Classroom Companion: Business

Multi-sided platform (MSP)
Cross-side
network effects
Same-side
network effects
Same-side
network effects
. Fig. 10.1 Network 
effects in a two-sided 
platform. (Authors’ own 
figure)
Box 10.1 Dynamics of Markets with Cross-Side Network Effects
A simple mathematical model for the 
temporal evolution of a two-sided plat-
form with a single cross-side network 
effect is shown in 
. Fig.
10.2
. This sim-
ple model can easily be extended to 
models with multiple same-side and 
cross-side network effects and more 
than two types of customers.
There are two types of customers, A 
and B. Initially there are N potential 
customers of type A and M potential 
customers of type B. Customers of type 
A adopts the platform service offered to 
them at a fixed rate p, symbolized by a 
valve controlled by the parameter p. 
The adoption rate of customers of type 
B is proportional to the number of cus-
tomers of type A, i.e., the adoption rate 
of type-B customers is qA. This is a 
cross-side network effect from cus-
tomer-side A to customer- side B. The 
flow parameters p and q are assumed to 
 
Chapter 10 · Multisided Platforms

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155
10
be constants; otherwise, the equations 
cannot be solved analytically. The flow 
parameters may depend on other fac-
tors such as price, service promotion, 
and visibility. This is not included in 
this simple model where the aim is to 
show how the growth in one customer 
segment may influence the growth in 
the other customer segment.
The flow rate of customers is equal 
to the number of users adopting the ser-
vice per unit time. This is, by definition, 
equal to the time derivative of the num-
ber of customers having adopted the 
service at given time. Hence, the flow of 
customers of type A = dA/dt = p(N − A) 
and the flow of customers of type 
B = dB/dt = qA(M − B), where N − A 
and M − B are potential customers who 
have not adopted the services yet. This 
gives the following set of coupled first-
order differential equations for the evo-
lution of the two markets:
dA
dt
p N
A
=
-
(
)
,
dB
dt
qA M
B
=
-
(
)
.
The first equation is solved immediately 
giving:
A
N
e
pt
=
-
(
)
-
1
.
Inserting this in the second equation 
gives:
dB
dt
qN
e
M
B
pt
=
-
(
)
-
(
)
-
1
with solution:
B
M
Me
q Nt A p
=
-
-
+
(
)
/
.
For small t, B increases as:
B
MNpqt
=
2
2
,
which is much slower than linear 
increase for small values of t.
The evolution of the relative market 
size (A/N and B/M) is shown in 
.
Fig.
10.3
. The abscissa is the time in 
years, the ordinate is the relative num-
ber of customers, and the flow param-
eters in the example are p = 0.17 and 
qN = 0.21.
Because of the feedback from cus-
tomer-side A to customer-side B, the 
growth of type-B customers will follow