The Gravity Model: a Study of Retail Goods Classification and Multiple Goods Shopping Effect
ABSTRACT - This theoretical paper deals with consumer trade area or store choice factors as reflected in spatial interaction models. Specifically, the Wilson Model is examined to attempt to explain what the coefficients a and b represent. The empirical manipulation points toward a strong tie between a and the traditional view of goods classification. Similarly, b is found to represent multiple goods shoPping effect.
Robert S. Ellinger and Jay D. Lindquist (1984) ,"The Gravity Model: a Study of Retail Goods Classification and Multiple Goods Shopping Effect", in NA - Advances in Consumer Research Volume 11, eds. Thomas C. Kinnear, Provo, UT : Association for Consumer Research, Pages: 391-395.
This theoretical paper deals with consumer trade area or store choice factors as reflected in spatial interaction models. Specifically, the Wilson Model is examined to attempt to explain what the coefficients a and b represent. The empirical manipulation points toward a strong tie between a and the traditional view of goods classification. Similarly, b is found to represent multiple goods shoPping effect. INTRODUCTION When examining consumer trade area or store choice behavior the factors of location, in terms of distance to be traveled or time for such a trip, and size, in terms of population or store counts, have often been addressed. A family of spatial interaction models have been developed to offer theoretical constructs that could be empirically evaluated. An early attempt at such a model was offered by Reilly (1929). His well-known thesis was that two cities attract patronage from an intermediate city in direct proportion to their population and in inverse proportion to the square of the distances from the intermediate city to the other two. This type of model has been referred to as a "gravity" model because of its relationship to Newton's gravity equation as found in the study of physical science. "Reilly's Law," as this theory became known, was modified by Converse (1949). The latter author suggested that the distance ratio no longer need be squared. Hence his thesis was that the proportional trade ratio was directly proportional to the population and inversely proportional to the distance. Although trade area bounding is of interest to those who study retail distribution, another question that arose from this body of work was concerned with the probability that a specific consumer would be attracted to one trading area over another. Huff (1962, 1963) addressed this latter question and proposed the model described algebraically in Equation 1. Where: Pij is the probability that consumers from each of the ith statistical units will go to specific shopping center t. Sj is the size of the shopping center j. Tij is the travel time to shopping center j. p
This theoretical paper deals with consumer trade area or store choice factors as reflected in spatial interaction models. Specifically, the Wilson Model is examined to attempt to explain what the coefficients a and b represent. The empirical manipulation points toward a strong tie between a and the traditional view of goods classification. Similarly, b is found to represent multiple goods shoPping effect.
When examining consumer trade area or store choice behavior the factors of location, in terms of distance to be traveled or time for such a trip, and size, in terms of population or store counts, have often been addressed. A family of spatial interaction models have been developed to offer theoretical constructs that could be empirically evaluated. An early attempt at such a model was offered by Reilly (1929). His well-known thesis was that two cities attract patronage from an intermediate city in direct proportion to their population and in inverse proportion to the square of the distances from the intermediate city to the other two. This type of model has been referred to as a "gravity" model because of its relationship to Newton's gravity equation as found in the study of physical science. "Reilly's Law," as this theory became known, was modified by Converse (1949). The latter author suggested that the distance ratio no longer need be squared. Hence his thesis was that the proportional trade ratio was directly proportional to the population and inversely proportional to the distance. Although trade area bounding is of interest to those who study retail distribution, another question that arose from this body of work was concerned with the probability that a specific consumer would be attracted to one trading area over another.
Huff (1962, 1963) addressed this latter question and proposed the model described algebraically in Equation 1.
Pij is the probability that consumers from each of the ith statistical units will go to specific shopping center t.
Sj is the size of the shopping center j.
Tij is the travel time to shopping center j.
pis a parameter estimated empirically for each product category.
The construct is used to estimate the probability that consumers in certain relatively homogeneous statistical areas, such as a neighborhood, will go to a certain shopping center to purchase a certain type of product.
Another viewpoint, tied to the probability that a consumer would patronize one of two or more market areas, was offered by Bucklin (1971). In this case the determining factors were relative attraction and distance the consumer had to travel to reach the market area. Equation 2 describes the Bucklin concept.
Pik is the probability that a customer located at point i will visit a market at point k.
"j is the attraction of area 3.
