http://weblab.com.cityu.edu.hk/blog/chengjun/files/2012/04/Bass-Diffusion-Model.pdf

### Introduction

I had made slides to understand the bass diffusion model which is proposed by Frank M. Bass in 1969. The model proposes that diffusion is motivated both by inovativeness and imitation: first, the innovative early adopters adopted the products, after which the followers will imitate them and adopt the products.

•The Bass model coefficient (parameter) of innovation is p.

•The Bass model coefficient (parameter) of imitation is q.

The

**slides** will show you how to deprive the equation of Bass diffusion model by both discrete-time model and continuous model (using Hazard rate).

“The probability of adopting by those who have not yet adopted is a linear function of those who had previously adopted.”

By solving the differential equation of bass diffusion model using Mathematica and Simulating the model using R code.

# basss diffusion model
# chengjun, 20120424@canberra
# refer to http://en.wikipedia.org/wiki/Bass_diffusion_model
# and http://book.douban.com/subject/4175572/discussion/45634092/
# BASS Diffusion Model
# three parameters:
# the total number of people who eventually buy the product, m;
# the coefficient of innovation, p;
# and the coefficient of imitation, q
# example
T79 <- 1:10
Tdelt <- (1:100) / 10
Sales <- c(840,1470,2110,4000, 7590, 10950, 10530, 9470, 7790, 5890)
Cusales <- cumsum(Sales)
Bass.nls <- nls(Sales ~ M * ( ((P+Q)^2 / P) * exp(-(P+Q) * T79) ) /(1+(Q/P)*exp(-(P+Q)*T79))^2,
start = list(M=60630, P=0.03, Q=0.38))
summary(Bass.nls)
# get coefficient
Bcoef <- coef(Bass.nls)
m <- Bcoef[1]
p <- Bcoef[2]
q <- Bcoef[3]
# setting the starting value for M to the recorded total sales.
ngete <- exp(-(p+q) * Tdelt)
# plot pdf
Bpdf <- m * ( (p+q)^2 / p ) * ngete / (1 + (q/p) * ngete)^2
plot(Tdelt, Bpdf, xlab = "Year from 1979",ylab = "Sales per year", type='l')
points(T79, Sales)
# plot cdf
Bcdf <- m * (1 - ngete)/(1 + (q/p)*ngete)
plot(Tdelt, Bcdf, xlab = "Year from 1979",ylab = "Cumulative sales", type='l')
points(T79, Cusales)
# when q=0, only Innovator without immitators.
Ipdf<- m * ( (p+0)^2 / p ) * exp(-(p+0) * Tdelt) / (1 + (0/p) * exp(-(p+0) * Tdelt))^2
# plot(Tdelt, Ipdf, xlab = "Year from 1979",ylab = "Isales per year", type='l')
Impdf<-Bpdf-Ipdf
plot(Tdelt, Bpdf, xlab = "Year from 1979",ylab = "Sales per year", type='l', col="red")
lines(Tdelt,Impdf,col="green")
lines(Tdelt,Ipdf,col="blue")
# when q=0, only Innovator without immitators.
Icdf<-m * (1 - exp(-(p+0) * Tdelt))/(1 + (0/p)*exp(-(p+0) * Tdelt))
# plot(Tdelt, Icdf, xlab = "Year from 1979",ylab = "ICumulative sales", type='l')
Imcdf<-m * (1 - ngete)/(1 + (q/p)*ngete)-Icdf
plot(Tdelt, Imcdf, xlab = "Year from 1979",ylab = "Cumulative sales", type='l', col="red")
lines(Tdelt,Bcdf,col="green")
lines(Tdelt,Icdf,col="blue")