Consumer Research Implications of Random Coefficient Models

The above relationship, modified with demographic variables, is the workhorse of econometric demand studies with aggregate data.

Probability Choice

Suppose that a product is such that consumers have a finite set of options and buy one such item on occasion to the exclusion of others. For example, a household may consider several different makes of autos, but it will usually purchase only one at a time. The time between purchases is usually substantial (e.g., more than a year). Other durable goods, such as refrigerators and televisions, are acquired in a similar fashion.

Now consider the problem of estimating the shares of various brands or models of such a product. It is well known that the linear specification given in Equation (1) misspecifies consumer behavior even if disaggregate data are used (Domencich and McFadden, 1975; Charles River Associates, 1973).

A class of models used to analyze consumer behavior under these conditions is based on the tenets of probability choice (Luce, 1959). They are widely used by transportation researchers (see Stopher and Meyburg, 1976, for a historical development of probability choice models in transportation). The most typical application is the use of the binary or multinomial logit model to estimate the probability of a traveler selecting one of several transport modes to make a trip. From a consumer research perspective, this decision can be generalized to the selection of one product or service over another. The probability of a consumer choosing alternative i (Pi) from n elements can be expressed as:

EQUATION   (2)

where:

h_kj = the attributes (which are denoted by k) of each of the n alternatives, and

Equation (2) is a multinomial logit model, and it de- scribes the probability of choice for alternative i as an S-shaped function of each of the product or service attributes. The sum of the P_i over individuals or households yields the demand, D_i, for the product.

Individual differences are not built into the underlying assumptions of the multinomial logit model. Sociodemographic descriptions of consumers can be inserted into Equation (2) to account for individual differences in alternative selection for cases in which the attributes of alternatives are invariant.

Even with the important theoretical advance of probability choice models, the problem of predicting product acceptance cannot be easily solved using aggregate data and assuming constant coefficients in the attributes of products. There are other options such as using special surveys or experiments, both of which may be costly. Alternatively, one can drop the assumption of constant coefficients and use widely available aggregate data. This option is explored below.

The Hedonic Demand Model

The hedonic demand model (HDM) allows using aggregate data to estimate market shares among very similar products. It is based on the theory of probability choice behavior while taking account of differences in individuals' tastes. The following develops the behavioral foundations of the model.

A basic assumption is that consumers select that alternative which yields the greatest utility. The utility (denoted by U) derived by a consumer (denoted by x) confronted with several discrete goods, D_i, and all other goods and services, Z, can be represented by:

U (D_i, Z|x)  (3)

It is useful to require that expenditures on Z and the selected alternative equal a consumer's income, I_x, so that:

where:

D_i = 1 for the selected alternative and 0 for all other alternatives,

P_i = alternative prices, and

Therefore, the choice situation being modeled can be rep- resented by:

U(D_i, Z|x) > U(D_i, Z|x) for all D_i, i/=j,   (5)

and the budget constraint of

It is also convenient to assume separability between utility based on D and Z so that the link to product attributes can be shown as:

U(h_j|x) > U(h_j|x) for all i/=j    (7)

where:

h_j = the vector of attributes of good j, and

x = the vector of taste characteristics.

Since Z does not vary when different D_i are evaluated. Equation (7) is equivalent to Equation(5) under the assumption of separability.

A further assumption of separability and additivity among the attributes makes it possible to define

EQUATION  (8)

where:

h_kj = attributes, such as size, horsepower, and price in the case of an auto, and

x_k = the tastes of a consumer with respect to those attributes.

There are m attributes under consideration.

Different consumers are represented by different x_k. The distribution of tastes across consumers and over attributes can be characterized by multivariate probability density functions.

It is useful for interpretive purposes to normalize the x_k so that the taste coefficient of price for each consumer is -1. This normalization permits Equation (8) to be rewritten as:

EQUATION  (9)

where:

x_k = marginal utilities which indicate the price a consumer would pay for another unit of the kth attribute.

