Classroom Companion: Business

Parameter
Description
N
Total population
B
Individuals that have bought the product or are using the service at time t 
(i.e., the current customers)
p
Coefficient of innovation
q
Coefficient of imitation
t
Time
 
Chapter 18 · Digital Market Modeling

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263
18
dB
dt
p N
B
qB N
B
p qB
N
B















.
This is the Bass diffusion equation. Note that the term p + qB is the intensity by 
which an item is sold in the infinitesimal period dt. dB/dt is, therefore, the demand 
for the good at time t since the demand is, by definition, the same as items sold per 
unit of time. The Bass equation is solved in 
7
Box 
18.1
.
Box 18.1 Solution of the Bass Equation
The Bass equation is separable since 
dB = (p + qB)(N − B)dt, resulting in the 
following equation where the depen-
dent variable B and the time variable t 
have been moved to each side of the 
equation:
dB
p qB
N
B
dt







.
Expanding the left-hand side of the 
equation and multiplying both sides by 
p + qN results in:
qdB
p qB
dB
N
B
p qN dt







.
Integrating both sides of this equation 
term by term gives:
ln
ln
ln
.
p qB
N
B
c
p qN t












By simple algebraic manipulations we 
find:
p qB
N
B
ce
p qN t






,
in which c is the constant of integra-
tion. Observe that for t = 0, B(0) = B
0
(the initial condition), so by setting 
t = 0 and B = B
0
in the equation, the 
constant of integration is easily found:
c
p qB
N
B



0
0
.
Inserting this and solving for B finally 
results in the solution of the Bass equa-
tion:
B t
pN
qNB
p N
B
e
p qB
q N
B
e
o
p qN t
p qN t
 



















0
0
0
,
in which B
0
= B(0) is the initial number 
of individuals possessing the good. 
These may be individuals who have 
attained the good as part of a marketing 
promotion, a product test, or together 
with a complementary product (e.g., the 
SMS attached to mobile phones). The 
significance of these customers is dis-
cussed in the main text.
18.2 · Bass Diffusion Model

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264
18
.
Figure 
18.2
shows the solution of the Bass equation for a total population of 
N = 10
6
individuals, B
0
= 10
4
initial customers, the coefficient of innovation p = 0.03, 
and the coefficient of imitation q = 3.8 × 10
−
7
. The values of p and q are based on 
the typical and average values found in the paper by Mahajan et al. (
1995
).
There are two special cases of the Bass equation:
5
If p = 0, there are only imitators, and the solution of the Bass equation is 
reduced to the logistic distribution (see, e.g., the Wikipedia article for the defini-
tion of the logistic distribution):
B
NB
B
N
B
e
qNt






0
0
0
.
0
1 00 000
2 00 000
3 00 000
4 00 000
5 00 000
6 00 000
7 00 000
8 00 000
9 00 000
1 000 000
0
2
4
6
8
10
12
14
16
18
20
Customers (B)
Year (t)
. Fig. 18.2 Plot of the Bass diffusion model. (Authors’ own figure)
 
Chapter 18 · Digital Market Modeling

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265
18
5
If q = 0, there are only innovators, and the solution of the Bass equation is 
reduced to the exponential distribution:
B
N
N
B
e
pt






0
.
Note that if there are no innovators (early adopters) but only imitators (i.e., 
p = 0), the solution of the differential equation is B = 0 for all t if B
0
= 0; that is, no 
one will ever buy the product. Therefore, B
0
> 0 for a non-zero solution to exist if 
p = 0; that is, a customer base must exist before the sales begin. This requirement is 
not necessary for the case of only innovators. In this case, customers will buy the 
good even if there are no initial customers.
.
Figure 
18.3
 shows the solution of the Bass equation for the two cases with 
only imitators (p = 0, q = 3.8 × 10
−
7
) and only innovators (p = 0.05, q = 0). For both 
graphs, the total population is N = 10
6
, and the initial number of customers is 
B
0
= 10
4
.
0
1 00 000
2 00 000
3 00 000
4 00 000
5 00 000
6 00 000
7 00 000
8 00 000
9 00 000
1 000 000 
0
2
4
6
8
10
12
14
16
18
20
Customers (B)
Year (t)
Only imitators (p=0)
Only innovators (q=0)
. Fig. 18.3 The Bass diffusion model with only imitators and only innovators. (Authors’ own 
figure)