HD28
.M414
-n
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Entry and Exit:
A Continuous Time Formation
Chi-fu Huang and Lode Li
WP# 3427-92EF&A
Last Revised, May 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Entry and Exit:
A Continuous Time Formation
Chi-fu Huang and Lode Li
WP# 3427-92EF&A
*
Last Revised, May 1992
Entry and Exit:
A
Continuous Time Formulation*
Chi-fu Huang^ and
Lode
Li^
Last Revised, April 1992
Abstract
We
study a class of continuous time optimal entry-exit decisions under uncertainty for a
and for a duopoly. Under very general hypotheses, the optimal policy for a single
single firm
firm exists and
is
unique.
This unique optimal policy
is
a barrier policy:
A
firm optimally
when the demand reaches certain barriers. In the context of a
duopoly, there may exist multiple subgame perfect equilibria. We demonstrate how to identify
a subgame perfect equilibrium in which both firms employ stationary barrier policies. In some
of these stationary equilibria, a firm may exit even when the demand has been rising on the
enters or exits from an industry
average.
aeMSt
"M.LT.
JUN
UBRAHIES
3 19921
'We thank David Kreps (the former co-editor) and Andreu Mas-Colell (the editor),
for many insightful comments and suggestions and an anonymous referee
Berry Nalebuff
proof for a proposition.
'MIT
Sloan School of Management, Cambridge,
'Yale School of Organization and Management,
MA
02139.
Box lA, New Haven,
CT
06525.
Jean-Francois Mertens, and
for
providing a
much
shorter
1
1
INTRODUCTION
1
Introduction
Many important economic
decisions are about the optimal timing to take certain actions.
Examples
of these decisions include the optimal timing to undertake a capital investment project, to exercise
an American
call option, to
buy a house, and to refinance a mortgage.
This paper focuses on
The methods developed
a firm's decision to enter or to exit from an industry.
here are general,
however, and can be applied to analyze optimal timing decisions in other contexts.
There
is
son (1990)
a vast economic literature on the entry and exit decisions of a firm, for which Wila good recent survey.
is
Much
of the literature in entry-exit decisions of a firm uses
The notable exceptions
discrete-time models.
Ghemawat and NalebufF
(1985). Fudenberg
are Dixit (1989), Fudenberg and Tirole (1986), and
and Tirole (1986) analyze a gaming situation between
two competing firms with incomplete information when there
Ghemawat and Nalebuff
is
no exogenous uncertainty; while
(1985) study a similar situation with complete information. Dixit (1989)
consider a single firm's entry and exit decision
when
the exogenous uncertainty
is
modeled by a
Brownian motion.
The purpose
of this paper
firm's optimal entry
Dixit (1989),
and
who drew
exit
twofold.
is
problem
in
First,
we use a general methodology
in solving a single
continuous time using the theory of martingales. Unlike
implications of his model using a particular parameterized example, we give
general qualitative characterizations of a firm's optimal timing decision. Using a continuous time
model when the choice variable
is
time
is
natural as time, after
all, is
continuous.
More important,
continuous time models are better able to yield clean analytic results. This has been evidenced in
particular by continuous time models in finance; see
get clean
and general analytic
results
Merton (1990). Here
which are usually
difficult to get
in this paper,
we
also
using discrete time models.
Second, we go beyond a single firm's problem to analyze a duopolistic entry-exit game with
complete information. The focus of our study here
in continuous
games. This
for
is
to
examine subgame perfect equilibria
directly
time without having to take limits of the outcomes of a sequence of discrete-time
in contrast to the
emerging literature on continuous time extensive form games;
see,
example, Simon (1987) and Simon and Stinchcombe (1988).
The
rest of this
in continuous
motion.
It is
time
demand
paper
for
is
organized as follows.
a single firm. The uncertain
Section 2 formulates the entry-exit problem
demand
demonstrated that the optimal entry-exit policy
firm should enter
the
is
if
the
demand
is
is
modeled by a standard Brownian
unique and
increases to exceed a certain level
is
and should
a barrier policy: The
exit Jifterwards
when
decreases to reach another level.
Although presented
in the context of
a Brownian motion uncertainty, our methodology can be
1
INTRODUCTION
used
2
any environment where the uncertainty
in
case, the optimal entry
enter the
first
wards the
first
and
exit decisions
is
may no
modeled by a Strong Markov process.
longer be barrier policies. Rather, a firm should
time the demand reaches a possibly time-dependent Borel
set
and should
time the demand reaches another possibly time-dependent Borel
of using a Brownian motion to model uncertainty
In such a
is
set.
exit after-
One advantage
that, besides the qualitative characterizations,
the entry and exit barriers can also be analytically expressed.