Even when one product dominates another on all attributes, it might be selected by a small fraction of consumers. In order to simulate this potential occurrence, it is convenient to allow a stochastic error term to be added to the utility function of Equation (9). It is usually assumed that the error term is distributed independently and identically across consumers. With the error term, the utility of alternative j can be represented as:

EQUATION  (10)

As a consequence of this error term, the probability of j being selected over i can be denoted by:

EQUATIONS  (11)  and  (12)

McFadden (1974a) has shown that in this case the probability of an individual choosing good j from among n goods given tastes x_k is:

EQUATION  (13)

EQUATION  (14)

EQUATION  (15)

TIES TO STANDARD CONSUMER RESEARCH TECHNIQUES

Random coefficient models are important for consumer research because of the wide range of topics to which they are relevant and the inherent power of the research strategy. In this section, we describe relationships between standard consumer research techniques and HDM in order to highlight the value of this recent econometric development vis-a-vis the analysis of consumer behavior. In particular, three techniques are compared and contrasted with HDM: multiattribute models; individual differences scaling; and market segmentation.

Multiattribute Models

A common form of the multiattribute model has been described by Wilkie and Pessemier (1973). They write a linear compensatory representation in these symbols:

EQUATION  (16)

where

i = attribute or product characteristic,

j = brand, and

k = consumer or respondent;

such that:

"_jk = consumer k's attitude score for brand j,

I_ik = the importance weight given to attribute i by consumer k, and

B_ijk= consumer k's belief as to the extent to which attribute i is offered by brand j.

Consumer researchers have extensively studied variations of this formulation. The original attitude conceptualizations were proposed by Rosenberg (1956) and Fishbein (1967). The relationships among A_jk, I_ik, and B_ijk have been examined under different measurement conditions and modeling frameworks.

A gap in the theoretical development of early multiattribute formulations was the nature of the relationship between A_jk and consumer behavior. The link between attitude and behavior is implicit with positive attitudes assumed to correlate with positive behavior, but it is known that better predictors of behavior are available.

The strength of the approach is diagnosis as opposed to prediction. Diagnosis results from the exposure of the salience of different product attributes as seen by consumers. To the extent that managers can change product attributes, they can use multiattribute modeling research to help them improve the salience of their product to consumers.

The utility equation (see Equation (10)) of HDM is defined in terms of coefficients for different attributes, in a similar manner to multiattribute models. However, the attributes, h_kj, are direct observations on products as opposed to respondents' judgments about product attributes. The utility concept is directly linked to consumer choice through utility maximization and probability representations (see Equations (13) and (14)). Finally, the taste coefficients, x_k in Equation (10) generally correspond to I_ik, of Equation (16), but they are estimated from aggregate data on market shares instead of being reported by respondents. Therefore, the random coefficient models do not require data from individual respondents. In the latter sense, they are more parsimonious in their data needs; this reduces the cost of analysis and thus adds to the desirability of HDM.

The theoretical basis of HDM is inherently reasonable, and its relationship to consumer behavior is explicit. Consumers are assumed to select the product with the largest utility to them, and utility is defined as a weighted function of attributes and a stochastic term. The weights for attributes are consumer taste coefficients which are estimated to maximize the likelihood of predicting aggregate market share data across respondents.

Individual Differences Scaling

Individual differences scaling follows from psychological research on perceptions and preferences. The notion is that the differences among individuals in preferences or perceptions can be used to quantify the similarities and differences among individuals. With respect to marketing research, it is assumed that these differences have implications with respect to consumer behavior. Among the models which have been commonly used for individual differences scaling are INDSCAL (Carroll and Chang, 1970), the vector model (Tucker, 1960), and variations of the ideal point model (Coombs, 1964).

