lntemational Journalof
International
EISEVIER
Industrial
Organization
Journal of Industrial Organization
12 (1994) 549-568
Entry into a new market
A game of timing
Steinar
Department of Economics,
Holden”
University of Oslo, P.O. Box 1095 Blinders,
Norway
Christian
N-0317 Oslo,
Riis’
Foundation for Research in Economics and Business Administration,
N-0371 Oslo, Norway
Gaustadalleen 21,
Final version received September 1993
Abstract
We study entry into a new market in a model where firms choose when to enter
the market. An early entry is profitable because it yields a strategic advantage in the
market; however, costs will also be larger owing to interest on the capital cost. It is
shown that rent equalization
need not occur, and that social welfare may be lower
under competition
than under pure monopoly.
Furthermore,
under some circumstances there is a strictly positive probability
that the firms enter simultaneously,
even in the limit when the period length converges to zero.
1.
Introduction
Recent theories in industrial
organization
attempt
to endogenize
the
structure
of market by analyzing entry and exit processes. However,
as yet
most of these theories use restrictive
assumptions,
among them that entry
into the market
occurs either simultaneously
or sequentially
in a pre* Corresponding author.
’ Thanks to Tor Jakob Klette, Karl Ove Moene, a referee, and participants at a seminar at
the Department
for valuable comments, and to Atle Seierstad for helpful discussions.
0167-7187/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved
SSDI
0167-7187(93)00433-O
550
S. Holden.
C. Riis I Int. .I. Ind. Orgum 12 (19944) S49-.S66x
determined
order [see Gilbert (1987) for an overview of the literature based
on the latter assumption].
While sequential
entry may seem closer to reality,
it is not satisfactory
to let the order of entry be exogenous.
In general, the
profits of a firm depend on when the firm enters, and this makes it important
also to model the process which determines
the order of entry. In this paper
we investigate
a model with endogenous
time of entry where firms may
achieve a strategic advantage
by building up capacity at an early stage when
the market is very small. By an early entry, a firm will incur larger costs
owing to interest on the capacity costs, yet the firm may be willing to do so
in order to obtain a larger market share.
Endogenizing
the time of entry has important
implications
for the market
structure.
Previous
studies (see references
below) have mostly concluded
that profits will tend to be equal in all firms, because the advantage of being
the first entrant will be dissipated in the fight to become the first. Fudenberg
and Tirole (1985) show however that rent equalization
need not occur in
three-player
games, and they also set up a highly stylized two-player
game
(‘grab-the-dollar’)
where firms may obtain different profits. In the present
paper we show that firms may obtain different
profits also in a standard
duopoly model.
The present paper draws upon and adds to two strands of the literature:
on entry deterrence
in models with endogenous
timing, and on pure timing
models. In the entry deterrence
literature,
our model is, to our knowledge,
the first that allows for simultaneous
entry when capacity decisions
are
endogenous.
Gilbert and Harris (1984), Mills (1988) and especially Anderson and Engers (1990) use industrial
organization
models rather similar to
ours, but without allowing for simultaneous
moves. As will be shown below.
in our model there will under some circumstances
be a strictly positive
probability
of simultaneous
entry, so ruling out simultaneous
moves is not as
innocuous
as it may seem.’
Dixit and Shapiro (1986), Cabral (1989) and Robson
(1990) consider
models where simultaneous
entry is possible. However,
Dixit and Shapiro
(1986) and Cabral (1989) only look at firms’ decisions of whether to enter or
leave the market,
while Robson
(1990) considers
a model where firms
commit to a certain price. Endogenous
capacities decisions complicate
the
model considerably
compared
with a pure entry model, but it also adds
significantly
to realism.
The timing model we use has similarities
with the model suggested
by
Fudenberg
and Tirole (1985) in their analysis of the adoption
of a new
our entry decision
also involves
the choice of
technology.
However,
capacity, and this requires a richer model than the one used by Fudenberg
’ The model itself is however
Sundaram
(1992).
entirely
non-stochastic.
in contrast
to, for example.
Dutta
and
S. Holden,
C. Riis I ht.
.I. Ind. Organ. 12 (1994) 549-568
551
and Tirole (1985). We believe that the timing model used in this paper also
allows for wider areas of application
within games of timing, and thus is of
independent
interest. A novel result is that there may be simultaneous
entry
even in the limit when the period length converges to zero. The reason for
this result lies in the possible discontinuity
of the payoff function
of the
follower. The discontinuity
arises as a consequence
of the leader’s strategy
shifting from ‘entry-accommodating’
capacity to ‘entry-deterring’
capacity as
the market grows.
In section 2, we present the basic industrial
organization
model of the
paper. Since the interaction
with the timing aspects complicates
the analysis
considerably,
we have deliberately
made the industrial
organization
model
as simple as possible. Section 3 analyzes the consequences
of endogenizing
the time of entry. Section 4 concludes.
2. The model
We consider a market for a new product. Let g(t) be an indicator for the
size of the market at time t, and assume that g’(t) > 0 for all t, SO that
demand
increases
over time. Furthermore,
g(t)
converges
towards g”“”
when t approaches
infinity,
so the market
stagnates
eventually.
Two
identical firms wish to enter the market. Each firm can enter the market at
any point in time after 0, and each firm is only allowed to enter once. There
is complete
information
regarding
the payoff functions
of the firms.
Furthermore,
if one firm enters,
the other will become
aware of this
immediately.
