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Traditional market share response (multiplicative competitive interaction or MCI) models have been gainfully employed in marketing research practice as an effective methodology for estimating competitive effects. Legions of books and articles on MCI models and their use have been published documenting the successful formulation and implementation of this class of models. In this spirit, this paper proposes a generalization of this class of models to a latent structure framework incorporating within-segment random brand effects. We apply and contrast this new formulation against the traditional aggregate MCI model formulations in an application involving physician prescription shares for three major brands of central nervous system (CNS) ethical pharmaceuticals (known as CNS drugs). We conclude the manuscript with managerial implications and suggestions for future research.

This new millennium will usher in an era of intense competition on both domestic and global fronts. Business firms of all sizes and varieties will become more concerned with the market share and profitability figures they achieve in the marketplace. Investors and respective stock market performance demand such attention. For product and brand managers in particular, a sense of urgency associated with the gains and losses of market shares for the products/services in their charge will continue to characterize business scenarios. It is clear that market indices such as market shares will command the attention of marketing managers as generic performance measures of a particular brand/service in the marketplace. It is therefore clearly desirable for those individuals concerned to have knowledge of the processes which generate such market-share figures, and to be able to calculate the impact of their own actions (and competitors' actions) on market shares, as well as respective profit implications. In this light, Cooper and Nakanishi (1988) and Cooper (1993) have produced comprehensive reviews of the major research performed in the area of market share models. Building on the research in Bell, Keeney, and Little (1975), Kotler (1984), Naert and Bultez (1973), Nakanishi (1972), Nakanishi and Cooper (1974), and others, Cooper and Nakanishi (1988) recommend the use of “multiplicative competitive interaction (MCI) models” for modeling market share based on the following attraction formulation. Let: According to Kotler (1984), a brand's market share is proportional to the marketing effort supporting it. Thus, equation(1) View the MathML source where: Mi=the marketing effort of brand i; c=a constant of proportionality. Since View the MathML source by definition of market share, View the MathML source or equation(2) View the MathML source and substituting for c using expression (2) in Eq. (1), one obtains: equation(3) View the MathML source implying that the market share of brand i is equal to the brand's share of total marketing effort (Kotler's fundamental theorem of market share; cf. Bultez and Naert, 1975). Note, if brands tend to differ in terms of their effectiveness of marketing effort, one can modify Eq. (3) above via: equation(4) View the MathML source where αi is the effectiveness coefficient for brand i's marketing effort, implying that even if two brands expend the same amount of marketing effort, they may not have the same market share. Here, the marketing effort of a brand is assumed to be some monotonic function of its marketing mix. Assuming that the elements of the marketing mix interact, Cooper (1993) suggests the multiplication function: equation(5) View the MathML source where: Pi=the price of brand i's product; Ai=the advertising expenditures of brand i; Di=the distribution effort of brand i; p, a, d, are estimated parameters reflecting the importance of each respective component of the marketing mix. Bell et al. (1975) derive the same representation/model using an attraction-based formulation. These authors posit that the primary determinant of market share is the attraction that consumers feel toward each alternative brand. Thus, the simple MCI model for the aggregate case can now be formulated (by substitution) as: equation(6) View the MathML source equation(7) View the MathML source where the Xik's reflect the marketing mix variables and εi is an error term. Mi can be either interpreted as marketing effort or attraction. βk reflects the parameters to be estimated which are common across brands in the simple version of the model. The point market share elasticities of this simple MCI model are: equation(8) View the MathML source Plotting this elasticity against Xki, one sees that share elasticity for this simple MCI model monotonically declines as Xki increases. Note, such elasticity formulations for the simple MCI model does not reflect differential marketing mix effectiveness by brand. Cooper and Nakanishi (1988) present a more comprehensive MCI model that explicitly reflects differential brand marketing mix effectiveness: equation(9) View the MathML source Obviously, the simple MCI model is a special case of this differential effects MCI model where all brands are equally effective: βik=βjk=βk, ∀i,j. The corresponding share elasticities are: equation(10) View the MathML source and the cross-elasticities are: equation(11) View the MathML source As Cooper and Nakanishi (1988) denote, Eq. (11) models cross-elasticity with respect to Xjk as constant for any brand (i≠j). Thus, the relative changes of other brands' share caused by brand j's actions are the same for any brand, though actual change in shares are different from one brand to another, depending on the current share level for each