Classroom Companion: Business

18.2 · Bass Diffusion Model

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18
The condition that the Bass equation produces an S-curve is that there is an 
inflexion point somewhere on the curve. An inflexion point is a point in which the 
tangent to the curve has a maximum (or minimum) value; that is, the increase in the 
growth rate changes from positive to negative (or vice versa). At this point, the 
second derivative of the curve is zero. How the inflexion point is found is shown in 
7
Box 
18.2
.
One of the most important issues when introducing a new product is the time it 
takes until enough customers have purchased the good so that the business has 
become profitable. This may be called the latency period for market penetration. It 
is reasonable to define the latency period to be the time it takes to reach 10% of the 
full market size (T
10
). For a market with only imitators (p = 0 in the Bass equation), 
we find by simple algebra that:
T
T
B
N
10
50
0
2 2
1












.
ln
/
,
in which T
50
is the time it takes to reach 50% of the potential market. 
.
Table 
18.2
 
shows the latency period for several values of B
0
(the number of customers that 
must be captured before the product is launched). In the table, the time to reach 
50% of the market, T
50
, is 5 years.
Box 18.2 Finding the Inflexion Point
The inflexion point is the point where the second derivative of the solution of the 
Bass equation vanishes, i.e., d
2
B/dt
2
= 0. The second derivative of the solution to 
the Bass equation is:
d B
dt
d
dt
dB
dt
d
dt
p qB
N
B
q
dB
dt
N
B
dB
dt
p qB
2
2
















The condition for the existence of an inflexion point is:
d B
dt
q
dB
dt
N
B
dB
dt
p qB
2
2
0








 .
Since dB/dt > 0, this leads to the linear equation q(N − B) − (p + qB) = 0 with solu-
tion:
B
qN
p
q
infl


2
,
in which B
infl
is the value of B at the inflexion point. Observe that there is an inflex-
ion point on the curve for B ≥ 0 provided that p < qN. If this condition is not ful-
filled, there is no inflexion point on the positive part of the curve.