There are at least four types of applications for individual differences scaling in consumer research. In common with aggregate multidimensional scaling, the technique finds how product attributes vary among one another from the user's perspective. Beyond this point, individual differences scaling is much more flexible. It is possible, for example, to stratify a sample along one or more sociodemographic variables to study how shifts in perceptions and/or preferences vary across strata. It is alternatively possible to cluster consumers according to their perceptions and/or preferences and examine shifts in sociodemographic variables over perceptual market segments. When it is more useful to find homogeneous perceptual segments than study variation along one or more prespecified sociodemographic variables, the latter procedure is to be preferred over the former. Both techniques, however, will reveal interrelationships of particular types of viewpoints with sociodemographic variables. A fourth application involves using an individual's weights or saliences for perceptual dimensions as the independent variables in a multiattribute formulation. The dependent variable is usually some indicator of consumer preference or purchase patterns. Illustrations of the above four applications are described in numerous sources (see for example, Dobson, Golob and Gustafson, 1974; Dobson and Kehoe, 1975; Green and Carmone, 1972; Green and Rao, 1972; and Dobson and Nicolaidis, 1976).

Individual differences scaling and HDM have complementary and contrasting relationships to each other. There are no guidelines within HDM to select attributes, but individual differences scaling can be used to highlight product attributes which are salient to major market segments. Physical indicators of these psychologically salient attributes could then be used in HDM. It is possible to use output from individual differences scaling in a multiple attribute representation which parallels that of HDM, but the interpretations of the two representations are substantially at variance. In the case of individual differences scaling, consumer tastes, as denoted by saliences for perceptual dimensions, are the independent variables. These tastes are then weighted to account for indicators of consumer preferences or purchase patterns. HDM, however, uses consumer tastes as weights instead of independent variables. Further, these consumer tastes are linked to behavior via a nonlinear, probability choice formulation rather than the linear, compensatory formulation frequently used with individual differences scaling output.

Both random coefficient models and individual differences scaling relate to market segmentation. The random coefficient models are attractive, in part because they can be estimated on aggregated market segmentation data. For HDM, the aggregate data are market shares for different products, a type of information which is frequently readily available. On the other hand, individual differences scaling will normally require a special data collection effort, but its output can be used to form market segments based on perceptual characteristics of respondents. For product positioning with respect to consumer segments in a broad, qualitative sense, the richness of individual differences scaling may be more useful, but for detailed, quantitative estimates of market shares, HDM may be more useful. In any event, there are opportunities for meaningful consumer research in comparing and contrasting the alternative methodologies.

Market Segmentation

Market segmentation in consumer research is predicated on two key concepts. It is initially assumed that meaningful groups of users of a product or service can be identified. The second concept revolves around the creative assessment of product and/or service requirements for different market segments. These concepts are jointly applied towards maximizing profits through rendering services and/or products which appeal strongly to lucrative market segments.

A variety of criteria have been proposed for assessing whether any particular group of market segments is appropriate for profit maximization. Kotler (1972) notes that it is important for segments to be measurable, accessible, and substantial. Wilkie and Cohen (1976) remark that "true" segments should be homogeneous within and heterogeneous between groups, useful as a behavioral correlate, and efficient to market from production, distribution, and pricing perspectives. A pivotal aspect of market segmentation is the selection of appropriate variables on which to divide consumers. The term segmentation basis is used to denote a set of variables selected to divide consumers. Three principal classes of bases are sociodemographic data (Carman, 1965; Assael and Roscoe, 1976), consumption patterns (Cunningham, 1955), and individuals' subjective judgments (Haley, 1968; Myers, 1976). A major summary and analysis of segmentation research and theory is provided by Frank, et al. (Frank, Massey and Wind, 1975).

