# Conceptual and Operational Problems With Market Share Models of Consumer Spatial Behavior

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David L. Huff and Richard R. Batsell (1975) ,"Conceptual and Operational Problems With Market Share Models of Consumer Spatial Behavior", in NA - Advances in Consumer Research Volume 02, eds. Mary Jane Schlinger, Ann Abor, MI : Association for Consumer Research, Pages: 165-172.

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http://acrwebsite.org/volumes/5760/volumes/v02/NA-02

The development of consumer spatial behavior models has been limited principally to those of the market share type. Such models have received widespread usage from academicians as well as practitioners. It is apparent from a number of these applications that the conceptual properties of this type of model are not well understood. This is evident from some of the problems to which this type of model has been applied, and from the questionable interpretations that have been given to the output.

As a consequence, the purpose of this paper is to re-examine the major behavior postulates associated with market share type models of consumer spatial behavior. In addition, some operational difficulties that are often incurred in implementing the model will be discussed. Finally, a recent development using standard econometric methods of parameter estimation for this type of model is discussed.

BASIC CHOICE AXIOM

Most models of the market share type depict consumer choice behavior in terms of Luce 1 5 basic choice axiom (Luce. 1959). The axiom is:

where:

PT(x) is the probability of an individual choosing alternative x from a finite set of T alternatives; T is a subset of some universal set U; and, v(x) is a positive real-valued function v on T.

Thus, the basic proposition of this axiom is that choice behavior is best described as a probabilistic, not an algebraic, phenomenon. When a consumer reaches a decision between alternative a and b, it is assumed that there is a probability P(a,b) that the choice will be a rather than b and this probability will generally be different from 0 and 1. The consumer is assumed to be able to evaluate the elements of U along some comparative dimension and to be able to select a finite subset of T alternatives. Furthermore, for situations in which pairwise choice discrimination is imperfect, this choice axiom implies that the ratio of the probabilities of any two choice alternatives is constant and independent of any assumptions about the structure of the set of alternatives.

In addition, the choice axiom complies with the notion of transitivity, i.e., if choice alternative a is preferred over b and b over c, then, a is preferred over c. While this preference ordering is subject to random fluctuations, it is assumed that such fluctuations are in keeping with the constant probability vector.

A CONSUMER SPATIAL BEHAVIOR MODEL

An initial attempt to model consumer spatial behavior in terms of Luce's basic choice axiom was made by Huff (Huff, 1962). The positive real-valued function v of x that was used in the model is: [The initial model specified only one parameter, an exponent on distance. However, subsequent refinements included an additional exponent on the attraction index variable.]

where

Aj = an attraction index of retail facility j;

Dij = the accessibility of a retail facility j to a consumer located at i; and

r,Y = empirically determined parameters.

The probability that a consumer located at i will choose to shop at retail facility j for a particular product or group of homogeneous products is:

The quotient derived by dividing Ay by Dyij is regarded as the perceived utility of retail facility j by a consumer located at i. Square footage of selling space is often used as a surrogate measure for the attraction variable Aj. Travel time, road and straight-line distance, and travel costs are measures that have been used for the accessibility variable Dij. The parameter associated with a variable reflects the sensitivity of the probabilities with respect to that variable. A comparison of the values of these parameters for different types of shopping trips is particularly meaningful. Various estimation procedures have been employed to estimate parameters of non-linear models (Wilson, 1974). In general, the procedures involve specifying some initial value for one of the parameters and then calculating a value for the other using some efficient search routine. This iterative procedure is continued until the lowest sum of squared differences is reached between the actual and expected number of consumers shopping at a specified set of retail facilities.

PROBLEM AREAS

Trip-type

Consumer spatial behavior as described by equation (2.0) pertains to single-purpose shopping trips. However, many shopping trips are multi-purpose in nature. These trips may involve multi-product purchase intentions in which different retail facilities are visited. Or, they may involve single-purpose shopping trips that are made in conJunction with non-shopping activities. In the case of multi-purpose trips, the proximity of a retail facility to a consumer's place of origin may not be nearly as important as the proximity of a retail facility to other retail facilities in which purchases are intended, or, to those non-shopping activity places that the individual intends to visit. Therefore, if the model is used, it is important that the set of retail facilities that are specified as choice alternatives be those that are most likely to be associated with single-purpose shopping trips.

