A Stochastic Formulation of the Bass Model of New-Product Diffusion

In the past several decades, new-product diffusion models has been an active area of research in marketing (see, e.g., Mahajan, Muller, and Wind 2000, and Mahajan and Wind 1986). Such models are useful because they can provide important insights into the timing of initial purchase of new products by consumers. Much of the work in this area has been spawned by a seminal paper of Bass (1969), in which it was postulated that the trajectory of cumulative adoptions of a new product follows a deterministic function whose instantaneous growth rate depends on two parameters, one of which captures a consumer?s intrinsic tendency to purchase, independent of the number of previous adopters, and the other captures a positive force of influence on a consumer by previous adopters. While Bass?s model, or the Bass Model (BM), yields an S-shaped cumulative-adoptions curve that has proven to provide excellent empirical fit for a wide range of new-product-adoptions data sets (especially for consumer durables), there also has been a common belief (see, e.g., Eliashberg and Chatterjee 1986) that it would be of interest to have an appropriate stochastic version of his model. The purpose of this paper is to formulate and study a stochastic counterpart of the BM. Inspired by a very early paper of Taga and Isii (1959), we formulate the trajectory of cumulative number of adoptions as a pure birth process with a set of state-dependent birth rates that are judiciously chosen to closely parallel the roles played by the two parameters in the deterministic BM. We demonstrate that with our choice of birth rates, the resulting pure birth process exhibits characteristics that resemble those in the BM. In particular, we show that the fraction of individuals who have adopted the product by time t in our formulation agrees with (converges in probability to) the corresponding deterministic fraction in a BM with the same pair of parameters, when the total number of consumers in the target population approaches infinity. Our formulation, therefore, supports and expands the BM by having explicit micro-level stochastic interactions amongst individual adopters.