Section 3 analyzes an exit
weak
is
game
of a duopoly.
The two
firms in the industry, one strong and one
a sense to be formalized, are seeking the best time to exit.
in
the one in which the strong firm does not exit unless the
One subgame
demand
is
perfect equilibrium
such that
it
cannot sustain
even as a monopoly and as a result the weak firm aJways exits before the strong firm does.
subgame
also identify other candidates for a
bound and lower bound on each
phenomena occur
firm's
in these equilibria.
for the
An
losses.
increasing
monopolist and
We
is
demand
subgame
entrant makes the
exit
when the demand has been
falls significantly.
after the
weak firm
The strong
a bad news. Thus
it
exits
when the demand
game
exits against the current
duopoly
become a
reaches a critical level from below.
of an incumbent versus a potential entrant.
We
perfect equilibrium. In this equilibrium, as expected, the existence of a potential
life
span of the incumbent shorter than when
incumbent may indeed remain as a monopoly throughout
it is
a monopoly even though the
its lifetime.
When
the possibility of future duopoly competition limits the potential future
consequence the incumbent
is
no potential entrant and thus
In addition, the entrant
may
less tolerant to
it
exits earlier
the
demand
monopoly
is
low,
As a
profits.
the current monopoly losses than a monopoly facing
than a monopoly
will
even before the entrant enters.
not enter the industry even though the
where both firms can be profitable
may be
firm then trades
increases the expected waiting time for the strong firm to
continue in Section 4 to consider a
exhibit a
may
interesting
strong firm, for example, because the weak firm
demand
becoming a monopolist
Some
perfect equilibrium strategies.
For example, either firm
plays tough and would not exits until the
off the potential of
perfect equilibria by characterizing the exact upper
subgame
on the average increasing. This happens
We
as a duopoly. This
able to enter after the incumbent exits.
And
is
demand
is
above the
level
so because by waiting longer, the entrant
the benefit from being a
monopoly
in the
future outweighs the losses in the current duopoly profit.
Section 5 outlines
exits.
how
the analysis of Section 2 can be generalized to allow re-entry once a firm
Concluding remarks are
in Section 6.
SINGLE FIRM PROBLEMS
2
Single
2
We
3
Firm Problems
consider a single firm's decision to enter or to exit out of an industry, which
is
characterized
by an uncertain demand. Formally we model the uncertain demand by taking the state space
from time
to be the space of continuous functions of time
to infinity.
complete description of one possible demand over the time horizon
at
time
t
if
the state
process X(u;,t)
=
is
u){t),
The information
u.
that
is,
"infinity"
demand
the
the firm has at
s
<
i} is
t
at time
all t.
Let
P
measurable and
is
to
P
That
this space
demand from time
will
we
to time
to theorem 2.3.4) that {Tt;t G
=
will
we
to
P
initial state
demand
u.
is
is
to time
with respect to
The information
the firm has at
the smallest sigma-field finer than
t
but also
C\s>t
^'
[0,
is
use
will also
the
all
t
is
fit -{
(7B(t),
where
fi
^
We
is
a
5
is
a
to denote the
random
Xt interchangeably with X{t).
(fi, 1"°)
with respect
a complete probability space. ^
We
will
to include not only the historical realization
demand
denoted by Tt.
^-
X
and a are
scenarios described in (fl,/",
P) that
^° with
respect
is
It
the "completion" of
follows from
Chung
(1982, corollary
oo)}, the increasing family of sub-sigma-fields of !F,
f^''
t.
fi
"complete" the measurable space
assume that the probability mecisure
family of conditional probability measures \^P^\x G
motion with
the state
= X(0) +
X{t)
is,
by {Q,,T).^ Thus {Q,T^P)
to the completed probability space and
J^t
the
u>{t) is
demands from time
J^°.
which
only happen with zero probability. This information
continuous in that
if
we have used X{t) and B{t)
also enlarge the information the firm has at time
of
represents a
a random variable independent of B, and
is
B{-,t), respectively. Often,
simplify the technical difficulty,
and denote
denoted by
is J^°,
constants. Note that in the above specification,
To
fl
be the probability on the measurable space {^,J^) under which
standard Brownian motion under P, X{Q)
and
and
represented by the smallest sigma-field on
is
(/z,(T)-Brownian motion starting from X(0).
variables X{-,t)
X{uj,t)
t is
contains the historical
from observing the demand over time
every J^° for
oo)
G
u;
The state-dependent demand can then be modeled by a coordinate
Mathematically, this information
which {X{iJ,s);0 <
[0,
Each
Q.
