework is developed which relates the shopping pro-
cess to utility theory. One of the implications of the utility theory framework
is that convenience goods shoppers should behave differently from shopping
goods shoppers. This hypothesis is tested.
The central place model and gravity model are both used to estimate
convenience goods sales. The central place model produces an R2 of .53 and
the gravity model an R2 of .06. The gravity model is then used to predict
only shopping goods sales and an R2 of .94 is obtained. A better model of
shopping behavior, predicting convenience goods sales using a central place
model and shopping goods sales using a gravity model, is posed.
Two additional hypotheses are examined. The first is, Shoppers choose
among proximate MRC's because of favorable attributes of the MRC's. The
MRC's within five-minutes driving time of one another are grouped together
and sales are allocated among the MRC's based on the relative abundance of
favorable attributes possessed by the MRC's. Adjusting estimated sales for
favorable attributes improves the results.
The second hypothesis examined is, Shoppers will not pass a comparably
sized center to shop at a center with fewer favorable attributes. This is termed
the interceptor effect and adjusting predicted sales to account for the effect
The final model produces an R2 of .99 and is significant at the .001
level. The estimates do not show unequal variance and the model is less
biased than the original gravity model.
Shopping centers are a large and important part of the retail goods
market process. Approximately $140 billion in annual sales in the shop-
ping goods category (general merchandise and appeal) occur in concen-
trated groups of stores loosely termed "shopping centers" (McKeever and
Griffin, 1977, p. 17). The 1972 Census of Retail Trade has recognized
the importance of shopping centers. The Census reports sales and
other information concerning shopping centers and has termed them
"Major Retail Centers" (MRC's). 1
The retail structure of most areas is dynamic in nature, requiring
many decisions to be made regarding retail facilities. As population
shifts and consumer demands change, new shopping areas must be con-
structed to meet these new areas of demand. New development requires
decisions by many different parties: the developers of the new shopping
1"MRC's include not only the planned suburban shopping centers but
also the older 'string' streets and neighborhood developments which meet
the above prerequisites. Frequently the boundaries of a single MRC include
stores located within a planned center and, in addition, adjacent stores out-
side the planned portion. In general, the boundaries of the MRC's have
been established to include all adjacent blocks containing at least one store
in the general merchandise, apparel, or furniture-appliance groups of
stores, and, where a planned center is involved, to include the entire center.
Downtown business areas (DBA's) in 122 cities of less than 100,000
inhabitants (the minimum size for CBD city) have been delineated along tract
lines in the same manner as the CBD's, although in all respects, other than
the method of delineation, they are treated in the same manner as any other
MRC." (The 1972 Census of Retail Trade Washington, D.C.: U. S. Depart-
ment of Commerce, Bureau of Census, 1973, p. 122.)
center, those who finance the new center, the prospective tenants of the
new center and land use planners who are concerned with allocating space
in the city where the new development is to occur. Shopping centers cur-
rently constitute about 1.2 billion square feet of gross leasable area, and
in 1970 construction of new centers accounted for 7 percent of all non-
residential construction (McKeever and Griffin, 1977, p. 17). Owners of
and tenants in existing centers must also make decisions about how best
to meet the competition of new centers.
Statement of the Problem
The structure of retailing in metropolitan areas throughout the United
States is changing. The Central Business District (CBD) does not dominate
the retail structure as it once did. Since 1958, more retail sales have occur-
red annually in the suburban ring than in the CBD (Casparis, 1967, p. 213).
McKeever and Griffin see the changing pattern of retailing as a result
of changing population and the rise in importance of the automobile as
society's principal form of transportation. They state:
The rise of the automobile, the rise of the suburbs,
and the rise of the shopping center are parts of a single
phenomenon. When cities spread beyond established
transportation lines, automobiles came into use to meet
a great variety of transportation needs. Retailing also
moved into the suburbs, in pursuit of the shifting pur-
chasing power; the present-day shopping complex
necessarily began as an innovation in retail location.
(McKeever and Griffin, 1977, p. 12)
It would appear, therefore, that a variety of forces, including shifting popu-
lation and changes in mode of transportation, are creating a demand for the
retail services offered by the suburban MRC.
The suburban shopping center is of recent origin but has grown
quickly in importance. According to the Urban Land Institute, there were
only 100 shopping centers in 1950. "The present edition of Shopping
Center Directory lists approximately 18,500 centers in the United States"
(McKeever and Griffin, 1977, p. 17). These figures indicate a dramatic
growth in the number of shopping centers. When the increase in the num-
ber of shopping centers is considered along with the increased percentage
of total SMSA sales occurring in the suburban ring, it becomes readily
apparent that the retail pattern of American cities is undergoing profound
A changing retail pattern requires decisions to be made. If new resi-
dential development is occurring in the urban or suburban fringe, then
new demand for retail goods may arise. The developer of retail space will
need to know (among other things) which location to choose for a new retail
center and what size the center should be.
At any given location, if the center is too large, then sales will be
inadequate to provide a competitive return on the capital of the investor.
This could be either from a generally substandard level of sales by all ten-
ants (and hence reduced rents) or from some tenants going out of business
and excessive vacancy rates reducing rents.2 From the viewpoint of society
as a whole, an underutilized or overbuilt center represents an inefficient
allocation of capital and improper land usage.
2The return on invested capital is a function of not only revenue, but
also expenses. However, reasonable and competent management cannot
lower expenses enough to compensate for inadequate revenues.
It is true that profits to the owner of a center (return on capital) may
not be perfectly linked to sales, but rents do usually vary with sales.
Rental agreements in shopping centers are usually structured with a base
rental rate (usually by the square foot of gross leasable area actually leaded)
and a second stratum of rents based on a percentage of the gross sales of
the leasee beyond some specified amount.
A center which is too small for a particular location and market area
also represents an inefficient use of capital. The undersized center will
not maximize the site value, so the site is not at its highest and best use.
Because of crowding or excessively long queues, consumers may be forced
to shop at a less desirable center and therefore bear an unnecessary cost
of travel to the less desirable center. Those consumers who do utilize the
undersized center may be forced to bear an added time cost because of exces-
sive waiting due to crowding.
There are questions for decision makers regarding existing shopping
centers too. How will new centers affect the level of sales in the existing
centers? What possible attributes might an older, existing center lack,
which would make that center more competitive with new centers? Gobar
Today, however, physically or technologically
obsolete shopping centers are becoming a problem
across the nation. In many cases, shopping centers
are no longer the wave of the future in terms of
retail development. Instead, they are becoming
an extension or a specialized version of the central
business district (CBD) problem. (Gobar, 1972,
Information regarding physically or technically obsolete centers would be
valuable to the investor-owner, if a relationship between technical or
physical improvement and the volume of sales of the center could be estab-
Predicting sales volumes for shopping centers before they are con-
structed and predicting the changes in sales volumes because of changes
in factors affecting existing centers are important to more decision makers
than just the investor-owners. The tenant must make a legal and financial
commitment in the opening of a new store and the signing of a lease. The
lender who finances the center would like to know the sales potential of the
center, since sales are of crucial importance in determining whether the
owner can meet his obligations to the lender. Land use planners must be
aware of demand for retail goods so that adequate space within the metro-
politan area is allocated to the retail function. Accurate predictions of sales
volumes will help to use capital efficiently, maximize the value of the de-
veloped site, safeguard the invested capital, and reduce the frictions of
One might expect that if reliable predictions of sales volumes for
shopping centers were so useful and valuable to so many people, accurate
and precise methods of predicting sales volumes would be well established.
However, when one reads the practitioner-oriented literature, articles
written by those who actually perform market and feasibility studies, one
feels that divine guidance rather than logic is the primary method employed
in the sales prediction process (see Tucker, 1974 and Brinkrant, 1970).
Several models for the determination of trade areas or sales volumes
have been developed by academicians. This study will analyze three such
models in detail. The models chosen were the central place model of Lasch
and Christaller, the gravity model of Reily and the zone of indifference
model of Devletoglov. These three models were chosen because they rep-
resented a fairly wide range of assumptions about consumer behavior,
they are cited fairly often in the literature and three models were considered
few enough so each model could be analyzed in detail. Each model is
explained in Chapter 2.
The problem under study may be stated: the ability to predict the
sales for a major retail center is extremely important to the decision process
of those involved with shopping centers. At this time there does not appear
to be one logical and effective model to aid such decision makers.
Objectives of the Study
The first objective of the study is to test the central place model,
the gravity model and the zone of indifference model as predictors of MRC
sales. Each of the three models was used to allocate total sales between the
19 MRC's in the Hartford SMSA. The predicted sales were compared to the
actual sales, and the performance of the models was analyzed.
A second objective of the study is to test for correlations between
attributes of the 19 MRC's and the level of sales of the centers. Various
authors mention factors such as adequacy of parking and attractiveness of
center as affecting the decision of the shopper. Comparing the predicted
sales to the actual sales and including certain attributes of the centers may
show a correlation between some of those attributes of shopping centers
and higher (or lower) sales than predicted.
A third objective of the study is to determine whether a utility frame-
work adds insight into consumers' decision making processes. If shoppers
are generally assumed to be rational and therefore utility maximizers,
then it should be possible to analyze their decisions by using utility theory.
The utility framework could add insight to the question of how the various
attributes of the centers actually affect the shoppers' decision.
A fourth objective of the study is to synthesize the information gained
from testing the three models with the information gained from testing the
correlation between MRC attributes and sales levels in the 19 MRC's, to
produce a model which is a better predictor of sales and is consistent with
the theory of utility maximization. This improved model was used to predict
sales in the Hartford SMSA, and its predictions were compared to actual
sales. The quality of the predictions of the improved model were compared
to the three previously tested models.
A fifth objective is to offer some possible explanations of shopping
behavior which all of the models overlook or fail to predict.
Significance of the Study
This study is significant for two reasons. First, it tests existing
models of sales volume determination for major retail centers within an
urban area. Major retail centers account for a large portion of retail sales
in SMSA's throughout the United States. The MRC's used in this study
vary greatly in size ($130 million to $5 million in annual sales in Hartford),
so any model for sales volume determination found to be useful here may
apply to a wide range of shopping centers.
The second reason this study is significant is that an improved model
for predicting sales of major retail centers is developed. The model has
been improved in two senses. The new model is framed in terms of utility
theory. Framing the model in terms of utility theory aids in understanding
how factors which are ignored in other models will impact on shoppers'
decision processes. No previous authors have related the shopping deci-
sion process directly to utility theory. This study frames the question
in a utility maximization context.
The new model is also improved in the sense that the predictions ob-
tained are more accurate than the predictions of the three previous models.
The improved model explains to a greater degree the variation in sales
among the 19 MRC's in the Hartford SMSA.
Setting of the Study
This study was conducted in the Hartford, Connecticut, Standard Met-
ropolitan Statistical Area. The predictions of the models were compared to
the sales data for each of the 19 MRC's in the Hartford SMSA. The Hartford
SMSA had a population in 1970 of 664,000. Annual sales at the 19 MRC's
ranged from $130 million to about $5 million. No contention is made that
Hartford is typical of other areas throughout the country. However, Hartford
is a fairly moderate sized SMSA with a reasonable number of MRC's. The
ideas tested here should be applicable to other areas but it will remain the
task of others to test how applicable the findings presented here are in
Organization of the Study
Chapter 2 is a review of the literature concerning the central place
model, the gravity model and the zone of indifference model. Both the theo-
retical and empirical work of various researchers is examined as it pertains
to each of the models. The shortcomings of the previous research, as
well as the contributions of the previous research to the problems of sales
volume determination in a metropolitan area, are discussed.
In Chapter 3 the central place model, gravity model and zone of indif-
ference model are expressed in mathematical form which allows them to be
used to predict sales for each of the 19 MRC's in the Hartford SMSA. Each
set of 19 predictions is compared to the actual sales of the 19 MRC's, and
the relative predictive ability of each of the models is evaluated.
In Chapter 4, factors not included in the three models are tested for
significance in explaining the level of sales for the 19 MRC's in the Hartford
SMSA. These new factors are also used to explain the residual or error term
produced by the predictions of the central place, gravity and zone of indif-
ference models. The method of analyzing those additional factors is step-
wise linear regression.
In Chapter 5 a theoretical basis for analyzing decisions which affect
both the monetary budget and time budget is developed. The analysis is
set in a framework which assumes the decision makers attempt to maximize
their utility. Shopping will be viewed as a decision making process requir-
ing utility maximization. The shopper may explore alternative purchases,
seeking to find a desirable good for the lowest possible price. Seeking out
all of the possible alternatives requires time. The shopper must decide
whether possible savings from obtaining a lower price or a product which
more exactly fits his needs, warrant a further expenditure of time. The
framework phrases the decision process as a tradeoff between the utility
of time and the utility of goods.
In Chapter 6 the results of testing the central place, gravity, and
zone of indifference model in Chapter 3 are combined with the results
from the testing of additional factors introduced in Chapter 4 and analyzed
in terms of the utility maximizing framework developed in Chapter 5.
A model to predict sales at MRC's is developed and tested by comparing
the predictions of the improved model to the actual MRC sales.
Chapter 7 summarizes the findings and contributions of the paper.
Additional research is suggested.
REVIEW OF THE LITERATURE
This chapter reviews the work which provides a theoretical and his-
torical basis for the three models tested. The three theories which provide
a basis for the respective models are the central place theory associated with
Lbsch and Christaller, Reilly's "law" of retail gravitation and Devletoglou's
hypothesis of a zone of indifference. The chapter discusses the various
assumptions of consumer behavior which underlie each of the theories or
propositions and discusses the work of other subsequent researchers who em-
pirically tested aspects of these propositions. The work of researchers who
examined specific characteristics of consumers, of travel and of the market
place are discussed and related to overall consumer behavior.
Central place theory states that a hierarchy of urban places (cities and
towns) exists. The highest order urban places have the most functions, the
most specialized functions and have the most extensive market area. The
lowest order urban places have a very limited market area of rural residents
and offer only the most basic goods and services. Consumers are assumed to
patronize the urban place which offers the good or service the consumers are
planning to purchase and is the shortest linear distance from their place of
Reilly's law of retail gravitation states that consumers are attracted by
the size of a center (the population of an urban place), but the distance
squared from the place of residence to the urban place acts as a frictional
force. Reilly's basic law was later modified and applied to metropolitan
areas first by Ellwood and later Huff.
