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The Bass Model The Bass Model was first published in 1963 by Professor Frank M. Bass as a section of another paper.1 The section entitled "An Imitation Model" provides a brief, but complete, mathematical derivation of the model from basic assumptions concerning market size and the behavior of innovators and imitators. The paper did not provide empirical evidence in support of the model. The classic Bass Model paper was published in 1969.2 It expanded the theory and provided empirical support. The paper became one of the most widely cited paper in marketing science. It was named by INFORMS as one of the Ten Most Influential Papers published in the 50-year history of it flagship journal Management Science. In both the 1963 and the 1969 papers, Professor Bass credited Peter Frevert (then a Purdue student, now retired from University of Kansas) with many of the ideas that led to the theory. As Professor Bass told the story, Peter came to his office one day to ask how one would express mathematically the idea of imitators and innovators. Professor Bass wrote out a precursor of the differential equation

, which is read "The portion of the potential market that adopts at t given that they have not yet adopted is equal to a linear function of previous adopters." The symbols in the equation are explained on the Bass Math page. There are other representations of the Bass Model using different symbols and what may seem to be a different equation, but they are all equivalent. One equivalent equation is shown below.

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The Bass Model is the most widely applied new-product diffusion model. It has been tested in many industries and with many new products (including services) and technologies. The Bass Model assumes that sales of a new product are primarily driven by word-of-mouth from satisfied customers. At the launch of a new product, mostly innovators purchase it. Early owners who like the new product influence others to adopt it. Those who purchase primarily because of the influence of owners are called imitators. The preferred Bass Model equation is the solution to the differential equation, mathematically it is

where A(t) is cumulative adoptions by time t. Adoptions are sales to first-time buyers. M is the potential market, the maximum number of cumulative adopters. F(t) is the portion of the potential market M that has adopted by time t. Mathematically it is

The above formula is the solution to the Bass model differential equation described on the Bass Math page. The three Bass Model parameters (or coefficients)are: •

M -- the potential market (the totol number of adopters)

•

p -- coefficient of innovation

•

q -- coefficient of imitation

The portion of adopters who adopt in time period t is

The above formula for f(t) is the Srinivasan-Mason4 form, which is preferred for estimation of Bass model parameters M, p and q as well as for forecasting. These formulae are implemented in the open-source Excel spreadsheet Bass Model Forecaster, which can be downloaded here (free). Adoptions at time t are

The Bass Math page has the complete mathematical derivation of the Bass Model from basic principles.