Bass diffusion model

The Bass model or Bass diffusion model was developed by Frank Bass. It consists of a simple differential equation that describes the process of how new products get adopted in a population. The model presents a rationale of how current adopters and potential adopters of a new product interact. The basic premise of the model is that adopters can be classified as innovators or as imitators and the speed and timing of adoption depends on their degree of innovativeness and the degree of imitation among adopters. The Bass model has been widely used in forecasting, especially new products' sales forecasting and technology forecasting. Mathematically, the basic Bass diffusion is a Riccati equation with constant coefficients.
In 1969, Frank Bass published his paper on a new product growth model for consumer durables. Prior to this, Everett Rogers published "Diffusion of Innovations", a highly influential work that described the different stages of product adoption. Bass contributed some mathematical ideas to the concept.
Model formulation.
Where:
Sales (or new adopters) formula_7 at time formula_8 is the rate of change of installed base, i.e., formula_3 multiplied by the ultimate market potential formula_10. Under the condition formula_11, we have that
We have the decomposition formula_14 where formula_15 is the number of innovators at time formula_16, and formula_17 is the number of imitators at time formula_8.
The time of peak sales formula_19
Explanation.
The coefficient "p" is called the coefficient of innovation, external influence or advertising effect. The coefficient q is called the coefficient of imitation, internal influence or word-of-mouth effect.
Typical values of "p" and "q" when time "t" is measured in years:
Derivation.
The Bass diffusion model is derived by assuming that the hazard rate formula_21 for the uptake of a product or service may be defined as:formula_22where formula_23 is the probability density function and formula_24 is the survival function, with formula_25 being the cumulative distribution function. From these basic definitions in survival analysis, we know that:formula_26Therefore, the differential equation for the survival function is equivalent to:formula_27Integration and rearrangement of terms gives us that:formula_28For any survival function, we must have that formula_29 and this implies that formula_30. With this condition, the survival function is:formula_31Finally, using the fact that formula_32, we find that the Bass diffusion model for product uptake is:formula_33
Extensions to the model.
Generalised Bass model (with pricing).
Bass found that his model fit the data for almost all product introductions, despite a wide range of managerial decision variables, e.g. pricing and advertising. This means that decision variables can shift the Bass curve in time, but that the shape of the curve is always similar.
Although many extensions of the model have been proposed, only one of these reduces to the Bass model under ordinary circumstances. This model was developed in 1994 by Frank Bass, Trichy Krishnan and Dipak Jain:
where formula_35 is a function of percentage change in price and other variables
Successive generations.
Technology products succeed one another in generations. Norton and Bass extended the model in 1987 for sales of products with continuous repeat purchasing. The formulation for three generations is as follows: 
where 
It has been found that the p and q terms are generally the same between successive generations.
Relationship with other s-curves.
There are two special cases of the Bass diffusion model.
The Bass model is a special case of the Gamma/shifted Gompertz distribution (G/SG): Bemmaor (1994) 
Use in online social networks.
The rapid, recent (as of early 2007) growth in online social networks (and other virtual communities) has led to an increased use of the Bass diffusion model. The Bass diffusion model is used to estimate the size and growth rate of these social networks. The work by Christian Bauckhage and co-authors shows that the Bass model provides a more pessimistic picture of the future than alternative model(s) such as the Weibull distribution and the shifted Gompertz distribution.
Adoption of this model.
The model is one of the most cited empirical generalizations in marketing; as of October 2020 the paper "A New Product Growth for Model Consumer Durables" published in "Management Science" had (approximately) 9725 citations in Google Scholar. 
This model has been widely influential in marketing and management science. In 2004 it was selected as one of the ten most frequently cited papers in the 50-year history of "Management Science". It was ranked number five, and the only marketing paper in the list. It was subsequently reprinted in the December 2004 issue of "Management Science".