Classroom Companion: Business

 
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Questions
1. Consider the Bass equation and the two graphs in 
.
Fig. 
18.3
 (B
0
= 10
4
, 
N = 10
6
, p = 0.05, and qN = 0.38). Assume that the graphs represent customer 
adoption to a social media service.
(a) How large is the initial market share (i.e., at time t = 0)?
(b) Consider the case with only imitators (p = 0). What is this distribution 
called?
(c) How can the Bass equation be expressed when p = 0?
(d) How many customers have adopted the service at the inflexion point when 
p = 0? And when p = 0.05?
(e) At what time does the inflexion point occur when p = 0?
(f) What is the rate of new customers joining the service at the inflexion point 
when p = 0?
2. Consider the Bass equation when there are only innovators (q = 0).
(a) What is the solution of the Bass equation for q = 0 and B
0
> 0?
(b) Find the inflexion point in this case (if any).
(c) Show that the market increases linearly for small t.
3. Consider two digital services satisfying the model for markets with competition 
and churning. Assume that c
1 = c2 ≠ 0 and d
2
> d
1
. Which of the following best 
describes the market state in the long run: B
1
= B
2
, B
1
> B
2
, or B
1
< B
2
? Explain 
why.
 
v
Answers
1. Bass equation.
(a) At time t = 0, the initial market share is B
0
/N = 10
4
/10
6
= 1%.
(b) The distribution is called the logistic distribution.
(c) When p = 0, the solution of the Bass equation is:
B t
NB
B
N
B e
qNt
 






0
0
0
,
(d) The number of customers at the inflexion point for p = 0 is 
B
infl
= N/2 = 500,000. When p = 0.05, the number is B
infl
= (qN − p)/2q = 434.211.
(e) Solving the Bass equation with p = 0 for t gives:
t
qN
B N
B
B N
B







1
0
0
ln
.
Inserting the value for B
infl
and the other parameters, we find that the inflex-
ion occurs at time t = 12.09.
(f) The rate of new customers is dB/dt = qB(N − B). When B = B
infl
= 500,000, 
this gives dB/dt = 3, 8 × 10
−
7
× 500,000
2
= 95,000 customers/year.