As noted above , HDM is organized around a segmentation structure, namely the market shares for different products. The primary thrust of this modeling orientation is to derive estimates of consumer tastes which are consonant with market share data. This consumer taste information can be used for product planning purposes. Also, market simulations can be performed which reveal the implications of an altered product mix. HDM assumes that the set of salient product attributes are known and that the purpose of a marketing analysis is to study substitution among products. When either of these conditions does not hold, another market segmentation analysis technique may be more appropriate. More traditional consumer research tools for market segmentation are directed at product positioning (i.e., what new product would consumers most like to have) and advertising targeting (i.e., what content should a promotional campaign have and to whom should it be addressed). A potentially useful purpose of the traditional approaches is the evaluation of marketing strategy effectiveness (Assael and Roscoe, 1976). It can be seen, therefore, that traditional consumer techniques and HDM complement each other with respect to market segmentation.

AN APPLICATION OF HDM TO AUTOS

The hedonic demand model was developed in a study of the impact of foreign trade controls on the import share of new car sales (Charles River Associates, 1976). The technique yields predictions of the effects of changes in prices and characteristics of any set of automobile models on the market shares of individual models. The predictions of share changes for individual models are then summed to obtain the predicted change in the share of a class of models resulting from any relative price changes.

The input data used are the price, selected attributes, and new car sales in a given time period of a set of automobile models. The attributes include available characteristics which either contribute to, or are correlated with, the quality of a car. Available attributes data for recent years include both the physical dimensions from Automotive News Almanac and performance data collected by the U.S. Department of Transportation and the U.S. Environmental Protection Agency. The physical attributes data include number of cylinders, headroom, height, horsepower, legroom, length, gas tank capacity, transmission type (automatic or manual), turning circle weight and width. The performance variables used were acceleration and fuel economy (miles per gallon). The sample period was the five-month period between April 1974 and August 1974; this period was selected to include the part of the 1974 model year subsequent to the period of acute gasoline shortage.

The results presented below should be viewed as first-run estimates only. Still, the results seem quite plausible and are very similar both to estimates from time series regressions and to a set of predictions of the impact of imported car price increases on market shares derived by questionnaire techniques (Market Facts, 1976).

The estimate of the taste distribution for characteristics which was used to derive the share elasticities with respect to relative price changes was based on five attribute variables: volume, passenger area, weight, turning circle, and gallons per mile. Volume is defined as the product of length, width, and height, and is a proxy measure for a combination of total seating capacity and trunk space. Passenger area is computed as the product of headroom and legroom, and represents a proxy for driver and front seat passenger comfort. Utility is positively correlated with volume, passenger area, and weight. Utility is negatively correlated with turning circle diameter (the smaller the turning circle, the more maneuverable the automobile) and with fuel consumption in gallons per mile.

Table 1 summarizes statistical information on the estimated distributions of the marginal rates of substitution between characteristics and price. The distribution of consumer tastes for auto attributes is assumed to be log-normal. The statistics reported in Table 1 were computed by taking exponents of the parameters of the normal distributions of the logs of the attributes. As the distributions are log-normal, the mean and median of the distributions are not the same.

TABLE 1

DISTRIBUTION OF ESTIMATED MARGINAL VALUES OF AUTOMOBILE CHARACTERISTICS (DOLLARS PER UNIT)

The numbers reported in Column (2) of Table 1, the median of the distributions, give the marginal value of an additional unit of the attribute in dollars; the units of the characteristics are defined at the bottom of Table 1. The coefficient attached to VOL 2 indicates that the median consumer is willing to pay $2.31 for an extra 10,000 cubic inches of volume. The coefficient attached to weight indicates that the median consumer is willing to pay an additional $2.78 for an additional pound. The median coefficient for turning radius is greater than for VOL 2, AREAG, and WEIGHT, but there is also the most differentiation among consumers on this attribute. Note the extremely large variance of consumer tastes for this attribute in Column (3). The median marginal utility for gallons per mile is the greatest of all the attributes. This coefficient, which was derived from data collected immediately after the oil embargo, shows the extreme sensitivity of consumers to gasoline availability. However, despite the time period for which the model is estimated, there is still substantial variability among consumers on the importance of this attribute.