The zone of indifference proposed by Devletoglou is a fairly new and
untested hypothesis. The zone of indifference hypothesis states that within
an urban area, many shoppers do not perceive the difference in travel time
between a journey from the residence to any of several shopping centers.
Shoppers may place a time limit on their trip from the residence to a shop-
ping center, but within that limit, one trip is seen as being about equivalent
Researchers from a wide range of disciplines have contributed to the
understanding of market areas by examining shopping center attributes and
consumer behavior, and the work of these researchers is related to the three
basic propositions already identified. To produce a more logically coherent
discussion of the literature, the propositions are treated separately and the
most sophisticated modifications and approaches are presented last.
Central Place Theory
August Losch (1954), 1 an economist, and Walter Christaller (1966), 2
a geographer, independently developed the basis of central place theory.
Losch approached the problem from the "micro" perspective of the individual
and showed how the forces of individual wealth maximization and the forces
of competition among firms lead to an orderly pattern of places of retail goods
distribution. Christaller chose the "macro" perspective of an entire nation
and showed how providing the goods required by the people of the nation
would lead to an orderly array of places offering retail goods. Although
Losch and Christaller each approached the problem by using the tools of his
1August Lbsch's original work was published in German in 1939; the
English translation used in this study was published in 1954.
2Walter Christaller's original work was published in German in 1933;
the English translation used in this study was published in 1966.
respective discipline, their results were very similar and both fall under the
general term of central place theory.
Losch and Christaller both predicted that an array of centers would
develop. Some centers would offer a wide variety of goods and have exten-
sive market areas. Some centers would have very limited market areas (only
the surrounding countryside) and offer only the most basic goods. Each
researcher also predicted that consumers would patronize the nearest center
(measured by straight line distance) offering a desired good. The predic-
tions of Losch and Christaller are built on four basic assumptions:
1. The goods are homogeneous. Shoppers cannot view different
articles of the same general type as different. This condition would
be violated, if, for example, the shopper is seeking a shirt but is not
precisely certain of what color, style, material, fit, pattern or price
he desires. The shopper may need to compare various shirts at
various market places and hence may need to cross market bound-
aries to make the comparison.
2. The population is scattered. A scattered population will insure
that if some individuals are uncertain which market area they are
within, then the affect of boundary crossing on market place sales
will be minimized. If population were concentrated near a market
boundary, then a misperception by the concentrated population
would have a much more dramatic effect on predicted market place
sales. It is possible that L'sch was anticipating what Devletoglou
later termed the zone of indifference. That is, if markets are
located within a relatively densely populated area, distances be-
tween markets might be so short that shoppers do not view that
distance as a deterrent to crossing market boundaries.
3. The population is numerous. If the population is not numerous,
there may not be enough individuals to sharply define a market area.
4. The center of each market area is treated as though the other
centers do not exist. If the consumers consider the centers of other
market areas as viable alternative places to shop, then the consumers
will cross the theoretical market boundaries and the actual market
boundary will then become indistinct, or different from the predicted
L'sch and Christaller developed their theories to explain the actions
of a disbursed, rural population (the people in an agriculturally-oriented
Germany). The population was scattered. The basic mode of transportation
was the horse and wagon, so the limitations of distance were acutely felt.
Probably most of the populace had strong ties with one relatively proximate
town. Strong ties with one town and the inconvenience of travel to other
towns could strongly encourage the shoppers to consider the center of one
market area and act as though the centers of the other market areas did not
exist. Probably the conditions L6sch and Christaller required for sharp
boundaries were fairly closely met by the area they were analyzing.
The conditions which L'osch and Christaller assumed and which are
necessary for sharp market boundaries, may not exist in modern urban
areas, such as the one in which this study was conducted. The population
of a moderately sized urban area would probably be judged numerous but
would certainly not be considered "scattered." The type of goods sold in
MRC's tend to be shopping goods (apparel and general merchandise). Shop-
ping goods are not homogeneous and vary greatly in style, color, size,
price and many other factors.
The automobile is the primary form of transportation in most American
cities, and the operator of an automobile may not view distance the same way
the driver of a horse and wagon would. The shopper in a modern city may
not perceive a disutility from driving to several shopping areas or bypass-
ing a center for one farther away. The required conditions of a scattered
population, homogeneous goods and the shopper considering only one mar-
ket center may not be as applicable to the consumer in a modern urban area
as those conditions were to the residents of rural Germany.
In analyzing the appropriateness of the approach of Losch compared
to the approach of Christaller, Brian Berry, a contemporary geographer
who has empirically tested various aspects of the central place hypothesis,
Christaller's formulation appears most relevant for
understanding the geography of retail and service
business, whereas that of Losch provides a framework
for analyzing the spatial distribution of market-oriented
manufacturing. Christaller's agglomerative require-
ment, for example, is compatible with the idea of travel-
cost minimization by rational consumers on multi-purpose
trips, a condition not satisfied by Losch, and the process
whereby Christaller locates smaller centers relative to
higher-order centers is not unlike the development
process observed in Iowa. (Berry, 1967, p. 73)
Berry views Christaller's formulation as the most useful for shopping
center analysis. He refers to Christaller's agglomerativee requirement," the
notion that a higher order center must offer all lower order goods, as being
compatible with "travel-cost minimization." Travel cost minimization means
the consumer will shop for an array of goods at one particular shopping point.
The shopper will combine trips to several low order centers into a single trip
to a high order center.
However, it does not appear that Christaller's system allows for the
travel cost minimization to which Berry alludes. If the consumer actually
practices travel cost minimization (and not impulse buying, which would
merely represent added consumption) then the consumer is shifting a portion
of his low order goods purchasing from the low order center's market area,
within which he resides, to a high order center. By shifting purchasing,
the consumer is reducing the total sales for the low order center nearest his
residence. The consumer is increasing total sales for the firms which sell
low order goods but are located at a higher order center. Christaller stated
that each low order urban place would have a market area equal to all other
urban places of the same order (assuming evenly distributed population).
Christaller also predicted that competition would reduce economic profits to
zero for all urban places. However, if residents of one urban place shift a
portion of their purchasing to another urban place, then the urban place
within whose market area they reside will not have a sales level sufficient
to meet costs while the market center benefitting from the shift will experi-
ence economic profits greater than 0. If the economic profits of any urban
place vary from 0, then Christaller's equilibrium conditions do not hold and
his geographical market array is not stable. So although Berry stated that
Christaller's market system was compatible with travel cost minimization
through combined trips, it does not appear to this author that such travel
cost minimization is compatible with Christaller's equilibrium conditions.
The setting for Lisch and Christaller's research was the rural country-
side of southern Germany. The question which seems relevant is, How well
do the assumptions regarding consumer behavior which fit the population of
a rural countryside fit the modern, urban shopper? Several authors, in-
cluding Berry, have tested aspects of central place theory. The remainder
of this section will review the empirical work of researchers who tested the
assumptions of central place theory and analyze how their research relates
to the problem of determining shopping behavior in contemporary urban
The Assumption of Zero Economic Profits
Losch and Christaller both hypothesized that the economic profits of
competing centers would be reduced to zero. Competition would cause new
centers to develop whenever the market demand of an area exceeded the
threshold level of sales necessary for a center to operate.
Berry and Garrison (1958) used the population of threshold centers
(those centers with only one supplier of a good) to estimate how many sup-
pliers of a good should be located in higher order centers. If the number of
suppliers was equal to or greater than the predicted number for a central
place, one could conclude that no economic profits existed and a high level
of "urban efficiency" had been obtained. The conclusion was that certain
sectors of the retail market system contain functions more amenable to
economies of scale than others. Increased size of establishment was more
likely "a characteristic of department stores than it was of apparel stores;
apparel stores were more likely to increase in size than were taverns"
(Berry and Garrison, 1958, p. 310). Even more importantly, Berry and
Garrison found that every market area was not as small as possible and
therefore excess profits did exist. If this finding is correct, then markets
might not reach the state of competitive equilibrium which L6sch and
Christaller hypothesized markets would attain.
Although Berry and Garrison found that excess profits did exist, they
did make two qualifying statements which seem especially applicable to the
problem of forecasting future sales volumes of MRC's in moderate and large
urban areas. First they found that as urbanization increases, the excess
Berry and Garrison contended, for example, that if there are 10
department stores in an area, the total demand in the area for department
store goods would need to increase by 10 percent before the demand would
support the existence of a new department store. If the area currently has
only one department store, then consumer demand must increase by 100
percent to support the existence of another department store. In the first
case, economic profits will tend to be closer to zero than in the second case
because large urban areas require a smaller percentage increase in demand
to justify new shopping establishments than do small urban areas (10 percent
or 100 percent in this example).
Berry and Garrison also found that urban growth changed urban
structure.4 Urban growth would encourage entrepreneurs to consider
entering the market and perhaps a more dynamic urban form would increase
opportunities for decision making on all levels. A growing urban area, for
example, would provide demand, so new stores could enter the market. New
stores would change urban structure (as opposed to old stores merely increas-
ing in size). In an area of static population, many of the retailers' costs are
3Berry and Garrison noted that the larger the urban center, the
less the excess profits per store for any given function. This would imply
that the larger the center, the more likely are Losch's equilibrium conditions
to be met or, put another way, as urbanization increases, the more likely are
equilibrium conditions to be met in the settlement system. But even this
notion is open to question, since it assumes that the nature of business ac-
tivities is more static than the size structure of urban centers--not a very
satisfactory notion" (Berry and Garrison, 1958, p. 316).
4"Clearly, there are close relations between urban growth and changes
in the structure of urban activities. It is only in lieu of more powerful notions
that the ideas expressed here are forwarded" (Berry and Garrison, 1958,
sunk costs, so the urban retail structure would probably tend to remain
more static. In other words, in a static area, demand would need to shift
radically to drive stores out of business or cause them to move. In a grow-
ing area, new demand encourages new entrants so the structure changes
whether or not the existing stores alter their size or location.
While the condition of no economic profits will probably not be met
exactly in any area, a large metropolitan area (and especially a growing one)
has conditions which are closest to those required for no economic profits.
The no economic profit requirement will not be considered to be significantly
at variance from reality so the market predictions of L~sch and Christaller
should not be rejected because of an argument that economic profits exist.
The more interesting and questionable assumptions of Losch and Christaller
are those involving consumer behavior. These assumptions are examined
The Assumption that Shoppers Patronize the Nearest Center
Both Losch and Christaller assumed that a consumer would travel only
to the nearest center offering a desired good. They judged "nearest" by
measuring the straight line distance from the place of residence to the urban
place. Several factors, such as lack of knowledge of the goods available at
a center, lack of a well defined need or the desire to compare prices, might
cause consumers to purchase a good at a center other than the nearest one.
Several researchers have attempted to test the nearest center hypothesis.
A detailed study of market areas and central places in southwestern
Iowa (Berry, Barnum and Tenant, 1962) showed a fairly clear pattern of
central places. A survey of the population showed that the choice of center
was a function of type of good (Food was purchased most often at the nearest
center), ease of travel (Paved roads were preferred to dirt roads even if the
paved road implied a somewhat greater distance) and the variety of goods
offered at a center. The variety of goods carried by a center appeared to
affect choice because some consumers would pass a center to shop at a more
distant one with a greater variety.
The Berry, Barnum and Tenant study has four important aspects.
First, the area chosen, southwestern Iowa, was a rural one which met the
condition of an evenly scattered population. Most of the population was en-
gaged in agriculture, so they may have had fairly similar demand curves.
The study was conducted to determine inter-urban trade areas, the trade
areas of cities and towns (not trade areas within cities or towns). The set-
ting appears to be fairly similar to the settings of the studies of L6sch and
Christaller, so one would expect fairly similar results.
Second, food was the good purchased most often at the nearest center.
Food shopping has several interesting characteristics. Since food is perish-
able, food shopping must be done more frequently than the shopping for dry
goods or other merchandise. Food is fairly homogeneous in the sense that
one carrot is pretty much like another. Shoppers may pick the "best" item
out of a group of those items, such as vegetables in a bin, but is is very dif-
ficult for one food retailer consistently to sell better quality items (nicer
carrots) than another food retailer. Most food retailers carry approximately
the same array of goods so the shoppers do not need to go to several stores to
find the item they need (such as just the right carrot). So shopping for food,
which is commonly referred to as a low order good, fits well with the assump-
tions of Losch and Christaller. Food is fairly homogeneous. Food shoppers
probably do not comparison shop; they consider the nearest center as the
only alternative and they are sure (from repeated shopping) which products
are available at the nearest center. One would expect, therefore, that food
shopping behavior would be predicted fairly well by the central place
hypothesis of shopping at the nearest center.
The third point of the Berry, Barnum and Tenant study to consider
is which transport factors affect the consumer's choice of center. It would
appear that the length of the journey in the strict sense of distance is not as
important as the length of the journey in the time sense. Also the pleasant-
ness of the trip appears to affect the choice of travel paths. The Iowa popu-
lation under study preferred paved roads to unpaved roads. The paved
roads would provide a more pleasant trip (less bumpy) and perhaps a faster
trip. Thus, ease of travel and time of travel appear to be part of the shop-
per's calculus as well as length of journey in terms of distance.
The fourth point is that a center offering a greater variety of goods
could draw consumers past a center offering a lesser variety of goods. Shop-
pers may view variety as important because they need to compare goods
before they can ascertain exactly what attributes they are looking for in a
good, or variety may be important because shoppers are not sure a retail
center will have the goods they desire. In central place theory, the number
of functions offered at a center is a result of the demand by the populace for
a wide range of goods (the higher order central place). Central place theory
does not recognize the phenomenon which appears to be occurring in the
Berry, Barnum and Tenant study, that a retail center offering a wide variety
of goods may induce the shopper to leave the market area predicted for that
shopper by central place theory and shop at a more distant center. Ap-
parently a center with a wide variety of goods may occur not just to meet a
demand by consumers, but may arise by design as a means of raising sales
(a strategy perhaps) and enable the center to create or focus demand.
