|
Article Excerpt
We propose a framework to investigate consumers' brand choice and purchase incidence decisions across multiple categories, where both decisions are modeled as an outcome of a consumer's basket utility maximization. We build the model from first principles by theoretically explicating a general model of basket utility maximization and then examining the reasonable restrictions that can be placed to make the solution tractable without sacrificing its flexibility. Comparing with prior models, we show why prior multicategory purchase incidence models overemphasize the role of the cross effects of a market mix of brands in other categories on the purchase incidence decision of a given category. Additionally, we show that prior single-category models are a special case of the proposed model when further restrictions are placed on the basket utility structure.
We estimate the model on household basket data for the laundry family of categories. We show (i) why prior single-category and multicategory models would systematically bias the estimates of the own--and crossprice/promotional purchase incidence elasticities; and (ii) how the market mix of each brand in each category affects the purchases across all categories, which can help retailers make promotional decisions across a portfolio of products.
Key words: multicategory brand choice and purchase incidence decision making; microeconomic theory of demand; basket utility maximization; simulated maximum likelihood
History: This paper was received March 8, 2005, and was with the authors 5 months for 2 revisions; processed by Tulin Erdem.
1. Introduction
When consumers visit a store, they typically purchase a basket of items that consists of brands from a set of product categories in the store. Such typical store visits involve two sets of decisions by consumers: first, which product categories to purchase in the store, and second, within the purchased categories, which brand to choose. An important question that arises in this context from a managerial and academic perspective is: How are these two sets of decisions made, and how are they influenced by the market-mix variables in the store?
To answer this question, we first need to understand the process through which consumers make the joint category purchase and brand-choice decisions. If we assume consumers to be utility-maximizing agents, it follows that their joint purchase decisions across all categories should be an outcome of their maximizing utility over their entire basket. Thus, a proper approach requires considering both brand-choice and category purchase decisions simultaneously across all categories in the store. Such an approach is theoretically appealing, and would also yield proper measures of impact of market-mix variables on consumers' purchase behavior. However, a drawback of this "proper" approach is that it is fairly intractable even for a moderate number of brands and categories.
Prior research, by and large, has followed one of two tracks. It has either addressed both brand-choice and category purchase incidence decisions in a single-category context (Chiang 1991, Chintagunta 1993) or addressed one aspect of the decision process in a multicategory context, which is brand choice (Ainslie and Rossi 1998, Erdem 1998) or category purchase incidence (Manchanda et al. 1999, Chib et al. 2002). However, a drawback of single-category models is that they ignore cross-category interrelationships by implicitly assuming that the basket utility maximization reduces to independently maximizing subutilities over each single category. Similarly, the implicit assumption in prior multicategory models is that the basket utility maximization reduces to independently maximizing utilities over brand-choice and category purchase incidence decisions. These assumptions, however, are not always well founded, and can lead to suboptimal solutions as well as biases in estimates of price and promotional elasticities.
The above discussion highlights the trade-off between optimality and tractability. To obtain a microeconomic theoretic specification that balances these imperatives, one needs to (a) start with a completely general basket utility maximization model from first principles, and (b) articulate the reasonable restrictions that can be imposed on such a model, while allowing flexibility in modeling the interrelationships between categories. Such an approach offers the following advantages. First, it will allow us to pin down the effects of market-mix variables of brands in any category on purchases across other categories. Second, it will yield significantly better estimates of these effects, which has significant implications for both retailers and manufacturers. Third, it will shed insight on the nature of additional restrictions that are required for the basket utility maximization to reduce to independent subutility maximizations as assumed in prior models. Articulating these additional restrictions will help us understand the nature of biases that can potentially exist in prior models.
1.1. Research Objectives
Our first objective is to build a tractable yet flexible model of multicategory purchase incidence and brand choice behavior. We start with a general framework of direct basket utility maximization that allows for all possible interactions among all brands in all categories inside the store. We discuss the factors that can be relaxed to get tractable solutions for brand choice and purchase incidence decisions. Specifically, we first show that under relatively mild assumptions of "weak separability" and "aggregation across unrelated categories," it is sufficient to consider only a family of categories that can possibly be used together during consumption. Second, we use the principle of "duality" to show how the joint purchase decisions can also be derived from an indirect basket utility. This result is crucial for empirical analysis because flexible functional forms lie mainly in the realm of indirect utilities, which allow for much greater flexibility in modeling the interrelationships between categories, while maintaining tractability (Pollak and Wales 1992).
