Bass diffusion model

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:[4]

• 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

The Bass diffusion model is derived by assuming that the hazard rate ${\displaystyle \lambda (t)}$ for the uptake of a product or service may be defined as:

$${\displaystyle \lambda (t)={f(t) \over {S(t)}}=p+q[1-S(t)]}$$

where ${\displaystyle f(t)}$ is the probability density function and ${\displaystyle S(t)=1-F(t)}$ is the survival function, with ${\displaystyle F(t)}$ being the cumulative distribution function. From these basic definitions in survival analysis, we know that:

$${\displaystyle f(t)=-{dS \over {dt}}\implies \lambda (t)=-{1 \over {S}}{dS \over {dt}}}$$

Therefore, the differential equation for the survival function is equivalent to:

$${\displaystyle {dS \over {S[p+q(1-S)]}}=-dt}$$

Integration and rearrangement of terms gives us that:

$${\displaystyle {S \over {p+q(1-S)}}=Ae^{-(p+q)t}}$$

For any survival function, we must have that ${\displaystyle S(0)=1}$ and this implies that ${\displaystyle A=p^{-1}}$. With this condition, the survival function is:

$${\displaystyle S(t)={e^{-(p+q)t}+{q \over {p}}e^{-(p+q)t} \over {1+{q \over {p}}e^{-(p+q)t}}}}$$

Finally, using the fact that ${\displaystyle F(t)=1-S(t)}$, we find that the Bass diffusion model for product uptake is:

$${\displaystyle F(t)={1-e^{-(p+q)t} \over {1+{q \over {p}}e^{-(p+q)t}}}}$$

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.[5]

This model was developed in 1994 by Frank Bass, Trichy Krishnan and Dipak Jain:

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

where ${\displaystyle \ x(t)}$ is a function of percentage change in price and other variables

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:[6]

${\displaystyle \ S_{1,t}=F(t_{1})m_{1}(1-F(t_{2}))}$

${\displaystyle \ S_{2,t}=F(t_{2})(m_{2}+F(t_{1})m_{1})(1-F(t_{3}))}$

${\displaystyle \ S_{3,t}=F(t_{3})(m_{3}+F(t_{2})(m_{2}+F(t_{1})m_{1}))}$

where

• ${\displaystyle \ m_{i}=a_{i}M_{i}}$
• ${\displaystyle \ M_{i}}$ is the incremental number of ultimate adopters of the ith generation product
• ${\displaystyle \ a_{i}}$ is the average (continuous) repeat buying rate among adopters of the ith generation product
• ${\displaystyle \ t_{i}}$ is the time since the introduction of the ith generation product
• ${\displaystyle \ F(t_{i})={\frac {1-e^{-(p+q)t_{i}}}{1+{\frac {q}{p}}e^{-(p+q)t_{i}}}}}$

It has been found that the p and q terms are generally the same between successive generations.

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.

The Bass model is a special case of the Gamma/shifted Gompertz distribution (G/SG): Bemmaor[7] (1994)

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[8] 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.