Classroom Companion: Business

18.3 · Model for Markets with Competition and Churning

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270
18
dB
dt
p
q B
N
B
B
B c
d B
B c
d B
1
1
1 1
1
2
1
1
1
2
2
2
2
1
















,
dB
dt
p
q B
N
B
B
B c
d B
B c
d B
2
2
2
2
1
2
1
1
1
2
2
2
2
1
















There is little hope to solve these nonlinear differential equations analytically, 
except in a few special cases. However, we may still draw some important conclu-
sions concerning the long-term evolution of the market without solving the set of 
differential equations as explained in 
7
Box 
18.3
.
In the case of two suppliers, there are two special cases that can be observed in dif-
ferent markets:
5
If there is only stimulated churning (c
1
= c
2
= 0) and d
1
and d
2
are independent 
of time, then the final state is B
1
= 0, B
2
= N if d
1
> d
2
, or B
1
= N, B
2
= 0 if 
d
1
< d
2
. These are, then, winner-takes-all markets leading to de facto 
monopolies (e.g., Facebook vs Myspace or VHS vs Betamax).
Box 18.3 Market Stability and Churning
In the long run, all potential customers 
have become customers of either 
Supplier 1 or Supplier 2, and there are no 
more potential customers left. This is, for 
example, the case in the mobile phone 
market in several countries (this has 
nothing to do with the sales of mobile 
phones but with the total number of 
mobile subscriptions). This means that 
B
1
+ B
2
= N. A steady state solution 
implies, moreover, that dB
1
/dt 
= 
dB
2
/dt = 0; that is, there is no net flow of 
customers in the steady state. In the 
steady state, there is, therefore, no net 
churning (i.e., C
12
= − C
21
= 0), which 
results in the solution of the quadratic 
equation B
1
(c
1
+ d
1
B
2
) = B
2
(c
2
+ d
2
B
1
), in 
which B
1
+ B
2
= N, for the final state of 
the market. This means that a potential 
customer has either become a subscriber 
of Supplier 1 or Supplier 2 and that the 
churning rates of the two suppliers are 
equal. The general solution is then:
B
c
c
d
d
N
c
c
d
d
N
d
d
c N
d
d
1
1
2
1
2
1
2
1
2
2
1
2
2
1
2
4
2





















B
N
B
c
c
d
d N
c
c
d
d N
d
d c N
d
d
2
1
1
2
2
1
1
2
2
1
2
2
1
1
2
1
4
2























.
From these observations, we draw some 
important conclusions in the main text.
 
Chapter 18 · Digital Market Modeling

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271
18
5
If there is only spontaneous churning (d
1
= d
2
= 0), then it follows most easily 
directly from the churning conditions (or from the above equations by letting 
(d
1
− d
2
) → 0) that the market ends up in the stable state with the following 
steady-state distribution of customers:
B B
c N
c
c
c N
c
c
1
2
2
1
2
1
1
2
,
,










.
This case may apply to mobile communications where competitors have rather 
stable market shares over long periods of time. We see that this state depends only 
on the churning parameters and is independent on how the market grows before it 
is saturated. In this simple model, the spontaneous churning coefficients are treated 
as constants. However, in actual markets, they may be complex time-dependent 
functions of prices, service content, user experience, user preferences, and so on. 
The market shares will then become fluctuating functions of time which, in some 
cases, may lead to winner-takes-all markets, for example, if churning only takes 
place from one competitor to the other (e.g., if c
1
= 0, c
2
> 0, then B
1
= N and 
B
2
= 0).
It is easy to extend the model to more than two competitors. If there is only 
spontaneous churning, it is feasible to find analytic expressions for the stable end 
state of the market for any number of competitors, though it is numerically cum-
bersome to calculate the exact values if there are more than three competitors. On 
the other hand, if there is only stimulated churning (and no spontaneous churn-
ing), then the market will eventually end up in a state in which one of the competi-
tors has captured the whole market. This is also a winner-takes-all market.
Note that in these models, the assumption is that the average churning proba-
bility is constant. In real systems, this is not the case, and it is reasonable to assume 
that the churning probability is a complex, fluctuating function of time depending 
on parameters such as price, loyalty, technical quality, customer laziness, or other 
mechanisms which may motivate the user to churn or not to churn to another sup-
plier. The motivation of this chapter is not to describe why users may churn but to 
show that churning may result in a number of final market states ranging from de 
facto monopolies to rather stable markets shared by two or more suppliers. The 
theory also shows that the long-term evolution is path dependent, where the path 
the market evolution will follow depends on all the parameters just mentioned (see 
also 
7
Chap. 
11
).