$4\mathrm{.}3$ Bass Diffusion Model: Predict New Product Sales and Product Rollover

You are the brand manager of Q$-$Phone, a new product based on quantum physics that is set to revolutionize the cell$-$phone market. Your firm predicts the market size for Q$-$Phone is $m=1000,000$ units. Initial sales are promising: the first year you sell $263,000$ units. Two years after the launch, however, you get worried. Although total sales are now over $615,000$ units, the monthly figures are declining for the past $10$ months, despite your hard work. The CEO is blaming you and wants you to resign.

To save your job, you must convince the CEO that the decline is not your fault. The CEO is a number guy, so mere talk won't work. To convince him, you must use the data and theory to demonstrate that the decline is due to the natural life cycle of Q$-$phone. Further, you want to predict when the Q$-$Phone market will saturate, and when the firm must develop and launch next generation of Q$-$Phone.

You first step is to predict when a consumer will buy. Let $t$ be the months after launching Q$-$Phone. Each consumer will buy only one Q$-$Phone. We call those who have bought Q$-$Phone adopters, and those who haven't prospects or potentials. In month $t,$ the market potential $($number of prospects$)$ is $M_t,$ sales is $x_t,$ and accumulative sales is $N_t.$ Hence,

$N_t={\displaystyle \underset{s=0}{\overset{t}{}}}{x}_s,M_0=m,M_t+N_t=m$

Moreover, the fraction of adopters in month $t$ is

$F_t=\frac{M_0-M_t}{M_0}=\frac{N_t}{m}$

A potential customer has probability $f_t$ to buy in month $(t+1)$; i$.$e$.,$ conditional on that consumer has not bought Q$-$Phone till month $t,$ he will buy with probability $f_t$ in the next month. The conditional probability $\frac{f_t}{1-F_t}$ follows the celebrated Bass diffusion Model:

$\frac{f_t}{1-F_t}=p+q*F_t$

where $p$ is the innovation rate, and $q$ the contagion $($imitation$)$ rate. Intuitively, purchasing decision is driven by two forces. First, a potential consumer may buy out of his intrinsic preference, independent of other customers. The innovation rate $p$ measures such force. Second, a potential customer may buy because of the social influence from other customers. For example, he may talk to other adopters, or read online reviews $($e$.$g$.,$ Facebook, Twitter, YouTube, Amazon, WeChat$).$ The more people bought Q$-$Phone, the stronger these word$-$of$-$mouth forces. The term $q*F_t$ measures such contagion, network effect.

By analyzing the data, you find market size $m=1{10}^6,$ innovation parameter $p=0\mathrm{.}01,$ and imitation parameter $q=0\mathrm{.}2.$

Simulate one life cycle of Q$-$Phone for $T=60$ months: find and plot the market potential $M_t,$ sales $x_t,$ and accumulative sales $N_t,$ for $t=0,1,$dots,$T.$

There are four stages of a product life cycle: introduction, growth, mature, and decline. The mature stage starts when the sales pass the peak; the decline stage starts when the fraction of adopters is over $80\%($i$.$e$.,F_{t}0\mathrm{.}80).$ Let $t_p$ and $t_d$ be the starting time of the mature stage and the decline stage. When will the mature stage start? How long will the mature stage last, i$.$e$.,d=t_d-t_p?$ Simulate $N=1000$ life cycles $($sample size$).$