Our second objective is to compare the specifications of our joint category purchase incidence conditions with those used in prior single-category brand choice purchase incidence models (Chiang 1991, Chintagunta 1993) and multicategory purchase incidence models (Manchanda et al. 1999). We show how prior multicategory purchase incidence models tend to overemphasize the role of cross effects of market mix of brands in other categories on the purchase incidence decision of a given category. We also show that prior single-category models are a special case of the proposed model when further restrictions are placed on the structure of the basket utility.
Our third objective is to empirically test the proposed model on household-level basket data. We choose the laundry family of categories--that includes detergents, washer and dryer softeners. We use a flexible functional form for the indirect basket utility to characterize the stochastic specification of the purchase decisions. We compare the proposed specification with (i) a nested specification that has no cross-category effects (that is, when the multicategory model reduces to multiple single-category models), and (ii) the statistical specification of Manchanda et al. (1999) that only models the purchase incidence decisions across categories. We show that our specification performs significantly better than the competing specifications, and why competing specifications systematically bias the estimates of the own- and crossprice/promotional elasticities. Furthermore, we show how market-mix variables of each brand affect purchases across all categories, estimates of which can help retailers make promotional decisions across a portfolio of products.
1.2. Related Literature
There are four streams of research relevant to this paper. Table 1 provides an overview of these four streams and the relative positioning of our work. The first stream consists of papers that have investigated brand choice and purchase incidence decisions in a single-category context (e.g., Chiang 1991, Chintagunta 1993). The second stream consists of papers that have investigated only brand choice decisions across categories (e.g., Ainslie and Rossi 1998, Seetharaman et al. 1999). The third steam consists of papers that have investigated only purchase incidence decisions across categories (e.g., Manchanda et al. 1999, Russell and Petersen 2000, Chib et al. 2002).
Apart from the methodological limitations discussed before, the usefulness of these three streams is also limited from a retailer's or multiproduct manufacturer's perspective--whose interest lies in understanding how the prices and promotions of each brand in a given category affect both brand choice and category purchase incidence decisions, not only in that category but also across other categories. For instance, the single-category models in the first stream do not model the cross-category effects of a brand's market mix. Similarly, the multicategory brand choice models in the second stream do not model the impact of each brand's market mix on category purchase incidence. Finally, the multicategory purchase incidence models in the third stream cannot pin down the effect of market mix at the brand level.
The fourth stream consists of papers that have modeled brand choice and purchase incidence decisions across categories: There are two recent papers in this stream. The first, by Song and Chintagunta (2006), uses a basket utility maximization approach to estimate the two decisions (along with the quantity decision) across categories. However, they start by choosing a specific functional form of the direct basket utility, which results in the purchase incidence probability of any category to be independent of the market mix in other categories. We show that their model is similar to a nested version of our model when further restrictions of "indirect additivity" are imposed on the utility. The second paper, by Chib et al. (2005), uses a statistical approach to specify the two decisions. A benefit of their approach is that it is not bound by theoretical restrictions, which allows more flexibility in modeling the interrelationships between categories.
We extend the four streams by building a model of basket utility maximization from the first principles that simultaneously specifies the purchase incidence and brand choice decisions across multiple categories, and properly accounts for the role of cross effects in the decisions. The rest of the paper is arranged as follows. In [section] 2, we discuss the theoretical specification of the joint purchase conditions based on a general specification of the basket utility. In [section] 3, we discuss the stochastic specification. In [section] 4, we discuss the data used for estimating the model, report the parameter estimates, and discuss the implications. In [section] 5, we conclude, and offer directions for future research.
2. Model Development: Theoretical Specification
In [section] 2.1, we propose a general framework of deriving the joint purchase decisions based on direct basket utility maximization. Following that, we discuss the assumptions that can be imposed on the structure of the direct basket utility that would yield tractable solutions for the joint purchase decisions.
In [section] 2.2, we derive the general specification of the joint purchase decisions after incorporating the assumptions discussed in [section] 2.1. Using the solution laid out in [section] 2.2, we show how the joint category purchase decisions can be derived from an indirect basket utility in [section] 2.3, the results of which will be used to characterize the stochastic specifications of the joint purchase decisions in [section] 3.
2.1. General Model of Basket Utility Maximization (1)
Consider a consumer who can possibly make a purchase in 1, ..., M product categories in a store on a shopping trip. We define product category l as a set of brands, 1, ..., Jr. Let the observed covariates on the consumer's shopping trip be: the total basket expenditure in dollars, y; unit price of a brand j in category l, [p.sub.l,j]; and the perceived quality of brand j in category l, [[psi].sub.l,j].