3?}
such that, under P^, A'
x or starting from x. All the almost surely statements
is
a
P
(/i,(T)-
will
is
right-
induces a
Brownian
be with respect
unless otherwise specified.
When
the firm
demand Xf, where
is
already in the industry,
7r(-) is
derives a profit rate of i^{Xi) at time
increasing^ and nonconstant.
'The procedure of completion with respect to
measure zero sets.
A probability space is said to be complete if
^We
it
P
all
is
We
t
given the
assume that £r[/o° ^''^'KC^OI^^l < °°
to generated a sigma-field
T using
T° and
all
the subsets of
P
the subsets of probability zero sets are measurable.
weak relations throughout. For example, increasing means nondecreasing, positive means nonnegative,
lower than means no higher than and etc. When a relation is strict, we will use, for example, strictly increasing.
will use
SINGLE FIRM PROBLEMS
2
and that the process {e
6
'"*/(A'(),<
3ff+} is
/(x)
bounded below and above by two martingales, where
= E,
f
e-''7r{Xt)dt
(1)
Jo
Ei[-]
is
is
the expectation under P^,
^+ =
be zero. By the fact that n
set to
When
and increasing.
the firm
is
is
SJ+
U{+<^}> ^^^ we understand that
X
increasing and
is
enter. (This
that
7r(2/)
for y
<
We
We
not necessary.
is
for the re-entry of
once enter,
q denote
will
show
in Section 5
a firm.) In order to avoid the
how our
assumed
> —r0
for y
>
and x{y) < —r(3
x^,
=
<
x°,
and
is
continuous
to be zero. Entering
and^
be the
prohibitively costly to re-
it is
results can be generalized to allow
trivial cases that the firm will either
for y
oo, e~'^* f{Xt)
the cost of entry
never exit, we assume that there exists a x° and y^ with
will
7r(j/)
> ra
— oo <
for y
>
i°
never enter
<
r/°
<
or,
oo such
y° and n{y)
< ra
2/°.
are interested in two problems a firm faces.
the industry
if
the firm
is
already in
exit time, to
E
denotes the collection of
is
the optimal time to exit from
problem the firm faces
(2)
,
optional times'*, E[-]
all
is
the expectation under P, and r
Markov property
of
of the firm can be written as^
_
rT
e-'-'Tr{Xt)dt
Given the discussion above,
(2)
to find an optimal
profits:
re-^'ir{Xt)dt-e-^T'p
the riskless interest rate. Note that, by the strong
i
is
is its
^0
T6T
T
first
maximize the expected discounted future
sup
what
First,
Second, given an optimal exit time of a firm, what
it?
optimal time to enter the industry? Formally, the
where
is
For simplicity, we also assume that once a firm exits,
cost of exit.
t
a Brownian motion, f{x)
not in the market, the profit rate
or exiting from the market incur a one-time fixed cost. Let
at
is
-rT,
- e-''
(3
X
,^
>
is
the objective function
,•,-,
f{Xo)-E[e-^'{fiXT) +
fi)].
equivalent to
(3)
Put
v{x)
*T
X
/"t,
:
u(
is
— »+
is
fix)
£[y'|J'r]
=
- mf.£,[e-^^(/(Xr) +
(4)
/?)].
if {w G O
T(u) < t} e J^, for ail 6 »+.
Markov property if for all optional time T and random variables
an optional time
said to have the Strong
we have
=
^[yiA'r]
a.s.,
<
:
where
/'r, the sigma-field of events prior to
Y
independent of
T, consists of events
A
so that
.4n(r <«}€/•.
^Throughout, we understand that E[e-'''^g(XT)]
tion g.
=
j^^^^^e-'''^'"^g{X(u!,T(ui))P{du) {or any real-valued func-
2
SINGLE FIRM PROBLEMS
By
the hypothesis that e~'^*f(Xt)
is
bounded below by a martingale, v(x)
continuous and A'
is
a Brownian motion, u
/
over, since
is
The second problem
that
the firm faces
is
finite for all x.
a continuous function.
is
to find the optimal time to enter the industry
is
More-
knowing
behave optimally afterwards:
will
it
5
-
sup E[e-''^iviXT)
a)],
(5)
TeT
where we have again used the strong Markov property of X. Put
=
i{x)
sup E^[e-''^{v{XT)
-
a)].
