# 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.