How to predict if a new product will catch on? / Bass diffusion model – part 2

I tried to postpone it but I guess it is time to start talking mathematics and dive into the Bass diffusion model in order to understand why the short head economy is happening. Therefor this post requires you to be focused and concentrated.

As said before the Bass diffusion model is a mathematical model that describes the “Diffusion of innovations”.

This is the Bass model principal:

These are the variables of the equation:

  • f(t) – The portion of the potential market that adopts the new product at time t.
  • F(t) – The portion of the potential market that have adopted up to and including time t.
  • a(t) – Adopters (or adoptions) at t.
  • A(t) – Cumulative adopters (or adoptions) at t.

And these are the parameters (coefficients):

  • M – The potential market (the ultimate number of adopters).
  • p – Coefficient of innovation (external influence such as PR, advertising etc.).
  • q – Coefficient of imitation (internal influence – interactions between adopters and potential adopters in the social system).

Another way to look at it is this:

The graph of a(t) looks like this:

The number of adoptors at time t - bass diffusion model

The number of adopters at time t

From the graph we can see the importance of the word of mouth influence (q). The flat blue line presents a case where there is no internal influence (meaning that the word of mouth doesn’t work at all)…

Let’s take a look at the Cumulative adopters graph (A):

The number of Cumulative adopters at time t - bass diffusion model

The number of Cumulative adopters at time t

As you can see, p (the external influence like advertising and PR) mostly affects the starting time of the adoption. q, the internal influence meaning the word of mouth) is the important parameter that influences the adoption and sales.

Here are another 2 examples:

  a4

a5  a6

are you still with us? that great! Get some coffee and let’s carry on…

 

How do we predict the future of the new product?

Well, it’s easy. After we have the real data of several time periods we can calculate the coefficients (p, q and m).

After we know the coefficients we can put them in the equation and predict the future adoption of the product.

Here is an example of how the model worked in the past:

Eight Generations of DRAM Chips Actual and Fitted 1974-2000 / same p's and q's - Data from Gartner Dataquest

Eight Generations of DRAM Chips Actual and Fitted 1974-2000 / same p’s and q’s – Data from Gartner Dataquest

 

Now let’s go back to the original equation:

On the left side of the equation is the portion of the potential market that adopts at time t (f(t)) given that they have not yet adopted (1-F(t)). It is known as a hazard rate. It also presents the probability that someone will adopt the innovation in this time period, given that he has not yet adopted it.

For example – let’s say that 10% of the potential market have bought the new product at time X, and till now (altogether including time X) 30% of the potential market have purchased the product, the result would be: 10%/70% = 1/7 = 0.14. That is also the probability that someone who has not yet purchased the product would do so in this time period.

If another 10% of the potential market bought the new product at time Y, and till now (altogether including time Y) 80% have already purchased the product, the result would be: 10%/20% = 0.5. Again, that is also the probability that someone who has not yet bought the product would do so in this time period – 50%. He can be one of the 10% that will buy the product (out of the 20% that still haven’t) or one of the 10% who would still not buy (out of the 20%). The chances are even in this case.

The right side of the equation is linear with respect to the number of previous adopters and the variables.

So the chances that someone would buy the product are influenced by the variables and the number of previous adopters.

The chances for adoption are higher when q, p and the number of previous adopters are higher.

all we have to do now is check if anything happened to these coefficients in the last several years.

to be continued…

 

For further reading – http://www.bassbasement.org/BassModel/BassMath.aspx

 

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