A study by Golledge, Rushton and Clark (1966) used data supplied
by the University of Iowa from a sample of 486 farm and 115 non-farm state
residents. The researchers measured the distance to the town of maximum
purchase5 and the distance to the town of nearest purchase. 6 By comparing
the mean distances of travel for the purchase of different types of goods, the
researchers determined that on the average people traveled only a short
distance for either maximum or nearest purchase and that there is substantial
variability in travel distance for different goods. Comparatively long trips
were generated by goods such as "girls clothing" (termed "spatially flexible"),
while "church" and "repairs to T.V. and appliances" generated relatively
short trips (termed "spatially inflexible").
A good such as "girl's clothing" has many attributes, such as style,
material, color, pattern and price. Even a particular type of clothing, such
as "dresses," is very heterogeneous in nature. Since both L6sch and
Christaller assumed homogeneous goods and that consumers knew exactly
which center offered a desired good, it is not surprising that central place
theory does a comparatively poor job of explaining the behavior of those
shopping for a heterogeneous good like girl's clothing. The shopper may not
know in advance exactly where the desired good can be purchased. The
shopper may not even know in advance exactly what he desires.
5"Maximum purchase" is defined on the basis of total dollars spent on
a good at alternative places in the study year. Where two towns are allo-
cated the same total expenditure, the nearest town is selected as the maxi-
6"Town of nearest purchase" is defined as the nearest town in which
a purchase was made. This may or may not be the spatially closest urban
Golledge, Rushton and Clark suggested--but did not test--two
hypotheses to explain the variability in travel:
1. The places of origin of some goods are less widely distri-
buted and opportunities for purchase occur at markedly dif-
ferent distances from the individuals in the sample.
2. The variability of travel is a result of the competitive power
of the good. (Golledge, Rushton and Clark, 1966, p. 264)
Without testing the hypotheses, the authors offered some comments on
the nature of "spatially flexible" goods and consumer travel behavior.
Thus goods such as 'girls clothing' [spatially flex-
ible goods] generate comparatively long trips, and,
as indicated by the standard deviation, encourage a
great range of trips over a variety of distances. Such
goods are extremely susceptible to competition from
alternative retail outlets, and some spatial variation
in prices could cause considerable variability of travel
distances. Under these circumstances consumers will
shop around to gain the maximum utility from their ex-
penditures. (Golledge, Rushton and Clark, 1966, p.
The Golledge, Rushton and Clark study appears to support the notion that
the purchase of certain goods encourages trips to centers other than the
nearest center offering the good and that shoppers may compare between
centers offering comparable goods.
A later study by Clark (1968) posed the nearest center question even
more explicitly. Clark hypothesized that consumers purchase goods at the
closest center offering the goods and that the market area of a particular
good is the same regardless of the order of the center in which that particular
good is offered. Clark's hypotheses are the same as Christaller's, so an
affirmation of Clark's hypotheses would also support Christaller's. Clark
found, however, that 34 to 48 percent of the sample did not patronize the
nearest store. Therefore, the nearest center hypothesis was not accepted.
When Clark assigned the centers to categories according to the num-
ber of functions of the center (class of center), the percent of the sample
using the nearest store was higher for lower order centers (centers with
few functions) than for higher order centers. However, all of the goods
tested were low order or convenience goods (grocery, vegetable, meat).
Clark found that 90 percent of the shoppers sampled used the central busi-
ness district exclusively when shopping for clothing and other hetero-
geneous shopping goods.
Clark appears to be affirming the work of Berry, Barnum and Tenant
and of Golledge, Ruston and Clark. Clothing and other shopping goods do
not fit the central place assumption of homogeneity and the shoppers search-
ing for these goods appear willing to travel farther in order to obtain the
advantages of a wide selection. Shopping for grocery type goods, which
must be purchased fairly regularly, appears to conform more closely to cen-
tral place theory in that shoppers tend to patronize the store nearest their
place of residence.
Clark's work adds insight to understanding consumer behavior. If he
had chosen an urban area with a greater number of higher order centers
capable of competing with the central business district, his study would
have been more useful. Clothing, furniture, appliances and other compari-
son type goods constitute the bulk of sales at MRC's. One might expect that
consumer behavior when shopping for these comparison goods is even more
complex than the behavior of the consumers in Clark's study, who were
shopping for convenience goods.
Another study, again primarily of low order goods, by Bishop and
Brown (1969) showed that many consumers travel farther than the nearest
store (60 percent), and that "Both store size and the centrality [proximity
to other retail facilities] of the store location had a positive and statistically
significant association with the distance traveled for grocery purchases"
(Bishop and Brown, 1969, p. 30).
From empirical studies, it appears that there are several factors
(type of good sought, ease of travel and the variety of goods available at
the point of purchase) which influence whether the shopper patronizes the
nearest center. Apparently the shopper is more apt to use the nearest center
to purchase low order, convenience type goods, than he is to purchase shop-
ping goods such as apparel or general merchandise.
The Single Trip--Single Order Assumption
Christaller assumed that a consumer purchased only goods of one order
on any particular trip. The market area for a good of any order, therefore,
is not influenced by a higher order center. Even though Christaller stated
that each higher order center provided an array of goods down to the lowest
order goods, he did not suggest an alteration of the market area of other
suppliers of a lower order good to account for residents of the other market
areas combining the purchase of a higher order good and a lower order
good in a single trip. An inhabitant of Christaller's plain would stop at the
nearest center providing a good and then travel to a higher order central
place to purchase a good of higher order even though a supplier of the
lower order good would be located at the higher order center. Clearly, the
distance traveled to the lower order center and then to the higher order
center will be at least as great as the distance from the residence to the
higher order center if the high order center, low order center and residence
are in a straight line and will be greater than the distance from the residence
to the high order center if the high order center, low order center and place
of residence are not in a straight line.
The study by Bishop and Brown (1969), previously cited, indicated that
the proximity of a store to other stores was an important attribute of the store
which affected the shopper's decision of where to shop. Several authors have
suggested that shoppers consolidate purchases of goods of different orders to
minimize travel distance or the number of separate destinations (Berry and
Garrison, 1958; Berry, 1967; Clark, 1968 and Bishop and Brown, 1969).
An interesting model to consolidate trips and minimize transport costs
was developed by Bacon (1971). He assumes shoppers value time, can fore-
cast their needs and visit only one center per trip. Joint economies may be
derived by increasing trips to more distant centers and decreasing trips to
nearer centers. Total time must be compared to ascertain whether the joint
purchase is economical.
Bacon's point is that if shoppers can forecast their shopping beyond
the next trip, it may be possible for the shoppers to combine trips by pass-
ing a low order center in favor of a higher order center where desired low
order goods can also be purchased. Bacon does not imply that shoppers are
comparing goods or choosing a more distant center of the same order. A
desire to minimize shopping time and the ability to forecast needs are all that
are necessary for the shopper to combine purchases at one stop. If shoppers
do combine purchases they will produce a market area other than the market
area central place theory predicts.
MacKay (1972) developed and tested a model of consumer behavior
based on a multi-stop assumption. He found a wide variation in the mean
number of stops made by shoppers when he grouped shopping trips by
store type (per 100 shopping trips). He also found that "On the average,
2.7 stops were made on shopping trips that included supermarkets" (MacKay,
1972, p. 137). MacKay's work seems to indicate that a multi-stop model of
consumer behavior may be closer to the consumer's actual behavior than a
model based on the one-stop assumption of central place theory.
The Homogeneous Demand Curve Assumption
To obtain the uniform market area which both L6sch and Christaller
predicted, one must have a uniformly distributed populace with homogeneous
demand curves. Utility theory suggests that an individual's demand curve
should vary directly with income. If income rises, the consumer will gen-
erally demand more goods. The threshold of a good, the minimum level of
sales necessary for the good to be offered, is dependent on sales, not
geographical area, so a market area with a relatively dense, high income
population should be smaller in area than the market area with a relatively
dispersed, low income population.
Getis (1963) compared the actual location of supermarkets in Tacoma,
Washington, to a theoretical distribution of centers based on central place
theory. At first comparison there appeared to be no correlation between the
theoretical and the actual locations. Then Getis applied a transformation
designed to adjust for the actual population distribution and the income of the
residents in the Tacoma area. After applying the transformation and allow-
ing for zoning, the locations predicted by central place theory were within
two city blocks (250 yards) of the actual supermarket locations. One can
conclude that adjusting for the demographic characteristics of an area and
the income characteristics of the inhabitants may greatly increase the rele-
vance of central place theory to actual spatial structure. One should also
note that Getis used supermarkets in his study. As was observed previously,
central place theory appears to be more applicable to low order goods,
such as grocerties, than to high order goods.
This section has reviewed empirical tests of several assumptions
associated with central place theory. The assumptions examined were
enterpreneurs cannot obtain economic profits, consumers patronize the
nearest center, consumers do not combine low and high order goods pur-
chases outside their market area and consumers have homogeneous demand
curves. The assumption of no economic profits seems to fit fairly well with
large sized (and even better with growing) metropolitan areas. Consumers
appear to comply more often with the nearest center assumption when shop-
ping for low order goods than when shopping for high order goods. Ap-
parently factors such as pleasantness of trip (or ease of travel) and the
variety of goods offered by a center may affect the choice of center. Several
studies indicated that the assumption of not combining low and high order
goods purchases in a single trip is not borne out by the evidence. The
homogeneous demand curve assumption does not fit reality, but apparently
can be reasonably easily adjusted to fit reality. It would appear that some
aspects of central place theory hold fairly well while others hold for the
purchase of only certain goods and other aspects do not appear applicable
While central place theory addresses the question of where stores will
locate, it does not explore the effect that store size may have on market area.
The gravity model is an attempt to include the effect of the store size on the
attractiveness or pull of the retail center.
Gravity models, as used in sociology and retail market analysis,
derive their name from their resemblance to Newton's Law of Gravity in
physics. Basically, gravity models state that the attractive force exerted
by any mass on a body in the surrounding area is directly proportional to
the size of the mass and inversely proportional to the distance squared of
the body from the mass. The first to apply this to retail market areas was
William Reilly, a geographer.
Reilly's "Law" of Retail Gravitation
Reilly (1929) sought to explain how rural Texas residents chose the
city in which they shopped. The data Reilly obtained led him to state his
"law" of retail gravitation as:
under normal conditions two cities draw retail
trade from a smaller intermediate city or town in direct
proportion to some power of the population of these two
larger cities and in an inverse proportion to some power
of the distance of each of the cities from the smaller
intermediate city. In any particular case, the expon-
ents used in connection with population or distance
are dependent upon the particular combination of retail
circumstances involved in that case. Typically, how-
ever, two cities draw trade from a smaller intermediate
city or town approximately in direct proportion to the
first power of the population of these two larger cities
and in an inverse proportion to the square of the dis-
tance of each of the larger cities from the smaller inter-
mediate city. (Reilly, 1929, p. 16)
Mathematically, Reilly's "law" may be stated:
B P Dbn
a aN bn
b b a
where B = The business which City A draws from intermediate Town T
Bb = The business which City B draws from intermediate Town T
P = Population of City A
P = Population of City B
D a = Distance of City A from intermediate Town T
Db = Distance of City B from intermediate Town T
n, N = Exponents of unspecified values
(Reilly, 1929, p. 48)
Reilly did not rely on a statistical technique to aid in the evaluation
of the data he obtained, nor did Reilly develop theoretical support for his
"law." In spite of this, Reilly felt his data spoke for itself and stated:
It is so readily acceptable that the amount of outside
trade which a city enjoys in any surrounding town is
a direct function of the population of that city and an
inverse function of the distance of the city from that
town, that the general law needs no support. (Reilly,
1929, p. 48)
Although Reilly may be correct in stating that outside trade is a direct func-
tion of the size of a city and an inverse function of the distance to the city,
one should note that Reilly does not specify the function. He states a formula
in general form, but by leaving the exponents of the formula (N and n) un-
specified, the formula can produce an unlimited range of values. There are
also many factors which could affect drawing power, and perhaps modify a
basic equation. Reilly, without empirically testing these other factors, dis-
misses them as unimportant. The very simplicity of Reilly's "law" may be
why other researchers (See Jung, 1959) have not accepted Reilly's "law"
quite as readily as Reilly did.
Even Reilly seemed to realize that factors other than population and
distance may be operative, as he compiled the following list of factors that
may influence the retail-trade territory of any given city:
1. Lines of Transportation
2. Lines of Communication
3. The Class of Consumer in the Territory Surrounding the
4. Density of Population in the Territory Surrounding the
5. Proximity of the Market to a Larger City Market
6. The Business Attractions of the City
7. The Social and Amusement Attractions of the City
8. The Nature of the Competition Offered by Smaller Cities
and Towns in the Surrounding Territory
9. The Population of the City
10. The Distance Which Prospective Customers Must Travel in Order
to Reach the Market, and the Psychology of Distance Prevailing
in that Part of the Country
11. The Topographical and Climatic Conditions Peculiar to the
City and Its Surrounding Territory
12. The Kind of Leadership Offered by the Owners or Managers of
Various Business Interests of the City
(Reilly, 1929, p. 21).
Some of these factors, such as the psychology of distance, are quite sophis-
ticated notions of consumer behavior and Reilly should be commended for
being cognizant of them, but he did not include these factors in his model.
In fact, Reilly felt that compared to size of center and distance to center,
these other factors held no explanatory significance.
One might assert that Reilly's model is useful because it reduces a very
complex situation to one in which the dominant factors may be analyzed.
Reilly's model certainly provides a simple method of determining retail mar-
ket areas. Even if one accepts the "law" on a very general basis--that size
and distance are both important and the stated directions of influence are
correct--one still must ask how accurate is the "law" when less aggregate
data are used? Are the factors which influence and alter the market area of
a particular store or city such that when aggregated with many other stores
and cities they tend to cancel out or are lost in averaging? One criticism of
Reilly's "law" is its apparent preciseness. An example of the apparent
preciseness of the "law" may be the exponent which Reilly used with the dis-
tance factor. In Reilly's model,
B P D
a aN bn
bb ab a
both the population exponent N and the distance exponent n, are left un-
specified. One would expect that the size of the exponents (N and n) would
have a profound influence on the size of the market area of a particular center.