Dij is the distance of the market from point i.
pis the exponent applied to the distance value and is related to the product category.
A model attributed to Gerry R. Wilson is similar to Bucklin's. However, size of market rather than "attraction" was an inherent part of the construct and instead of a single exponent only applied to the distance parameter, he introduced exponents for both distance and size. Equation 3 reflects his Posture.
Pijm is consumer j's preference for good i in market m.
Sm is the size of market m.
djm is the distance from consumer j to market m.
ai is referred to as the spatial attractiveness of good i and is the exponent to which the distance measure is raised.
bm is termed attractiveness of the good i and a m the exponent to which the market size is raised.
When the model is examined, the terms that pertain to size and distance cause little concern. However, the empirical determination of the values of a and b and their subsequent interpretation calls for attention.
The meanings of empirically-determined variables can be inferred from observation of causes and effects among the variables. This could be tone in one of two ways. First, one could demonstrate that there is some other observable variable whose effects on the interpreted variables of the model are similar to those effects produced by the uninterpreted variable. Second, one could demonstrate that the variation of the uninterpreted variable produces a result that we expect from the motel. This latter approach will be followed by the authors.
For the model at hand, relationships between the values of a and retail goods classifications and b and multiple goods shopping effect will be explored. The logical question is why look at these two theories to find meanings for a and b. First, let us look at the potential a - goods classification tie. The convenience, shopping and specialty classifications have been linked to effort levels (American Marketing Association 1960; Holton 1958) and probable gains associated with comparison shopping (Holton 1958). Both of these factors have implications for distance to be traveled ant/or the size/concentration/variety of establishments. This then may tie directly to the Wilson construct. The classification of goods as convenience, shopping or specialty is controversial. Copeland (1932) first suggested these three terms and defined them basically as a function of effort linked to the consumer's desire for the goods. Minimum effort was associated with convenience goods, additional effort was used for shopping goods and "special purchasing effort" was exerted for specialty goods. Holton (1958) challenged these definitions and asserted that consumer buying went beyond strength of desire to include the perceived costs of shopping. Simply put, the consumer continues to shop for all types of goods until the costs outweigh the incremental satisfaction gained from further search.
Bucklin (1963) offered a new classification that incorporates the concept of the "preference map." (The preference map was suggested earlier by Bayton (1958)). Bucklin (1963) summarizes the construct as a rough ranking of the relative desirability of the different kinds of products that the consumer sees as possible satisfiers for his needs. His classification of goods is as follows: Convenience Goods are those goods for which the consumer, before his need arises, possesses a preference map that indicates a willingness to purchase any number of known substitutes rather than to make the additional effort required to buy a particular item; Shopping Goods are those goods for which the consumer has not developed 8 complete preference map before the need arises, requiring him to undertake search to construct such a map before purchase; Specialty Goods are those goods for which the consumer, before his need arises, possesses a preference zap that indicates a willingness to expend the additional effort required to purchase the most preferred item rather than buy a more readily acceptable substitute. It was also suggested by Bucklin (1963) that the classification of goods may be extended into the retailing area where outlets could be labeled as convenience stores, shopping stores or specialty stores.
It is appropriate at this time to acknowledge that a good may have differing classifications for the same individual over time based on such factors as need urgency, availability of substitutes, ease of want satisfaction, desirability and the like. Further the same good wag generally be in different categories for different people. However, such caveats should not be the basis for reluctance to pursue the whole classification of goods dilemma. Let us now return to an examination of the relationship between a and the classification of goods.
The empirical determination of a and the theoretical interpretation of its values have been the source of controversy. The controversy arose when empirical determinations of a turned out to vary at differing scales or regions.
Clark and Rushton (1970) indicated that the distance between alternative markets has a major impact on the value of a for the same goods when a gravity model is used to compare intraurban and interurban regions. Bucklin (1971) in a more theoretically-oriented article asked whether shoppers would be more likely to move across "normal" trading area lines if the distances were short than if they were long. The reply to his own question was. "Yes."
