# A Generalized Utility Model of Shopping Behavior

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Jimmy E. Hilliard, Ronald L. Vaughn, and Fred D. Reynolds (1975) ,"A Generalized Utility Model of Shopping Behavior", in NA - Advances in Consumer Research Volume 02, eds. Mary Jane Schlinger, Ann Abor, MI : Association for Consumer Research, Pages: 157-164.

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http://acrwebsite.org/volumes/5755/volumes/v02/NA-02

[Data for this study were provided by the Urban Affairs Institute, Bradley University.]

Since the 1920s there has been a continuing interest in building and testing formal models of consumer spatial behavior. This paper continues the model building tradition by developing a generalized model of spatial behavior and by presenting a preliminary test of the model in the context of intraurban clothing shopping.

Consumers engage in many decisions and activities in many different settings and situations. One almost inescapable activity, however, is spatial mobility -- an acknowledgement to Detroit. In the 1930s and 40s the auto liberated the shopper. And, free to roam, shoppers have traveled to the stores, shopping centers, and even cities that offered them what they wanted when they wanted it.

Spatial mobility, geographical population redistribution and other consumer and competitive factors have caused the framework of retailing to maintain an almost constant state to flux. To survive and prosper, retailers must attract shoppers and convert them to buyers. To do so requires, in part, an understanding of consumer spatial behavior or at least the ability to predict that behavior -- the subject of this paper.

THE SITUATION

The opening of Northgate, a regional shopping center in Seattle, in 1950 launched a retailing innovation that rapidly diffused to other metropolitan suburbs during the 1950s and 60s. Given new shopping alternatives, consumers changed their spatial behavior and it became necessary for students of that behavior to explore new bases for predicting it. Some researchers tried Reilly's formula, substituting square footage of selling space and driving time for population and distance. Generally, however, the results were mixed and unsatisfactory. Other researchers have followed the lead of Huff who formulated a spatial model based on Luce's choice axiom (Huff, 1962; and Luce, 1959).

Huff's central thesis is that consumer spatial behavior is best described as a probabilistic phenomenon, which can be modeled in terms of consumers' perceived utilities or benefits of alternative shopping centers. That is, the probability of a consumer choosing a given shopping center is equal to the ratio of the utility of that center to the combined utilities of all centers under consideration.

SOME PROBLEMS

The spatial model developed by Huff has several appealing features: (1) it is based on a theory of individual choice behavior; (2) it incorporates variables that are readily obtainable, and (3) it can be applied in a straightforward manner to predict intraurban choice and to calculate sales potentials for existing and proposed center locations. Nevertheless, there are several problems related to the full utilization of the model including appropriate measures of utility, appropriate parameter estimation procedures, population heterogeneity, and inaccurate prediction in certain situations. Only the latter is examined in this paper. [A more comprehensive examination of the problems is found in Hilliard and Reynolds (1973).]

Huff noted in his pilot test that the model consistently obtained poorer predictions in situations where more than one shopping center were in close proximity. This problem may arise because the model assumes independence between centers. That is, the model does not allow for the perceived utility of one center to be affected by its position relative to another center -- a notion inconsistent with the concepts of complementary and substitute centers.

This paper attacks the problem of center independence by presenting preliminary findings of a comparative study of the Huff model and a more general utility model that allows for any existing interaction effects of centers to affect the calculation of consumers probabilities of shopping center choice. The test obviously is limited in that the other problems associated with operational choice models are ignored.

THE GENERALIZED UTILITY MODEL

In thinking of generalized utility, consider first the somewhat analogous n-body problem in planetary attraction. Here the force vector in a given direction acting upon a point mass depends not only on the mass and distance of the body lying along that vector but upon the mass (and distance) of all surrounding bodies. Specifically, in the generalized n-body model, forces generated by all bodies are summed in vector fashion to yield the direction and magnitude of the net force vector and from this components of force in any direction may be obtained. Thus the concepts of planetary attraction are consistent with interdependence among surrounding bodies. We do not propose to represent the shopping process as being completely analogous to planetary attraction. However, consideration of the n-body model suggests an important generalization of the two-body model as posited by Huff.