Reilly chose 1 as the exponent for the population or drawing power and 2 as
the exponent for the distance or friction factor. To obtain the distance
exponent, he tested the model with exponents which ranged from 0 to 12.5
and found that the model predicted sales well for the greatest number of
cases when the exponent was between 1.5 and 2.5.7 Squaring the distance
has become the accepted practice (see Ellwood, 1954), but there is evidence
that many factors may influence which power provides the best fit to the data
and there is no single "correct" exponent (see Pfanner, 1940 and Thompson,
Reilly's model is a significant departure from central place theory.
The notion of the importance of the size of a retail area as an attractive force
is certainly an intriguing one. Reilly's model appeared to do a fairly good
job of predicting the shopping patterns of rural Texans. An important aspect
of Reilly's "law," which is sometimes ignored, is that Reilly conducted his
research in a rural area and was seeking to determine shopping behavior for
the residents located between cities and not within cities. Thompson (1966)
points out the importance of and the possible misuse of Reilly's work:
7 No. of Cases Value of n
45 0 1.5
87 1.51- 2.5
35 2.51- 3.5
24 3.51- 4.5
15 4.51- 5.5
14 5.51- 6.5
6 6.51- 7.5
5 7.51- 8.5
12 8.51- 9.5
"In other words, a clear mode occurs in the range 1.51-2.5 which
shows that the exponent of the distance is nearer to the second power than
to any other even power" (Reilly, 1929, p. 49).
Despite widespread concern over the conceptual
and empirical basis of the Reilly approach, marketing's
first, most-publicized, and possibly, last, "law" has
been applied to a variety of situations. One author
[Nelson] quite recently suggested that the law be
used for defining the trading areas of neighborhood
shopping centers. This ignores the fact that the
original law was never intended to quantify the move-
ment of persons within such narrow confines or for
such a broad assortment of items. Both Reilly and
Converse argue that the concept is only applicable
to cities of quite some size and a good many miles
apart. (Thompson, 1966, p. 5)
Regardless of comments such as Thompson's or Reilly's own caveat
concerning the use of his "law," the gravity law has been frequently applied
to intra-urban shopping behavior.
The Gravity Model Within an Urban Area
An early explanation of how the gravity model could be used to deter-
mine market areas within a single urban area was provided by L. W. Ellwood
(1954), who described how to forecast sales for urban and suburban shopping
centers. Ellwood saw Reilly's work as being directly applicable to shopping
within urban areas. Both to justify the application of Reilly's work and point
out what he considered to be the relevant factors in the analysis of intra-
urban shopping, Ellwood addressed several factors affecting shopping center
sales and consumer behavior.
Ellwood stated what he saw to be pertinent facts regarding market
areas and sales potential.
First, new stores do not create a penny's worth of new
retail business. All they can do is spread the existing
volume thinner by dividing it among a larger number of
places to trade. The population expected to patronize
each new store now buys its requirements from existing
establishments. The new store will succeed only if it
offers enough added convenience and special appeal to
pull enough customers away from their present sources
of supply to justify the cost of the new store. An estimate
of potential volume for a proposed new market place must
start with information as to the existing volume of business.
(Ellwood, 1954, p. 581)
The idea that new stores do not create new business is an interesting
point in Ellwood's statement which frequently appears in the practitioner-
oriented literature. Economists would suggest that a new store opening
anywhere within the urban area must enable at least some individuals to
lower their transport costs. These individuals would then have more spend-
able income which would shift their demand curves and actually create more
demand for all goods, some of which could be sold by the local retail stores.
Therefore, a new store could create new expenditure, but probably not a
significantly large amount. A behavioral psychologist might suggest that
a new store would lower the cost of travel in both time and money, so that at
least some consumers might visit local retail facilities more frequently. More
frequent visits, even if they are social in nature, would present the shopper
with more advertising, displays of merchandise and other factors which
could lead to more impulse buying or a greater awareness of needs. Theoret-
ically, this process of greater exposure of the shopper to the goods could
create more demand, but practically it is probably not significant.
Ellwood continued his discussion by analyzing the trade area bound-
aries within an urban area.
Secondly, there are no definite boundaries for the trade
area of a retail district. The pull of every successful re-
tail district overlaps that of every other retail district
within its general market area. The general market area
is virtually world-wide with respect to some classes of
merchandise. Some retail business gravitates to the
salons of Paris from the ranches of Texas. An estimate
of the potential volume of a proposed new market place
must allow for the composite pull of all other competing
retail districts. (Ellwood, 1954, p. 581)
Ellwood's contention that there are no definite boundaries for trade areas was
supported by L6sch. He stated that in an urban area if buyers and sellers
are not scattered, if buyers are aware of alternative places to make purchases
and if goods are not homogeneous "markets overlap and boundaries become
indistinct" (L6sch, 1954, p. 13). Probably, within most urban areas, the
conditions Losch suggests do exist, so market boundaries are indistinct. In-
distinct or fuzzy trade area boundaries are a radical departure from the well-
defined regular hexagons hypothesized in central place theory.
Ellwood's third and last statement referred to a logical consequence of
the gravity model--that at some point between two centers the attractive forces
of the centers will be equal. The point of indifference, or equal attraction in
two directions, is termed the breaking point. Ellwood stated that:
despite the lack of definite boundaries and
despite the fact that people pass each other going in
opposite directions on streets and highways to buy the
same kinds of merchandise, there is a breaking point
between each two competing retail districts where the
natural pull of one district is equal to that of the other."
(Ellwood, 1954, p. 581)
Traffic studies of inter-urban travel indicate that Ellwood is correct
and breaking points can be observed (Suefellow, 1967). Whether these same
breaking points exist within an urban area is not clear. Certainly the idea
proposed by Devletoglou, that a zone of indifference exists within which
distance is not relevant, would suggest that ideas contrary to Ellwood's exist.
More recent researchers using gravity models and computers have developed
techniques which do not require the use of breaking points (Huff, 1962) and
this later research is analyzed in the next section.
Ellwood used the gravity model to aid in choosing the location of shop-
ping centers. He felt the problem was basically one of estimating the opti-
mum size of a store by testing potential sites for a proposed shopping center.
The most important factors governing the potential sales volume were the
existing volume of trade for all competing centers, the location, size and
other competitive aspects of the existing centers and the accessibility of the
proposed new center location to existing and future population.
Ellwood suggested certain modifications be made to Reilly's "law" of
retail gravitation and then stated the new "law" as:
The principal retail districts within a metropolitan
trading area attract trade from the residential sections
of the area approximately in direct proportion to the
size of the retail districts and in inverse proportion
to the square of the driving-time distance from each
residential section to the retail districts. (Ellwood,
1954, p. 583)
So apparently Ellwood not only accepted Reilly's basic formulation of the
model, but also the values of the exponents N and n which Reilly obtained
through testing his basic formulation. Using N = 1 for the exponent for the
drawing power or attractive force and n = 2 for the exponent for the distance
or frictional force certainly makes the model resemble Newton's law of gravity
but there is no particular reason why those specific values (1 and 2) should
be chosen for the exponents for a model to be used anywhere other than
Texas (where Reilly used his model). Ellwood does not address the problem
of "calibrating the exponent" for a particular area and seems to accept 1 and
2 without verification or theoretical support.
Ellwood suggested the attractive pull of each center on the population of
the urban area could be calculated and lines of equal attraction drawn. The
lines of equal attraction defined a market area for each center, and the poten-
tial sales volume of the center was a function of the current expenditures of
the population within the market area.
There is an interesting contradiction between Ellwood's statements on
market areas and the market areas produced by his application of the gravity
model. Ellwood stated "There are no definite boundaries for the trade area
of a retail district" (Ellwood, 1954, p. 581). Yet, his application of the
gravity model produces quite distinct breaking points between centers, and
there is no provision in the model to account for shoppers who cross the
breaking point to shop.
Ellwood did not empirically test his proposed application of Reilly's
"law" for shopping center sales allocation within an urban area. Ellwood
merely explained how the "law" might be applied. One of the significant
aspects of Ellwood's article is that it was published in a very practitioner-
oriented journal (The Appraisal Journal) at a time (1954) when shopping
centers were beginning to proliferate. It would be interesting to know how
many shopping center feasibility studies performed by real estate profes-
sionals use some sort of gravity model based on Ellwood's article.
Ellwood was certainly not the only person to apply Reilly's "law" to
shopping within an urban area. In 1962, Huff combined Reilly's "law" with
computer technology to produce a model which did not require distinct trade
area boundaries to produce forecasts of shopping center sales.
The Huff Probability Model
Huff (1962) believed the use of breaking points (by Reilly and Ellwood)
implied exact market area boundaries and that actual market areas are not
delineated by exact boundaries. To estimate markets without using breaking
points, Huff developed a model which allocated sales based on probabilities.
A shopper residing past the breaking point may shop at the center exerting
the smaller attractive force, but will shop more frequently at the center with
the greater attractive force.
When Huff framed Reilly's gravity model in a probability context, he
ij n S.
where P.. = The probability of a consumer at a given point of origin i
13 traveling to a given shopping center j;
S. = The size of a shopping center j;
T.. = The travel time involved in getting from a consumer's travel
3 base i to shopping center j; and,
A = A parameter which is to be estimated empirically to reflect
the effect of travel time on various kinds of shopping trips.
(Huff, 1962, p. 15)
Huff felt his model was superior to Reilly's since probabilities better reflect
the shopper's decision process. 8
Huff empirically tested his model by surveying the residents of three
neighborhoods to determine how often those residents chose each of 14 differ-
ent shopping areas. As Reilly empirically determined the value of the dis-
tance exponent n (and used n = 2) so did Huff empirically determine the
value of the distance exponent (which he termed A) for two different types of
goods, clothing and furniture. The three neighborhoods produced different
8Huff (1962, p. 15) also included a utility component,
13 = 9_
n n S.
Z u.. Z _L
j=1 13 j=1 T.
which stated that the probability of shopping at center j was equal to the
utility of shopping at center j divided by the sum of the utility of shopping at
all centers. However, Huff did not develop this utility framework further.
X's for furniture and clothing. Huff observed that the values of the distance
exponent A were generally larger for clothing than for furniture and that
this was the expected result, that shoppers would not travel as far (distance
raised to higher power) for clothing as they would for furniture (distance
raised to a lower power so distance relatively less important). Huff did not
explain how his model might be applied to the wide range of goods offered
by shopping centers in order to allocate total sales between centers.
It is unfortunate that the only goods which Huff analyzed, clothing and
furniture, were both relatively high order, shopping goods, since it would
be interesting to compare the X's of high order goods to those of low order
goods. Huff did ask in his questionnaire for information on food (a low
order good) purchases, but he did not report what A he obtained for food
Ellwood and Huff both felt a gravity model was appropriate for alloca-
ting sales between shopping centers in an urban area. Neither Ellwood, Huff
nor anyone else has empirically tested a gravity model on a wide range of
goods within an urban area. Empirical verification of the gravity model is
still left with Reilly who used the gravity model to examine the shopping
behavior of residents who lived outside urban areas, not inside urban areas.
One might expect that if the use of gravity models within urban areas
lacks a sound empirical basis that its use is founded on a sound theoretical
basis. However, it is interesting to note that central place theory has a
sound theoretical base (for the given assumptions), but little empirical jus-
tification, while gravity models have shown fairly good empirical results,
9See Reilly (1929), Pfanner (1940), Huff (1962), Alcaly (1967) and
Young (1975) for further information on this topic.
(at least outside urban areas) but lack a sound theoretical basis. A sound
theory should relate the consumer's decision process to the actual level of
patronage of the shopping centers. One would certainly expect this theory
to be couched in a utility maximization framework that would show the con-
sumer to be a "rational person."
Berry (1967) credited Huff with the first theoretical basis for the
gravity model. Berry stated that Huff's probability model stemmed:
not from Reilly's empirical rules, but from a
reasonable theory of individual choice behavior, it has
sound foundations. Most important, this recent con-
tribution from marketing science provides, in a single
consistent frame, a basis for the practice of marketing
geographers, which it replicates. (Berry, 1967, p. 129)
The "reasonable theory of individual choice behavior" Berry refers to is
Huff's equation for the probability of center j being chosen from J (all alter-
natives). This equation is reprinted by Berry as:
and Z = 1.0 with 0 < pj < 1 and the "utility" of store j is directly pro-
portional to where S. is the attractiveness of store j and T.. is the
travel time from i to store j (Berry, 1967, p. 129).
However, it is not clear what Huff meant by u., the utility of store j.
He stated that "The utility of a shopping center to a consumer is based upon
a host of different factors" (Huff, 1962, p. 10). Huff then stated that the
"host of different factors" was too numerous to mention, but that empirical
evidence (the same type of evidence Reilly used) suggested two factors
important enough to dominate the rest. One such factor is variety of goods
and Huff asserted:
the greater the number of items carried by such
centers, the greater is the consumer's expectation that
his shopping trip will be successful. Therefore, con-
sumers will show a willingness to travel further distances
for various goods and services as the number of such
items available at various shopping centers increases.
(Huff, 1962, p. 17)
The other factor influencing utility is travel time and Huff stated:
The utility of a shopping center to a consumer is also
influenced significantly by the effort and expense that
is perceived to be involved in traveling to various shop-
ping centers. The anticipated costs of transportation,
the effort involved in preparing for as well as making
the trip, and other opportunities that must be foregone,
tend to detract from the utility of a shopping center the
farther it is from the consumer's travel base. (Huff,
1962, p. 17)
Huff's empirical observations are interesting and may even be reason-
able approximations of certain shopping behavior. His formulations are not a
theoretical basis for the gravity model. Huff merely substituted the word
utility for the impact of the attractive force and distance factors. Utility, in
an economic sense, may be defined as "the satisfaction a person obtains from
the goods and services he consumes" (Ferguson and Maurice, 1974, p. 66).