Product-Type

The model of consumer spatial behavior is formulated to specify the likelihood of a consumer choosing a given retail facility to purchase a specific product, e.g., an automobile, or, a group of related products, e.g., groceries. Therefore, it is important that the attraction values that are specified for those retail facilities comprising the set of choice alternatives be in keeping with the product(s) purchase intentions of the consumer. This point is often overlooked by those employing the model. Frequently, no product or group of related products are specified except in a very aggregative way, e.g., retail goods and services. Similarly, in those cases where a specific product(s) is designated, the measure of attraction for the retail facilities is often an aggregative measure that reflects the attraction for other products as well, e.g., total square footage of a department store when only purchases of women's jewelry are being analyzed.

Spatial Equilibrium

The essence of a market share model of consumer spatial behavior is that a consumer may share his patronage among feasible alternatives and his preference ordering will be in keeping with a constant probability vector of the form (.60, :30, .10). The model does not indicate how the market-sharing principle was established, and it provides an equilibrium solution without knowing if a consumer is, in fact, in equilibrium. Furthermore, the probability vector is derived from cross-sectional data which may not be a true indication of spatial equilibrium conditions (Golledge, 1970). It seems reasonable that for most resident populations there will be some consumers who are in the initial search phases of behavior, some with partially-formed preference patterns, and others with well-formed patronage habits.

Choice Alternatives

Problems may also arise in using the model if the subset of choice alternatives is not defined correctly. Two types of errors can occur. First, if a choice alternative which is actually part of a consumer's set has been excluded, then although the ratio of probabilities between any pair of choice alternatives will not be affected, errors of prediction can occur. In this case, expected values (e.g., annual expenditures) for those choice alternatives which were included in the specified subset will be over-estimated.

Secondly, if a choice alternative which is not actually in a consumer's set of choice alternatives is included, then expected value errors will depend on model specification. For example, if an individual does not consider some particular choice alternative then there is probably some characteristic of the choice alternative that distinguishes it from those alternatives which are considered. If this characteristic is reflected as a variable in the model specification, then the model will be able to discriminate effectively between choice alternatives. Consequently, differences between the observed and expected values will not be the result of defining the set improperly. Conversely, if the differentiating variable is not specified in the model, then the effect of not defining the choice alternatives properly can result in marked differences between observed and expected values.

Group Behavior

Another potential problem arises when the model is used to examine the choice behavior of groups of consumers. The original choice axiom applies specifically to individual choice behavior. However, most applications of the model are oriented toward the analysis of group behavior. Typically, a geographic area is divided into sub-areas and the proportion of times consumers patronize places is averaged for each of these areas. [An excellent study exemplifying this approach was conducted by Haines, et. al. (Haines, Simon and Alexis, 1972).] The assumption is that the smaller the geographic area the more homogeneous are the residents and thus the more similar their spatial behavior. The parameters of the model are estimated for each sub-area and then averaged across all sub-areas for prediction purposes. If the parameter estimates for each of the sub-areas vary considerably, then the average of these values will be misleading and result in incorrect predictions.

Choice Determinants

A consumer's perceived utility of a retail facility is unquestionably determined by a myriad of factors. However, equation (2.0) specifies only two variables even though these variables are regarded as surrogates for a number of different variables. While other variables could be included in the model, there is some difficulty in estimating the parameters associated with such variables. Furthermore, the techniques traditionally used in estimating the parameters do not assure that a global maxima or minima will be obtained nor are the statistical properties of the estimates known. Due to the nonlinear formulation of the model most researchers believed that the parameters could not be estimated by standard econometric methods. However, recently Nakanishi and Cooper demonstrated that the parameters can be estimated using least squares (Nakanishi and Cooper, 1974). A generalization of equation (2.0) with respect to consumer spatial behavior is as follows:

where:

pij = the probability that a consumer located at i will choose a retail facility i;

Xkij = the kth of g variables describing a retail facility j in terms of a consumer located at i;

Bk = the sensitivity of wij with respect to variable k; and

m = the number of retail facilities.

