A piecewise-diffusion model of new-product demands

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.