In order to use utility theory to provide a theoretical basis for the gravity
model, one would need to begin with this basic definition of utility and show
how shoppers choose shopping centers in order to maximize their utility.
Clearly, Huff did not do this.
Another attempt to develop a theory which would support the gravity
model was made by Niedercorn and Beckdolt (1969). They divided the
gravity model into three components: (1) an origin factor (the capacity of
an origin, or the person at the origin to interact with a destination), (2) a
destination factor (the drawing power or attractiveness of the destination)
and (3) a linkage factor (the effect of the space between the origin and the
destination). Their hypothesis was that the utility to an individual was a
function of the number of trips that person took, per unit time, to destination
kUij = f(kTij)
where kUi = Net utility of individual k at origin i of interacting with
Persons or things at destination j, per unit time, and
kT.. = Number of trips taken by individual k from origin i to
ik destination j, per unit time.
(Niedercorn and Beckdolt, 1969, p. 275).
Stating that utility is a function of the number of trips taken is an interesting
approach. One might expect the utility to be an inverse function of the num-
ber of trips--the fewer the trips the better. Certainly travel distance and
travel time must be somewhat negatively related to the number of trips taken.
A fairly typical view of trip taking and travel distance is that consumers
attempt to minimize the frictions of space, by making fewer trips (see Hoover,
1948). However, Neidercorn and Beckdolt assumed a direct positive relation
between the number of trips and utility, so the more trips made by the con-
sumer, the greater the gain in utility.
Niedercorn and Beckdolt also assumed that more than one person or
possible interaction existed at destination j, but the tripmaker could experi-
ence only one interaction per trip. When the utility was summed over all
destinations, these researchers obtained:
kU = a Z P f(kTij
where kUi = Total net utility of individual k at origin i of interacting with
persons or things at all destinations, per unit time,
P. = Population of destination j, and
a = Constant of proportionality
(Niedercorn and Beckdolt, 1969, p. 275).
One interaction per trip is an interesting assumption. Generally, re-
searchers maintained that it was the possibility of multiple interactions which
caused an area to have drawing power. Huff (1962) said that attractiveness
of a center was a function of the variety of goods offered at a center, which
implies a shopper is interested in more than one good. Reilly (1929) stated
that business attractiveness and social attractiveness affected the trade area
and his statement certainly implied multiple interactions. In fact, the notion
of time minimization--because of the ability to make all of the required pur-
chases at one destination--is basic to the concept of the shopping center (see
Stoll, 1967). Based on Reilly and Huff's statements on the advantages of re-
tail agglomeration, that shoppers can combine necessary purchases into one
trip, one would certainly not expect a theoretical basis of the gravity models
to be based on an assumption of only one interaction per trip.
Rather than assuming that the shopper minimizes travel costs (time and
money), Niedercorn and Beckdolt assumed the shopper placed a budget con-
straint upon travel expenditures (time and money). The budget constraint
assumption may be quite different from the way consumers actually view
shopping costs. An important aspect of retailing is comparison shopping or
comparing the prices of alternative purchases. One would expect that a con-
sumer would make a decision about travel based in part on the expected sav-
ings from purchasing a more distant, less expensive good, instead of a more
convenient more expensive substitute good. Total travel costs would then
be some function of possible price savings (or increased utility from the
Niedercorn and Beckdolt maximized utility within the budget constraint
and summed over all individuals at origin i. They obtained:
T.. = E kT..
Ur kml k
=) ) ( 7)( kMA)
where T. = Total number of trips taken by all individuals from origin i
to destination j, per unit time, and
Mi = Total amount of money that all individuals at origin i are will-
ing to spend for travel to all destinations, per unit time.
If travel time, rather than money, is the relevant constraint,
then Mi/r in Equation (11) is simply replaced by sHi, where Hi is
total time allocated by all individuals at origin i for travel to all
destinations, per unit time. The equations for T.., using either
the money or the time constraint, are similar to tNe version of the
"gravity law" showing that Ti is inversely proportional to di raised
to the first power (Niedercorn and Beckdolt, 1969, p. 277).
This has the form of a basic gravity model. -n is the attractive force
which is the relative population (or size) of j compared to total population (or
total size). d is the distance from i to j.
The theoretical base developed by Niedercorn and Beckdolt is not as
useful as it might be, because the assumptions they propose differ so greatly
from the actual behavior of the consumers whose behavior is supposedly being
explained. Niedercorn and Beckdolt do not make clear how the number of
trips yields utility. 10 They merely state that utility is a direct function of
the number of trips. They assume one interaction per trip and do not relate
this to how a shopper may combine a variety of tasks into one trip. Finally,
they set a budget constraint for travel costs and maximize utility rather than
relating travel costs to the possibilities of saving money through comparison
In summary, Niedercorn and Beckdolt manage to obtain a formula
which looks like a gravity model, but they do not frame their model so the
consumer decision process is made more understandable. Their assumptions
regarding utility and per trip interaction do not fit the evidence of how the
shopper views travel or the size of a center. Certainly, any theoretical ex-
planation of shopper's behavior should flow logically from accepted theory
regarding utility maximization. Theory which does not flow logically does
not help the user of the model to understand how the model predicts behavior
and how changing some aspect of the problem will affect the final outcome.
The model becomes a "black box" which may be useful in ordering data but
not in obtaining reliable predictions under varying circumstances.
Although many market analysts use some form of a gravity model to
predict trade areas (Kelly, 1956; Cohen and Lewis, 1967; Applebaum, 1961;
Thompson, 1964; Brinkrant, 1970), there apparently is no convincing
theoretical basis for the gravity model.
Ellwood (1954) felt that gravity models were superior to other methods
of sales volume prediction because gravity models had been tested and had
10See the comment on Niedercorn and Beckdolt by Mathur (1970) and
the reply by Niedercorn and Beckdolt (1970).
strong empirical support. Reilly (1929) and Converse (1949) are two of the
researchers whom Ellwood cited as giving support to the use of gravity
models.11 However, both Reilly and Converse studied the attractiveness of
and the trade between urban areas (cities, towns, etc.) and did not study the
market areas within the urban areas.
Berry (1967) pointed out what he termed the difference between the
countryside and the city when referring to markets in rural areas. He
S. the tyranny of distance [in rural areas] is so
great that areas of indifference are strictly limited.
The transition from one market area to another is
rapid and complete. However, within the city, at
high population densities, a variety of centers can
achieve at least threshold sales within the minimum
distance consumers are willing to travel on a shop-
ping trip. Areas of market area overlap are great.
(Berry, 1967, p. 87)
Bucklin (1967) also noted that distance and mass may be insufficient to
analyze shopping on a micro-level. He found that income, social position,
perception of the center and the type of shopping affected the choice of cen-
ters within an urban area.
Even though much has been said regarding whether Reilly's "law"
should be applied to choices within an urban area, one should note that Reilly
(1929) himself did not mention applying his "law" to any problem other than
determining trade between metropolitan areas. He did not suggest applying
his "law" to determining trade areas within metropolitan areas. Reilly's
"law" of retail gravitation and the various gravity type models of Ellwood
and of Huff are intriguing, partly because they reduce the shopping decision
11Both Reilly (1929) and Converse (1949) have conducted much re-
search and are recognized authorities in the area of gravity models. (See
Ellwood, 1954, p. 587.)
process to a very simple model. However, the use of gravity models for
allocating sales within an urban area has three serious caveats. First,
Reilly's "law" of retail gravitation was not intended to be used within an
urban area. Second, a gravity model has not been tested as a sales alloca-
tion model within an urban area. Third, the gravity model lacks a theoretical
basis relating it to utility theory.
Central place theory and Reilly's "law" were both set forth to explain
the behavior of rural residents who had to choose an urban place to obtain
needed goods and services. While both central place and gravity models have
been applied within urban areas, they were not specifically designed to be so
applied. If the shopping behavior of residents within an urban area is truly
different from the shopping behavior of those residents outside an urban area,
then one might expect a theory designed specifically to explain the shopping
of urban residents to be applicable to this study. A theory of shopping within
an urban area was proposed by Nicos Devletoglou, who hypothesized that
shoppers may be indifferent to differences in distance between alternative
shopping centers within the confines of an urban area.
The Zone of Indifference Hypothesis
The zone of indifference hypothesis states that the differences in driv-
ing time from the place of residence to various alternative shopping centers
in an urban area are small enough that shoppers do not perceive a difference
in utility between the various trips. In other words, once one is in an auto-
mobile and driving, the difference between say a 15-minute drive and 20-
minute drive is negligible. If shoppers do not include the distance to a store
in their decision process and if that distance is less than some limit on dis-
tance the shopper has set, then a zone of indifference will exist within which
shoppers will not discriminate between shopping centers because of distance.
Since the shoppers are indifferent to the distance factor, they must choose
their place of shopping because of factors other than distance. The notion of
shopper indifference to distance is certainly different from the assumptions of
either the central place or gravity models, both of which assume distance is
very important in the shopping place selection process.
Devletoglou (1965) presented his hypothesis as a contrary view to the
classic treatise "Stability in Competition" by Harold Hotelling (1929). To pro-
vide a better basis for the Devletoglou viewpoint, Hotelling's article is ex-
Hotelling analyzed how two competitors (duopolistic competition), each
attempting to gain a locational advantage over the other, would locate. As-
sume that two competitors sell products that are identical in every respect
(including price), shoppers are totally indifferent as to which of the two com-
petitors they patronize. The one and only method of selection employed by
shoppers is that shoppers will always patronize the nearest competitor. To
make the spatial analysis simpler, assume the competitors can locate anywhere
along a straight line (similar to stores on main street in a town or to an ice-
cream vendor on a beach). Also assume that customers (with homogeneous
demand curves) are evenly distributed along the same straight line. Hotelling
now posed the questions, If the two competitors are mobile, what final loca-
tions will the competitors choose and will the assumed system ever achieve
stable locations? Call the competitors A and B and let the line on which they
are located have length D. Let A pick any initial location dl which is dl units
from the left or zero end of line D.
Competitor B will now attempt to locate so that he will be closer to a
greater number of customers than A. B will compare the lengths of line
segments dl and (D-d) Since customers are evenly distributed along D,
choosing the longest line segment is synonymous with locating nearest to the
most customers. If (D-d1) > d (as it appears in our diagram) then B will
locate immediately adjacent to A, between A and D. If B locates any distance
away from A, then B must share evenly with A the customers between B and
A. B will locate:
1 0 distance
A will now benefit by relocating between B and D and thereby gaining
a locational advantage. This relocation process will continue until dI = (D-d1),
at which point it will be impossible for either A or B to relocate and gain a
locationed advantage. The point where d1 = (D-d1) is the center of the line,
D. Hotelling points out that locating both competitors in the center of the mar-
ket does not minimize the travel costs for society (society meaning the con-
sumers on the line D). Locating one competitor one-quarter the distance from
the left end of the line and one competitor one-quarter the distance from the
right end of the line would minimize travel costs for society.
Devletoglou's Dissenting View
Develtoglou correctly points out that the competitors must be totally
indistinguishable to the customers and that shoppers must be completely
averse to traveling one inch beyond the point necessary to obtain the
desired good. If either competitor were to lower price, improve the appear-
ance of his store, expand his merchandise line, give green stamps or do any-
thing to distinguish himself from the other competitor, then some shoppers
might be willing to travel farther to shop at the competitor of their choice.
One aspect of Hotelling's example is that in order for the two compe-
titors to obtain as much of the market as possible, they must locate adjacent
to one another. If the competitors are adjacent, then by definition, a very
small (or zero) space separates them. If the extra distance to the more dis-
tant competitor is very small, then customers will require only a small incen-
tive to bypass the closer competitor for the more distant.
Devletoglou also points out that even if the competitors are essentially
identical, customers may still choose the more distant of the two. Shoppers
may be indifferent when choosing between two competitors based on the
distances to the competitors if the distances are almost the same. Devletoglou
hypothesizes that trip lengths must vary by what he terms a "minimum sensi-
ble difference" if the shoppers are actually going to discriminate because of
Devletoglou stated that consumers will attempt to minimize traveling
distance (transport costs) subject to some "minimum sensible" constraint of
Quite simply, the principle put forth is that some posi-
tive minimum, however small or large, exists for the con-
sumer where the difference in distance between patronizing
one store rather than the other can be axiomatized as in-
consequential, and thus too weak a criterion for practical
choice or revealed preference. The consumer's postulated
apathy toward applying the traditional economic calculus
'to the letter' beyond a given stage may be conveniently
interpreted as the effect of discontinuities in the utility
functions. (Devletoglou, 1965, p. 142)
Devletoglou also suggested that minimum sensible distance may be a
function of trip length, longer trips implying a greater minimum sensible
difference than shorter ones. For example a five-minute trip or a 15-minute
trip to alternative centers means a difference in distance of 10 minutes. The
10-minute difference may exceed the minimum sensible difference in distance
for trips in the five-minute category. A 25-minute trip or a 35-minute trip to
alternative centers also means a difference in trip length of 10 minutes, but
may not exceed the minimum sensible difference in distance for trips in the
25-minute category. The implication is that minimum sensible difference is
relative to trip length and does not have a fixed value.
If minimum sensible difference in distance does vary directly with trip
length, this increased value of minimum sensible difference for longer trips
has two important implications for the present study. First, several previous
studies (Berry, Barnum and Tenant, 1962, for example) have indicated that
consumers tend to travel a greater distance for high order shopping type
goods than for low order convenience type goods. If the purchase of shopping
type goods can be generally associated with a longer trip, then the zone of
indifference implied by the size of the minimum sensible difference will be
larger for those longer shopping goods oriented trips than the zone of indif-
ference associated with the shorter convenience goods trips. If the zone of
indifference for trips associated with shopping goods is larger compared to
the zone of indifference associated with convenience goods, then consumers
may perceive more feasible alternatives when selecting a shopping goods
center than when selecting a convenience goods center. Relating shopping
goods purchasing and convenience goods purchasing to the size of the
zone of indifference may add insight to understanding the two different shop-
ping processes. Shoppers may act as though they have a relatively small
zone of indifference for convenience goods and a fairly large zone of indiffer-
ence for shopping goods. Distance would affect the shoppers differently,
depending on the type of goods sought.
