The Bass model: Diffusion of new products

Predicting the adoption of new products by consumers can be a very difficult task. With no historical data, it’s hard to make solid predictions about the viability of a product still in development, or on the brink of launch. Fortunately, the Bass model of diffusion offers a scientific approach to this problem. What is the theory behind this model, and how does it work?

The Bass model

The bass model was developed by Frank Bass in 1969, and is meant to predict the rate of adoptions over time for new products. This helps in getting a picture of how the amount of new customers will develop. The model assumes that there are two ways of communication raising awareness of the product: mass-media communication (such as newspapers, magazines, TV, radio etc.) and word-of-mouth communication (oral communication about the product between people). This distinction is important because it affects different kinds of adopters: either the innovators or the imitators. Innovators are the early adopters. They are not influenced by the behavior of others and can decide to adopt the product based solely on mass-media communication. Initially, the innovators will be the largest part of your customers, and are very important to influence the second group: the imitators. Imitators are adopters that choose to try a new product based on word-of-mouth communication by others and the amount of previous adopters. Just like the name suggests, they mimic the behavior of others: the more adopters there are, the more imitators will also decide to adopt the product. To clarify things, here’s a graph of the adoption process: (please note: displayed are the new adopters) 

The bass model number of new adopters

As you can see, initially most adopters are innovators, but as time progresses more and more imitators will adopt the product. Summing the number of new adoptions gives the following graph with the cumulative number of adopters: 

Cummulative number of adopters bass model

Why does the graph have an s-shaped curve? Just after the launch of the new product the amount of adopters is still small, yet innovators will start to adopt the product. Gradually more and more innovators adopt the product, and as a consequence the number of imitators will start to rise. This follows an exponential process: the more adopters there are, the more new imitators will adopt the product. There comes a point in time however were most individuals in the market that are interested in the new product, and have the willingness to pay the set price, have been reached trough communication. The size of a market is just not unlimited. Therefore after the initial exponential development of adoptions, the number of new adopters will start to fall.

Now that the theory behind the model is explained, let’s take a look at the formulas used to determine the number of adopters and the adoption rate.

How does the bass model work?

Three parameters are needed to be able to fill in the formulas. Needed are the coefficient of innovation, the coefficient of imitation and the size of the market, the latter of which we will call the number of potential adopters. How to determine these parameters? For the number of potential adopters (m), market research will have to be conducted. The coefficients of innovation (p) and imitation (q) are set by looking at historical data. When launching a new innovative product, often one does not have the luxury of historical data for this product itself. Therefore it is common practice to look at the coefficients of similar products that previously have been launched within the same product category and/or market. But what are these coefficients exactly? The coefficient of innovation stands for the chance that an innovator will adopt the product at time t. Likewise, the coefficient of imitation represents the chance that an imitator will adopt the product at time t. Now on to the formula. The rate of adoption (called P) at time t is given by the following formula:

bass model formula

(Y(t-1) represents the cumulative number of adoption so far.)

As you can see in the formula, the amount of imitators depends on the proportion of the market potential that has already adopted the product. The formula reflects the theory previously described: the more adopters, the more imitators. At the same time, the coefficient of innovation is independent of the previous amount of adopters.

Now that we have a formula for the amount of adoptions at time t, we can create a formula for the number of new adoptions (S) at time t. Only consumers that have not adopted before can become a new adopter. Therefore the number of new adopters is determined by multiplying the rate of adoption, for which we have established the formula, with the amount of potential adopters that is left:

Bass model formula

With the aid of these formulas one can easily make sound predictions about the amount and rate of product adoptions to expect, without having access to any historical data of your own.

Source: Mahajan V., Muller E., & Bass F. M. (1990). ‘’New product diffusion models in marketing: A review and directions for research’’. Journal of Marketing, 54(1):1–26

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