At entry, a firm i builds up capacity x,, and this choice is
irreversible.
The capacity can either be high, x,,, in which case subsequent
entry is deterred,
or low, xd (in a previous version we considered
a model
with a continuous
choice of capacity).
The investment
costs, which are
incurred
at entry, are either c(x,) or C&Y,,), where c(x,) < c(x,).
If only one firm has entered in capacity xi, the net revenues at time t are
R(x,, O)g(t) >O for all f, i = d, m. If the other firm also has entered,
in
capacity x,, j = d, m, the net revenues of a firm that enters in capacity x, at
time r is R(x,, x,)g(t).
We assume that R(xd, 0) > R(x,, xd) > 0.
If a firm does not enter the market at all, it obtains zero profit. We assume
that no firm can earn a positive profit by entering at time 0, that is.
R(x,, 0)
I
g(s) emrs ds - c(xi) < 0,
i = m, d,
(1)
where r is the real rate of interest.
We first consider
the decision problem
of the last firm to enter (the
follower) when the other firm (the leader) has already entered. If the leader
552
S. Holden,
C. Riis I Int. J. Ind. Organ. 12 (1994) S49-568
best off by not entering.
enters in capacity x,, the follower is by assumption
possible profit of the
If the leader entered
in capacity xd, the maximum
follower (discounted
to time 0) is
FD =max
R(xd, xd)
I
(2)
g(s) em” ds - c(x d) emrr
We assume that FD > 0 [this is obviously
fulfilled for suitable values of
R(x,, xd), g(t), r and c(x~)], so that if one firm enters in capacity xd, the
other firm will eventually
also enter. The optimal
time of entry of the
follower, T, is given by the first-order condition
(3)
R(XdY xlJ)g(r) = +CJ.
As g(t) is increasing
in t, (3) determines
a unique optimal time of entry r
of the follower.
The leader is assumed to be able to perfectly forecast the ensuing action
taken by the follower, represented
by the optimal time of entry, 7. We first
investigate
the leader’s choice between entry deterrence
and entry accommodation
when the time of entry of the leader, t, is taken as exogenous.
The profit under the entry deterrence
(monopoly)
strategy is
M(t) = qx,,
The profit
under
0)
J g(s) eprs
the entry
em”.
-
accommodation
(4)
(duopoly)
D(t) = R(xd, 0) i g(s) e-rs ds + R(xd, xd) /g(s)
strategy
is
emrs ds - c(xd) emrr.
7
I
(5)
Given the time of entry, t, the leader clearly chooses the strategy that
gives the higher profit. (If both strategies give the same payoff we assume
that the leader chooses the entry accommodation
strategy.) Thus the profit
of the leader as a function of time of entry is
L(t) = max[D(t),
M(t)].
(6)
As shown in the appendix,
either one of the strategies
is the more
profitable
for all t, or there exists a unique point in time tM for which
M(tM) = D(t”),
M(t) < D(t) for t < t”, and M(t) > D(t) for t > t”.
The crucial issue in the model is how the relationship
between L(t) and
F(t) depends
on time.
Lemma
1. There
exists
a unique
point
in time
t* such that
if the leader
S. Holden,
C. Riis I Int. .I. Ind. Organ. 12 (1994) 549-568
553
enters before t* it is more profitable to be the follower, while if the leader
enters after t* it is more profitable to be the leader, that is, F(t) 2 L(t) for all
t d t* [and F(t) > L(t) for all t < t*], F(t) < L(t) for all t E (t*, T) and F(t) G
L(t) for all t 2 T. L(t) is continuous
for all t, while F(t) is continuous
except
at t”, where the leader changes strategy from duopoly to monopoly.
All
proofs are in the appendix.
Since entry is uncoordinated,
there is a risk of simultaneous
entry. We
assume that if simultaneous
entry occurs, the payoffs are independent
of the
planned
capacities.
This is clearly a restrictive
assumption.
However,
without this assumption
the choice of capacity will depend on the risk of
simultaneous
entry, which will complicate
the analysis considerably.
A
possible justification
of the assumption
is that although the commitment
to
enter (which could take the form of the signing of a contract)
is made at a
single point in time, the actual process of installing
capacity takes some
time. This enables the firms to choose the duopoly capacity if simultaneous
entry occurs.’ A simultaneous
entry at t gives
w>=WXd,Xd)
I
g(s) e _ ” ds - c(xd) em”‘.
(7)
On comparing
(2) and (7), it is clear that S(t) is increasing monotonically
until T, where S(T) = FD. Thus, S(T) = F(T) = FD if the leader has chosen the
duopoly strategy and S(r) > F(T) = 0 if the leader has chosen the monopoly
strategy.
Note that F(f) is non-increasing
in t before 7. Furthermore,
we
know that sufficiently
early in the game S(t) < 0, and thus S(t) < F(t)
irrespective
of which strategy the leader chooses. Thus, there exists a unique
point in time t’ such that F(t) > S(t) for all t < t’, and F(t) s S(t) for all t b t’.
If the leader chooses the duopoly strategy, then t’ = 7, while if the leader
chooses
the
monopoly
strategy,
then
t’ CT.
In
this
case,
t’ =
min[T, max[t”, t”]], where t” is given by S(t”) = 0.
The structure of the market depends on the intersections
of the F. D and
M curves. There are three different cases, depending
on how profitable the
monopoly
strategy is compared with the duopoly strategy. It turns out that
in equilibrium
entry will occur at t” in all cases. In case 1 (Fig. l), the
’ More formally, assume that time is discrete (cf. section 3 below) and that it takes one period
to build capacity.