In developing this model we denote first the intrinsic drawing power of center j by I-, assumed to be independent of other centers. However, a center's net drawing power potentially is enhanced by substitute and complementary contributions from neighboring centers. Also, the assumed contribution is attenuated by the disutility or cost between centers. Thus, the mean utility model is now hypothesized to be

U

_{j}= a[I_{j}+ S_{i=j}I_{i}/D_{ij}]/D_{j}(1.0)

where:

Dij is the perceived cost of travel between centers i and j,

Dj is the perceived cost of travel from origin (home) to site j,

and Ij + S_{i=j}I_{i}/D_{ij} is the net drawing power of center j.

The general model (1.0) can be operationalized with a number of empirical measures. Intrinsic drawing power, Ij, typically has been approximated by a size proxy such as the square footage of selling space in a center while cost, Dj, has been measured by driving time or geographical distance. The impetus for using these measures has been, of course, that they are obtainable on an objective, non-obtrusive basis at less cost than other measures which require field surveys. It should be clear, however, that other measures could be used to operationalize (1.0). For example, measures of consumers' perceptions and preferences through non-metric multidimensional scaling or conjoint measurement could possibly be used to find utility scales that would provide a more epistemic isomorphism between the theory and its empirical operants. Furthermore, the parameter estimation scheme which we develop can in theory accommodate any composition of measures making up Ij or Dj. For further exposition, however, we operationalize (1.0) using the more traditional measures as this allows a more direct comparison with current models.

Thus, using square footage (S) as a measure of Ij and distance (d) as a measure of Dij and Dj yields the operational estimation

U

_{j}= a[S_{j}+ S_{i=j}S_{i}/d_{ij}^{l2}]/d_{j}^{l1}(2.0)

Using the more traditional measures makes it easier to see that this formulation is a generalization of the two-body gravitational model of consumer shopping behavior. For example, suppose l2 is large. This suggests a high disutility or cost in traveling between centers (assuming dij>1), and in such a case Uj reduces approximately to the two-body formulation (exactly as dij or l2 -> *. Empirically, the n-body model could be used as a test statistic for the two-body model in the sense that large estimates of l2 would tend to support the two-body formulation and small values would tend to refute it (assuming that Sj and dj are effective proxies for utility and disutility).

The chief advantage of generalizing the two-body model to a multiple center one is that the n-body model must have at least equivalent predictive power and yet it allows for cases where consumers perceptions of the utility of a center cannot realistically be assumed as independent of other centers. The primary disadvantage is the incorporation of the additional parameter l2 and attendant parameter estimation complications.

THE STUDY

Data for this study was obtained from the Peoria, Illinois trading area. The central business district and five shopping centers were specifically focused on with shoppers selected systematically from each center. The square footage estimated for the CBD was 2,100,000 while corresponding estimates ranged from 100,000 to 750,000 square feet for the centers. The sample size for each center varied from 350 to 625 respondents. By combining selected census tracts, a total of 25 residence areas were defined. Spatial disutility was approximated by Euclidian distance (units in miles). The basic responses modeled were clothing intentions data, e.g., "I came here to shop for clothing" obtained via personal interview.

Estimates of l1 and l2 were obtained in this survey by applying the maximum likelihood technique to the probability model

P

_{j}= U_{j}/S_{j}U_{j}, (3.0)

where Uij is given by (2.0). Operationally, the minimum of the negative of the log likelihood function was obtained by a conjugate gradient method (the FMCG subroutine in the IBM Scientific Subroutine Package). Thus the function

L = -S

_{i}S_{j}N_{ij}log_{e}[P_{ij}(l_{1}l_{2})]. (4.0)

was minimized with respect to l1 and l2 where Nij represents the observed number of shoppers from area i ln center j and Pij (l1, l2) is the theoretical probability function given in (3.0).