Another phenomenon which the zone of indifference hypothesis may
help explain is the shopping behavior of outshoppers. The outshopper is
one who travels outside his local area to do his shopping. Since, by defini-
tion, the outshopper is leaving the local area, one may safely assume the
outshopper's trip is long enough to carry him to a non-local area. One would
therefore expect the outshopper's trip to be fairly long. If the outshopper
travels a fairly long distance to shop, he should also have a fairly large
"minimum sensible difference" and hence a large zone of indifference. A few
researchers have analyzed how shoppers perceive the trips they take, and
other researchers have examined the characteristics of outshoppers. These
two areas of research are discussed in the next section.
Thompson (1966) suggested that shoppers may compare the closeness
of stores using a subjective distance factor. The actual or geographic dis-
tance is modified by the consumer's perception of the space through which
he must travel.
When distance is measured by a traveler's perception of how great that
distance is rather than by measuring the actual distance or using a map or
time measurement, then the distance is termed psychological distance.
Thompson suggested that the pleasantness of a trip may modify the traveler's
perception of the distance. A pleasant trip would produce a psychological
distance less than the actual distance while an unpleasant trip would cause
the psychological distance to be greater than the actual distance. A psycho-
logical view of space is supported by Horton (1971) who found that people
develop an "action space," an area with which they are most familiar, and
tend to remain in that action space. The action space may be oriented toward
work or some other point to which they frequently travel. Distances within
the action space are viewed as shorter than equal distance outside the action
space. Horton also hypothesized that a psychological barrier may exist
which discourages any trips outside the action space.
It appears that factors other than distance or driving time may affect
how a traveler views the "length" of his trips. Travelers may develop a
psychological view of distance. The psychological distance a traveler attri-
butes to various trips may not correspond with the actual distance or even
the actual relative distance (The traveler may perceive the trip which is
shorter, in miles or average time, to be the longer).
In another article, Thompson (1964) suggested that:
the larger the unit of purchase or the amount of
money involved and the greater the element of style
inherent in the merchandise, the farther the consumer
will travel to buy it. (Thompson, 1964, p. 10)
Thompson's ideas are supported by Young (1975) who found the market area
for regional shopping centers, which tended to offer very stylized goods, to
be substantially larger than the market areas of community centers, which
offered more convenience type goods. Young found driving times of up to
65 minutes for the regional centers and only 25 minutes for the community
Some authors have examined the characteristics of the shopper who
travels outside his local area (termed an outshopper) and the goods he buys.
In comparing outshoppers to non-shoppers, Reynolds and Darden (1972) found
the outshoppers to be better educated, higher income earners and more
"urban oriented" than non-outshoppers. Herrmann and Beik (1968) found
outshoppers to be most frequently in the higher income groups and most
frequently to purchase style type goods (women's coats, men's suits, rugs,
women's dresses, curtains and drapes and women's sportswear). Using a
different data set than that of Herrmann and Beik, Thompson (1971) found
that outshoppers tended to be in higher income groups and purchased
style goods (women's coats, curtains and drapes, rugs and carpets, men's
suits and women's fancy dresses).
It is interesting to compare the list of goods which Herrmann and
Beik stated were most often associated with outshoppers to the list which
Thompson associated with the outshoppers. The goods on the two lists are
very similar. All items on both lists are either fairly large ticket items
(rugs, coats, curtains and drapes) or are very style oriented (women's
fancy dresses, men's suits) or both style oriented and large ticket items.
In another study Collazzo (1966) found that high income shoppers were
less loyal to a particular store, had a more precise idea of what they wanted
and would pay more for quality and service. One might deduce from the
aforementioned studies that high income shoppers may be able to perceive
more differences between goods and stores than low income shoppers. A
higher level of differentiation may partly explain why high income people tend
to be outshoppers. High income shoppers may need a larger number of pos-
sible choices in order to find exactly what they want and they may be willing
to travel farther to obtain exactly what they desire. Since outshoppers and
high income shoppers tend to travel greater distance for their purchases,
they may be expected to have greater minimum sensible differences in dis-
tance. A greater minimum sensible difference in distance would imply a
fairly large zone of indifference. If outshoppers and high income shoppers
are a significant part of the market for shopping goods, their behavior may be
important in allocating total MRC sales. If outshoppers and high income
shoppers tend to have large zones of indifference, they may be affected by
shopping center factors other than distance to center. Since the central place
and gravity models rely heavily on distance as an explanatory factor in
shopping behavior, these models may fail to predict the behavior of out-
shoppers or high income shoppers because these two groups are not so dis-
tance sensitive. As a significant part of the population, the higher income
shoppers may have a reasonably large zone of indifference. Since high income
shoppers are also the people who purchase the greatest amount of goods from
shopping centers, the zone of indifference effect may be even more important.
In this chapter, the literature concerning three different theories of
consumer behavior is analyzed. The central place theory proposed by
August Lisch (1954) and Walter Christaller (1966) is summarized as "the
shopper patronizes the nearest center offering a desired good." Later
authors such as Berry and Garrison (1958), Berry, Barnum and Tenant (1962)
and Getis (1963) discussed and tested central place theory. Central place
theory appeared to explain fairly well the shopping patterns for low order,
convenience type goods.
Central place theory did not predict well the shopping behavior of those
seeking higher order shopping type goods (girls dresses, coats, furniture,
etc.). Shoppers appear to be willing to travel farther to obtain a wider selec-
tion when purchasing heterogeneous goods and appear to place greater impor-
tance on proximity when shopping for homogeneous goods.
The Law of Retail Gravitation developed by William Reilly (1929) is the
second theory discussed. Reilly's theory may be summarized by noting that
the shopper is attracted by the size of the center but discouraged by the dis-
tance to the center. Reilly tested his "law" using rural residents of Texas.
His work was later applied to shoppers residing within an urban area (by
Ellwood, 1954 and Huff, 1962). The probability gravity model developed by
Huff was judged to be the most sophisticated and appropriate form of the
gravity model. The gravity model lacks empirical support or a sound theore-
tical basis. No evidence could be found of an empirical test of the gravity
model using a wide range of goods and shopping centers within an urban area.
Several proposed theories are discussed but all of those theories have seri-
ous flaws which make them unacceptable as a basis for the gravity model.
The third hypothesis was one proposed by Nicos Devletoglou (1965) as
an alternative to the center clustering theory proposed by Hotelling (1929).
Devletoglou felt that distances to shopping centers in an urban area may be
short enough so that shoppers do not perceive a difference in utility because
of a difference in trip length. Devletoglou's notions are recent and untested,
but offer an interesting contrast to either central place theory or the gravity
In Chapter 3, each of the three models is tested as a predictor of shop-
ping center sales. The tests are conducted using the same data so the relative
ability of each of the three models can be judged. This study also proposes a
model specifically designed to predict shoppers' behavior within an urban area.
The gravity model and central place model were not specifically designed to be
used within an urban area.
TESTING THE MODELS
In this chapter a mathematical expression for each hypothesis dis-
cussed in Chapter 2 is presented in model form. The central place model,
gravity model and zone of indifference model are then used to predict the
level of sales for the 19 major retail centers (MRC's) in the Hartford,
Connecticut, Standard Metropolitan Statistical Area. The predictions of
each model are then compared to the actual MRC sales. The accuracy or
quality of the predictions are analyzed using regression analysis.
The testing of the three hypotheses within an urban area is deemed
important because none of the three hypotheses has been tested in an intra-
urban setting using centers offering a variety of goods. When the central
place hypothesis was tested within an urban area, the tests were performed
using only low order goods (for example food, see Getis, 1963). The
gravity model was tested within an urban area by Huff, but he tested the
model for the purchase of only furniture and clothing, used only three
neighborhoods for the population of consumers and did not obtain a pre-
diction of total sales for the shopping centers in his study. The tests in
this study produce an estimate of total sales and compare the quality of the
predictions of the three models.
The Method of Testing
In testing the usefulness of any model, one must first confront the
question, Which variable will be the predicted variable? This study is
concerned with the decision maker who must appraise the feasibility of a
shopping center. The relevant question for this decision maker is, What
level of sales can I expect a shopping center at a specified location to
obtain? The decision maker would certainly be interested in how
frequently the various residents of the surrounding area might be expected
to visit a particular center, but the decision maker would view frequency
of visits only as they relate to the expected overall level of sales. The use-
fulness of the models presented in this study will be judged by their ability
to predict total sales.
The Development of the Models
The usefulness of the central place, gravity and zone of indifference
hypotheses are judged by how well these hypotheses predict the total
sales for the 19 MRC's in the Hartford SMSA. In order to compare the use-
fulness of the hypotheses, it is necessary to express each hypothesis as a
model that will produce a prediction of total sales for a shopping center.
The next three sections are devoted to expressing each hypothesis as a
The Central Place Model
Assume that each consumer will patronize the nearest center offering
a desired good. In this study, nearest center means the center which is the
shortest driving time from the place of residence. The method of comput-
ing driving time for the testing of all three models is explained later in this
chapter in the section labeled "Data." Americans are very automobile
oriented, so using driving time as the "distance" measure seems logical
The model's task is to compare the driving time from each census
tract i to each of the 19 MRC's j. The model must select the MRC as-
sociated with the shortest driving time and allocate the potential sales from
the census tract to that MRC. The model must perform the trip comparison
and sales allocation for each of the 167 census tracts in the Hartford SMSA.
The basic form of the model is:
Sales = i I. (1..) such that
1.. = 1 if d.. < d when k = (1, ...J) but k j
13 13 ik
j1 = 0ifdij. >dik for anyk = (1, ...J) but k j
1] 1] J k
Ii = Spendable income in census tract i
.ij = 1 or 0, 1 if census tract i is closer to MRC j than to
any other MRC and 0 if census tract i is not closer to
MRC j than to all other MRC's
Sales. = Sales at MRC j
i = 1 through 167 since there are 167 census tracts in the
k = Each MRC other than MRC j
d = Driving time
The Gravity Model
The model assumes that consumers are attracted by the size of the center
and that the distance to the center acts as a frictional force which discourages
shopping at that center. This model emphasizes the importance of variety to
the consumer (assuming larger size implies more variety), but recognizes the
inconvenience of travel. In testing the model driving time from the place of
residence is used as the distance or frictional factor and the gross leasable
area in square feet of each MRC is used as the force of attraction. The model
P.. = (dii) 2
Size = Size in S.F. of gross leasable area of MRC j
dij = Driving time from census tract i to MRC j
P = Probability of $1.00 of spendable income from census tract
1 i being spent in MRC j
Applying the above equation to the data produces an array of prob-
abilities of the spendable income from each census tract being spent at a partic-
ular MRC. The sum of the probabilities for each census tract is 1.0. The
probabilities are interpreted as the probability of $1.00 of spendable income
in census tract i being spent at MRC j. Spendable income means the total
amount of income which will actually be spent in the 19 MRC's. So, by defi-
nition all of the spendable income will be allocated between the 19 MRC's.
The above equation performs the allocation process by assigning
probabilities to the census tracts but does not produce a sales estimate for
the MRC's. The next step converts the probability from the above equation
into a dollar amount and allocates those dollars to the appropriate MRC's.
This step allocates the spendable income between the MRC's.
Sales. = Z P.. I.
J i=1 1] 1
Pij = Probability of $1.00 of income from census tract i being
spent in MRC j
Ii = Spendable income in census tract i
Sales. = Sales at MRC j
Total sales for each MRC are obtained by multiplying the spendable income in
each census tract i by the probability of a dollar from census tract i being
spent at MRC j and then summing these amounts for all census tracts i.
The model used here is essentially the same model used by Huff (1962).
The value of X, the distance exponent, in the first tests of the model is 2.
Other values of I are analyzed and tested later in this study.
The Zone of Indifference Model
Basically this model is a simple statement of Devletoglou's hypothesis.
Intraurban distances may be short enough that shoppers do not really perceive
a disutility from driving a few minutes longer to reach a particular center.
Stated another way, if shoppers have a 15-minute minimum sensible difference
in distance, then they will be indifferent between trips to any shopping center
within 15 minutes of the place of residence. The model compares the travel
time from census tract i to all MRC's. If the trip length to some of those MRC's
is less than some set limit (say 15 or 20 minutes), then the model will divide
the spendable income of census tract i equally between the MRC's within the
driving time limit.
The time limits of 15 and 20 minutes were chosen somewhat arbitrarily.
No previous tests of this model could be found in the literature, so this test
is exploratory in nature. One justification for the limits chosen was a study
by Brunner and Mason (1968) in which they examined the driving times of
the patrons of five MRC's in the Toledo, Ohio, SMSA. Approximately 70 per-
cent of the patrons of the five MRC's lived within 15 minutes driving time
from the centers.
The model used in this study may be expressed as:
Sales. = Z i (lij
1 i=l J
1.. = 0 ifd.. > d (limit)
1.. = 1 if d.. < d (limit)
Ii = Spendable income in census tract i
.. = A factor whose value is 1 if the driving time from census
11 tract i to MRC j is less than a limit and 0 if over the limit.
Sales. = Sales at MRC j
The portion of the model:
j=l 1 1
represents the number of MRC's within the driving time limit. The spend-
able income of each census tract i must be divided by this number.
The portion of the model:
then compares dig for census tract i to the limit d and if d. is less than d
a portion of the spendable income I. is allocated to MRC j.
The portion of the model:
sums the spendable income allocated to MRC j from all the census tracts i.
Three basic types of data were used in the testing of the models. Infor-
mation was obtained about the level of sales and type of sales for each MRC
and about the physical characteristics of each MRC. This information was
termed "MRC characteristics." The second type of data concerned the distance
and type of road used in travel to each center; this information was termed
"linkage characteristics." The third type of information was about the con-
sumers and was termed "buyer characteristics."
The 19 shopping centers used in this study were the MRC's, including
the Central Business District (CBD), reported in the 1972 Census of Retail
Trade for the Hartford, Connecticut, Standard Metropolitan Statistical Area.