(6)
TeT
The hypothesis
that e~'''/(A't)
X. Similar to v, v
A
is
bounded above by a martingale implies that v{x)
solution exists for either (3) or (5)
We
and
will
time
is
once
it is
the
is
the
time that the demand
in the industry
and when
never enter to begin with and u*
The next theorem shows
=
/*
T'
for
some
Proof.
gale,
is
/'
6
K
Theorem
A
and
it.^
We
and
and are characterized by two barriers
(5) exist
time that the demand
When
will
when
u*
inf{<
<
If
first
to the sigma-field on
f2
level of
— —00,
>
:
to (3)
v{Xt)
/*
and the entry
the firm will never exit
demand and
thus
it
will
the firm enters at any level of
is
a barrier policy.
and
+
/3
=
0}
=
inf{<
>0:Xt<
I'},
(7)
—rfi, is a solution.
Li (1990)
shows that there
time that e~^*(f{Xt)
+ 0)
is
^+}
is
bounded below by a martin-
exists a solution to (3)
and
this solution
equal to the largest regular submartingale
these two processes never equal in certain state of nature, the firm will then
regular submartingale {Yt\
times
lower than
never enter.
claim that this largest submartingale
for all optional
is
= -oo,
/*
+oo, the firm exits at any
Similarly,
X
the existence of a solution to (3) and this solution
Huang and
of
characterized by the
never exit.
(3)
the hypothesis that the process {e~^^f{Xt),t 6
1
dominated by
=
with 7r(/*)
From
exists an optional time that attains the
higher than u".
= +00.
There exists a solution
2.1
first
is
demand; and when u' = +00, the firm
Theorem
means there
show below that the solutions to
first
for all
respectively.
The aptimaLexit time
u*.
is finite
also a continuous function.
infimum or supremum,
/*
is
S >T,
t
S 3?+}
is
E[Ys\J'^t]
is
e~^^<j){Xt)
=
e~^^{f{Xt)
—
v{Xt)).
an optional process so that for any bounded optional time T, E[Y.f] < c»
a.s., where an optional process is a process measurable with respect
< Yt
x [0,oo) generated by
all
the processes adapted to /" having right-continuous paths.
SINGLE FIRM PROBLEMS
2
First
we show
that e~'''0(A'()
easy to see that v{x)
optional times
>
-/3 for
is
a regular submartingale dominated by e~'''(/(,\'()
Thus
all x.
<
e~'''0(A'()
+
e~'"'(/(.Y()
0). It
Note that
a.s.
f3)
+
for
is
any
S > T,
+ P)\XsUt
e-^^ inf E[e-^^^-^HfiXr)
reT
E[e-^^4>iXs)\J'T]
a.s.
T>S
>
e-^^inf £[e-^(^-^)(/(X.)
+
/?)|^r]
a.s.
r>r
=
e-^^(/(XT) - KXt))
=
e"""
a.s.
a.s.
</)(Arr),
where we have used the strong Markov property of X. Thus
e~'^*(/){Xt) is
a regular submartingale
dominated by e~''*f(Xt).
Second we claim that
e~^^(j)(Xt)
Suppose otherwise and
0).
Y{u;,t)
bility.
>
let
y
is
the largest regular submartingale dominated by e~'^^{f(Xt)
be a regular submartingale dominated by e~'"'(/(Xt)
whose projection onto
e~^'<i){X{u,t)) on a set of {u,t)
+
/3)
+
and
of a strictly positive proba-
17 is
Define
=
Sn
mi{t >
:
Yit)
> e-'^VC^O + -}•
n
It is
easily verified that
Pr{5n < oo} >
last
e~''^"(^(Xs„)
+
is
an optional time and by the hypothesis there
Since e-'"'(/(Xt)
0.
e-'-nXsJ =
where the
Sn
^^
+
/?)
>
m^f^^^ Eie-^-^ifiXj)
Y{t),
+
>
0)\Ts„]
is
^^
F
strictly positive probability.
positive probability, which
an n >
so that
we have
equality follows from the fact that
^ with a
exists
W^^^
E[YiT)\Ts„]
a submartingale.
is
Thus F(5„) >
= y(5„)
a.s.,
However, F(5n) >
e~''^"<f){Xs„) with a strictly
a contradiction.
Third, we want to show that
r- =
is
a barrier policy, that
that f{x)
—
4>{x)
=
is,
T*
is
v{x) and v{x)
inf{<
the
is
>
:
first
f{Xt)
+
/?
-
time that Xt
increasing since
tt
is
<i>{Xt)
is
less
=
0}
than or equal to a
/*
so that
T* =
inf{t
Note
increasing and nontrivial and a Brownian
motion has stationary and independent increments. Recalling that v
there must then exist a
level /*.
