A Piecewise-Diffusion Model of New-Product Demands

Abstract

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 purebirth-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.

How can you capture cultural dynamics?

“…Taga and Isii (1959) may be one of the earliest attempts to formally model a diffusion process. Since then, some have developed into more sophisticated formal models especially in the context of market penetration by new products (e.g., Niu, 2006). However, it was Nowak et al (1990) that marked the recent upsurge of interest in simulations of cultural diffusion processes.…”

A Sales Forecast Model for Short‐Life‐Cycle Products: New Releases at Blockbuster

“…The first force is due to an intrinsic interest in the given product, independent of the number of previous adopters, and the second is due to a positive influence from previous adopters. This leads to a differential equation (involving two parameters denoted by p and q that correspond to these two forces) for the fraction of potential adopters who would have adopted by time t , for t ≥ 0 (Bass , p. 217, or Niu , p. 679). For a given “large” population size, the solution of this differential equation then yields an S ‐shaped cumulative‐sales curve.…”

“…(), Niu (, ), Rogers (), Schmidt and Druehl (), Tanny and Derzko (), and Van den Bulte and Joshi (). As will be seen in section 3 below, the formulation of our model, which extends Niu (), is also related to this notion (see the committed‐ and potential‐buyer populations in section 3).…”

“…In this article, we will develop a sales model that incorporates all three features above. The construction of our model relies closely on recent stochastic extensions of the BM by Niu (, ). The model is presented in the next section.…”

“…The core of our formulation is the development of a set of recursive schemes that iteratively compute, for every i , a reasonable estimate for each of the means , , and and for the variance , based on the historical estimates of the means of all sales components prior to period i . This approach is similar in spirit to the expected‐history piecewise (i.e., period‐by‐period) diffusion model in Niu , p. 686, section 3.2), and is an extension of the formulation there to the present setting of a vector series of dependent sales components. We will develop the scheme for in and then in , that for .…”