The tables from the Census of Retail Trade which apply to the Hartford
SMSA are summarized in Table 1. The most useful information reported,
for this study, is the level of sales for each MRC.
The total reported sales for each of the 19 MRC's is shown on Table 1,
"Reported MRC Sales." The sales information for some categories has been
deleted (indicated by capital D) to comply with disclosure rules.
The disclosure rules require that enough information be omitted so that
the reader cannot obtain the actual sales for any particular establishment. In
order to use the census data for this study, the missing data had to be esti-
The first estimates were made in the "all other goods" category. This
category includes automobile dealers, gasoline service stations, building
material suppliers, mobile home dealers and hardware stores. The estimates
of sales in this category were made by first obtaining the average sales by
store for all stores in the "all other goods" classification in the 19 MRC's
(deleting the CBD). The reported sales in this category were $37,684,000,
and there were 152 stores reporting for an average of $247,920 or approximately
$248,000. The nonreporting stores were assumed to be similar to the re-
porting stores and therefore to have average sales of $248,000 per year. The
number of stores in the "all other goods" category for each center for which
the total sales had been deleted was multiplied by $248,000 to obtain an estimate
of total sales in this category. Estimating sales in the "all other goods" category
often produced a residual estimate in another nonreporting category.
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The category "all other goods" represents a group of goods which one
does not generally associate with shopping centers, such as automobiles
and mobile homes. The analysis of the consumer decision process for such
goods is beyond the scope of this study. Since "all other goods" does not in-
clude the type of goods one generally associates with a shopping center, it
was decided to eliminate the sales in this category from the analysis and hence
from the total sales reported for each MRC. The total sales minus "other goods
sales" is termed "adjusted total sales" and is the last column in Table 2,
"Adjusted MRC Sales." The term actual sales is used to indicate "adjusted
Sales for the CBD, West Hartford Center and Enfield Malls were broken
into more reporting categories than were the other MRC's and by estimating
the sales in a missing category, total sales for all the categories for these
centers could be obtained. The data reported were:
CBD West Hartford Enfield Malls
Convenience Goods Sales* # Stores Sales* # Stores Sales* # Stores
Food Stores $2,987 17 $6,399 13 $11,026 6
Eating & Drinking 6,790 42 1,582 6 3,173 10
Drug Stores D** 2 D** 4 D* 4
*All sales are in thousands.
**D represents deletions.
By estimating the sales for the drug stores, total sales of convenience
goods can be obtained. Average sales for drug stores were $100,000 per year
so that figure was substituted for the deleted sales to produce the following
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CBD West Hartford Enfield Malls
Convenience Goods Sales*- #Stores Sales*- #Stores Sales* #Stores
Food Stores $2,987 17 $6,399 13 $11,026 6
Eating & Drinking 6,790 42 1,582 6 3,173 10
Drug Stores 200E** 2 400 E** 4 400 E** 2
Total $9,977 $8, 381 $14,599
*All sales are in thousands.
**E represents estimates which were used to produce Table 2.
The usefulness of this study depends heavily on the accuracy of the
sales data reported as actual sales. Sales data were estimated in one category
or another for eight of the 19 MRC's. Several factors help minimize the effect
of estimating sales.
The Bureau of the Census attempts to comply with nondisclosure and
still report as much useful information as possible. This attempt to minimize
the effects of nondisclosure is apparent from the data which were deleted.
Information was deleted in the "all other goods" category when there were
very few stores in this category. In three of the cases there was only one
store in this category, in two cases there were two stores and in one case there
were four stores. Even a large error in estimating these sales would produce
only a relatively slight error in total sales. The case is the same for drug store
sales. First there were very few drug stores in the centers with deleted informa-
tion (2,4 and 4). Second, the typical drug store sales were small, averaging
only $100,000. Even a large error in estimating these sales would produce only
a small error in total sales. While no estimating procedure is perfect, and
certainly not the one used in this study, the procedure was deemed acceptable,
and the final sales totals are considered sufficiently accurate for useful analysis
of the shopping center sales allocation process.
Each MRC was visually inspected, and a profile of each center is in-
cluded in the Appendix. During the inspection of each center, information
was obtained about the attributes of the center.
There is a potentially unlimited amount of information, or number of
attributes for each MRC. In any type of research some criteria must be
established to decide which information is most relevant and useful and which
information may be disregarded. The following assumptions were made re-
garding the typical MRC customer and the various states of his shopping trip.
First, the general assumptions will be briefly stated. Next, the factors about
which information was obtained will be stated and the justification for includ-
ing those factors will be explained.
The typical MRC customer is assumed to drive an automobile to the MRC.
If the customer drives, his decision process for choosing an MRC should be
affected by how convenient and expensive it is for him to reach the MRC from
a road and park his car (how the trip affects his time budget and income
budget). The customer should be affected by how pleasant the atmosphere of
the MRC is and how safe the customer feels (what are his chances of being
robbed or mugged).
Most attributes of the MRC's were rated using a value system. Typical
ratings might be attractive-unattractive or adequate-inadequate. Any subject
rating system lies prey to the biases of the person performing the rating.
However, because of the nature of the attributes being judged, no other sys-
tem of judging those attributes appeared workable.
Information was obtained on the following attributes.
Size. The area in square feet of the gross leasable area for each MRC
was measured and is termed total square feet. The area of stores selling "all
other goods" was estimated and subtracted from total square feet to obtain
"total adjusted square feet." The total adjusted square feet were allocated
between convenience goods stores, and shopping goods stores (These cate-
gories will be discussed in Chapter 5).
Size was considered an important factor because the gravity model re-
quires the size variable to be numerically stated. Size may also be a good
proxy for the variety of goods offered or the degree of specialization in retail
stores within a center. Throughout the study the assumption will be made that
large shopping centers increase the probability of any shopper finding either
a specific good he requires or a variety of goods he requires.
Since size was considered a crucial variable and could not be determined
by inspection alone, a great deal of effort was put forth to obtain accurate esti-
mates of the size of each center. Three sources of information were used.
First, letters were sent to local tax assessors requesting information on the
size of the MRC's. Second, each MRC was "paced off" and the size estimated
from the pacing. Third, aerial photographs were obtained for each MRC and
measured to obtain an estimate of the size of the MRC.
Accessibility. This factor addresses the question, How easily can the
shopper reach the shopping center parking lot from the intra-urban road
network? If the shopper must leave the primary road network and wind his
way through a maze of smaller streets in order to reach the center parking lot,
then this center would be described as having difficult access. A similar case
occurs for downtown shopping. The downtown area may be easily reached,
but if the parking is remote or difficult to reach, the accessibility of the down-
town area would be rated difficult. The important point is that accessibility
is rated in terms of the accessibility of the parking which complements the
the shopping area. Each MRC was rated as having "difficult access," "aver-
age access" or "easy access."
Parking adequacy. After the shopper reaches the parking area, he
must find an empty space convenient to his final destination. If an MRC does
not have adequate parking spaces, the shopper may leave because no space
is available, the search for a space is unduly long or the only available spaces
are an inconveniently long distance from the shopping area. Parking was
judged as inadequate or adequate. This study will assume that shoppers
will, over time, prefer MRC's with adequate parking to those with inadequate
Parking cost. First, in an economic sense, no parking is free. The
suburban shopping center, which is typically associated with "free" parking
simply raises the rents of the tenants to cover the costs of a parking lot. The
tenants must in turn charge higher prices so the consumers are charged based
on the size of purchase rather than on the length of parking time. However,
shoppers may perceive a direct charge (through metering) as less desirable
than an indirect charge (through higher prices). The assumption is made in
this study that shoppers prefer "free parking" (financed by an indirect charge)
to "metered parking" (financed by a direct charge). Each MRC was rated as
offering metered or free parking.
Type of center. After parking, the shopper must walk to the various
stores in which he wishes to shop. The length and pleasantness of such walk-
ing would be affected by the layout and construction of the MRC. The first
inclination was to include enclosed malls in one group and all other centers
in another. However, of the 19 centers, only two are of the enclosed mall
type. The final solution was to divide MRC's into two categories based on
the dispersion of the stores comprising the MRC. An MRC with stores
separated by nonretail uses (offices, streets, etc.) was termed dispersed.
An MRC with contiguous stores was termed compact. The assumption is
made that shoppers prefer compact MRC's (few intervening uses) to dis-
persed MRC's (many intervening uses). Some of the early tests per-
formed in this study used the open-enclosed categorization rather than
the dispersed-compact categorization. The category used is clearly
stated at the appropriate point.
Appearance. While the customer is shopping within an MRC, the
attractiveness of the MRC should affect how pleasant the shopping experi-
ence is. Each MRC was rated using the very subjective categories of
"unattractive," "average" and "very attractive." It is assumed that shop-
pers prefer a more attractive shopping place to a less attractive shopping
place. Attractiveness is very difficult to define. The form and decorat-
ing of the MRC were important, but perhaps more important in the rating
were the cleanliness of the center and the level of upkeep.
Neighborhood. The type of neighborhood in which the MRC is
located may affect how "safe" the shopper feels. If the shopper perceives
a neighborhood to be bad or dangerous, he might feel there is an in-
creased probability of being robbed or mugged. Many MRC customers
are women (and women may be more subject to robbing or mugging)
and a large portion of the MRC's sales occur after dark (a time when
more robbing and mugging occurs). A large number of women shoppers
and night shoppers may make the type of neighborhood a significant factor
in the shopper's decision process. The neighborhood in which each MRC
is located was rated "bad" or "good."
Age. The age, in years, of each MRC was recorded. Age is not
intended to be a proxy for level of upkeep. The age of the center could be
very important for a new center. The new center may not have established
itself or penetrated the markets of older more established centers.
The factors listed above are deemed to be relevant to the shopper's
decision process. These factors are tested in Chapter 4 for correlation with
MRC sales. In Chapter 6 these factors are incorporated into an improved
model for MRC sales allocation.
All three models required an estimate of the driving time from each
census tract to each MRC. The estimate of driving time was obtained by first
estimating a likely or convenient path to each MRC from each census tract.
The convenient path was chosen by inspecting a map of the Hartford SMSA.
No attempt was made to choose a best route based on scenery, congestion,
traffic signals or similar factors. Generally, the path which appeared to be
the shortest was chosen. An attempt was made to make use of main roads
and limited access highways.
The trip from each census tract to each MRC was divided into three
1. Distance from census tract to major road
2. Distance along major road
3. Distance from major road to MRC.
The three segments of the shopping trip were then categorized according
to the type of road used during the trip segment. The three categories of
roads were: (1) expressway, (2) arterial or (3) street. An average rate of
speed was assigned to each category as follows:
1. Expressway 51 m.p.h.
2. Arterial 30 m.p.h.
3. Street 25 m.p.h.
The average speeds were applied to be measured distances and summed for
each trip so total driving time from each census tract to each MRC was ob-
The speeds were obtained from the Connecticut Highway Department
which divided travel by road type and the section of the SMSA in which
travel occurred. For this study, the "suburban" speeds were chosen as a
proxy for all speeds.
There are 167 census tracts in the Hartford SMSA and 19 MRC's. This
study required a calculation of the driving time from each census tract to each
MRC or 3,173 different trip calculations.
The quality of the models as predictors of MRC sales is judged by com-
paring the actual sales of the MRC to the predicted total sales obtained from
each model. Part of the prediction process requires each model to sum the
sales attributable to the residents of each census tract. To estimate the expected
sales for each census tract, it was necessary to estimate the portion of total
income each individual should spend on the type of goods sold at an MRC.
Information on family spending patterns has been reproduced as Table 3,
"Family Spending Patterns."
1 Expressway Arterial Street
Urban 43 m.p.h. 23 m.p.h. 21 m.p.h.
Suburban 51 m.p.h. 30 m.p.h. 25 m.p.h.
Rural 55 m.p.h. 35 m.p.h. 32 m.p.h.
Source: Connecticut Interregional Planning Program (Hartford, CT: Trans-
portation Planning Section, Division of Planning, Connecticut Highway
Department, Technical Summary #3), 1963.
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Each model allocates spendable income so an estimate of the spendable
income in each census tract was required. First, it was necessary to esti-
mate the portion of income each individual might spend on the two categories
of goods sold at an MRC, The family expenditure information obtained was
reproduced as Table 4, "Income Allocation as Percent of Mean Income." The
sales of MRC's are generally composed of convenience goods and shopping
The category "food" was chosen from the "Family Spending Patterns"
in Table 3 as representative of expenditures in the convenience goods category.
"House furnishings" and "clothing and accessories" were chosen from Table
3 as representative of expenditures in the shopping goods category. The
next step was to determine whether the percentage of income spent for con-
venience goods or shopping goods varied with income level. The average
annual expenditure for each category was divided by the mean of each income
category as shown in Table 4. Expenditures for convenience goods tend to
decline as a percent of income as income increases while the percentage of
income spent on shopping goods tends to remain fairly constant (see Table 3).
The model developed in this study is not intended to relate the demand
curves of consumers to a level of retail sales. The model is not predictive in
the sense that it will translate a given income level to a level of retail sales.
The model developed in this study is intended to allocate an existing level of
retail sales among a number of retail centers. The allocation process assumes
a level of total sales for the retail centers as a group and then allocates the ap-
propriate portion of those sales to each center.
The allocation process for all three of the models tested allows for the
possibility of some consumers spending more than other consumers. Dif-
ferent income groups may allocate different amounts for purchasing retail
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goods, but a uniform percentage of that amount is spent at MRC's. The rela-
tive amount allocated to retail goods was obtained using "Family Spending
Patterns," as shown in Table 4. The figures obtained were:
Percent Available for the
Total Family Income Purchase of Goods at an MRC
0 $ 5,000 38%
$5,000 $10,000 34%
$10,000 and above 28%
The 1970 United States Census for the Hartford Connecticut SMSA
was used to obtain the number of families in each income category in each
census tract. The income categories used were:
Less than $ 4,999
$ 5,000 to $ 9,999
$10,000 to $14,999
$15,000 to $24,999
$25,000 to $50,000
$50,000 or more
To allocate the income of the residents of the Hartford SMSA, the
models summed spendable income as though all of the income were to be
spent at MRC's. The predicted sales for the MRC's were summed, and that
sum was compared to total actual sales. A multiplier which would correct
the predicted sales was determined. The multiplier was then applied to
the predictions for each MRC so the total predicted sales would equal the
total actual sales.
