Page 73 - ITU Kaleidoscope 2016

ICTs for a Sustainable World

000
Where g [b] depends on the capacity of CP to find more clients in
000
their business model. We make the assumption g [b] < 0 because
in the first step, CP works with a very defined clients base. CPs max π isp = (p isp − c isp ) α + (p z − c z ) (β + (1 − α))
2 t
improve their strategies in the further steps. As result d π isp < 0.
dt 2 (20)

(p isp − p z ) (1 − α)
Proof theorem 2: Optimal data traffic CP in terms of g [b] , on + 1 + t · b [t] (21)
β + (1 − α)
page 4
s.t. t ≥ 0
00
−b·r · g [b]
0
g [b] =
r Under universal service condition p z is defined as the net cost to
0
g [b] provide a limited bandwidth Internet service over the network. p z =
b = −
00
g [b] c z. Replacing in (20), we get:

The price elasticity of CP is defined by: (p isp − p z ) (1 − α)
π ISP = 1 + t · b [t]
β + (1 − α)
∂b cp [t]
b cp [t]
t = (17)
∂t
t
∂π ISP (p isp − p z ) (1 − α)
= 1 + b [t] (22)
Replacing (15) and (11b) on (17), we obtain: ∂t β + (1 − α)

(p isp − p z ) (1 − α) 0
+ 1 + t · b [t] (23)
β + (1 − α)
1
r·g 00 [b] 1 t t
t = b = b·r·g 00 [b] = − = − (24)
−b·r·g 00 [b] −b·r·g 00 [b] (b · r · g [b]) 2 t ∗ 2
00
t t ∗
We obtain t from equation 22:
Proof of Theorem 3 : optimal termination fee (t) from ISP under ∗ b [t]
Zero rating price as an option to subscriber, on page 4 t = − 0
b [t]
max π isp = (p isp − c isp ) α + (p z − c z ) β + t · b [t] Proof of Welfare Analysis, on page 5
t
(18) Under this scenario, Internet users utility is defined by:
s.t. t ≥ 0
termination fee t is measured in function of c z and c isp correspond- u z [b] = f [b] − p z [b]
ing respectively to the cost per-subscriber of a limited Internet ser-
vice and to the normal Internet service. p z and p isp corresponding Corresponding the universal service policy with the legal definition
to each service price. Under the universal service legal definition, p z = c isp the price’s subscription to this Internet service will de-
price is measured as the net cost of provide the service, p z = c z . fined as the net cost of delivering the limited Internet technology.
Simplifying 18 we obtain: Welfare superior is present in all the cases and highly positive as
result of p z < p . Variation of social welfare is ∆CW > 0
π isp = (p isp − c isp ) α + t · b [t] The benefits of CPs and ISP are defined as:
∗
where: π cp = r · g [b] − c cp − (t) b [t]
∂π isp 0
= b [t] + t · b [t] (19) z ∗
∂t π ISP = (p z − c z ) β + (t) b [t]
∗
From 19 we obtain t : Under a monopolistic ISP that provides a limited Internet service,
b [t] the total welfare is determined by:
∗
t = −
0
b [t] z
Wz = π cp + π isp
∗
∗
∗ = (r · g [b] − c cp − (t) b [t]) + ((p z − c z ) β + (t) b [t])
The optimal termination fee t does not change under the distribu-
tion of a normal Internet service. = r · g [b] − c cp + (p z − c z ) β (25)
As a second case, the optimal termination fee (t) ISP under the con- Under universal service condition p z is defined as the net cost to
sideraton that monopoly benefits lost by the Internet users shifting provide a limited bandwidth Internet service over the network. p z =
is defined as : c z.

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