brand. To resolve this last limitation, Cooper and Nakanishi (1988) present the most general MCI model, the fully extended MCI model: equation(12) View the MathML source Here, the attraction for brand i is now a function not only of the firm's own marketing mix actions, but also of all other brands' actions as well. The βijk parameters for i≠j are the cross-competitive effects parameters, while those for i=j are the direct effects parameters (as in the differential effects MCI model above). The direct and cross-elasticities for this model are:As these authors denote, this fully extended MCI model involves the estimation of much more parameters than the simple and differential effects version:Because of this explosion of the number of parameters, the fully extended MCI model has been rarely implemented. Note, since these models are nested, one can utilize a likelihood ratio test for model selection in the aggregate case. By using a log-centering operation (cf. Cooper & Nakanishi, 1983 and Cooper & Nakanishi, 1988), these three MCI models can be estimated via multiple regression with time series data: equation(14) View the MathML source equation(15) View the MathML source equation(16) View the MathML source where: Alternatively, one can also employ a maximum likelihood estimation framework. This second approach is more difficult to implement, but allows the errors (i.e., brand-specific random effects) to have different variances across brands and non-zero correlations. Problems arise in estimating such aggregate models in the presence of substantial sample heterogeneity. Estimates of such MCI aggregate-level model parameters may be biased under such conditions. Worse is the potential to mask the real underlying effects of what is taking place in the market. Consider the case of two relatively equal sized market segments who respond to price in radically different ways. For segment 1, price may be utilized to signal quality. Here, higher prices may be preferred and related to higher shares for this particular market segment. For segment 2, economy and value may be a primary driver. Here, low prices may drive higher market shares. Unless a disaggregate (segmented) MCI model is estimated, an aggregate analysis may mask the price effect showing it to be insignificant overall. The primary contribution of this research is to generalize this MCI framework to account for sample and parameter heterogeneity. Rather than modeling market shares over time, our application deals with modeling prescription shares for different brands of central nervous system (CNS) drugs for individual physicians (see also Leeflang et al., 1992 and Parsons & Abeele, 1981). Given the well-documented difficulties associated with aggregate sample models constructed over a heterogeneous sample (cf. DeSarbo & Cron, 1988, Leeflang et al., 2000 and Wedel & Kamakura, 2001), we devise hybrid random effects and latent structure MCI models that simultaneously estimate market segments of physicians (both segment size and membership), as well as the segment level MCI parameters (see Allenby & Lenk, 1994 and Gupta & Chintagunta, 1994, for somewhat related work on market structure and logistic regression). We formally test these models against their aggregate counterparts (described above) and depict their superiority based on well-known model selection heuristics.

Understanding what drives market share will con- tinue to be a primary concern of marketing managers worldwide. In some markets, such as the pharmaceut- ical industry, where share data for individual custom- ers are available with increasing frequency (e.g., both monthly and weekly behavior can be purchased from data suppliers in pharmaceuticals), understanding what drives share at either the individual customer level or among groups of customers has become increasingly important. By understanding which cus- tomers to target promotions, especially expensive marketing mix elements like sales forces, and which parts of the marketing mix to utilize, marketing managers will be better able to maximize the return on their promotional investments. The models devel- oped in this paper provide additional insight to mar- keting managers because they simultaneously uncover the key determinants of market share for competitorsas well. As a result, they should be able to develop stronger marketing strategies that allow them to obtain the competitive advantage required for maximal mar- ket success. Future research in this methodological area can develop along a number of alternative paths. One, further commercial applications are needed along a wider range of products and services. Two, more complex models can be constructed which allow for continuous heterogeneity within segments across all coefficients utilizing hierarchical Bayesian methods (cf. Lenk & DeSarbo, 2000 ). Three, comparative studies tracking predictive validity across a wide range of models and products is needed. Four, finite sample properties of the derived estimates need to be exam- ined carefully, especially in saturated fully extended MCI models and their generalizations. Five, future research effort should be allocated toward the develop- ment of a generalized latent structure MCI model with varying forms of coefficient heterogeneity, together with nested model statistical tests for model selection. Finally, empirical comparisons with latent structure models (e.g., finite mixtures of multivariate log-nor- mal distributions) modeling the volume (not shares) of prescriptions which would more closely represent the financial revenue obtained from each physician might prove beneficial in terms of better evaluating and targeting the derived market segments