The Bass diffusion model is commonly used to forecast adoption of new technologies. It rests on three assumptions: that there is a fixed population of users (who will eventually adopt the new technology), that innovators will adopt in proportion to the current number of adopters, and that there are imitators whose adoption is proportional to the product of the current adopters and the remaining potential adopters. The origins of the Bass diffusion model lies in epidemiology: infections may spread through a population in a similar fashion (to new technologies or services): innovators are people prone to getting a disease, while imitators catch the disease from the innovators. The number of imitators over time is affected by two factors, workingin opposite directions. There are more innovators to infect future imitators, but there are fewer potential adopters left to infect. Equation4.1 reflects the Bass diffusion model, with the assumption that the rate of innovation (p) and the rate of imitation (q) are constant over time

where nt is the number of new adopters during period t, Nt1 is the cumulative adopters are the beginning of period t, and N is the total pool of potential adopters. Equation 4.1 states that the new adopters will come from the innovators (a constant fraction of the remaining non-adopters) and the imitators (a constant proportion of the product of the current adopters and remaining fraction of non-adopters). Equation 4.1 is not in the best form to estimate from actual data, however. Multiple regression can be used to estimate the diffusion curve when it is rewritten in the form of Equation 4.2:

Equation 4.2 only requires data on total adopters at each point in time, and using multiple regression analysis to estimate Y=a+bX+cX2 will permit estimation of the parameters of the Bass diffusion model.

a. Exercise4-5.xlsx provides annual data for the number of active Facebook users (measured in millions) over its 2004 2018 history (derived from Facebook annual financial filings and other data sources). The total number of users is given, along with its squared value, the number of new users, the lag of the number of users, and its squared value. Estimate the Bass diffusion model by fitting a multiple regression model to this data, and then deriving p and q from the estimated coefficients of that model. Assume that the total number of potential adopters is 3,500,000,000. Typical estimates of the Bass diffusion model for consumer manufactured goods result in estimates of the coefficient of innovation (p) around 3% and estimates for the coefficient of imitation (q) of around 35%. Compare your estimates for Facebook with these values and interpret the differences.

b. Simulate the 20192021 end of year total user number, using theregression model and the uncertainty in the data. Provide a 95% confidence interval for your prediction. (Hint: Use the bootstrap technique for this, as well as solely using the uncertainty embodied in the standard error of the regression equation.)

c. Now assume that the saturation level of users is uncertain, using a Pert distribution with a minimum of 3 billion, most likely value of 3.5 billion, and maximum value of 5 billion users. Simulate the end of year total users for 20192021. Compare your answers in parts (b) and (c).