A second problem with the determination and interpretation of a is caused by the way it is calculated. This leads to variations based on market patterns. Ewing (1974) noted that when a is calculated in the regression form of the gravity model, if the distances from the intermediate location are equal, the migration probabilities (proportions attracted) will rarely be equal. Also, since there are a diversity of origin points in the typical aggregate gravity model, any value of a may be a distortion, the size of which may be unknown. Hence the value of a found may be in question. Cesario (1975, 1976) also found that the model form used affected the magnitude of a. Until this issue is resolved, attempts to tie * values to specific products may be of little value. However, the gravity model in all studies (Golledge 1966, Johnson 1971, Clark 1970, x 11 1974, etc.) has been a better predictor of consumer activity than the classical least effort model. The latter model is based on the assumption that a consumer will always go to the closest location to purchase or receive goods. It was used as a simplifying assumption by all early regional-economic models of the 19th Century and continued to be used in such major works of normative modeling as Losch's The Economics of Location (1945). In most of these works the assumption was used by implication only; but it made sense When travel costs, both in time and money. were significant. The least effort model is a submodel of the gravity model. Using the Wilson version, it would be the construct that results from an a value of infinity and a b value of 1.
Though the tie between a specific a and a specific product is difficult to find at this time (as noted earlier), a tie of a values to classes of goods seems feasible. The research presented in this article was designed to do just that
Let us now turn to the potential b - multiple goods shopping effect. The reader will recall that b refers to market attractiveness and is attached as an exponent to the size of the market. Bender (1964) noted that there are a number of "secondary costs" that can be associated with a market that will increase or decrease its attractiveness. Among the costs he noted were time-travel, wait and search and psychological costs. Examples of the latter would be the cleanliness of a market and the friendliness of the sales clerks. One of the best known of these "costs" is the reduction in cost caused by the multiple goods shopping effect. The consumer exhibits this type of behavior when he or she proposes to purchase more than one good in a single trip. If this consumer can go to a market and purchase two or more goods rather than one, he or she can then average the total transportation cost over several purchases rather than just one good. Another view is that travel costs for the second good are zero because the trip had to be made for the first good anyway. If comparison shopping is to be done and several comparisons can be made in the same market, search time is saved. This can also be deducted from the total cost of all the goods purchased on that trip. In this case the consumer may desire to go to a larger market that is farther away rather than to a closer, smaller market. This type of behavior was noted by Golledge, Rushton and Clark (1966) in their study of Iowa's rural population. Similarly, Berry (1967) observed that, "...the larger places with greater cumulative accessibility draw in consumers from longer distances" and "...the market areas of higher level centers are greater than those of lower level centers for the same order of goods." These clues from the literature point toward a possible link between multiple goods opportunities for shopping and b.
The first objective of this study is to attempt to determine if there is a relationship between the Spatial Attractiveness of a Good, a , and the concept of Classification of Consumer Goods. The second objective is to seek the relationship between the Market Attractiveness of a Good, b, and the multiple goods opportunities concept.
The hypotheses to be tested are:
H-1: As the value of Spatial Attractiveness, a , increases (holding Market Attractiveness, b, constant at a value of 1) consumers will exhibit behavior that moves from that typical of specialty goods effort toward that typical of convenience goods effort.
H-2: As the value of Market Attractiveness, b, increases (holding Spatial Attractiveness, a, constant at a value of 1) consumers will exhibit an increase in responses typical of those people interested in multiple goods opportunities.
The methodology used to test these hypotheses does not require a direct measure of a consumer's behavior. Rather, the spatial effects of that behavior can be compared with the spatial effects of the consumer's behavior predicted by the model to at least partially validate the hypotheses, thus the model.
Initially, the predictions of the mathematical formulation of the model will be compared with the predictions from the hypotheses; if they don't match then the hypotheses related to the model may be rejected. If the predictions match, then the next step is to tie the predictions to logical patterns of consumer behavior. If no logical explanation of the hypotheses can be found then the model should be rejected. Finally, if the hypotheses prove to be mathematically and logically sound, they should be tested against the best available evidence, found in the literature, to determine if the model is in contradiction to any current empirical findings.
To accomplish the task of checking the predictions of the mathematical model against the hypotheses a computer program of the model was constructed. Output from the program was plotted into the figures that follow in this article. The particular model used in this test was two dimensional, one spatial and one economic. Several assumptions were made; first, there are two markets one hundred units apart; second, there are no other markets; third, one center is twice the size of the other; fourth, the consumers are evenly distributed over the landscape. These assumptions were made to simplify the situation sufficiently to allow the spatial outcomes of the consumer's behavior, classified by the model, to become self-evident. At most sites these variables (population, transportation and topographic features) are nonuniform and the economic noise generated by them can mask the predictions of the model. For these first steps in the evaluation process it is relatively important to see how the model works in complex situations, if it does not work in simple situations. In future work these environmental assumPtions can be relaxed allowing the model to be compared with actual results.