>0:Xt<
/'}.
is
continuous and v{x)
> —0,
2
SINGLE FIRM PROBLEMS
we are
Finally,
T'
left
a solution to (3)
is
7
to demonstrate that
if
and only
/*
G
J?
and
< — r/3. To
7r(/*)
see this,
we observe that
a solution to
if it is
mf,£[e-V(Xr)],
where
/•oo
= E
fix)
e-'-'(x(Xt)
/
+
r0)dt
Jo
Given
That
T*
this,
is,
T*
is
is
a solution to (2)
sup
E /
TeT
Jo
also
that
=
7r(2/)
+00,
7r(j/)
> —r0
+
e-'\7:{Xt)
if it is
a solution to
= fiXo)-\nf^E[e-^'fiXT)].
r(3)dt
must be
for y
>
profit function
clear that the firm will not exit unless
it is
+ rP <0. Hence we have
If /*
and only
an optimal exit time when the firm's
exit cost. In this case,
7r(l')
if
ir{l*)
<
= — oo, we must
/*
x°. Similarly, if
y and this contradicts the hypothesis that Tr{y)
the
one can show that solution to (5)
demand Xt
Theorem
2.2
is
some u* G
There exists a solution
5f
= infO >
with 7r(u*)
Combining Theorem
the firm
is
Theorem
r/3
and there
loss;
that
is,
is
no
when
< — r/5
for
j/
<
also characterized
is
y.
i°.
is
This contradicts the hypothesis
7r(j/)
> — r/3
Hence
/*
6
for arbitrarily small
5?.
I
bounded from above by a mar-
by a barrier: enter the industry
if
greater than or equal to a given level u*.
T'
for
making a
it is
have
Using similar arguments and the hypothesis that e~'''/(Xt)
tingale,
+
-rf3.
than —r(3 for arbitrarily large
less
is tt
2.1
>
and
v{Xt)
:
ra,
to (5)
is
=
and
v{Xt)
- a] =
inf{<
>
:
Xj > u*},
(8)
a solution.
2.4, the
optimal entry time and exit time for the firm given that
currently outside the industry are recorded below:
2.3 Suppose that the firm
for the firm
is
T^ defined
in (8)
and
is
currently outside the industry.
the optimal exit time for the firm
T^ =
With the continuity and unbounded
that the barrier policy characterized in
inf{<
>T^ :Xt<
2.3
is
the optimal entry time
is
/'}.
variation properties of a
Theorem
Then
(9)
Brownian motion, we next show
the unique optimal entry-exit policy.
2
SINGLE FIRM PROBLEMS
Theorem
2.4 The T^ and T^ defined in (8) and (9) are the unique optimal entry and exit times.^
We
Proof.
8
will
show that T' defined
in (7)
is
the unique solution to (3).
The arguments
for
proving the uniqueness of the optimal entry time are similar.
Proposition 2 of
Thus
a.s.
if /*
Assume
Huang and
= -oo, T" must
Li (1990)
=
Tn
By
if
5
r„
[
T*
P-a.s.
> T'
Put
0.
in{{t
>0:Xt<l'--}.
n
the fact that a Brownian motion has continuous and
easily seen that
another optimal exit time, then S
is
be the unique optimal exit time.
P{S > T'} >
therefore that
shows that
unbounded
Thus {5 > T*} = Uni'^ > T^]-
If
variation sample paths,
we can show
it is
that, for every n,
we have
E[e-^^<j>{Xs)\TT^
> e-^^"0(XTj
a.s.
on {S > T„},
(10)
then we are done as this will imply
E[e-'^<{>{Xs)\J^T']
>
e-''^' 4>(Xt')
a.s.
on {S > T'}
and hence
E[e-^^{f{Xs)
and 5
is
f3)]
> E[e-^^4>iXs)] > E[e-^^' 4>iXT')] =
sub-optimal, where the
To show
Then
+
for
i
=
(10) let r
=
inf{<
first
inequality follows as e~'''</)(X()
> r„
:
Xj =
and
/*}
set
So
=
E[e-^'^'
is
this,
<^ is
+
f3)],
dominated by e~'^\f{Xt)+(3).
Tn, S2
= SV
T„, and 5i
=
52 A
r.
0, 1,
e-'^'<j>{Xs.)<E[e-'^'^^<l>{Xs„,)\Ts.],
since
{/{Xt')
a submartingale. Inequality (11) for
i
=
is
strict
a.s.,
(11)
on {^i > 5o}
= {5 >
r„}.
To
see
we note that
e-^^''<A(X5o)
=
-
e-^'%fiXso) +
E[e-'^'<l>iXs,)\J'so]
/?)
