Here, we introduce Bass diffusion model which is a classic way to predict sales for newly launched product in the market. It is an effective way to know the overall sales of the product in its lifecycle and make strategy accordingly.
After launching a new product in the market, sales prediction for it has always been a difficult task due to lack of historical data. However, having an accurate prediction is extremely important not only from a marketing standpoint, but also for managing overall product life cycle. It helps in making overall strategy for the product like determining the time for markdown, promotion and introduction of an updated version of the product in the market.
Bass diffusion model is a classic model for this purpose and it has successfully been used for sales prediction of many such new products very accurately. It consists of some simple differential equations describing the process of how a product gets adopted in the market.
The model makes two basic assumptions about the potential buyers:
The other assumptions are:
Next, let’s have a look at the equations first and formulate them:
Let’s define the cumulative probability of a customer buying the product till time t is F(t). Then the probability of purchase for that customer at time t is f(t) = F´(t). The rate of purchase at time t is:
Which basically signifies the ratio between probability of purchasing the product at time t given that the customer has not bought it till time t.
As per Bass, this purchase rate can be defined as:
Where p is the rate of innovative adoption and q is imitation rate. Further, q is multiplied by F(t) as imitation can happen only based on the current innovative adoption. So, the above equation gives probability of total adoption by a customer at time t, both from innovative adoption and imitation adoption.
Now, if we solve the equation, we get as follows (If you are not interested in Maths, you can skip this part and jump to the final values for f(t) and F(t)):
From t = 0, we get,
F(0) = 0 and
Substituting the above in the main equation and solving for F(t), we get,
Also, if the total market size is m, then the total adoption at time t will be m*f(t). Below are the sales curves for different values of p and q considering the market size to be fixed at 100,000. Usually, in real life, p is smaller than q.
Now, the immediate next question is how to estimate p, q and m. It can be estimated from first few week’s sales of the product or past sales of similar product (eg older version of the product or product category). Here, we will see how we can obtain the estimate using OLS. However, the same can also be obtained using NLS.
If the market size is m then total sales in any period is given by,
s(t) = m*f(t)
And cumulative sales up to time t is S(t) = m*F(t). From Bass equation, we get,
With simple algebraic calculation, the above can be re-written as:
So, the Bass equation essentially becomes a regression of sales at time t on cumulative sales till time t. And the coefficients β0, β1 and β2 can easily be estimated using OLS. Once, the estimation is done, p, q and m can be back calculated from the coefficients as below:
The above is a quadratic equation in m and can easily be solved to get a value for m in terms of β0, β1 and β2. Substituting the value of m in p = β0/m and q = -mβ2, we can also get the value for p and q.
Although, it is an excellent method of predicting sales of a newly introduced product in the market and has been used for many use cases in past, there are some limitations of it:
To overcome the above limitations, lots of researches have been conducted and the model has been extended to accommodate some of the other factors affecting sales. Like the below one takes into account the price and other features of the product (denoted by x(t)):
In conclusion, in the article, we saw how Bass diffusion model can be used to estimate sales of a new product in the market, how to estimate the coefficients and some of its limitations. This is a well researched model and lots of extensions of the model are also available. The key issue, however, is the estimation of the coefficients. Nevertheless, the model can still give a fair prediction of sales which can be used for many business decisions regarding the product strategy.