One reason for choosing to have the two markets not be of the same size was to show variations in the consumer behavior patterns resulting from difference in market size. A second reason is the recognition that the normal scenario for a consumer is to choose between markets of differing sizes and giving the choice between two "equal" sized markets would be inappropriate.
RESULTS AND CONCLUSIONS
Figure 1 shows the effect a change in a has on the probability that a consumer will go to a certain market. Two things should be noted. First, as a increases, the general slope of the line increases. Second, as a increases the sloping line has a tendency to drift to the midway point between the two markets; which position say be viewed as "shopping in the home area." One say also see from the plot that with a large value of a (a = 10) and a constant b (b = 1), the curve begins as a horizontal line at
100 percent (of customers going to Market 1) extending to near the mid-point where it drops precipitously to zero percent. This type of finding is virtually the same as that found if a least effort model were applied to the situation.
CHANGE IN CONSUMER DEMAND WITH A CHANGE IN ALPHA
Figure 1 also shows that as a increases the consumer becomes more aware of distance in his or her spatial preference for a good. Be or she will be less willing to travel a greater distance to purchase the good, that is, the nearest market will be sought. In light of the Bucklin (1963) goods classification definitions one may theorize that a good with a high value of a could be classified as a Convenience Good. Again referring to Figure 1, one notes that as the value of a decreases, the consumer becomes increasingly less distance sensitive. Bence the acceptance of greater levels of effort begin to emerge. One could theorize from these findings that goods with lower a values would be classified as Shopping Goods or Specialty Goods. Most likely, goods with the lowest Spatial Attractiveness values wouLd be in the specialty category and those with higher values in the shopping category. This leads to the conclusion that Hypothesis B-L may be accepted.
Figure 2 shows the effects that changes in b have on the probability that a consumer will go to one or the other of the two markets with a being constant (a = 1). As the value of b increases, the probability that a consumer will go to the larger market increases. Also, as the slope of the curve increases the zone in which the consumer exhibits indecisive location behavior decreases. This means that the delimitation of the market region boundaries for these goods is easier and the transition zone between markets will be reduced. By the time b becomes large (b > 10), the probability of the consumer going to the smaller market is near zero. In summary, as b increases the consumer exhibits increasing attraction to larger markets. Such markets generally allow for multiple goods shopping as one of their assets. This leads to the conclusion that Hypothesis H-2 should be tentatively accepted.
CHANGE IN CONSUMER DEMAND WITH A CHANGE IN BETA
One informal spatial-economic theorem can be derived from the demonstration of the correlation between the effect of a on the model and convenience-comparative shopping behavior. Market researchers have long noted that where consumers exhibit convenience shopping behavior for a particular good, the firms producing that good tend to disperse; while when the consumers exhibit comparative shopping behavior, the firms producing and/or selling that good tend to congregate. If this were interpreted in terms of a it would indicate that for a good with a high a, an entering firm would try to find a new location. This would result in or reinforce a pattern of one firm or outlet per market. For a good with a low value of a, new firms would attempt to locate near existing firms forming agglomerations. Thus the consumer's behavior in the gravity model could strongly affect the pattern of firm locations in markets being dependent on the value of a.
Multiple goods shopping has major consequences for spatial marketing because it is one of the basic forces causing the development of markets. Since the consumer will spend less on transportation, he or she can afford to travel slightly farther and purchase more of the goods (and possibly still save money). For example, assume that a consumer needs two goods; if he goes to a market that offers both of these goods, he will spend less on transportation, and he may spend more for the goods 80 his demand function could be higher. If two firms offering these goods can locate near one another then they will significantly increase their market demand, it could be argued. Christaller (1933) asserted this and pointed to higher profits or at least the ability to stay in business because of this proximity phenomenon. Hence locating two or more firms near one another which will then form a larger market has the potential to benefit both the customer and the complex of firms. We see from this that firms producing/selling goods in an environment where consumers exhibit behavior similar to that of the model with a high b, would tend to agglomerate.
