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Article Excerpt
The Bass Model (BM) is a widely-used framework in marketing for the study of new-product sales growth. Its usefulness as a demand model has also been recognized in production, inventory, and capacity-planning settings. The BM postulates that the cumulative number of adopters of a new product in a large population approximately follows a deterministic trajectory whose growth rate is governed by two parameters that capture (i) an individual consumer's intrinsic interest in the product, and (ii) a positive force of influence on other consumers from existing adopters. A finite-population pure-birth-process (re)formulation of the BM, called the Stochastic Bass Model (SBM), was proposed recently by the author in a previous paper, and it was shown that if the size of the population in the SBM is taken to infinity, then the SBM and the BM agree (in probability) in the limit. Thus, the SBM "expands" the BM in the sense that for any given population size, it is a well-defined model. In this paper, we exploit this expansion and introduce a further extension of the SBM in which demands of a product in successive time periods are governed by a history-dependent family of SBMs (one for each period) with different population sizes. A sampling theory for this extension, which we call the Piecewise-Diffusion Model (PDM), is also developed. We then apply the theory to a typical product example, demonstrating that the PDM is a remarkably accurate and versatile framework that allows us to better understand the underlying dynamics of new-product demands over time. Joint movement of price and advertising levels, in particular, is shown to have a significant influence on whether or not consumers are "ready" to participate in product purchase.
Subject classifications: marketing: new products, buyer behavior, pricing; probability: diffusion; inventory/production: stochastic, nonstationary demand.
Area of review: Manufacturing, Service, and Supply Chain Operations.
1. Introduction and Summary
Operational decisions of firms often depend critically on demand information; indeed, demand can be taken as what drives entire supply chains. Conversely, production and marketing strategies also directly impact on demand growth. The appropriate modeling of demand is therefore of importance. This is particularly true for new products, because demands for such products have a highly dynamic and hence less predictable growth behavior. The Bass Model (BM), introduced over three decades ago (Bass 1969), is a new-product sales-growth model that has been widely used in marketing (see, e.g., Mahajan and Muller 1979, Mahajan and Wind 1986a, Mahajan et al. 1990, Parker 1994, Rogers 1995, and Mahajan et al. 2000). The model was originally developed and tested for consumer durables, but it has been found to be applicable to many other product categories as well. Furthermore, it has been generally recognized that the BM, in addition to its usefulness in marketing, can serve well as an integrated component in formulating production, inventory, and capacity-planning models (see, e.g., Cohen et al. 2000, p. 256, Table 10.1). The formulation of the BM is, however, deterministic. Consequently, it does not offer the often sought and possibly critical information on the variability of the sales trajectory of a new product. This is one of the primary motivations that prompted the initiation of work on a stochastic generalization of the BM in Niu (2002). This paper is a continuation of Niu (2002); it is concerned with the formulation, sampling-theory development, and empirical examination of a suite of stochastic new-product demand models that are further extensions of the BM.
To properly motivate the model development, we begin with a description of the BM. For t [greater than or equal to] 0, let N(t) be the cumulative number of adopters of a product by time t in a large target population; then, it is postulated that the trajectory of N(t) is deterministic and that N(t) can be approximated by the solution of the differential equation
[d/dt]N(t) = [m - N(t)][p + [q/m]N(t)], (1)
where N(0) [equivalent to] 0, m is the size of the population, and p and q are two parameters that are called, respectively, the coefficient of imitation and the coefficient of imitation. That is, N(t) is assumed to grow at a rate that equals the product of m - N(t), the number of individuals who have not yet adopted the product by time t, and p + (q/m)N(t), a linear function in the number of existing adopters. The parameter p can be interpreted as reflecting the extent of a consumer's intrinsic propensity to purchase the product, and the parameter q the extent of a positive influence from an existing adopter on the entire population.
The BM can also be formulated in terms of the cumulative fractions of individuals who have adopted the product over time. For t [greater than or equal to] 0, let
F(t) [equivalent to] [N(t)]/m; (2)
then, Equation (1) is equivalent to
[d/dt] F(t) = [1 - F(t)][p + qF(t)]. (3)
Note that it is implicit in (2) that m is essentially "at infinity," so that the ratio N(t)/m is independent of m. It follows that the fraction in (2) can intuitively be thought of as the "probability" for a randomly-selected individual in an infinite population to have adopted the product by time t. Equation (3), together with this adoption-time-distribution interpretation for (2), is in fact often taken as the primitive definition of the BM.
While the S-shaped solution of (3) (Bass 1969, p. 218),
F(t) = [1 - [e.sup.-(p+q)t]]/[1 + (q/p)[e.sup.-(p+q)t]], (4)
has often been found to fit well with observed sales data, the question of whether or not there exists a suitable version of the BM that incorporates randomness into the model has been raised by many researchers (see, e.g., Jeuland 1979, Mahajan and Muller 1979, Mahajan and Peterson 1985, Eliashberg and Chatterjee 1986, Boker 1987, and Skiadas and Giovanis 1997). Stochastic formulations are desirable because they could allow greater realism in terms of micro-level interactions between consumers; they allow the sales trajectory to be dynamic, or history dependent (note that (4) does not have a dynamic dependence on the sales history; see, e.g., Trajtenberg and Yitzhaki 1989, p. 36, paragraph after Equation (4) in particular); and, most importantly, they could lead to sampling theories that are needed to support parameter estimation and to conduct interval sales forecasts (see, e.g., Schmittlein and Mahajan 1982, p. 74, top two paragraphs; Eliashberg and Chatterjee 1986, pp. 192-194; and Boker 1987, pp. 64-65).