Furthermore,
between periods there is an interim moment where firms can
costlessly change their choice of capacity.
Finally, assume that payoff functions are such that
under simultaneous
entry it is more profitable
to choose the duopoly capacity irrespective
of
which capacity the opponent
chooses [this is obviously fulfilled for suitable values of R(x,. x,),
R(x,, x,). R(x,, x,) and R(x,. x,,)]. In this setup, if simultaneous
entry occurs, the unique
Nash equilibrium
is that both firms choose the duopoly capacity.
554
S. Holden.
C. Riis
I ht.
J. Ind.
Organ.
12 (1994) 549-568
L(t)
F(t)
t-
t’
T
t
t”
Fig. 1.
duopoly strategy is more profitable than the monopoly strategy at t*, so that
the leader will enter in the duopoly quantity.
Here, t* < tM (or tM does not
exist .)
In order to develop some intuition,
we shall loosely explain some of the
results that will be derived formally below. Consider Fig. 1. For f < t*,F(t) >
-W) = max[W),
M(t)], so that both firms will wait in the hope that the
other firm enters first. For t> t*, L(t) > F(t) so that both firms will try to
preempt the other firm. However, in the interval (t*, t’), the leader chooses
the duopoly strategy as D(t) > M(t). Thus, F(t) > S(t), so that both firms will
rather be the follower than enter simultaneously.
Hence, in a symmetric
equilibrium
the firms cannot enter with certainty
within this interval.
For
f 2 t’, however, M(t) > D(t) and thus S(t) > F(t). Here, both firms will enter
with certainty,
as simultaneous
entry is better than being the follower.
In case 2 (Fig. 2), the monopoly strategy is more profitable relative to the
duopoly strategy, so that it is more profitable to be the follower than to be
the leader immediately
before the point in time where the monopoly
strategy becomes more profitable than the duopoly strategy. Here, tM = t* <
t’. In this case the payoff function of the follower is discontinuous
at t*, and
there will be a strictly positive probability
of simultaneous
entry at t*. The
intuition
is that after t* it is strictly better to be the leader than to be the
S. Holden,
C. Riis I ht.
J. Ind. Organ. 12 (1994) 549-568
555
Fig. 2.
follower,
thus both firms are so eager to preempt the opponent
that they
accept the risk of simultaneous
entry. Case 2 has two subcases. In subcase
(a), which is illustrated
in Fig. 2, S(f*) < 0 so that tM = t* < t’. In subcase (b)
s(t*) 2 0, so that t* = t’.
In case 3 (Fig. 3), the monopoly
strategy
is even more profitable
compared
with the duopoly strategy, so that the monopoly strategy is more
profitable than the duopoly strategy even at the first point in time where the
leader can obtain positive profits. Here, tM < t* <t’ (if tM exists at all).
3. A game of timing
The model in section 2 was, for convenience,
developed
in continuous
time. However, as emphasized
by Fudenberg
and Tirole (1985), strategies in
continuous
time do not contain enough information
to represent
all the
equilibria
that exist in discrete-time
models [see also Simon and Stinchcombe (1989)]. For example, in a model in continuous
time, if both firms
enter with certainty
at the same point in time, then simultaneous
entry is
certain to occur. In a discrete-time
model, however, one can also include the
‘intensity of entry’, that is, the probability
that a firm enters in each period.
S. Holden,
C. Riis I Int. J. Ind. Organ.
12 (1994) 54%St%
Fig. 3
In the limit when the period length converges
to zero, an entry will occur
immediately
irrespective
of whether the per period probability
is 0.9 or 0.1.
The probability
that simultaneous
entry occurs will however vary with this
probability.
As will become apparent below, we need the ‘intensity of entry’
to adequately
represent
aspects of real economic significance.
Thus, in this
section time is assumed to be divided into periods of exogenous length. The
focus is on the limit when the period length approaches
zero.
The type of strategies we adopt in this section is based on much of the
same intuition
as the strategies of Fudenberg
and Tirole (1985). However,
the different economic
setting causes an important
technical difference.
In
both models, there is a certain point in time t” from which it is advantageous to be the first to enter. In Fudenberg
and Tirole (1985) the payoff of the
follower
is increasing
over time, so the players will not consider
the
possibility
of entering before r*. In our setting, however, the strategy of the
leader may shift from entry accommodation
to entry deterrence
at t*, so the
payoff of the follower drops at t*. Hence, the players may choose to enter
with a certain probability
before t *, in order to reduce the risk of becoming
the follower
if entry is after t*. Thus, we cannot use the strategies
of
Fudenberg
and Tirole, where entry ‘immediately’
before a certain point in
time is not possible.
S. Holden,
C. Riis I Int. .I. Ind. Organ. 12 (1994) 549-568
557
We assume that a possible entry in a period k takes place at the beginning
of the period, at time t,. Let pi be the probability
that firm i enters in
period k, conditional
on the other firm not having entered before, and let
G;(t) be the cumulative
probability
that firm i has entered
by time f,
conditional
on the other firm not having entered
before. Thus, Gi(tk) =
C,& P1.
Let k* be the last period before t*, k’ the first period after t’ and K the
first period after T. (In case 2(b), t* = t’, so that k* + 1 = k’.) To simplify
notation,
let subscripts
denote periods, so that we use L,, F, and S, for
L(t,), F(t,) and S(t,T). Thus, Fk 3 L, for all k c k* (and Fk > L, for all
k < k*), Fk <L, for all k E (k*, K) while Fk <L, for all k 2 K. Moreover,
S, < Fk for all k <k’, while S, 2 Fk for all k 3 k’.
Proposition 1. In a subgame perfect equilibrium (SPE), in the limit when the
period length approaches zero, we have for any E > 0 and u > 0,
G,(t* + e) > 1 - (T
and/or
G,(t* + e) > 1 - u,
G;(t* - .e) = G,(t* - e) = 0.
That is, no firm will enter before t* certainly enter before t* + E.
E,
and one of the firms will almost
Proposition
1 applies to all SPE, no matter whether they are pure or
mixed. Let us now consider the equilibria with pure strategies. Assume that
firm i plans to enter with certainty in period k. If k > k* + 1, the best reply
of firm j is to enter in period k - 1, as L,_, > Fk.3 If k 6 k*, the best reply of
firm j is to wait and let firm i enter first, as L,_,, < Fk for all k c k* and
s > 0. Proposition
2 follows immediately
from this reasoning.
Proposition 2. The unique pure strategy SPE is given by the following
strategies. One firm plans to enter with certainty in period k* + 1. The other
firm enters either in period k*, or plans to enter in period k* + 2, depending
on what gives the higher payoff. (Zf the two alternatives of the second firm
give the same payoff, then there exist two equilibria, one for each alternative.)
(There is clearly also an equilibrium where the identities of the firms are
reversed).
A problem
with the pure strategy
equilibria
is that they must be
asymmetric
(otherwise
simultaneous
entry would occur with certainty).
Thus, there is a need for some sort of coordination
device to determine
3 If period
of L(r).
k is after L(t) has reached
its maximum,
firm j will instead
enter
at the max point
558
S. Holden,
C. Riis I Int. J. Ind. Organ. 12 (1994) 549-568
which firm chooses which strategy. However,
if there exists a coordination
device it should ideally be incorporated
in the model. This is particularily
important
in situations where the firms obtain different profit levels, as both
firms then would wish to be the one obtaining
the higher profit.
We then proceed to characterize
the equilibria
with mixed strategies.
Lemma 2. If there is no entry before period k’, both firms will enter
certainty
in period k’, that is, Gi(tk,) = G,(tk,) = 1.
Lemma 3. In an SPE with
probability
of entry in period
(k”, k’].
with
symmetric
mixed strategies,
the common
k, pk, will be strictly positive for all k E
As will become apparent
below,
strictly positive probability
of entry
in cases 1 and 2 there
before and at k”.
will also be a
Proposition 3. The unique symmetric SPE strategies are given by the
following vector of probabilities of entry (p, , pz,. . , p,,. . . pkCm,,pk. ,. .),
where p, = 0 for s < r and s > k’, while p,, > 0 for s E [r, k’], and where r is
the last period where the following condition holds:
The relationship
between
Pktl =~#-Sk)l(Lk
the probabilities
-&+,I+
is given
c
.s>k+l
by
PAL,+, -L,)I(L,-Sk+,),
k= I,..., k’ - 2,
px, =pk._,(Fk,m,
-S,.p,)i(L,.m,
(9)
-S,.),
(IO)
and
k’-1
Pk’
=
I-
c
k=r
pk.
(11)
Proposition 4. When the period length converges to zero, the expected profits
of the firms in an equilibrium with symmetric strategies converges to L(t*).
Note that if FD > L(t*) > 0, then
differ even if their expected payoffs
the realized payoffs
are the same.
of the firms
may
Proposition 5. When the period length converges to zero, we have: In case I,
where F(t*) = FD = L(t*) > 0, and in case 3, where F(t*) = L(t*) = 0, the
probability of simultaneous entry goes to zero. In case 2(a), where F(t*) =
S. Holden,
C. Riis I ht.
559
J. Ind. Organ. 12 (1994) 549-568
FD > L(t*) > 0 > S(t*), the probability of simultaneous entry before or at t*,
p-(S),
is given by
p-(S)
= L(t*)(FD - L(t*))I[(2FD
- L(t*))(L(t*)
+ FD - 2S(t*))]
> 0.
The probability
H(t*)
The
before
(12)
that
an entry
= 1 - [l - Gi(t*)]’
= L(t*)/(L(t*)
In case 2(b),
p-(S)
where
before
= [L(t*)(2FD
risk of simultaneous
entry
or at t”, is given by
p+(s)
occurs
or at t* is given by
- L(t*))]/(FD)2 > 0.
t”, p’(S),
after
conditional
- 2S(t*)) > 0.
(13)
on no entry
(14)
F(P) = FD > L(t*) > S(t*) 2 0, we have
= [L(t*)) - (1 - H(t*))S(t*)
- H(t*)(L(t*)
{[S(P) - (L(t*) + FD))/2]H(t*)}
+ FD)12]I
>O,
(15)
H(P) = 1 - [(FD - L(t*))/(FD - S(t*))]’ > 0,
(16)
p+(s)
(17)
and
= 1.
Cases 1 and 3 are very similar to the standard
results in timing games
[Fudenberg
and Tirole (1985), Gilbert and Harris (1985)]. In both cases,
both firms obtain the same payoff, and there is no conflict as to who enters
first. Thus, they will mix with a low probability
of entry, and the risk of
simultaneous
entry will be zero in the limit when the period
length
converges
to zero. Indeed,
if there were a non-negligible
risk of simultaneous entry, both firms would prefer to wait and let the other firm enter
first. Note also that a comparison
of cases 1 and 3 shows that the existence
of an entry-deterring
technology
induces an early entry, and thus lowers
industry profits.
Case 2 (Fig. 2), on the other hand, is in contrast to the standard results.
Here,
the firms are not indifferent
as to which firm enters first. In a
symmetric
equilibrium,
the firms still mix over entering
in the periods
immediately
around t*. However, before t*, L(t) <F(t) = FD and both firms
will ‘wait longer’ in the hope that the other firm enters first. After t*,
L(t)>F(t)=O
[ su b case (a)] and the firms will ‘hurry more’, trying to
preempt
the other firm. The probability
mass of entry will be much more
concentrated
close to t* in case 2 than in the other cases. As the probability
mass is more concentrated,
there will also be a non-negligible
probability
of
simultaneous
entry. Indeed, it is only because there is a risk of simultaneous
560
S. Holden,
C. Riis I Int. J. Ind. Organ.
12 (1994) S49-S68
entry that the probability
mass is not completely
concentrated
in the last
period before f* and the first period after. The expected payoff of the firms
is L(t*). This implies that the possible gain from F(t”) > L(t*) is dissipated
through
the risk of simultaneous
entry or an entry after t”. A further
difference
between case 2 and most two-firm games with endogenous
timing
is that the firms obtain different profits, unless simultaneous
entry occurs.
Compared
with Fudenberg
and Tirole, the difference
in results derives
from the fact that we have a richer model, where firms not only decide when
to enter, but also in what capacity. This additional
feature implies that the
optimal response
of the follower may be a discontinuous
function
of the
time of entry of the leader, which again implies that entry can also take
place at a point in time where the profits of the follower differ from the
profits of the leader.
4. Concluding
remarks
If firms enter sequentially
into a market, the first firm will claim the bigger
market share and thus obtain greater profits. But which firm will be the
first? We have analyzed
this problem
in a dynamic
model where firms
themselves
choose when to enter. As an early entry is more costly, firms
weigh the additional
costs against the benefit of obtaining
a larger market
share.
The actual outcome
of the game will depend on the cost and demand
structure
of the market under consideration.
We show that under standard
assumptions
in duopoly theory, the firms will in most cases obtain equal
profits, but it is also possible that their profits differ. In contrast to most of
the previous literature.
we explicitly allow for the possibility of simultaneous
entry. It turns out that under some circumstances
there is a strictly positive
probability
that simultaneous
entry occurs, which emphasizes
the importance of allowing for this possibility.
The reason for this result lies in the
discontinuity
of the payoff function
of the follower at the point in time
where the leader shifts from ‘entry-accommodating’
to ‘entry-deterring’
capacity. Note that the discontinuity
of the payoff function of the follower is
caused by the change in strategy of the leader, and does not depend on the
fact that there are only two available capacities in our model.
The timing model we use can be viewed as an extension
of the model of
Fudenberg
and Tirole (1985). We believe that this part of our model can be
used in a large variety of timing games, and thus be of independent
interest.
Case 3 shows that under endogenous
timing, social welfare may be lower
under competition
than if one firm has been given the monopoly rights. The
reason for this is in the spirit of Posner (1975), that the monopoly profits are
dissipated
by the early entry induced
by the competition
to become the
S. Holden,
C. Riis I Int. J. Ind. Organ. 12 (1994) 549-568
561
monopolist.
In a previous version of the paper with a continuous
choice of
capacity,
this result is even stronger,
by the fact that the early entry (to
obtain monopoly)
also leads the monopolist
to choose a lower capacity than
what would have been chosen by a firm that had been given the monopoly
rights from the outset.
Appendix
The relationship
between M(t) and D(t). The possibility
that one of the
strategies
is the more profitable
for all t is trivial, so we focus on the
existence
of t”. Here we show that M(t) = D(t) implies that M’(t) > D’(t)
which, as M(t) and D(t) clearly are continuous,
ensures that there exists at
most one point in time tM for which M(t”) = D(t”). Furthermore,
M(t) <
D(t) for all t < t”, and M(r) > D(t) for all t > t”.
The slopes of the functions
D(t) and M(t) are
D’(t) = [ - R(xd, O)g(t) + rc(xd)] eer’,
M’(t) = [ - R(x,,
From
(Al)
[R(x,,
it follows
O)g(t) + rc(x,)]
eP”.
that M’(t) > D’(t)
0) - R(Xd, O)] q
(Al)
if
< c(x,> - C(X,).
(A4
We have
Rkll~ 0)
1 g(s) emrv ds -
= R(xd, 0)
< R(.xd, 0)
c(x,)
emr’
1g(s) emrs ds + R(xd,
I
xd) [g(s)
eers ds - c(x,) ePr’
(-43)
g(s) e-rr ds - c(x,) em”,
where the equality follows from M(t) = D(t),
from R(xd, xd) < R(xd, 0). (A3) implies that
J,“g(s) emrs ds
[R&n, 0) - R(xd, 011
Since g’(t) > 0, we have
e-,l
while
the inequality
<4%1>- 4%).
follows
(A41
S. Holden,
562
C. Riis I Int. .I. Ind. Organ. 12 (1994) 549-568
j”,“g(s) emrs ds > g(t)Jcmem”.’ds
e
-
rl
e
g(t)
--r,
r
.
It is clear that (A4) and (A5) imply (A2).
QED
Proof of Lemma 1. First consider the case where no entry has taken place
before the optimal time of entry of the follower, T. If the leader chooses the
duopoly strategy, the follower will enter immediately,
and both firms obtain
the same payoff. However,
if the monopoly
strategy is more profitable
and
is thus chosen by the leader, the follower obtains zero payoff.
Then consider the possibility of entry before 7. If entry occurs early in the
game,
the leader
obtains
negative
profit by assumption,
whereas
the
follower always obtain a non-negative
profit. If entry occurs immediately
before T, however, the profit of the leader is greater than the profit of the
follower [compare (2) and (5)]. Furthermore,
inspection
of (5) shows that
[as g’(t) > 0] if L(t) > F(t) for t = s, then L(t) > F(t) for all t E (s, T).
The continuity
of L(t) and F(t) follows from the continuity
of payoff
functions,
except at the point in time where the leader changes strategy.
QED
Proof of Proposition 1. We first prove that G,(t* + F) > 1 - u and/or G,(t* +
E) > 1 - u. The intuition
in the proof is that after t*, the first firm to enter
will obtain the higher profits. Thus, both firms will try to preempt the other
firm. Define 7r, = max{(L(t) + F(t))/2lt b t*}.’ Thus, given that no firm has
the expected
profits of at least one of the
entered
by t*, in equilibrium,
firms, say firm i, is less than or equal to n,. Then define t, by L(t,) = TI-,, and
consider the interval (t, + ~/2, t, + E). Let K denote the set of all periods
within this interval. Observe that L, > TI-,for all k E K. Thus, the only thing
that can prevent firm i from entering with certainty in this interval is the risk
of simultaneous
entry. To obtain a more precise condition,
let q), denote the
probability
that firm i enters in period k conditional
on no entry before k [SO
4; =A/(1
- G,k-l))l. Thus, firm i will enter unless the expected profits of
entering
in period k, qiSk + (1 - q’JLk < 7~,, or equivalently
qi 2 (L, r,)l(L,
7 S,). Define zk = (L, - rI)l(L, -S,),
so that firm i will enter
unless qi 2 zk for all k E K. To prevent firm i from entering
within this
interval,
the probability
that firm j enters during this interval must be
1-
n
kEK
(1-
4:) a I-
rI
(1 -zk).
kEK
’ This maximum clearly exists, as L(t) is continuous,
and at this point F(t) < F(t”) for all t > t”. Moreover,
approaches
infinity, because of the discounting.
while F(r) is continuous
except at t”,
L(t) + F(t) converges
to zero when I
S. Holden,
C. Riis I Int. J. Ind. Organ. 12 (1994) 549-568
563
When
period length
to zero,
number of
in K
goes towards infinity. As the zk’s are fixed [as L(t) and S(t) are continuous]
and strictly greater than zero, it is clear that to prevent firm i from entering,
the probability
that firm j enters during the interval (ti + ~12, t, + &) must
approach
unity. Hence we know that the probability
that an entry occurs
before t, + F approaches
unity.
Then define 7r2 = max{(l(t)
+ F(t))/2lt* s t < t, + E}. In equilibrium,
the
expected profits of at least one of the firms, say firm i, must now be less than
or equal to v~, as we know that an entry will occur before t, + E. Then
define t, by L(t2) = 7~2, and consider the interval (t2 + e/2, f2 + E). Just as
above we can show that to prevent firm i from entering with certainty in one
of the periods in this interval, the probability
that firm j enters during this
interval must approach unity. Hence we know that the probability
that an
entry occurs before t, + E approaches
unity.
This reasoning can be applied recursively as long as L(t) > F(t). Each time
the difference
between
L(t) and F(t) will be approximately
halved. The
procedure
will not stop until we have reached t*, which proves that the
probability
that an entry has occurred immediately
after t* will approach
unity.
We now proceed to prove that Gi(t* - .Y)= G,(t* - E) = 0. Consider
the
choice of firm i of whether to enter at t* - F or somewhere
in the interval
(t* - ~/2, t*). When the period length converges
to zero, the number
of
If the conditional
periods
in the interval
(t* - e/2, t*) goes to infinity,
probability
that firm j enters in period k, conditional
on there has been no
entry before k, qi, does not approach zero for any period k in the interval
(t* - ~/2, t*), then the probability
that j enters within this interval
will
certainly go towards unity (as shown in the first part of the proof). In this
case it is better for firm i to wait and obtain FD than to enter at t* - E, as
L(t) < FD for all t < t*. On the other hand, if qi does approach zero for
some period k in the interval (t* - ~/2, t*), then it is better for firm i to
enter in this period and obtain L(t* - ~/2) with certainty,
than to enter at
period L(t* - E), as L’(t) > 0.
QED
Proof of Lemma 2. Note that S(t) s F(t) for all t 3 t’, so that if the other
firm enters, then the best reply is to enter. Under duopoly [D(t) > M(t)],
L(t) has already reached its maximum at t’, so entry is the best strategy also
if the other firm does not enter [as L(t) 2 F(t)]. Under monopoly,
we must
consider the period where L(t) has reached its maximum
(we know that a
maximum
will exist, as the market stagnates and the profits are discounted
to time zero). When L(r) has reached its maximum,
entry is the best strategy
irrespective
of whether the other firm enters or not. Thus, if no entry has
occurred before that, there will be simultaneous
entry with certainty.
Using
backwards
induction
from this period, it is clear that (as L,_, > S, for all
564
S. Holden,
C. Riis
I Int. 1. Ind.
k > k’ under monopoly)
entry
after k’, and thus also at k’.
Organ.
12 (1994) 549-S@
will be the dominant
strategy
in all periods
QED
Proof of Lemma 3. Observe first that in a symmetric
equilibrium,
the firms
will not enter with certainty
before k’, as Sk < Fk for all k <k’. Combined
with Lemma 2 this ensures that pks > 0.
Assume
then that pk > 0 for all k E [s, k’]. We want to show that this
implies that pr_, > 0. pk > 0 for all k E [s, k’] implies that the expected
profits of entering
are the same for all periods within this interval.
The
expected
profits of entering
in period k’ are at most FD, because
this
strategy either leads to the firm being a follower or simultaneous
entry at k’.
Thus, if the probability
that the other firm enters in period s - 1 is zero,
then it is strictly better to enter in period s - 1 and obtain L,,_, > FD. Thus,
we cannot have p,%~, = 0. Using induction,
starting with s = k’, it is clear that
pk > 0 for all k E (k*, k’].
QED
Proof
of Proposition
3. The proof is based on a construction
of the
equilibrium
strategies,
which proves both existence and uniqueness.
From
Lemma
3 it follows that the firms mix over all periods in the interval
(k*, k’]. Consider
two periods, k and k + 1, within this interval.
[In case
2(b), consider the two periods, k* = k’ - 1 and k’.]
As both pk>O
and pk+, >O, both firms must be indifferent
between
entering
in period k or period k + 1. The expected payoff of firm j if it
enters in period k is
,ZkP.,F,
while
+ PA,
the expected
c p,Fs +PJ~
r<k
By setting
+ Pk+lLk + ,,,T+,
payoff
of j if it enters
+~k+,Sk+,
+
c
,>A+1
(A6) and (A7) to be equal,
P k+, =~k(Fk-Sk)l(Lk-Sk+l)+
W)
PsL,,
in period
k + 1 is
P$L,+I.
(A7)
we obtain
c
F.>k+I
PAL,+,
-Ld’(L,
-%+,I.
(9)
From
Lemma
2, p, = 0 for s > k’. Thus,
pk’ =pk.-*(Fk’_,
- S,._,)I(L,._,
pkc is given
by
-s,.>.
Moreover,
let r be the first period with a positive
there is entry with certainty at k’, it follows that
(10)
probability
of entry.
As
S. Holden,
C. Riis I ht.
.I. Ind. Organ. 12 (1994) 549-568
56.5
k’-1
pk’=
1 -
c
k=r
(11)
Pk.
We can use (9) recursively,
and (lo), to find the relationship
between the
probabilities
of entry in all periods with strictly positive probability
of entry.
For a given first period with strictly positive probability
of entry, (9), (10)
and (11) give us the necessary equations
so that we can solve for unique
values of the p’s.
It then remains to find r. This can be done by the following iterative
procedure,
where we set r = k’ initially.
(1) Assume that r is the first period with strictly positive probability
of
entry, thus p, > 0 for all s 2 r, while p, = 0 for all s <r.
(2) Construct
the equilibrium
strategies by using (9), (10) and (11).
(3) Check whether the following condition
holds:
If (A8) holds, both firms will strictly prefer to enter in period r - 1. Thus,
we set r = r - 1 and return to (1). If (AS) does not hold, then we know that
(8) holds. As L’(t) > 0 in the relevant interval (t < t*), (8) also implies that
L, cp,S,
+ c p,L,,
J>r
all
k < r.
(A9)
In this case no firm will profit by entering
before period r, and we have
found the first period with strictly positive probability
of entry.
We know that the procedure
will stop, and a unique first period will be
found, because if we come close enough to time zero then, the expected
profits of being the leader will become negative. As the expected profits in
equilibrium
are positive, (A9) must hold before this point.
QED
Proof of Proposition 4. That the expected
profits converge
to L(t*) is
equivalent
to expected profits being in the interval (L(t*) - CT,L(t*) + a).
We first show that the expected
profits are at least L(t*) - (T. From
Proposition
1 we know that a firm can obtain at least L(t* - E) by entering
at time t* - F, as there is no risk of simultaneous
entry then. Since L(t) is
continuous,
any firm can obtain at least L(t*) - cr, for sufficiently
small F.
We then show that expected profits cannot be more than L(t*) + cr. In cases
1 and 3 this is immediate
from two observations:
(1) L(t) and F(t) are
continuous,
and (2) Proposition
1 ensures that entry occurs sufficiently close
to t*. In case 2, F(t) is discontinuous,
and we may have FD > L(t*).
Consider first the possibility that the probability
of entry before t* is zero,
H(P) = 0. In this case the expected profits must be lower than L(t* + E) by
Proposition
1, and thus less than L(t*) + (T for sufficiently small E. Consider
S. Holden. C. Riis I Int. J. Ind. Organ. 12 (1994) 549%-568
566
then the possibility that H(t*) > 0. This implies that the firms have a positive
probability
of entering before t*. In a mixed strategy equilibrium,
firms are
indifferent
to each of the alternatives
they mix over. The expected profits of
entering
in the first period with a positive probability
of entry cannot be
higher than L(t*).
QED
Proof
of Proposition 5. In case 1, where F(t*) = FD = L(t*),
the probability
of simultaneous
entry can be inferred directly from Proposition
4. As the
expected
payoffs of the firms are equal to the payoffs each of the firms
receive if there is not simultaneous
entry (L(t*)), whereas the payoff under
simultaneous
entry is lower, it follows that the probability
of simultaneous
entry must converge to zero with the period length.
In case 3, where L(t*) = 0, neither of the players will enter before t* as
L(t)<0
for t <t*.
Thus, the probability
of simultaneous
entry can be
inferred directly from Proposition
4, just as above.
We now turn to case 2, where F(t*) = FD > L(t*) > 0. We first show that
the probability
that the entry occurs before or at t* is strictly positive.
Assume the opposite, that Gi(t*) = G,(t*) = 0, and consider firm i’s decision
in the last period before or at t*. If firm i deviates by entering it will obtain
L(t*), whereas in a symmetric equilibrium
it can at most expect L(t*)/2 if it
waits to after t*. Thus, it is optimal to deviate, and we know that G,(t*) > 0.
First consider subcase (a), where S(t*) < 0, where we first find H(t*), the
probability
that an entry occurs before or at t*. In a mixed strategy
equilibrium,
firm j must be indifferent
as to whether it will enter ‘early“‘ and
become
leader with certainty
[payoff L(t*) when the period length converges to zero], or wait and let the other firm enter first, i.e. plan a ‘late’
entry (in the hope that the other firm enters before t*).The expected payoff
of firm j if j plans to enter so late that i is almost certain to have entered, is
(A 10)
G,(t*)FD,
where Gi(t*) is the probability
that firm i enters before or at t*. As firm j
mixes over both these alternatives,
we must have L(t*) equal to (AlO),
which is equivalent
to
G;(t*) = L(t*)IFD.
Thus,
in a symmetric
(All)
equilibrium,
H(t*) = 1 - [l - G&*)1* = [L(t*)(2FD
- L(t*))]/(FD)*
> 0.
(13)
We then proceed to find the probability
of simultaneous
entry after t*,
conditional
on no entry having taken place before t* [p+(S)]. Observe that
if no entry has taken place by t*, the expected payoff of entering in a late
period is zero, as this entails becoming the follower (almost) with certainty.
S. Holden,
561
C. Riis I Int. J. Ind. Organ. 12 (1994) 549-568
As this is one of the alternatives
firm j mixes over, the expected payoff of
firm j, given that no entry has occurred before t*, must be equal to zero.
The expected payoff of j can however also be calculated in another way. If
there is no simultaneous
entry, j has probability
one-half of becoming
the
leader, and one-half of becoming the follower (as the firms use symmetric
strategies).
Thus the payoff of firm j if no entry has occurred before t*, is
[l -p’(S)]L(t*)/2
and by setting
p’(S)
this expression
= L(t*)l[L(t*)
Observe
that
becoming
the
is strictly less
We are now
denoted p-(S).
equal
to zero we obtain
- 2S(t*)] > 0.
(14)
it is only because simultaneous
entry is strictly worse than
follower [,S(t*) < 0] that the probability
of simultaneous
entry
than one.
ready to find the probability
of simultaneous
entry before t*,
The ex ante expected payoff of firm j, is
-p-(S))(L(t*)
H(t*)[(l
(A121
+p+(S)S(t*),
+ FD)/2
+p-(S)S(t*)]
+ (1 - H(t*))O,
(A13)
where the first (last) term is the payoff if the entry takes place before (after)
t*, and where we in the last term have inserted that the expected payoff is
zero if no entry has taken place before t*. From Proposition
4 it follows that
(A13) is equal to L(t*), and we can solve for p-(S) to obtain
p_(S)
= [H(t*)(L(t*)
+ FD) + (1 - H(t*))O - 2L(t*)]/
[H(t*)(L(t*)
= L(t*)(FD
+ FD - 2S(t*))],
- L(t*))I[(2FD
- L(t*))(L(t*)
+ FD - 2S(t*))]
>o.
(12)
As all terms are strictly positive, it is clear that
In case 2(b), where F(t*) = FD > L(t*) > S(t*)
2 that if there is no entry by t”, then there will
certainty
in period k’, the first period after
expected payoff of firm j if it plans to enter in
G;(t*)FD
which,
that
+ (1 - G,(t*))S(t*),
by the same
argument
equilibrium,
(Ala)
as above,
G;(t*) = [L(t*) - S(t*)]/[FD
In a symmetric
p-(S) is greater that zero.
3 0, it is clear from Lemma
be simultaneous
entry with
t*, so that p+(S) = 1. The
period k’ is
must
-S(P)].
it follows
that
be equal
to L(P),
implying
(AN
568
S. Holden,
C. Riis I Int. J. Ind. Organ. 12 (1994) 549-568
H(t*) = 1 - [l - Gi(t*)]*
= 1 - [(FD - L(t*))I(FD
The ex ante payoff
N(t*)[(l
- &s(t*))]’ > 0.
(16)
of firm j is
-pP(S))(L(t*)
+ FD)/2
+p_(S)S(t*)]
+ (1-
H(t*))S(t*),
(‘416)
which
must
p_(S)
be equal
to L(t*).
This yields
= [L(t*) - (1 - H(t*))S(t*)
- H(t*)(L(t*)
+ FD)12]I{[S(t*)
- (I&*)
+ FD)/2]H(t*)}
>o.
Substituting
out for H(t*) from (16) shows that p-(S)
(15)
> 0.
QED
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