User:Apdevries/Bass diffusion formula

The Bass diffusion model was developed by Frank Bass and describes the process how new products get adopted as an interaction between users and potential users. The model is widely used in forecasting, especially product forecasting and technology forecasting.

Frank Bass published his paper "A new product growth for model consumer durables" in 1969 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.

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.

Model formulation
$$\frac{f(t)}{1-F(t)} = p + {q}F(t)$$

Where: $$\ f(t) $$ is the rate of change of the installed base fraction $$\ F(t) $$ is the installed base fraction $$\ m $$ is the ultimate market potential $$\ p $$ is the coefficient of innovation $$\ q $$ is the coefficient of imitation

Sales $$\ S(t) $$ is the rate of change of installed base (i.e. adoption) $$\ f(t) $$ multiplied by the ultimate market potential $$\ m $$:

$$\ S(t)=mf(t) $$ $$\ S(t)=m{ \frac{(p+q)^2}{p}} \frac{e^{-(p+q)t}}{((1+\frac{q}{p})e^{-(p+q)t})^2} $$

The time of peak sales $$\ t^* $$

$$\ t^*=\frac{Ln \frac{q}{p}}{(p+q)} $$

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:
 * The average value of p has been found to be 0.03, and is often less than 0.01
 * The average value of q has been found to be 0.38, with a typical range between 0.3 and 0.5
 * q typically takes values between 0.5 and 0.9, with 0.7 the average



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 variable, e.g. pricing and advertising. This means that decision variable can shift the Bass curve in time, but that the shape of the curve is always similar.

Although many extensions of the model has 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:

$$\frac{f(t)}{1-F(t)} = p + x(t){q}F(t)$$

where $$\ x(t) $$ 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:

$$\ S_{1,t} = F(t_1) m_1 (1-F(t_2)) $$ $$\ S_{2,t} = F(t_2) (m_2 + F(t_1) m_1 ) (1-F(t_3)) $$ $$\ S_{3,t} = (m_3 + F(t_2)) (m_2 + F(t_1) m_1 ) $$

where $$\ m_i = a_i M_i $$ $$\ M_i $$ is the incremental number of ultimate adopters of the ith generation product $$\ a_i $$ is the average (continuous) repeat buying rate among adopters of the ith generation product $$\ t_i $$ is the time since the introduction of the ith generation product

$$\ F(t_i) = \frac{1-e^{-(p+q)t_i}}{1+\frac{q}{p} e^{-(p+q)t_i}} $$

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 first special case occurs when q=0, when the model reduces to the Exponential distribution.
 * The second special case reduces to the logistic distribution, when p=0.