Classroom Companion: Business

9.6 · Estimating the Value of Networks

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142
9
in which the binominal coefficient is the number of groups of k people that can be 
formed by n people. For large n, n
2
≪ n
D
≪ 2
n
. The value of the network is then:
V
n
N
n
Reed
D

 
mod
~
~
.
To see that this is a reasonable assumption, note that by Reed’s law, the value of the 
network doubles when a new customer is connected to the network because V
Reed
(n + 1) ~ 2
n + 1
= 2 × 2
n
= 2 × V
Reed
(n). This vastly overestimates the value that a 
single person may have on the network. In the modified case, the value increases 
more modestly:
V
n
D n V
n
Reed
Reed








 
mod
mod
~
/
.
1
1
Box 9.5 Derivation of the Modified Law
The formula follows from the observation that:
N
n
k
n
k n k
n n
n k
k
k
D
k
D
k
D






 







 








1
1
1
1
1
!
!
!
!

and that 
n
k
n
D





 






for k < D and D < n/2 (Pascal’s triangle). Hence,
N
n
k
D
n
D
D
n n
n D
D
D
n
D
n
k
D
D
D






 





 



 





1
1
1

!
!
~
using the “big O notation” and the fact that D ≪ n.
9.6.5 
 Summary and Comparison of Network Laws
.
Table 
9.1
 summarizes the network laws presented above. These laws cover a 
broad array of different networks with very different underlying value production 
mechanisms.
Reed suggests that Sarnoff’s law, Metcalfe’s law, and Reed’s law may be used to 
analyze the effect of merging two network companies. The value of the merged 
company may then either be proportional to:
5
The sum of users of the two companies (Sarnoff), leading to a linear increase in 
value
5
The increased number of possible interactions between users enabled by the 
merger (Metcalfe), leading to a quadratic increase in value
5
The number of groups that can be formed among users in the new company 
(Reed), leading to an exponential increase in value
 
Chapter 9 · Network Effects

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143
9
This simple analysis may then, in some cases, uncover otherwise hidden values and, 
in other cases, avoid overoptimistic valuations of the new company.
.
Figure 
9.13
 shows numerical examples of Metcalfe’s, Odlyzko-Tilly’s, and 
Sarnoff’s law using a linear scale. Observe the significant differences in value as a 
function of the number of users for the three laws. Reed’s law and the modified 
Reed’s law have not been plotted since they both increase so fast that they are sig-
nificantly steeper than Metcalfe’s law and would, therefore, almost overlap with the 
vertical axis.
. Table 9.1 Network laws. (Authors’ compilation)