A second consequence of multiple goods shopping is the development of content hierarchies for those firms benefiting from the multiple goods shopping effect. A content hierarchy of goods for a set of markets was a concept first discussed by Christaller (1933). Essentially, it states that there is a hierarchy of services -- a village will contain all the goods and services of a hamlet plus some additional; a town will contain all of the goods and services of a village plus some additional, etc.
The content hierarchy develops, in part, because of the threshold demands of the various firms in the market. If a firm sells enough to cover its costs it is breaking even. In order to "sell enough" it has to have enjoyed some minimum demand for its goods--this demand is the "threshold demand."
Because multiple goods shopping increases market demand, a firm with a large threshold demand can sake the greatest profit when it locates near a firm or firms with slightly lower threshold demands. Consequently, a content hierarchy will develop. If firms with lower order threshold demand subsequently locate in the market where the firs with the higher threshold demand is located, then the content hierarchy will remain unchanged. A large number of studies can be cited in support of the existence of content hierarchies. Berry (1967) found a content hierarchy in Iowa and a tendency toward a spatial hierarchy. Skinner (1964) found in a study of China that not only did spatial and content hierarchies exist, but rural periodic markets (the smallest markets) also formed part of the system. Abiodum (1967) found a content hierarchy for central goods, particularly for public goods, in Nigeria. The spatial hierarchy, however, was missing or at least masked in the latter country. Carter et al. (1970) found the content hierarchy extant in Welsh towns, but again failed to find spatial hierarchy. They traced this lack of spatial hierarchy to the population distribution and to historical and physical disturbances. Johnson (1971), concentrating on dental services, found overall uniformity in the content hierarchy of central places in New England and Preston (1971) found a content and spatial hierarchy in the northwestern United States. One reason that there were difficulties in finding spatial hierarchies in some of these studies could be that consumers need not exhibit multiple goods shopping and market attractiveness behavior for all goods and markets. Bell, Lieber and Rushton (1974), in a study of location of firms in Minnesota and Iowa, posited and empirically supported just such a pattern. They suggested that certain types of goods and services would be found in a state of agglomeration, forming a content hierarchy, while others would be located so as to minimize distance only. In gravity model terms, some goods would have a low a and the market or firm a low b, or a low a and a high b. Firms producing/selling these goods or services would ten to agglomerate and form a content and even a spatial hierarchy. Consumers could have a high for some goods. In this case firms producing/selling goods with a high a and a low b would always locate as far as possible from competing firms leading to the hexagonal pattern found under classical ideal conditions. They say locate at a market near the center of the pattern without much loss of market ares, as discussed by Goodchild (1955). This may happen for a variety of reasons associated with costs, but not with consumer demand. It is conceivable that in locating, some of these types of firms will actually fall into one order of the content hierarchy, but it is not necessary. It is equally possible, and perhaps more probable, that some of these firms will seem to be random in the content hierarchy, that is, sometimes location will be in large towns and sometimes in small towns.
A final case of the gravity model, where both a and b are high remains to be considered. Prediction of where firms will locate under these conditions is immensely difficult. As can be seen from Figure 3; with a high a , i.e. the consumer going to the nearest market, large changes in b appear only to move the location of maximum slope slightly toward the smaller market. This suggests that the firms producing or selling the goods will not be a part of the content hierarchy because the effect of b seems to dominate. But much more investigation of the gravity model would be needed to establish when the effects of * and when the effects of * would dominate. However, from the evidence presented above, the need for a * in the gravity model has been demonstrated.
CHANGE IN CONSUMER DEMAND WITE A HIGH ALPHA AND A CHANGE IN BETA
Little reason has been found for rejecting this model, rather much support has been found in several seemingly disparate lines of research. However, additional work is called for. For example, the mathematics of and need careful scrutiny to establish the fact that no two combinations of and will give exactly the same results. Empirical work should be pursued to determine and values for given classes of goods. This could be done through research on the spatial patterns of goods purchase within varying cultural regions and situations. Also, cultural variance in spatial preference for various classes of goods could help to determine how successful the potential location of a new type of retail firm would be. Additional work needs to be done on the refinement of the threshold demand concept and what its impact would be on how closely establishments should be placed together. These are but a few of the possible options for future endeavor.
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Robert S. Ellinger, Western Michigan University
Jay D. Lindquist, Western Michigan University
NA - Advances in Consumer Research Volume 11 | 1984
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