After making a log transformation and rearranging terms, the model is of the form:

where

EQUATION and pi is the geometric mean of the m different probabilities corresponding to each of the consumers;

EQUATION and Xki is the geometric mean of the m different values of Xkij describing the retail facilities in terms of variable k for a consumer located at i; and

Eij = the difference between the model estimates of yij and the actual values of yij.

Thus the original two-variable model can be extended to q variables and the parameters associated with these variables can be estimated using least squares techniques. For example, if the effects of price, distance and size on consumer spatial behavior are to be assessed, the model would be expressed as follows

where

pij = the probability that a consumer in sub-area i will choose location j;

Sij = the square footage of selling space in retail location j;

Bs = the sensitivity of pij with respect to store size;

Dij = the distance between sub-area i and location j;

BD = the sensitivity of pij with respect to distance;

Lij = the price level for a given consumer at i with respect to retail location j;

BL = the sensitivity of pij with respect to the price level; and

m = the number of retail locations.

The transformed model would be:

where:

BS, BD, BL, Sij, Dij and Lij are defined as before;

Yij = EQUATION and pi is defined as the geometric mean of the m different probabilities associated with a consumer at i;

Si = the geometric mean of the m different retail square footage values;

Di = the geometric mean of the distances between the location of a consumer at i and the m different retail locations: and

Li = the geometric mean of the price level for the m different retail locations for a consumer located at i.

There are distinct advantages to this approach.

1. Additional variables with potential explanatory power can easily be added to the model and the associated sensitivity parameters estimated.

2. Using alternate forms of full and restricted linear models, the importance of each variable can be tested.

3. The solution for the least-squares estimates of the sensitivity parameters assures a global minimum.

4. If the appropriate least-squares technique is used, the parameter estimates will be unbiased and of minimum variance.

5. Computer algorithms for least-squares estimation are readily available.

However, it should be noted that there still exist certain limitations.

1. If any one of the proportions are zero, the geometric mean will be zero and that particular proportion cannot be transformed.

2. The traditional assumptions necessary for tests of significance of alternate linear models apply.

CONCLUSIONS

Considerably more empirical research of consumer spatial behavior is needed in order to assess the validity of the Luce choice axiom. Such research should be structured in keeping with the conceptual properties underlying the choice axiom. Examples of needed research include:

1. What particular products or groups of related products involve single-purpose shopping trips?

2. How many retail outlets normally comprise a consumer's set of choice alternatives with respect to the purchase of a given product or group of related products?

3. Does the number of outlets comprising the set of choice alternatives vary not only by product but also in terms of the socio-economic characteristics of consumers?

4. How stable are consumer patronage patterns over time with respect to different product purchases? Do these patterns vary among different types of consumers?

5. What are the perceived attributes as well as the objective equivalences that determine the utility of a retail facility to a consumer? [Research pertaining to the spatial aspects of this question has been done by Briggs (Briggs, 1969), Burnett (Burnett, 1973), and MacKay (MacKay, Olshavsky, and Sentell, 1975).]

6. How similar are these attributes, and their relative importance, among differing types of consumers?

REFERENCES

Briggs, R. The scaling of preferences for spatial locations: An example using shopping centers. Unpublished M.S. thesis, The Ohio State University, 1969.

Burnett, P. The dimensions of alternatives in spatial choice processes. Geographical Analysis, 1973, 5, 181-204.

Golledge, R. G. Some equilibrium models of consumer behavior. Economic Geography, 1970, 46, 417-423.

Haines, G. H., Simon, L. S., & Alexis, M. An analysis of central city neighborhood food trading areas. Journal of Regional Science, 1972, 12, 95-105.

Huff, D. L. A probabilistic analysis of consumer spatial behavior. In W. S. Decker (Ed.), Emerging Concepts in Marketing. Chicago: American Marketing Association, 1962(b).

Huff, D. L. The delineation of a national system of planning regions on the basis of urban spheres of influence. Regional Studies, 1973, 7, 323-329.

Luce, R. D. Individual Choice Behavior. New York: John Wiley & Sons, 1959.

MacKay, D. L., Olshavsky, R. W. & Sentell, G. Cognitive maps and spatial behavior of consumers. Geographical Analysis, in press.

Nakanishi, M. & Cooper, L. G. Parameter estimation for a multiplicative competitive interaction model--least squares approach. Journal of Marketing Research, 1974, 11, 303-311.

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