Bass Model and Diffusion

The Bass model or Bass diffusion model^[1] was developed by Frank Bass. It consists of a simple differential equation that describes the process of how new products get adopted in a population. The model presents a rationale of how current adopters and potential adopters of a new product interact.

The basic premise of the model is that adopters can be classified as innovators or as imitators, and the speed and timing of adoption depends on their degree of innovation and the degree of imitation among adopters.

Bass Model

Parameters:

• $p$ is the coefficient of innovation
• $q$ is the coefficient of imitation

The coefficient $p$ is called the coefficient of innovation, external influence or advertising effect. The coefficient $q$ is called the coefficient of imitation, internal influence or word-of-mouth effect.

Model formulation

$$ \frac{f(t)}{1-F(t)}=p+q F(t) $$

Where

• $F(t)$ is the installed base fraction
• $f(t)$ is the rate of change of the installed base fraction, i.e. $f(t)=F^{\prime}(t)$

Indeed, $F(t)$ can be interpreted as a distribution representing the fraction of adopters. And ${f(t)}/({1-F(t))}$ is its failure rate, i.e., the likelihood of a non-adopter adopting the product at time $t$.

It can be expressed as a differential equation $$ \begin{cases} & \displaystyle\frac{\mathrm{d} F}{\mathrm{d} t}=p(1-F)+q(1-F) F=(1-F)(p+q F) \\ & F(0) = 0 \end{cases} $$

The solution is $$ F(t)=\frac{1-e^{-(p+q) t}}{1+\frac{q}{p} e^{-(p+q) t}} $$

• when $q=0$, the model reduces to the exponential distribution.
• when $p=0$, the model reduces to the logistic distribution.

Because new product sales data are commonly available in discrete time intervals (such as quarters or years), the discrete-time version of the BM is commonly estimated by using the nonlinear least-squares procedure, first proposed by Srinivasan and Mason (1986)^[2], by minimizing the following criterion function. $$ S S E=\sum_{t=1}^T[\hat{N}(t)-N(t)]^2 \tag{NLS} \label{NLS} $$ where $T$ stands for the total number of periods for which new product sales data are available, $N(t)$ represents the observed sales of the new product in period $t$, and $\hat{N}(t)$ represents the predicted sales of the new product in period $t$ and is given by $$ \hat{N}(t)=M \ast[F(t)-F(t-1)] $$ Minimizing the criterion function in Equation $\eqref{NLS}$ yields estimates of $p, q$, and $M$ for the new product.

Generalized Bass model

The generalized Bass model^[3] takes into the effect of pricing.

If the price sequence is $r_t$, $x(r_t)$ reflects the effect of price. $$ \frac{\mathrm{d} F}{\mathrm{d} t}=p(1-F)+q(1-F) F=(1-F)(p+q F) x(r_t) $$

Kalish (1983) and Feichtinger (1982) have assumed that market potential $m$ can be expressed as a function of price. $$ \frac{\mathrm{d} F}{\mathrm{d} t} = (m(r_t) - F)(p + q F) $$ where $m(r)$, the market size, is limited by the number of consumers who are willing to purchase the product at price $r$.

Other extensions

「Multigeneration Product Diffusion in the Presence of Strategic Consumers」extends the Bass Model to multi generation products.

Diffusion-Choice Model^[4]

Consider a firm selling $n$ products concurrently and maximizing profit within a given planning horizon of $T$ time periods. Define $M$ as the total market potential. Let $Z_{i t}$ be the sales of product $i$ in period $t$ and $Y_t$ be the cumulative sales (including all products) by the end of period $t$. Thus, $$ Y_t=\sum_{s=1}^t \sum_{i=1}^n Z_{i s} $$ The demand for product $i$ in period $t$ is given by $$ \begin{aligned} & Z_{i t}=\left(M-Y_{t-1}\right)\left(\alpha+\beta Y_{t-1}\right) q_{i t,} \quad i=1, \ldots, n, \\ & \text { where } q_{i t}=\frac{\exp \left(a_{i t}-b_t p_{i t}\right)}{1+\sum_{j=1}^n \exp \left(a_{j t}-b_t p_{j t}\right)} \end{aligned} $$ The firm’s price optimization problem is given by $$ \max _{\substack{p_{it} \\ i=1, \ldots, n \\ t=1, \ldots, T}} \sum_{t=1}^T \sum_{i=1}^n\left(p_{i t}-c_{i t}\right) Z_{i t} $$

Define $H_t:=M-Y_{t-1}$ and $F_t:=\alpha+\beta Y_{t-1}$ for $t=1, \ldots, T$. Then sales of product $i$ in period $t$ can be rewritten as $$ Z_{i t}=F_t H_t q_{i t} $$ Define cost-to-go function as $$ J_t=\max _{\substack{p_{i s } \\ i=1, \cdots, n ; \\ s=t, \ldots, T}} \sum_{s=t}^T \sum_{i=1}^n F_s H_s\left(p_{i s}-c_{i s}\right) q_{i s} $$ Rewrite the price $p_{i t}$ as a function of the purchase probability vector $\mathbf{q}_t=\left(q_{1 t}, q_{2 t}, \ldots, q_{n t}\right)$ : $$ p_{i t}\left(\mathbf{q}_t\right)=\frac{1}{b_t}\left(a_{i t}-\log q_{i t}+\log q_{0 t}\right) $$ where $q_{0 t}=1-\sum_{i=1}^n q_{i t}$. Subsequently, $$ J_t=\max _{\substack{q_{i t} \in[0,1] \\ i=1, \cdots, n}}\left[F_t H_t \sum_{i=1}^n\left(p_{i t}\left(\mathbf{q}_t\right)-c_{i t}\right) q_{i t}+J_{t+1}\right] $$

Diffusion in Social Networks^[5]

• Multigeneration Product Diffusion in the Presence of Strategic Consumers

• Dynamic Pricing for New Products Using a Utility-Based Generalization of the Bass Diffusion Model