# The Bass Diffusion Model

Total sales of a product in any given period = (Sales to Innovators) + (Sales to Imitators)

Innovators are people who are early adopters of the product. They are people who buy a product because they like it; not because other people are buying.

Imitators are people who have heard about a product from someone else and are likely buying it because other people are.

(Sales to Imitators) = (Total Market – Previous Buyers) * (% of Imitators) * ([Previous Buyers] / [Total Market])

The Bass Diffusion Model states that:

Qt = p(M – Nt-1) + q*(Nt-1/M)*(M – Nt-1)

Where:

Qt = the number of adopters during time t
M = ultimate number of adopters (market size)
Nt-1 = cumulative number of adopters at the beginning of time t
q = effect of each adopter on each non-adopter. This is the coefficient of imitation.
p = individual conversion rate absent adopters’ influence. This is the coefficient of innovation.

The innovation effect is described by:

p(M – Nt-1)

And the imitation effect is described by:

q*(Nt-1/M)*(M – Nt-1)

We can rewrite the Bass Diffusion Model as follows:

Qt = pM + (q – p)Nt-1 – (q/M)N2t-1

Notice that this is a quadratic equation. A regression analysis can be performed on the data to find coefficients such that:

Qt = a + bNt-1 – cN2t-1

where:

a = pM
b = q – p
c = -(q/M)

M can be found by applying a variation of the Quadratic formula:

M = [-b +/- sqrt(b2 – 4ac)] / 2c

Finally, since a is known from regression and M has been calculated, p can be found. Then, p can be used to find q.