The Bass model is used in marketing in order to forecast the diffusion of sales of a new product and takes the following form :
dN(t)/dt= (P + Q / M x N(t)) x (M − N(t))
N(t): number of adopters of new product at time tSolving the above differential equation gives the Ricatti function:
M: total market size
P: External influence, e.g. advertisement
Q: internal influence, e.g. word of mouth
N(t)= M x (1 - exp(-(P + Q) x t)) / (1 + Q / P x exp(-(P + Q) x t))
dN(t)/dt= M x (P + Q) ^2 / P x exp(-(P + Q) x t)) / (1 + Q / P x exp(-(P + Q) x t))^2
- innovators which are not influenced in the timing of their purchase by the number of people who have already bought the product;
- imitators which are inlfuenced by others
Residual_Market(t)= M - N(t)Note that if you set P= 0 the above equa. diff. is exactly the logistic function:
Innovators(t)= P x Residual_Market(t)
Word_of_Mouth(t)= N(t) / M
Accessibility_of_Imitators(t)= q x Word_of_Mouth(t) x Residual_Market(t)
dN(t)/dt= Innovators(t) + Accessibility_of_Imitators(t)
dN(t)/dt= Q / M x N(t) x (M − N(t))
The Stochastic Bass Model
P(Adopter(t+1) | Susceptible(t))= p + q / M x N(t)
That's it for now, in a next post we will explore the use of this stochastic Bass model for the stochastic "tracking" of an oil production curve.
 Georg Holtz, An individual level diffusion model, carefully derived from the Bass-Model. Working Paper. September 2004.
GUSEO, R., DALLA VALLE, A. and GUIDOLIN, M. (2005). World Oil Depletion Models: Price Effects Compared with Strategic or Technological Interventions
 GUSEO, R. e DALLA VALLE, A. (2004). Oil and Gas Depletion: Diffusion Models and Forecasting under Strategic Intervention, Atti della XLII Riunione Scientifica della Società Italiana di Statistica, Bari 9-11/6/2004, Vol. Sessioni Spontanee, 733-36, CLEUP, Padova.
peak+oil bass+model US oil+production