Mathematical Derivation of the Bass Model
The equation, which we will refer to as the "Bass Model Principle"
that directly expresses the basic principles of the Bass Model[1] is
.
This is read "The portion of the potential market that adopts at time
t given that they have not yet adopted is equal to a linear function of
previous adopters."
The following paragraphs explain each of the variables in the above equation.
The 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 Bass Model parameter representing the potential market, which is the
ultimate number of purchasers of the product, is constant. It is
denoted by
.
Time intervals are numbered sequentially with the first full time interval
(usually year) of sales at t = 1 in the Srinivasan-Mason^{2} form
of the Bass Model equations. A time interval is denoted
.
The Bass model coefficient (parameter) of innovation is
.
The Bass model coefficient (parameter) 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
.
F(t) is a
cumulative distribution function (CDF); therefore, as t increases,
F(t) will approach 1. Its
companion probability density function (PDF) is
f(t).
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 paper^{,1} the above equation is not the
best choice today for forecasting or for parameter estimation.
The discussion about which equation is the best choice -- and why -- is
somewhat involved, but for those in a rush, we have a short answer.
More algebraic manipulation yields another form of the Bass model differential
equation, which is convenient for
finding the solution (solving the differential equation). The
following equation is in the classic form of a
differential equation; that is, the equation relates a variable,
in this case F(t), to its
derivatives of various orders, in this case first order,
dF(t)/dt, which is the rate of change of F(t) at time t
.
The solution to the Bass Model differential equation above is
.
.
And, of course,
.
As beautiful as they are, in discrete time models where time can only take values such as 1, 2
3, ...,, the two equations above should not be used together for estimating
parameters and forecasting because they are inconsistent. The obvious
question is "Which equations should
I use for estimating parameters and forecasting?"
- 1.
Bass, Frank M. 1969. A new product growth for model consumer durables.
Management Science 15 215-227.