Tuesday, January 31, 2006

The Stochastic Bass Model

Background

The Bass model is used in marketing in order to forecast the diffusion of sales of a new product and takes the following form [1]:
dN(t)/dt= (P + Q / M x N(t)) x (M − N(t))
where the parameters have the following meaning:
N(t): number of adopters of new product at time t
M: total market size
P: External influence, e.g. advertisement
Q: internal influence, e.g. word of mouth
Solving the above differential equation gives the Ricatti function:
N(t)= M x (1 - exp(-(P + Q) x t)) / (1 + Q / P x exp(-(P + Q) x t))
with the following derivative (equivalent to the oil production):
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
New sales can be seen as a combination of two set of buyers:
  1. innovators which are not influenced in the timing of their purchase by the number of people who have already bought the product;
  2. imitators which are inlfuenced by others
Intermediate quantities can be computed:
Residual_Market(t)= M - N(t)
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)
Note that if you set P= 0 the above equa. diff. is exactly the logistic function:
dN(t)/dt= Q / M x N(t) x (M − N(t))
Application to Oil Depletion

In the case of oil depletion, N(t) will be the cumulative oil production, Q will be the growth rate (usually noted K) and M the Ultimate Recoverable Ressource (URR). The Generalized Bass Model has been investigated by Guseo et al. [2,3] for the modeling of oil depletion (see thread on peakoil.com: Peak prediction based on the Riccati equation: 2007!). The diffusion by internal influence controled by P plays a minor role in the oil production and generally has a very small value. We can interpret that small value by the fact the oil demand is driven by the economic growth and imposed by the maturity of the oil infrastucture without any innovations. The figure below gives an example of the Bass model applied to the US production (EIA data). We can see that the Bass model curve is very similar to the logistic model because of the small value of P. The fitting was performed using the Marquardt's method for non-linear least squares which smoothly switches from a simple downhill search to Newton's method as it approaches the minimum.



The Stochastic Bass Model

Recently, some have proposed a time-discrete stochastic interpretation of the Bass model which is seen as a pure birth random process [1]. At time t, for each M - N(t) individual susceptible buyer, the probability of an acquisition at time t+1 is the following:
P(Adopter(t+1) | Susceptible(t))= p + q / M x N(t)
You can demonstrate that realizations of the Stochastic Bass Model (SBM) will converge in mean toward the Ricatti function. Applied to the oil production (or consumption), the above transition probability is the probability of a produced barrel to be consumed at time t. The figure below gives an example applied to the US production where three different SBM are shown with different values of p. Each curve is the average value of 2,000 runs. We can see that the parameter p controls in fact the time shift of the curve.




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.

[1] Georg Holtz, An individual level diffusion model, carefully derived from the Bass-Model. Working Paper. September 2004.

[2]GUSEO, R., DALLA VALLE, A. and GUIDOLIN, M. (2005). World Oil Depletion Models: Price Effects Compared with Strategic or Technological Interventions

[3] 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.