Mathematical Formulation of the Bass Diffusion Model

The Bass Diffusion Model mathematically represents the adoption process through a differential equation that describes the rate at which new adopters enter the market. This rate depends on both the proportion of innovators and the influence of previous adopters on imitators.

The basic equation of the Bass Diffusion Model is as follows:

    f(t) = (p + q * F(t)) * (1 - F(t))
  

Here:

• f(t): The probability density function, representing the rate of adoption at time t.
• F(t): The cumulative distribution function, representing the cumulative proportion of adopters at time t.
• p: Coefficient of innovation (rate of initial adoption).
• q: Coefficient of imitation (social influence factor).

The cumulative number of adopters at time t, N(t), can be expressed as:

    N(t) = m * F(t)
  

where m represents the market potential, i.e., the total number of potential adopters.

The model reveals that adoption grows over time due to two effects: the intrinsic attractiveness of the product (p) and the influence of previous adopters (q). Early in the product's lifecycle, adoption is driven mainly by innovators, but as the product gains traction, imitators play a larger role in driving growth.

The point at which adoption reaches its maximum rate is known as the inflection point, occurring when the proportion of imitators begins to exceed that of innovators. This creates the characteristic S-shaped curve of the adoption process.

Understanding the Bass Diffusion Model helps companies strategize product launches, marketing efforts, and forecast long-term product performance. It provides valuable insights into how and when a product is likely to gain market traction.