The Predictions of the Models
Four sets of predictions were obtained, one each from the gravity
model, the central place model, the zone of indifference model with a 15-
minute driving time limit and the zone of indifference model using a 20-minute
driving time limit. The predictions are on Table 5, "Predicted MRC Sales Us-
ing The 3 Basic Models."
Analysis of the Predictions
By inspecting Table 5, "The Predicted MRC Sales Using The Three
Basic Models," one can judge that the gravity model was the best sales
predictor. In order to obtain a more exact comparison of the models and to
determine how "good" the models are, the predictions were analyzed using
an ordinary least squares regression equation. The equation used had the
form Y. = a + bx. + E.. Each of the predictions was substituted for x. and
1 1 1 1
(a + bxi) was then compared to Y.. Values of a and b were obtained which
minimized the sum of the squared error terms, (Ei)2. A regression coefficient,
R2, an F statistic and the level of significance of the regression equation were
calculated for each of the four sets of predictions. The results were:
Model R2 F Significance
Gravity .80 65.73 P < .001
Central Place .42 12.09 P < .003
Zone of Indifference .09 1.61 P = .222
Zone of Indifference .06 1.51 P = .299
The zone of indifference model yielded the least accurate predictions.
The model explained less than 10 percent of the variation in sales between
centers and the predictions were not significant at the five percent level.
This first test considered only driving time in allocating sales. One would
expect that if only driving time were considered, the zone of indifference
model should yield useless predictions.
Analysis of the zone of indifference model is less straightforward than
the analysis of the gravity and central place models. The basic premise of
the zone of indifference model is that shoppers are indifferent to shopping
centers within a certain driving time if only driving time is considered.
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Therefore, one would not expect the zone of indifference model to have any
explanatory validity if only driving time is considered. In a later section, at-
tributes of the MRC's which might affect the shoppers decision will be intro-
duced into the analysis and tested for effect on the predictions of the zone of
The central place model and the gravity model both produced predic-
tions which were significant at the five percent level. However, the gravity
model appeared to be a much better predictor of sales. The gravity model
explained 80 percent of the variation in sales while the central place model
explained only 42 percent of the variation.
Since both the gravity model and the central place model appear to
explain a portion of the sales variation between centers, it might be that
each model represents some aspect of the shopper's decision process. Two
possibilities seem plausible and will be briefly discussed.
The central place model emphasizes convenience, by assuming shoppers
use the nearest center. The gravity model emphasizes the size of the shop-
ping center, by assuming shoppers are attracted by size. Perhaps shoppers
make trips to the nearest center offering a good when they are certain of what
they want and are certain of what the nearest center offers. These same
shoppers may patronize the larger centers to purchase a variety of goods
(combine trips) or to explore a wide range of possible alternative purchases
when the shoppers are unsure of precisely what they plan to purchase. To
summarize, the shopper may be viewed as patronizing the nearest center for
day to day needs and then making special trips to the larger centers.
An alternative view of the decision process would be that the shopper
always purchases a certain set of goods at the nearest center and always pur-
chases another set of goods at the larger center. Expressed another way, one
might say the shopper is most strongly influenced by convenience when
purchasing one type of good and most strongly influenced by the size of
the center (or perhaps the variety of goods offered) when purchasing
another type of good. This hypothesis proposed and tested in Chapter 5 is
the more useful one for explaining MRC sales.
In summary, the zone of indifference model did not yield useful
predictions, but the model is modified and tested in a later section. The
gravity model was clearly superior to the central place model as a predictor
of MRC sales, but the central place model did appear to produce significant
results. The fact that both models seem to explain some shopping behavior
is examined more closely later in this study. Since the gravity model seemed
to be the most promising predictor, it was necessary to examine more closely
the model, and particularly the value of the distance exponent, X. The value
of A used in comparing the three basic models was 2, but this value was chosen
somewhat arbitrarily. The next section compares various values of X on a more
Analysis of the Distance Exponent, A
Reilly (1929) used 2 for the value of the distance exponent, A. He
obtained 2 by examining the number of cases explained by using various
values of A and noticed a clear mode around 2. Reilly did point out that the
value of A should be determined empirically and that A was not a theoretical
constant, as it is in Newton's law of gravity.
Ellwood (1954) used 2 as the value of the distance exponent. However,
Ellwood was applying Reilly's model to an urban area shopping center analysis
and was either unaware of the true empirical nature cf X or simply failed to
In order to test how altering the value of the distance exponent A would
affect the predictive ability of the gravity model, four different values of X
were used to predict sales in the 19 MRC's. The four values used were 0.5,
1.0, 2.0 and 3.0. The predictions obtained using different values of A are
shown on Table 6, "MRC Predictions with Varying Values of A."
The same regression equation, Y. = a + bx. + E. that was used to
analyze the predictions of the three basic models was used to analyze the
predictions of the gravity model with varying values of A. The results
A R2 F Significance
0.5 .898 150.36 P < .001
1.0 .849 95.32 P < .001
2.0 .795 65.73 P < .001
3.0 .744 49.37 P < .001
The predictions obtained with X = .05 are clearly superior to those
with X = 2.0. The respective R 's are .90 and .80, so if a value of 0.5
is used for A, approximately 90 percent of the variation in sales is explained
while using A = 2.0 explains only 80 percent.
A value of A of 0.5 would indicate that distance does not have as great
a frictional affect as it would if A = 2.0. One might expect distance to have
less effect within an urban area than in open rural country such as where
Reilly tested his model. In other words, urban residents may not discrimi-
nate between a three-mile trip and a six-mile trip the way rural residents
would discriminate between a 25-mile journey and a 50-mile journey.
The analysis of the predictions of all of the models depends heavily
on the accuracy of the statistical measures employed. The accuracy of the
statistics depends on how well the analysis complies with the assumptions
on which the statistics are based.
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Examination of the Statistical Measures
Pindyck and Rubinfeld (1976) state the three basic assumptions of
the classical normal linear regression model with the following require-
i. The model is specified as follows:
Y. B +B X2i +BXi B Xni + E
1 1 2 2i 3 3i nni 1
ii. The X's have finite mean and variance and are uncorrelated
with the errors in the model. No linear relationship exists
between two or more of the independent variables.
iii. The errors are independently distributed from a normal popu-
lation with 0 expected value and constant variance. (Pindyck
and Rubinfeld, 1976, p. 94)
Each of these assumptions was related to the regression model used in
this study to test the predictions.
Assumption i, correct model specification
For the model used in the study B1, or a, is the Y intercept of the
regression line and was determined by the least squares estimation of the
regression model. B2, or b, is the slope of the regression line and was also
fitted by the regression model. The model used complies with assumption i.
Assumption ii, no correlation between independent variables or error terms
There is only one independent variable X in the regression model so
there is no correlation between X's (no multicollinearity). The X.'s are the
predictions of the models. They are real numbers over a limited range so
they have finite mean and variance.
A problem does arise when one examines the assumption of no cor-
relation between X. (the independent variable) and the error terms. The
violation of the assumption of no correlation is also related to assumption
iii, constant variance in the error terms. The violation of the constant
error term, no error term correlation with X assumption, is termed
heteroscedasticity. Pindyck and Rubinfeld cite an example of hetero-
scedasticity that is applicable to this study:
For example, if one is examining a cross section
of firms in one industry, there may be reason to
believe that error terms associated with very large
firms have larger variances than error terms as-
sociated with smaller firms; sales of larger firms
might be more volatile than sales of smaller firms.
(Pindyck and Rubinfeld, 1976, p. 95)
Large MRC's may be analagous with large firms so a check for heterosced-
asticity was performed.
The sales information and the predictions using X -0.5 were listed
in order of descending predicted sales. If heteroscedasticity is present,
then one should be able to detect it by examining the error term, or the
standardized residual. If the standardized residual tends to be larger
for large MRC's and smaller for small MRC's then heteroscedasticity is
present. The error terms were compared by using the Goldfield-Quandt
The Goldfield-Quandt test compares the residual sum of squares
(ESS2) for the centers with highest predicted sales (in this case eight
centers) to the residual sum of squares (ESS ) for the centers with the
lowest predicted sales (again eight). In this case:
If the Goldfield-Quandt statistic (8.40) exceeds F8,8 (which is 3.23 at the
5 percent level), then heteroscedasticity is present. Since 8.40 > 3.23, the
model does exhibit heteroscedasticity. The final model was also examined
for heteroscedasticity and in Chapter 6 that model is compared to the basic
Assumption iii, uncorrelated error terms
Correlated error terms (auto correlation) are generally associated
with time series data. The predictions obtained using the models do not
involve time series estimations, so auto correlation was deemed not to be
The zone of indifference hypothesis states that factors other than
distance will influence the shopper's choice of center. The model of the
zone of indifference hypothesis did not include factors other than distance.
Since the model did not include factors other than distance, the model
should yield insignificant results. The zone of indifference model did not
produce estimates of sales that were significant at the 5 percent level.
The central place model and the gravity model produced estimates
which were significant at the 5 percent level. The central place model
had an R2 of .42, and the gravity model had an R2 of .80. Both the central
place model and the gravity model seem to predict part of the shopper's
behavior. In Chapter 6, this study tests the hypothesis that the central
place model is the best model for predicting the sales of one type of goods
sold at MRC's (convenience goods) and the gravity model is the best model
for predicting the sales of the other type of goods sold at MRC's (shopping
The value of the distance exponent, X, used in the gravity model was
2.0. This value of the distance exponent was obtained empirically by Reilly
(1929), but would not necessarily apply to gravity models used with other
data sets. To obtain an appropriate value of X for the data used in this
study, the gravity model was used with varying values of X. Each set of
predictions obtained was tested for explanatory significance and the A,
which produced the estimate with the highest R2, was chosen as the "best"
value of X. Using X = 0.5 produced estimates with the highest R2 .90,
and using the traditional A = 2.0 produced estimates with an R2 of .80.
A possible explanation for the lower value of A is that distances within an
urban area are short and travel is relatively easy so the urban shopper
views distance as a relatively weak frictional force. The rural shopper,
who Reilly (1929) studied, traveled greater distances, used a less conven-
ient mode of travel and, therefore, might view distance as a relatively
strong frictional force. Obviously, a lower value of A will cause distance
to have less impact on the prediction of the model.
THE WEIGHTED GRAVITY MODEL
In this chapter some attributes of MRC's which have not been included
by other researchers into the basic gravity model are examined. A model is
developed and tested which uses a weighted square footage as the force of
First, the MRC attributes which were discussed in Chapter 3 were
tested for correlation with MRC sales level. Next, these attributes were used
to "weight" the size of each MRC so that MRC's which rate high in attributes
show a relative increase in size (and hence in attractiveness). Finally this
new model was tested and the results analyzed.
Additional MRC Attributes
In order to test the proposition that factors not included in the models
could explain the variations in sales, other factors were introduced into the
analysis. These factors were defined in the section of Chapter 3 labeled "MRC
Attributes." The attributes considered were:
Size Area in square feet
Number of stores Convenience and shopping goods stores
Type of center Open or enclosed
Parking cost Metered or free
Parking adequacy Inadequate or adequate
Appearance Unattractive, average, or attractive
Accessibility Difficult, average, or easy
Neighborhood Bad or good
Regressions were run to test the importance of the added variables.
The dependent variables in the regressions were:
1. Actual sales
2. Actual sales minus predicted sales from the gravity model
3. Actual sales minus predicted sales from the central place model
4. Actual sales minus predicted sales from the zone of indifference
model (15-minute limit)
5. Actual sales minus predicted sales from the zone of indifference
model (20-minute limit)
Three different sets of independent variables were used in the regressions
against the dependent variables. These independent variables were:
List I Size, type of center, parking cost, parking adequacy,
appearance, accessibility, neighborhood and number
List II The same variables as List I each variable (except size
and number of stores) were multiplied by the size of the
List III Type of center, parking cost, parking adequacy, ap-
pearance, accessibility, and neighborhood.
The variables in List I were chosen to include all variables obtained in the
inspection of the MRC's and to ascertain which variables were most important
in explaining the variations in sales. List II was chosen to explore the effect
of center size on the explanatory power of the variable. List III was chosen
to eliminate completely size of MRC from the analysis.
The variables were assigned values as follows:
Size Actual adjusted square feet in MRC
Number of stores Actual number of stores selling convenience
goods and shopping goods
Type of center Open = 0 enclosed = 1
Parking cost Metered = 0 free = 1
Parking adequacy Inadequate = 0 adequate = 1
Appearance Unattractive = 0 average = 1 attractive = 2
Accessibility Difficult = 0 average = 1 easy = 2
Neighborhood Bad = 0 good = 1
Table 7, "Analysis of Regressions," indicates the results of the regressions.
Analysis of the Regression
The first regressions used actual sales as the dependent variables.
Using List I for the independent variables produced an overall R2 of .96.
Size produced the highest individual R2 and accounted for .88 of the varia-
tions. Only two centers were of the enclosed type, which may account for the
unexpected negative sign. The regression using List II produced an R2 of
.98 and again size was the most important factor, even though all of the other
variables were multiplied by size. The most interesting aspects of the first
two regressions are the tremendous impact of size and the relatively high
R2's obtained (.94 and .95) after entering only three variables.
The variables in List III are interesting because no measure of size was
included with these variables (Both size and number of stores were deleted).
Parking cost and neighborhood both produced negative betas. The negative
betas were both unexpected and are not easily interpreted. A possible
explanation of the unexpected signs may be related to the small number of
cases of 0 for each variable (6 of 19 for neighborhood and 3 of 19 for parking
cost). The other four variables (accessibility, type, parking adequacy and
appearance) have the expected sign and do contribute to the explanatory
power of the regression equation. The regression using List III does show
vividly the importance of size as a variable. Without a size variable the
regression obtained an R2 of only .48 while the regressions which included
size reached R2's of .96 and size alone produced an R2 of .88.
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