- E[e-^'^{f{Xs,) +
=
E[
e-^'{n{Xt)
+
r(i)dt\J^So]-E[
JSo
=
E[f
P)\J'so]
TOO
fOO
e'^'iniXt)
+
rP)dt\J^s,]
JSi
'
e-^\ir{Xt)
+
rP)dt\Tso]
<
0,
JSo
We used a different argument
by an anonymous referee.
in
an earlier version to prove this theorem. The current proof
is
suggested to us
SINGLE FIRM PROBLEMS
2
where the
first
9
<
inequality follows because 7r(X()
i
=
1)
—r(3 for
In
and
t
is
G [5o,5'i).
= f(Xs,) +
The two
for
i
=
0,1,
inequalities in (11)
(t
and the
=
and
=
E[e-^'^<l>{Xs,)\:Fso]
=
E[E[e-^^<j>iXs,)\J's,]\rs,]
>
e-^^"<^(.YrJ,
(10).
sum, we have shown that there
this solution
is
exists a
Markov property and time and
the latter of which allows us to conclude that v
and the
barriers are time independent.
to the cases where the
that the barriers
demand
is
homogeneity of a Brownian motion,
increasing and time independent, thus the barrier
Thus these two theorems can be generalized
may be time dependent and may
the uniqueness in
to the Strong
is
spatial
when
Theorem
the
not be half-lines any more. Rather, the optimal
demand
enters or leaves a time-dependent Borel set.
2.4 uses the fact that a
Markov property, continuous and unbounded
Brownian motion has,
variation sample paths.
not use the distributional property of a Brownian motion, namely that
distributed.
Thus the proof
The advantage
statements
made
strated below.
work
will
easily
any optional process^ having the Strong Markov property except
policy will prescribe entry or exit
The proof of
unique solution to a firm's entry and exit decision
a barrier policy. Note that in our demonstration of Theorems 2.1 and 2.2, we
only used the Strong
policy,
T„}, 4>(Xs,)
imply that on {5 > Tn),
E[e-^'4>{Xs)\TT^
which
>
equality follows since, on {S
for
its
in addition
But we did
increments are Normally
any (nondegenerate) diffusion process.^"
of using a Brownian motion to model uncertainty
is
that, besides the qualitative
above, we can explicitly calculate the optimal exit and entry barriers as demon-
To determine the optimal
/'
barriers
and u*, we
counted profits for any barriers, using Harrison (1985, chapter
3).
first
calculate the expected dis-
Let T{y)
=
inf{(
>
:
Xt
=
y}.
Then
exp[-a,(x - J/)]
«(x,y).^.[e-^(v)j^|-pi-«:;--y;{
exp[-a*(j/ - x)]
\i
X
if
y
>y,
;-^^>
(12)
X,
and
fix)
= E,
/
7o
See footnote
7 for the definition of
the existence results of
Huang and
A diffusion process is
diffusion process also has
=1
e-'\{Xt)dt
J
niy)e{x,y)dy,
(13)
J-oo
an optional process and we need the optionality of the
demand
in
order to use
Li (1990).
a process having continuous sample paths
unbounded
variation sample paths.
and the Strong Markov property.
A
nondegenerate
SINGLE FIRM PROBLEMS
2
10
where
a.
=
a-'^[^J^i^ -\-2o^r^- n],
a*
=
a-'[^/i2
+
and
(15)
2a2r-;i].
(16)
Let a discrete variable indicate whether the firm has entered the market (/) or not (0), and
v(xj\l) and v{x,u,l;0) be the expected future
and
u
when the
v{x,u,l;0)
current
is a;.
It
under a barrier policy with parameters
profits
follows from the strong
(
f(x)-EAe-^^^'HfiXT^,)) +
{
-fi
Markov property
= f{x)-eixJ)ifil) +
f3)]
^,[e-'-^W(t;(XT(„),/;/) -a)]
f
=
demand
=
let
^(x,u)(t;(t.,
/;/)-«)
X
that
for
x >
/,
for
i<
/;
of
(3)
for
x
<
u,
for
X >
u.
/
<
v{x,
{
l\
- a
I)
Thus, the values of the optimal barriers can be determined by finding an
/*
maximizing v{x,l;I)
and an u' maximizing v{x,uj*; 0). The optimal expected
=
v{x,l'\ I)
in
the industry or not.
profits are v{x)
v{x,u'J'';0), respectively, depending upon whether the firm
These
is
already
=
0,
and v{x)
=
results are recorded in the following proposition.
