Classroom Companion: Business

11.3 
 Impact of Churning
In markets with several competitors offering similar services, users may, from time 
to time, change their affiliation with one supplier for another. This is called churn-
ing. Churning is common in the mobile phone market. Churning causes fluctua-
11.3 · Impact of Churning

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tions in the market shares of each mobile network operator but has kept the 
distribution of average market shares rather stable over long periods of time. This 
is so because the churn from one operator to another has on average been equal to 
the churn in the opposite direction.
In other markets—for example, VHS vs Betamax and Facebook vs Myspace—
this is not the case, and a net churn in favor of one supplier takes place.
7
Box 
11.1
contains a simple mathematical model for competition between 
two technologies (e.g., VHS vs Betamax) showing the impact churning has on the 
growth and decline of the two technologies.
Box 11.1 Mathematical Model for the Temporal Evolution of Markets 
with Churning
This is a simple mathematical model 
for the temporal evolution of a 
winner-takes- all market (e.g., VHS 
vs Betamax). For simplicity, assume 
that the adoption rate (p) for new 
customers is the same for both tech-
nologies. This does not change the 
validity of the arguments: it just 
makes the computation simpler.
There is a bandwagon effect 
causing a net churning flow from 
technology 2 (e.g., Betamax) to 
technology 1 (e.g., VHS). The 
model is shown in 
.
Fig. 
11.1
.
The differential equations for 
the dynamics of this system are:
dA
dt
p N
A B
rB






,
dB
dt
p N
A B
rB






,
in which dA/dt and dB/dt are the 
change in the number of users of 
technology 1 and technology 2, 
respectively, N − A − B is the num-
ber of users that have not adopted 
any of the technologies at time t, 
and rB is the flow of churners tech-
nology 1 receive from technology 2. 
Since p is the adoption rate, 
p(N − A − B) is the total number of 
new adopters per unit time adopt-
ing either technology 1 or technol-
ogy 2.
Adding the two differential 
equations gives:
dA
dt
dB
dt
d A B
dt
p N
A B










2
.
This is a separable differential equa-
tion in A + B with solution:
A B
N
e
pt






1
2
.
Inserting this in the second equa-
tion yields a linear differential equa-
tion for B:
dB
dt
rB
pNe
pt


2
with solution:
B
pN
p r
e
e
rt
pt







2
2
.
This gives for A: