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Article Excerpt
Utility maximizing solutions to economic models of choice for goods with either discrete quantities or nonlinear prices cannot always be obtained using standard first-order conditions such as Kuhn-Tucker and Roy's identity. When quantities are discrete, there is no guarantee that derivatives of the utility function are equal to derivatives of the budget constraint. Moreover, when prices are nonlinear, as in the case of quantity discounts, first-order conditions can be associated with the minimum rather than the maximum value of utility. In these cases, the utility function must be directly evaluated to determine its maximum. This evaluation can be computationally challenging when there exist many offerings and when stochastic elements are introduced into the utility function. In this paper, we provide an economic model of demand for substitute brands that is flexible, parsimonious, and easy to implement. The methodology is demonstrated with a scanner panel data set of light-beer purchases. The model is used to explore the effects of price promotions on primary and secondary demand, and the utility of product assortment.
Key words: Bayesian analysis; econometric models; pricing research; product management History: This paper was received June 25, 2001, and was with the authors 3 months for 3 revisions; processed by Pradeep Chintagunta.
1. Introduction
Economic models of brand choice must deal with two complexities when applied to the study of packaged goods. First, consumers are restricted to purchasing discrete quantities of brands. Demand for brands is defined on a grid of available brand-pack combination and not on the entire set of real numbers. Second, the unit price of a brand often depends on the quantity purchased. Quantity discounts are typically available where the per-unit price declines with the size of the package. These two complexities invalidate the use of first-order conditions to identify utility maximizing solutions. First-order conditions may not hold at the available package sizes. Furthermore, constraints imposed by the nonlinear budgetary allotment can lead to first-order conditions that identify the minimum rather than the maximum utility. In this paper, we propose a random utility choice model capable of dealing with discrete quantities and quantity discounts.
Quantity discounts are common in marketing. They occur with packaged goods when the unit price declines with larger quantities, and with services when the price declines as usage increases (e.g., cell phones). Quantity discounts afford manufacturers the ability to price discriminate between high-volume and low-volume users (see Dolan 1987). The depth of price promotions for a particular quantity results in temporary substitution away from other brands and an increase in demand for the product category. An important issue in managing discounts is in determining the expected increase in sales from increased product usage versus substitution from competitive brands. This requires a comprehensive model that can handle a large number of distinct brand-pack combinations that are typically available in many product categories.
Researchers in marketing who are studying consumer demand for packaged goods have tended to ignore the complexities associated with quantity discounts, and have used one of three strategies for dealing with unit prices that depend on the purchase quantity. Table 1 provides a selective review of the literature. The first approach to dealing with nonlinear prices is to explode the number of choice alternatives and model each brand-size combination as a distinct choice alternative. In this approach, consumers are assumed to derive utility from having the brand bundled into different quantities. Although various parameters and constraints could, in theory, be introduced into the analysis so that the estimated parameters exhibit desirable properties (e.g., diminishing returns to quantity), a drawback is that it conditions on the product class expenditure and therefore does not provide insight into the trade-off between the product class and other goods. This approach confounds the expenditure decision for the product category and utility for the brand. A consumer who regularly purchases a particular quantity will be seen as having low utility for larger quantities. However, this may be because of the consumer's budget constraint and not because of a lack of preference for larger quantities. The second approach to dealing with nonlinear prices has been to restrict analysis to a particular package size. Whereas such analysis provides valid measures of brand preference, it usually rules out the study of purchase quantity and category expenditure. The third approach uses an average per-unit pricing variable in the analysis. The validity of this approach depends on the degree to which the actual price schedule is linear, which is often not the case.
In this paper, we propose a random utility model for consumer choice that deals with discrete quantities and quantity discounts. The model does not treat different brand-size combinations as different multinomial choice alternatives. Instead, we derive the likelihood specification for the observed demand data from more fundamental assumptions about random utility. This approach results in a parsimonious likelihood specification that facilitates the study of consumer demand across package sizes and does not assume that prices are linearly related to quantity. Our model is not affected by the presence of quantity discounts that are frequently encountered with packaged goods data and deals directly with constraints imposed by discrete package sizes.
In [section]2, we discuss problems encountered with discrete quantities and quantity discounts encountered in the study of packaged goods. Our approach to dealing with these issues in the context of a brand-choice model is discussed in [section]3. We note that our model can accommodate the presence of quantity discounts, discrete quantities, or both, and therefore has wide application. Furthermore, both standard linear utility and nonhomothetic utility structures (Allenby and Rossi 1991) can be incorporated into the framework. In [section]4, we describe a scanner panel data set of light-beer purchases in which there are more than 80 brand-pack combinations available to consumers in the market. Parameter estimates of our parsimonious demand model are discussed in [section]5. Pricing implications are explored in [section]6, and conclusions are offered in [section]7.
2. Problems Encountered with Discrete Quantities and Quantity Discounts
Consider a brand-choice model arising from a linear utility structure (u([chi]) = [psi]'[chi]), where the price of the brand depends on quantity. In this case, the vector of marginal utility, [chi], is constant, and utility, u, is maximized subject to the budget constraint [[summation of].sup.k.sub.k=1] [p.sub.k]([[chi].sub.k]) < y, where [p.sub.k]([[chi].sub.k]) is the price of [[chi].sub.k] units of brand k. Assume that the price function [p.sub.k] ([[chi].sub.k]) reflects quantity discounts. For example, the price schedule could be a marginally decreasing function of quantity:
(1) p([[chi].sub.k]) = [[chi].sup.1/a.sub.k]; a > 1; [chi] [greater than or equal to] 0.
The identification of the utility maximizing solution using first-order conditions typically proceeds by differentiating the auxiliary function:
(2) L = [psi]'x - [lambda]([k.summation over k=1][p.sub.k]([x.sub.k]) - y),
setting first derivatives to zero, and solving for the values of [x.sub.k], at which the vector of marginal utility is tangent to the budget curve. We note immediately that, if demand is restricted to available pack sizes, first-order conditions will not necessarily apply at the observed quantity demanded. However, even when quantity is not restricted to a fixed number of values, the use of first-order conditions can lead to a solution identifying a point of utility minimization, not utility maximization. As illustrated in Figure 1 for the case of linear utility and price quantity discounts, the point of tangency is not associated with the utility maximizing solution. Utility is greater at the intersection of the budget curve and either axis.
[FIGURE 1 OMITTED]
The reason that first-order conditions are associated with a point of utility minimization rather than utility maximization is because the auxiliary function in Equation (2) is convex, or has a positive second derivative. The second derivative of Equation (2), when prices follow Equation (1), can be shown to be equal to:
(3) [[differential].sup.2]L/[differential][x.sup.2.sub.i] = -[lambda](1/a)(1 - a/a)[x.sup.(1-2a)/a.sub.i],
which is positive for a > 1 and x > 0. More generally, whereas the concavity of the auxiliary function is guaranteed when the utility function is concave and the...
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