Proposition 2.1
/'
is
number
the unique
satisfying
TOO
h{n = and u'
number
the unique
is
+
r
e^-'[Tr{z)
- ra]dz +
— oo <
/*
<
<
x°
j/°
,V^^
^(^^
I
^
''^'''
Now we
policy
is
<
u*
<
0.
(18)
oo. In addition, the optimal expected profits are:
-
j 0{x,u'){v{u')-a)
j v{x)-a
_
~
j
\
—(a + (i)e''''' =
ifx<u*,
\ix>u*;
-0
fix)-eix,nif{n +
if
(3)
.
The
firm should enter the industry
u* and should exit afterwards
7r(/')
< — r/3, once
when
the
demands
falls
if
the
X < r,
,„„.
^''>
ifx>/-.
demand
.
^^'
have completely solved the optimal entry and exit problem of a firm.
very simple.
that since
(17)
a.
Jl*
Moreover,
rP]dz
satisfying
= -
h(u')
e-^''[n(z)
rises to or
to or below the threshold
in the industry, the firm will not exit until the
The optimal
above the
number
/*.
level
Note
instantaneous profit
2
SINGLE FIRM PROBLEMS
becomes
rate
demand
11
significantly negative. This
because there
is
be higher and produce positive
will in the future
in the industry in anticipation of the rise in
is
a strictly positive probability that the
Thus the firm chooses
profits.
demand. This together with the
to remain
exit cost
and the
discounting cause the firm to optimally delay the exit decision. Similarly, the firm does not enter
the industry unless
instantaneous profit rate
its
demand
probability that the
So the firm waits
cost.
The
will
until the
explicit expressions for
Proposition 2.2 Let
>
Xi
1:2
A
later.
fi,
cr
and
o-nd
a strictly positive
an entry
is
comparative
also allow us to derive the following
>
(j\
""^
02:
and
and increasing
f3
is
sufficient high to enter.
is
and u'
barriers for these two profit functions
decreasing in
significantly positive as there
soon decline to make the profit negative and there
demand
/*
is
'^' '1
ond
I2
be the corresponding optimal exit
exit costs, respectively.
in r,
and u*
firm with a uniformly higher profit rate for
all levels
l\
<
/2-
Moreover,
a and a and decreasing
demand than another
of
demand, the
Also, the higher the expected increase in the
Then
increasing in
is
static:
I'
is
in r.
firm will exit
later a firm will exit; while the
The former
higher the interest rate, the higher the exit barrier and the earlier the firm exits.
is
obvious and the latter follows since the firm does not exit immediately after the instantaneous
profit
becomes negative
makes future
in the anticipation of future profits
is
demand, the lower
so since the exit option of the firm limits the downside risk of the uncertain
demands and thus the added upside
suffer current losses. Finally,
The comparative
in the interest rate
In addition, the larger the volatility of the
profits less valuable.
the exit barrier. This
and an increase
potential by an increase in
an increase
statics for the
in the exit cost
a makes the firm more
makes the firm stay
optimal entry time are
less intuitive
in the
market longer.
because a change of the
An
parameters also affects the optimal exit time on which the optimal entry time depends.
in
a has two
and
in the
effects:
the increase makes
mean time
firm wiU be
it
it
more
likely that the firm will suffer loss in the
making a negative instantaneous
profits.
the volatility of the demands,
may
or
may
not increase the
the riskless interest rate also has two effects. First,
it
increases the exit barrier and thus
suffer losses shorter.
The
latter
incentive to enter earlier.
since the optimal exit time
be longer.
makes the firm
The combined
is
In anticipation of the latter
entry barrier and enters the industry later.
its
Second,
increase
near future
depresses the exit barrier and thus increcises the time span over which the
the former, the firm increases
firm.
willing to
also earlier,
it
life
So an increase
span of the firm.
An
makes waiting to enter more
afford to enter earlier
in
increase in
costly for the
makes the time span over which the firm
effects are that the
it is
and because of
will
and the former gives the firm
optimal entry time
unclear whether the total
life
is
earlier.
But
span of the firm wiU
THE EXIT GAME
3
There
is
no
barrier
change
clear direction of
higher profit rate or
other hand,
12
when
^i
in the
On
increases.
optimal entry time when the firm has a uniformly
the one hand,
it
makes waiting more
costly.
On
the
increases the time span over which the firm will suffer losses by decreasing the exit
it
and thus creates an incentive
These are two opposing
To
effects.
for the firm not to enter until the
illustrate the point,
Let tt{z,6) be the instantaneous profit parameterized by
continuously differentiable, and d'K{z,6)/d6
as 6 increases. For simplicity also
>
assume that a =
/3
=
is
we consider a change
6.
for all z.
demand
Assume
Thus, the
0.
profit rate
first
in the profit rate.
continuous,
Tr{-,6) is
From the
sufficiently high.
