4.1 Possible equilibria under open access policy
The follower takes one of three kinds of strategies: access strategy, bypass strategy and access-to-bypass strategy. Hence, the bypass equi- librium or the access equilibrium can occur in service-based competition under open access policy. In the analysis below, however, we mainly focus on “the access-to-bypass equilibrium” in which a leader firstly enters a market by building a new network, a follower then enters with access to the leader’s network, and in the future the follower will construct a bypass.
The access-to-bypass equilibrium has two features: it is a leader–
follower equilibrium, and a follower undertakes a sequential investment.
The reason that we restrict our attention to the access-to-bypass equilib- rium is that the equilibrium can generally occur under a stochastically growing demand environment, and it shows some peculiar characteristics of network industries.
4.2 A follower’s choice of strategy
In this subsection, we specify the condition under which a follower chooses access-to-bypass strategy over the other two strategies. Wefirst examine when the follower makes the transition from the access project to the bypass project. Suppose a follower already enters the market by accessing a leader’s network. Then, a follower earnsYΠ(2) instead of YΠ(2)−vin every period after building a bypass at the cost ofIn. Solving the follower’s bypass investment problem, we obtain thefirm value after the investment;Yt∆Π(2)/(r−α)+(v/r−In) where∆Π(2)≡Π(2)−Π(2). The firm value is the expected values of the incremental profitflow generated from access-to-bypass and the saving of the access charge payment minus the investment cost of a bypass.
We defineVT(Y) as the option value of the investment in a bypass.
From the standard procedure andVT(0)=0, we haveVT(Y)=GYβ.5 The parameterG and the trigger point YB∗ are the solutions that satisfy the following value-matching condition and the smooth-pasting condition.
(11) VT
YB∗
= ∆F YB∗
−In
(12) VT
YB∗
= ∆F YB∗
Here,∆F(Y) ≡ Y∆Π(2)/(r−α)+v/ris the value ofthe transition project from access to bypass(i.e., the difference in the values between the bypass project and the access project). From the above two conditions, we derive a trigger pointYB∗at which the follower builds a bypass.6
(13) YB∗= β
β−1 r−α
∆Π(2)
In−v r
Next, we consider the follower’s decision problem of whether it should enter the market by accessing the leader’s network. Note that, since there is an opportunity to build a bypass after access to the leader’s network, the effective value of the entry by access includes not only its value, but also the option value of the investment in a bypass. In fact, using the standard procedure again, we can derive the effective value of the entry by access that has a similar form toVT(Y). Then, using the value-matching and the
5The reason for the requirement thatVT(0)=0 stems from the observation that ifYgoes to zero, it will stay at zero.
6We also obtainG=(YB∗)−β(ββ1
1−1In−β11−1vr).
smooth-pasting conditions and the effective value of the entry by access, we derive the trigger pointYA∗.7
(14) YA∗= β
β−1 r−α Πˆ (2)
Ip+v r
Now we can characterize the follower’s choice of strategy. It is appar- ent that the follower does not adopt the access strategy, since∆Π(2)>0 and the demand is stochastically growing.8 Hence, we have the following lemma:
Lemma 4.1. The follower adopts the access-to-bypass strategy (or bypass strategy, respectively) if and only if
(15) Ip+v
r ≤(>)Π(2)
Π(2)(Ip+In).
Proof. Observe that, when YB∗ < +∞ and (0<)YA∗ ≤ YB∗, the follower adopts the access-to-bypass strategy. Equation (15) is derived by rewriting the condition thatYA∗ ≤ YB∗. WhenYA∗ > YB∗, only the bypass strategy remains for the follower. In fact, this is confirmed, because
YB∗≡ β β−1
r−α
∆Π(2)
In−v r
< β β−1
r−α Π(2)
Ip+v r
≡YA∗
⇔Π(2) (Ip+In)<Π(2)
Ip+v r
. (16)
From equation (15), we note that, if the follower adopts the access-to- bypass strategy, thenv/r≤In, becauseΠ(2)/Π(2)<1. That is, the present value of the access charge cannot exceed the investment cost for a network if the follower adopts the access-to-bypass strategy. Otherwise, too large an access charge induces a follower to enter by building a bypass rather than to enter through access to a leader’s network.
When the follower chooses the access-to-bypass strategy, its value func- tion is derived as follows.
7The derivation of the effective value of the entry by access and the trigger pointYA∗is omitted, since the procedure is almost the same as that stated above.
8From the assumptions that∆Π(2;v)>0 andYevolve according to a geometric Brownian motion such that it has the expected growth rateα, there necessarily exists a trigger pointYB∗
at which the follower starts a bypass project.
(17) VFAB(Y)=
⎧⎪⎪⎪⎪
⎪⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪⎪⎪⎪
⎩
Y
YA∗
β
YA∗Π(2)ˆ
r−α −vr −Ip +
YA∗
YB∗
βYB∗∆Π(2)
r−α +vr −In i f Y<YA∗
YΠ(2)ˆ
r−α −vr−Ip +Y
YB∗
βYB∗∆Π(2)
r−α +vr −In i f YA∗≤Y<YB∗
YΠ(2)
r−α −(Ip+In) i f YB∗≤Y
The trigger pointsYA∗andYB∗are given by (14) and (13), respectively.
