The name is absent

A.3 Investment effect on outgoing traffic

Let us first derive the change in the investor’s profits from outgoing traffic due
to a change in investments. Deriving _j π_iⁱ_,j with respect to ki yields:

∂    πⁱ

∂. “ = ^-ci(ki)y X s sj (a ^- bi(ci ⁺ tj^{)) 1 1 -}

∂ki              2

With i ∈ {-j} the last term is non-negative. By increasing efficiency profits from out-
going calls also increase.

For si close to 0 or close to 1 the investment effect on outgoing minutes is smaller which
causes an inverted u-shaped relationship for the investment effect dependent on the own
market share. Both if the investor is comparably small and if the investor is comparably
large a lower number of calls originate from the investor’s network.

In a nutshell, two effects on the investor’s profits exist: As the absolute investment effect
on retail prices for outgoing calls is only half the absolute investment effect on origi-
nation costs and termination rates the per-unit mark-up for outgoing calls increases.
Additionally, by reducing off-net prices customers are more willing to call customers in
competitive networks.

Next we derive the effect of investments on the competitors’ profits from out-
going traffic to the investor: We know from (4) that i’s investment reduces its
termination rate. Furthermore, we know from 2 that j ’s off-net retail price for calls to
i is reduced only to half the extent as the reduction in termination rates. Thus, not
only i’s per-unit mark-up for outgoing calls to j but also j’s per-unit mark-up for out-
going calls to i increases as the change in i’s off-net price is lower than the termination
rate change. By reducing off-net prices the call length for outgoing calls to network i
increases in total raising j’s profits for outgoing calls to i:

ji = ^- i2 sisj ⁽a ^- bj (cj ⁺ ti)) > 0

Finally, let us derive the investment effect on profits from outgoing traffic to
competitors’ which are not the investor. j’s profits from outgoing calls to any
other providers -j 6= i are given by ^P_-j π_j^j_,-j = ^P_-j (pj - t-j - cj)sj s-j (a - bj pj,-j).

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