THE RESULTS

Table 1 depicts observed probabilities and those estimated by the two-body and n-body models. The natural log of square footage was used in the n-body model as a proxy for utility. While there are substantial deviations from the observed probabilities for both models, the n-body model clearly fits much better. For example, the sum of squared deviations from observed probabilities (SSD) was 7.442 for the two-body model and 2.045 for the n-body model. Obviously, the disproportionate size of the CBD distorts true utility. Thus the SSD differential is due both to the additional parameter of the n-body model plus the log transformation of utility.

Table 2 is a comparison of the two-body and n-body models where log square footage is used for both models. The results here are quite similar, though the n-body model exhibits a slight superiority by the SSD criterion -SSD = 2.045 versus 2.790 for the two-body model.

The estimates of the parameter l1 were quite similar for both models. When the log transformation was used, l1 was 1.552 for the two-body model and 1.514 for the n-body model. The estimate for the two-body model without the log transformation was 2.272. This is consistent with expectations since the log transformation attenuates the numerator and the smaller value of l2 reduces the value of the denominator.

The estimates obtained for l2 were unexpected. According to the rationale developed earlier, the added contribution to utility by a center increases as distance decreases (since 1/dij l2 increases as dij decreases). However, our estimates of l2 were negative. For the log n-body model l2 was estimated to be -1.803. The most obvious explanation for this result is that, in effect, center interation reduces utility. Indeed, we posited that U(A+B) > U(A) + U(B), where, for example, U(A) is the utility of center A. Instead, this suggests the possibility that, essentially, U(A+B) < U(A) + U(B). Strictly speaking, nearby centers must add utility under our formulation, but the negative exponent implies that more utility is added by distant centers -thus, except for an additive constant, the implication is that nearby centers subtract more utility than distant centers.

EMPIRICAL, TWO-BODY, AND N-BODY PROBABILITIES: N-BODY SQUARE: FOOTAGE LOG TRANSFORMED

EMPIRICAL, TWO-BODY, AND N-BODY PROBABILITIES: SQUARE: FOOTAGE LOG TRANSFORMED FOR BOTH MODELS

While the interaction effect explanation is plausible, the role of the CBD in our estimates perhaps offers the more likely interpretation. It is likely that even under the log transformation the utility of the CBD was overstated. In attempting to compensate for this, the role of l2 was to further minimize the effect of the CBD. Since the CBD is near the centroid of the shopping area, the distance to other centers is, on the average, smaller than the analogous statistic for other centers. Thus, a negative exponent l2 makes distance a positive utility factor and the effect is to increase the utility of outlying shopping centers relative to that of the CBD.

Some intuitive modifications may possibly correct the overstated utility of the CBD. For example, the hours of operation were much shorter than those in shopping centers. Thus a likely utility proxy is square footage times hours per day of customer access; i.e., square footage shopping hours. Also, the ratio of parking space to square footage is a possible utility attenuating factor. Taken together, these modifications may provide a better estimate of utility and suggest the role of l2 in quantifying center interdependence.

MODEL EXTENSIONS

A more comprehensive three parameter model is presently under development. The probability equation in this model follows from the Luce choice axiom. However, utility is cast in the form

U

_{j}= log_{e}(S_{j})[p_{i=j}exp(-alog_{e}(S_{i}))/d_{ij}^{l2}]/d_{j}^{l1}= log

_{e}(S_{j})[1/p_{i=j}S_{i}^{a/dijl2}]/d_{j}^{l1}. (4.0)

The rationale for this model is that each multiplicative term 1/p_{i=j}S_{i}^{a/dijl2} either attenuates or magnifies the utility of center j depending on the value of a. That is, a < 0 gives a value greater than one for this term while a > 0 yields a value less than one. a = 0 is consistent with center independence since the multiplicative term reduces to one in this case. Increasing distance, dij, and l2 diminishes the center interaction effect.

REFERENCES

Hilliard, J. E., and Reynolds, F. D. A multiple center gravitational model of consumer shopping behavior. Unpublished paper. University of Georgia, 1973.

Huff, D. L. Determination of intra-urban retail trade areas. Los Angeles: Division of Research, Graduate School of Business Administration, University of California, 1962.

Luce, R. D. Individual choice behavior. New York: John Wiley and Sons, 1959.

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