Mathematical Derivation of the Bass Model

The equation that directly expresses the basic principles of the Bass Model¹ is

This is read "The portion of the potential market that adopts at t given that they have not yet adopted is equal to a linear function of previous adopters." The following paragraphs will explain each of the variables in the above equation. This equation is a differential equation because it contains both the quantity F(t) and its time derivative f(t) as will be explained below.

The potential market, which is the maximum number of cumulative adopters is constant. It is denoted by

.

Time intervals are numbered sequentially starting at 1 in the Srinivasan-Mason² form of the Bass Model equation. A time interval is denoted

.

The Bass model coefficient of innovation is

.

The Bass model coefficient of imitation is

.

The portion (fraction) of the potential market that adopts at time t is

.

The portion (fraction) of the potential market that has adopted up to and including time t is

.

f(t) is the time derivation of F(t), which is expressed

.

The number of adopters (first-time buyers) at time t, which is sometimes called "sales" at t, is

.

The cumulative number of adopters up to and including time t is

.

Because the total number of adopters is 100% (or 1) of the potential market, the number of adopters at time t who have not yet adopted is

.

The portion of the potential market to adopt at t divided by the portion that have not yet adopted, which is sometimes read "the portion that adopts at t given that they have not yet adopted" is

.

The above quantity is know as a hazard rate.

A conveniently chosen constant (one that makes the equations work out nicely) is the constant imitation coefficient divided by the constant potential market M:

.

When Professor Bass first wrote out the equation for the Bass Model he represented this constant with a single letter (e.g., q). Only later after some algebraic manipulation did he see that the equation could be simplified by letting the constant be the quantity q/M.

Now we can write again the initial equation with a more complete understanding of the equation constitutes. To repeat, "The portion that adopt at t given that they have not yet adopted is equal to a linear function of previous adopters" is represented

.

A little algebraic manipulation yields one form of the Bass Model differential equation. In this equation adoptions a(t) is a function of cumulative adoptions at t.

 .

Notice that in the above equation, a(t) (adoptions or sales at t) is a function of the cumulative number of adopters, not t. Although used in the original Bass Model paper¹, the above equation is not the best choice today for forecasting or for parameter estimation because:

• In forecasting it is most convenient to use a formula for adoptions or sales that is a function of time t, not cumulative adoptions.
• When estimating Bass Model parameters using the above equation, there is likely to be error in the independent variable (cumulative adoptions), which is highly undesirable. When using the solution to the Bass Model differential equation (below), time t is the error-free independent variable.

More algebraic manipulation yields another form of the Bass model differential equation, which is convenient for finding the solution.

 .

The solution to the Bass Model differential equation above is

 .

Although there is a continuous-form expression for the f(t) Bass Model function given in Bass 1969¹, the Srinivasan-Mason form² of the discrete Bass Model f(t) function is preferred for parameter estimation and forecasting. It is

 

and

           .

For parameter estimation, the equation is the best choice for estimating parameters M, p and q using nonlinear regression and historical adoptions data. If only cumulative adoptions A(t) data are available, it should be "differenced" to obtain adoptions a(t) = A(t)-A(t-1).

1. Bass, Frank M. 1969. A new product growth for model consumer durables. Management Science 15 215-227.

2. Srinivasan, V. Seenu and Charlotte Mason. 1986. Nonlinear least squares estimation of new product diffusion models.  Marketing Science, 5 (2), 169–178.