Given the observed covariates on a shopping trip, the consumer's problem is to choose which categories to purchase and, among the purchased categories, which brands to choose. In order to specify the solution to this problem, we define the consumer's direct basket utility as a function of her perceived qualities, {[[psi].sub.l, j]}, and purchased quantities, {[x.sub.l,j]}, of all brands j = 1, ..., [[J.sub.l] in all categories l = 1, ..., M. We start with a very general specification of the direct utility, which allows for all possible interactions between brands across all categories, (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.1)
where [X.sub.i, j] is a function of consumer's perceived quality and purchased quantity of brand j in category l. Thus, the consumer's problem can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
subject to
[[J.sub.1].summation over (j=1)] [p.sub.1, j][x.sub.1, j] + [[J.sub.2].summation over (j=1] [p.sub.2, j] [x.sub.2, j] + [[J.sub.M].summation over (j=1)] [p.sub.M, j][x.sub.M, j] = y (2.2)
(budget constraint)
[x.sub.l, j] greater than or equal to] for all l, j (nonnegativity constraints).
The utility maximization in (2.2) is subject to the budget and nonnegativity constraints. The nonnegativity constraints are critical--they show how the optimal solutions of quantities of all brands across all categories, {[x.sup.o.sub.l, j]}, are related to category purchase incidence and brand choice decisions. They imply that if the optimal solution is such that: (a) [x.sup.o.sub.l, j] = for all brands j = 1,..., [J.sub.l] in category l, then category l will not be purchased; (b) for any brand k in category l, [x.sup.o.sub.l, k] > 0, then category l will be purchased and brand k will be chosen in category l.
The derivation of the joint purchase conditions entails simultaneously solving the first-order conditions that follow from the utility maximization. (3) Because the direct basket utility in Equation (2.1) allows for all possible interactions between all brands across all categories, the first-order condition for any brand will be a function of the quantities of all brands across all categories. Note that such an approach is analytically and empirically intractable. As the number of brands across all categories increases, simultaneously solving the first-order conditions across all brands and all categories becomes analytically intractable; additionally, it leads to an explosion in the number of parameters that capture the pairwise interactions between all brands, thus making it empirically intractable. Therefore, we will next make some reasonable assumptions on the structure of the utility that will yield relatively more tractable solutions to the brand choice and category incidence decisions.
ASSUMPTION 1 (WEAK SEPARABILITY AMONG ALL M CATEGORIES). Weak separability is the least restrictive way to group brands in one product category and separate them from brands in other categories (Pollak and Wales 1992). This is done by simplifying the basket utility in
Equation (2.1) in terms of subutilities for each category as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.3)
where [v.sub.l]([{[X.sub.l, j]}.sup.h.sub.j=1]) is category l's subutility, which is a function of the qualities and quantities of all brands in category l only. Equation (2.3) simplifies the utility maximization problem by allowing the interactions between brands across different categories to occur through a common channel--the interactions in their respective category subutilities. Thus, instead of taking into account all pairwise interactions between brands across different categories, we only need to consider the interactions between the subutilities of different categories.
ASSUMPTION 2 (PERFECT SUBSTITUTES SPECIFICATION OF THE SUBUTILITIES). We assume that in any product category, at most one brand can be purchased by a consumer on a given purchase occasion. We thus specify the subutility for any category l as (Hanemann 1984) as
[v.sub.l]([{[X.sub.l, j]}.sup.h.sub.j=1]) = [h.summation over (j=1)] [[psi].sub.l, j][x.sub.l, j]. (2.4)
Although it is a reasonable assumption for categories such as detergents, softeners, ketchup, etc., in which consumers typically purchase at most one brand on an occasion, it may not hold true for categories like soft drinks, in which consumers do purchase more than one brand. For such cases, in a spirit similar to Bell and Lattin (1998) and Chib et al. (2005), we can further divide such a category into two or more categories such that in each of the subcategories consumers typically purchase at most one brand. By doing so, we can specify a perfect substitute subutility for each subcategory, as in (2.4). (4)
ASSUMPTION 3 (AGGREGATION OVER UNRELATED PRODUCT CATEGORIES). If we are only interested in studying the purchase decisions for a subset {1,..., L} of the M categories, then under certain conditions we can assume that the remaining categories, L + 1,..., M can be aggregated into a composite...
|
