Bass Diffusion Model: Innovation and Imitation
Model 1 2
$$ \dot{N} = \left( p + q {{ N } \over { K }} \right) \left( 1 - {{ N } \over { K }} \right) $$
Variables
- $N(t)$: Represents the number of entities in a population at time point $t$.
Parameters
- $K$: Carrying Capacity, describes the size of the environment that can support the population. The number of entities cannot grow beyond the carrying capacity.
- $p$: Coefficient of Innovation or Global Growth Rate, describes the driving force for growth regardless of the group’s size. In business, this could be the advertising effect of launching new products.
- $q$: Coefficient of Imitation or Local Growth Rate, describes the driving force for growth proportional to the size of the group. In business, this could be word of mouth or trends. In the context of business, the number of entities in a group is the number of consumers. While most readers will find a consumer-oriented explanation more accessible, in reality, the Bass diffusion model can be applied wherever it is advantageous to change the status quo with innovation and follow trends.
Explanation
- Global Growth: Initially, whether it’s a new product or technology, appealing with some innovative elements attracts the attention of consumers and leads to global growth according to the global growth rate $p$. Especially in the early stages, when there are almost no consumers, assumed as $N \approx 0$, the Bass diffusion model reflects a linear trend as $\dot{N} = p$. The growth due to PR is independent of how many people currently use it, and it is assumed that the effect of innovation does not change.
- Local Growth: As the user group begins to grow, diffusion into the non-user group occurs. This diffusion is proportional to the size of the group and has a strong influence as the group increases its share in the environment, especially in the later stages of growth. It reflects not only the envy to use what others are using but also naturally becoming a consumer because everyone else is.
Derivation
Logistic Growth Model: $$ \dot{N} = {{ r } \over { K }} N ( K - N) $$
The Bass diffusion model is, in fact, a complete generalization of the commonly used linear logistic growth model. Simply substituting $\displaystyle r := {{ q } \over { K }}$ with
$$ \begin{align*} & \dot{N} = {{ r } \over { K }} N ( K - N) \\ \implies& \dot{N} = r N \left( 1 - {{ N } \over { K }} \right) \\ \implies& \dot{N} = \left( q {{ N } \over { K }} \right) \left( 1 - {{ N } \over { K }} \right) \end{align*} $$
So far, nothing has mathematically changed, and this is precisely the Bass diffusion model itself when exactly $p = 0$. Let’s modify the first term $\displaystyle \left( q {{ N } \over { K }} \right)$. To ensure that the group has at least a minimum growth rate, regardless of the number of entities, simply adding the constant term $p \ne 0$ to this term does the trick.
$$ \dot{N} = \left( p + q {{ N } \over { K }} \right) \left( 1 - {{ N } \over { K }} \right) $$
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Limitations
The limitation of the Bass diffusion model is that it explains only single innovations. In reality, innovations that can change the ecosystem’s panorama can continuously emerge, but the Bass diffusion model assumes the existing trend will never leave the throne. To overcome this, a generalized model is the Norton-Bass Model, which reflects the continuous emergence of innovations 3.