The upper and lower bounds of the 95 percent confidence interval (see Columns (4) and (5)) indicate the degree of dispersion in tastes for each attribute among consumers. The alternative degrees of variability across attributes reveals the ability of HDM to characterize individual differences among consumers. It should also be kept in mind that the model was estimated on physical attributes data and aggregate market share statistics. In any event, it can be seen that the most widely dispersed values are for the attribute "turning circle." Apparently some consumers place a high value on this attribute for parking and general maneuvering ability, but others have practically no preference for it at all. It is likely that a substantial portion of the variation for gallons per mile reflected different expectations among consumers as to gasoline availability and price. The range of variability for the remaining three attributes is substantially narrower than for TURN and GPM.

Table 2 compares the predicted shares consistent with the utility function estimated in the model with actual market shares of different automobile types. The model was estimated using two different price variables --the one year old price in August 1974 of 1974 models reported in National Market Reports, Inc., Red Book and the estimated new transactions price of 1974 models. The new transactions price variable was corrected by applying "typical" discount factors for different types of cars reported in the annual automobile buying issue of Consumer Reports (April 1974) to list prices reported in the 1974 issue of Automotive News Almanac. The data reported in Table 2 show that the predicted shares from the model conform fairly closely to the actual shares; the model fits the data well. For example, the actual share for all domestics was .86798 and the share estimated by the random coefficient model with adjusted new car prices was .86183. Predicted import shares are slightly higher than actual shares. There is little difference in model estimates across the two price indicators which suggests the model is robust in the face of varying data quality.

TABLE 2

COMPARISON OF PREDICTED SHARES TO ACTUAL SHARES BY AUTOMOBILE TYPE, APRIL-AUGUST 1974

SUMMARY AND CONCLUSIONS

A major purpose of this report was to present a recent econometric development which has potential value as a consumer research tool. The historical antecedents and underlying assumptions were initially described. These assumptions were subsequently compared and contrasted with those of three standard consumer research techniques. Finally, an application of the approach to auto demand was summarized. It is clear that HDM parallels the functions of other consumer research techniques, that it requires little data which can be obtained without survey collection costs and time delays, and that it has the potential to produce accurate results which are subject to rich interpretation.

The logic of HDM is based on a simple assumption of utility maximization at the level of individual consumers. Furthermore, the formulation of this model assumes a multiattribute structure, but it is based on physical product attributes. Individual differences among consumers are represented by taste coefficients for the attributes. These coefficients are estimated in a probability choice framework to be consistent with market share data.

As mentioned above, HDM is multiattribute in structure and therefore has a common link with the Fishbein (1967) and Rosenberg (1956) models. A principal difference between the two approaches is the use of physical attribute data versus subjective judgments in the psychological models. The individual differences scaling models, such as INDSCAL, allow for variations among consumers in a fashion which is similar to HDM. However, neither individual differences scaling nor Fishbein models, as initially proposed, were designed to facilitate the estimation of the probability of consumer choice, but HDM is directed toward this objective. Market share data, a type of market segmentation, provide a basis for the estimation of HDM. In the latter respect, a tie is established to a pervasive consumer research tool.

When the model was estimated on auto market share data, its properties were revealed. It was shown that variability among consumers could be represented. It is important to recall that this variability among consumers is described without a reliance on surveys. Only readily available data, such as market shares for models and physical product attributes, were required. In addition, it was possible to estimate a model representation which accurately fitted the data. This ability to accurately describe variability among consumers while correctly estimating market share data is an attractive property, particularly when it is combined with the minimal data requirements of the random coefficient models.

Further research is justified with random coefficient models on the basis of the results described above. These models can have rational structures which show a relationship between utility and probability choice at the level of the individual consumer. They also have extremely simple data requirements. HDM is compatible with standard consumer research functions. Therefore, its use in marketing studies will achieve normal objectives and concurrently extend the state of the art in consumer research.

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