Building on a very early model of Taga and Isii (1959), Niu (2002) recently proposed a stochastic version of the BM, called the Stochastic Bass Model (SBM). The idea behind the formulation of the SBM is simply to replace the deterministic trajectory N(t) defined in (1) by a state-dependent pure birth process (see [section]2.1 for a formal definition). Paralleling the BM, the birth rates in the SBM are assumed to depend on two parameters that are conceptual equivalents of p and q. It is shown in Niu (2002) that for sufficiently large m, the fraction of individuals who have adopted a product by time t in the SBM is, with probability close to 1, located within any given small neighborhood of the corresponding fraction (2) in a BM with the same pair of parameter values. In other words, the SBM has been shown to be consistent with the BM in the limit.
There is, however, an important difference between the SBM and the BM, namely, that the former provides a well-defined model instance for every finite m, whereas the latter, defined in (2) and (3) with m intended to be at infinity, does not have sufficient structure to lend itself to detailed modeling when m is finite (although it does work well as an approximation). Thus, the SBM properly "expands" the BM by covering the full spectrum of population sizes. The expansion naturally suggests the possibility of using the SBMs as building blocks in the rigorous construction of new-product demand models that allow varying population sizes over time. Because the active sales-growth periods of most successful products span more than a decade, the proper development of such models has long been recognized as a challenging problem in the literature (see, e.g., Mahajan and Wind 1986b, pp. 15-16).
In this paper, we exploit the finite-population expansion in the SBM formulation and introduce a Piecewise-Diffusion Model (PDM) in which sales of a product in different time periods (years, typically) are assumed to be governed by stochastic outcomes of successive runs of a history-dependent family of SBMs. More specifically, the SBMs in the PDM are constructed to have appropriately-linked parameters and to have differently-sized populations of "ready" (or "genuine") potential buyers of a product, both of which are explicitly tied to the cumulative sales history. The PDM is thus, in contrast with the BM, a fully-dynamic model.
The PDM is in fact a versatile umbrella that covers a host of versions that progressively incorporate more marketing-mix variables, i.e., price, advertising, promotion, distribution, product quality, and other information (see, e.g., Kalish and Sen 1986, pp. 90-91). At the base level, we assume that in every period, the number of ready potential adopters is a constant fraction, called the participation fraction, of those in the entire target population who have not yet adopted the product. The constant-fraction assumption is intended to approximately model the aggregate impact of all underlying marketing-mix variables. This results in a basic version of the PDM that can be useful in scenarios where detailed marketing-mix information is not available. Following the well-established spirit of a stream of prior research (see, e.g., Robinson and Lakhani 1975; Bass 1980; Horsky and Simon 1983; Kalish 1985; Kalish and Sen 1986; Jain and Rao 1990; Bass et al. 1994, 2000), we also develop versions of the PDM that explicitly account for the marginal impact of two most important marketing-mix variables, namely, price and advertising. Specifically, in the context of the PDM, we assume that the effects of price and advertising can be captured through a "response function" that for every given time period, takes as input the current price and the current advertising spending and returns as output a prescribed participation fraction for that period; in other words, we assume that price and advertising jointly modulate successive participation fractions. This results in a couple of more sophisticated versions of the PDM, with a set of parameters that reflect current-period effects of price and advertising. Finally, we assume that advertising also has a cumulative effect and formulate into the PDM a parameter that explicitly reflects the extent of such impact.
A rigorous sampling theory for the PDM is also developed in this paper. The theory is built on the solution of a stochastic differential equation that, when m is sufficiently large, accurately describes the movement of the deviation (properly scaled) between the actual adoption trajectory and the expected adoption trajectory in the SBM. This results in a central limit theorem, on the basis of which an explicit likelihood function for successive sales in the PDM is derived. The likelihood function is then used to conduct a careful empirical validation of the PDM.
The PDM has been tested against and found to be solidly supported by data for a variety of products. In this paper, we will discuss empirical results only for a single product, namely, room air conditioners. (A thorough empirical study for seven other products will be reported in a companion paper; see Niu 2006.) This particular product is chosen because its sales trajectory has a very generic pattern, and because it has been studied, and thus benchmarked, extensively in previous literature. For this product, we also compare the empirical performance of the PDM with those of the BM, which does not explicitly include price and advertising, and the Generalized Bass Model (GBM; see Bass et al. 1994 and Krishnan et al. 1999), which is a well-established extension of the BM that includes price and advertising (the majority of other extensions of the BM includes either price or advertising, but not jointly). The PDM is found to deliver superior performance, in terms of both model fit and step-ahead forecasts. The accuracy of multiperiod step-ahead forecasts (up to five steps in a single set of forecasts without parameter update) of the PDM over that of the GBM is particularly noteworthy, with marginal reductions in mean squared errors that are in the neighborhood of 99%.
The PDM is highly microscopic. Beyond fit accuracy, the PDM yields a substantial amount of surgical information that allows us to better understand the dynamics of a host of underlying characteristics that drive the demand trajectory. Some of the conclusions from our empirical analysis are highlighted next.
It is well known that parameter estimates of the BM, most critically that for m, are not stable (see, e.g., Srinivasan and Mason 1986, p. 177, Table 6, the "m" column in particular; and Putsis and Srinivasan 2000, pp. 268-269). This strongly suggests that the BM, despite its acknowledged fit performance, is not well specified. In the PDM, m is considered generic and static, typically set according to the number of households; its traditional parameter status (Bass 1969) in the BM is shifted to the participation fractions. This results in remarkably stable estimates both for the model parameters and for successive levels of cumulative market penetration, the latter of which is of particular managerial interest.
For a product that has existed for a sufficiently long duration prior to the first period of an empirical study, a consumer's intrinsic/initial interest in the product (p in the BM) is found to be (typically)...
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