A stochastic formulation of the Bass model of new-product diffusion

• Main Author: Niu Shun-Chen
• Format: Article
• Language: English
• Published: Hindawi Limited 2002-01-01
• Series: Mathematical Problems in Engineering
• Subjects: Pure birth processes; Diffusion models; New-product adoptions; Epidemics
• Online Access: http://www.hindawi.net/access/get.aspx?journal=mpe&volume=8&pii=S1024123X02001977

<p>For a large variety of new products, the Bass Model (BM) describes the empirical cumulative-adoptions curve extremely well. The BM postulates 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 an individual&#39;s intrinsic tendency to purchase, independent of the number of previous adopters, and the other captures a positive force of influence on an individual by previous adopters. In this paper, we formulate a stochastic version of the BM, which we call the Stochastic Bass Model (SBM), where the trajectory of cumulative number of adoptions is governed by a pure birth process. We show that with an appropriately-chosen set of birth rates, the fractions of individuals who have adopted the product by time <math alttext="$t$"> <mi>t</mi> </math> in a family of SBMs indexed by the size of the target population converge in probability to the deterministic fraction in a corresponding BM, when the population size approaches infinity. The formulation therefore supports and expands the BM by allowing stochastic trajectories.</p>