4.3 A leader’sfirm value
Next, we derive a leader’sfirm value. As mentioned before, we focus on the case when a follower takes the access-to-bypass strategy. In that case, the value function of the leader can be derived as follows.9
(18) VABL (Y)=
⎧⎪⎪⎪⎪
⎪⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎨⎪⎪⎪⎪
⎪⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎩
YΠ(1) r−α
1− Y
YA∗
β−1 + Y
YA∗
β
YA∗Π(2)ˆ r−α
1−YA∗
YB∗
β−1 +vr
1−YA∗
YB∗
β +YA∗
YB∗
βYB∗Π(2) r−α
−(Ip+In) i f Y<YA∗
YΠ(2)ˆ r−α
1−Y
YB∗
β−1 +vr
1− Y
YB∗
β +Y
YB∗
βYB∗Π(2) r−α
−(Ip+In) i f YA∗≤Y<YB∗
YΠ(2)
r−α −(Ip+In) i f YB∗≤Y 4.4 The equilibrium
Before describing the equilibrium, we derive eachfirm’s equilibrium strategy. Afirm invests in the production facility atYA∗and the network atYB∗ if it is a follower (i.e., if a rival has already entered the market). If a rival has not entered the market, it invests in both facilities atYwhere Y<YA∗andVABL (Y)>VFAB(Y). Irrespective of the rival’s decision, it never invests atYwhereVLAB(Y)<VFAB(Y).
We can now derive the access-to-bypass equilibrium as a subgame perfect Nash equilibrium. Although we have not found the leader’s trigger pointY∗Lwhere the leader immediately invests, the equilibrium definesY∗L. As Fudenberg and Tirole (1985) discussed,YL∗ is the smallest value of Y
9Given YA∗ and YB∗, the leader’s firm value is defined as follows: VABL ≡ Et[TA
t e−rτYτΠ(1)dτ]+Et[e−rTATB
TA e−rτ
YτΠˆ(2)+v
dτ]+Et[e−rTB][Ytr−αΠ(2)]−(Ip+In) where TAis thefirst timeYtreachesYA∗andTBis thefirst time it reachesYB∗.
that satisfies VABL (Y) = VFAB(Y). If VABL (Y) < VABF (Y), two firms want to be the follower, hence neither firm will invest. On the other hand, if VLAB(Y) > VFAB(Y), both firms want not only to invest immediately, but also to invest at smaller level of Ymotivated by preemption to the rival firm. Therefore, one of the twofirms invests, i.e., becomes the leader when VLAB(Y)=VFAB(Y).
Proposition 4.1. There exist two access-to-bypass equilibria differing only in the identities of the two firms under the conditions that (In/2) < vr ≤ Π(2)/Π(2)
(Ip+In)−Ip. A unique leader’s trigger point Y∗L ∈ 0,YB∗
in the equilibria is characterized by
VLAB(Y)<VABF (Y) i f Y<Y∗L VLAB(Y)=VABF (Y) i f Y=Y∗L VLAB(Y)>VABF (Y) i f Y∈
Y∗L,YB∗ VLAB(Y)=VABF (Y) i f Y≥YB∗. Proof. See Appendix.
The access-to-bypass equilibrium is as follows. The twofirms do not enter the market whenY∈
0,Y∗L
, whereY∗Lis the trigger point at which the leader enters the market. WhenY∈
0,YA∗
, the leader enjoys monopoly profits. When Y ∈
YA∗,YB∗
, the follower operates with access to the leader’s network. Finally, when Y ∈
YB∗,+∞
, the follower serves its customers by building a bypass.
The conditions in Proposition 4.1 suggest that there exist lower and upper bounds of v. Ifv is too small, the condition that (In/2) < vr will be violated. This is because v that is too small gives the two firms the incentive to be the follower rather than the leader. Thus the access-to- bypass equilibrium cannot exist. On the other hand,vcannot be too large compared withInas Lemma 4.1 states.
We have two observations to make on Proposition 4.1. First, two asym- metric leader–follower equilibria appear in this game. The two equilibria differ only in the identities of the twofirms: firm 1 becomes the leader in one equilibrium, andfirm 2 becomes the leader in the other equilibrium.
Hence, we use equilibria rather thanequilibrium in Proposition 4.1. The equilibrium outcomes are the same between the two equilibria.
Second, although thefirms face uncertainty and irreversibility of in- vestment, an incentive for preemption to a rival firm still exists. While being the follower is motivated by the value of waiting before making the
investment and the opportunity of making sequential investments, being the leader is motivated by earning not only the monopoly rent but also the access profit in an open access environment. Since the motivation for being the leader is stronger than the motivation for being the follower (i.e., VABL (Y) > VFAB(Y)) forY ∈
Y∗L,YB∗
in the access-to-bypass equilibrium, the twofirms participate in the race to become the leader. This point can be confirmed when comparing the game in which the roles of leader and follower are exogenously preassigned to thefirms. In that case, afirm pre- assigned as the leader makes use of an option value to wait, which implies that its optimal trigger point becomes larger thanY∗L.