7r(2,-) is
uniformly higher
is
order condition (17),
we
obtain:
since k{1',8)
and dw{z,S)/d6 >
<
0.
It
then follows from (18) that
'<'••*%-/." '••"-''^"-''^=
06
Tr{u-,6)
Note that n{u',S) >
Thus,
0.
above expression inside the parentheses, the
in the
first
term
represents the effect due to the increased profit loss after entry because of the reduced exit barrier,
which
calls for
an increase
profit loss while waiting,
z
6
which has a negative
and dTr{z,S)/d6 >
[^',u']
The second term
in u*.
for
some
represents the cost of waiting due to an increased
effect
on the entry barrier u*.
U dn{z,8)/d8 =
non-trivial interval in (u*,oo), then
du'/d6 >
0, i.e.,
for
the
firm will enter later with a higher profit rate. Notice that the negligible increments of the profit
rate in the region [/*,u'] implies an insignificant waiting cost but a significant profit loss after entry
if
there
is
a significant decrease in the exit barrier which could be caused by a higher profit rate in
the region (u*,oo).
The
3
We
Exit
investigate in this section an duopolistic exit game. There are initially two firms in the industry
facing a stochastic
firm
0ij
i
if
>
the corresponding exit cost, where
x°^
and
7r,j
is
7r,_,(j/)
{e~^^fij{Xt),t G
(1)
demand modeled by a Brownian motion. Denote by
there are j firms in the market and the total
cLSSume that
y
Game
i
=
1,2 and j
=
demand
1,2.
As
in
in
ff,_,(A't)
the industry
^+}
is
by replacing k with
for y
< x^y Moreover, E[J^
The
profit for
a firm
in a
Xt, and by
we
> — r/3,j
for
e~'^'\iT,j{Xt)\dt]
<
duopoly situation
is
7r,j(j/)
oo and that the process
bounded below and above by two martingales, where
7r,j.
is
the single firm context,
increasing and nonconstant, and there exist i°, so that
< -r/3
the profit rate for
/,j is
as defined in
naturally less than that
THE EXIT GAME
3
13
a monopoly situation, and exit
in
>
^iiiy)
A
^i2{y) for all y,
and
is
>
(Sn
usually
when
firm will exit from the industry
depends on whether
rate of a firm
depend on the
analysis of this exit game,
strategies
made by
exit decisions
we
its
we designate
is still
at
specification of the
and
t
in
situation occurs.
In the
game
its
will
assume that
Note that once a firm
prohibitively costly to re-enter.
it is
We
game.
optimal strategy afterwards should simply be
any time
profit
decision will certainly
its exit
Thus a gaming
established in Section 2 in a single firm context. Therefore, the
if
As the
to remain.
our attention on subgame perfect Nash equilibria in pure
and therefore need an extensive form
becomes a monopoly,
that
x°2-
no longer profitable
the other firm.
will focus
once a firm exits from the industry,
<
i°j
a monopoly or a duopoly,
is
it
is
it
Thus we assume
costly for a monopolist.
then follows that
It
fi,2-
more
unique optimal exit time
be completely specified
will
any state u, the strategy a firm follows given that
its
opponent
in the industry.
Formally,
time with
let
</>{
4),{u>,t)
:
>
fi
X
K+
P-a.s.
t
i->
S?+ be the strategy of firm
and we
recall that 3(f+
where
i,
</>,(-,<)
:
fi
—*
J?+
is
denotes the extended positive real
an optional
line.
We
also
impose the following regularity conditions:
1.
the set
A =
is
2.
{{u;,s)
enx^+: (f>i{uj,s)
=
4>t{uj,t) is right- continuous in
opponent are both
will not exit
where
(21)
t.
an optional time S. Our interpretation of
i
e »+}
progressively measurable;^^
For brevity of notation, we will often use 0,(5) to denote
its
s, s
purposes
Given
</>
S.
The two
become
will
satisfy these
immediately
=
4>iiS)
in the industry
</>,
and
its
where
in the states
opponent
4>,{S)
clear later.
> S and
Denote by $ the space of
if
firm
i
and
continue to be in the industry, firm
will
regularity conditions are about
a random variable for
At any optional time S,
as follows:
is
(/>,(a;,5(u;)) as
will exit
how
all
(^,(f)
immediately
in the states
changes over time and their
the mappings
(f)
:
Q
X
S?^. t-+
Jf+ that
above conditions.
6
$ and an
optional time S,
T(5;
<l>)
we put
=
mf{t > S
:
4>{uj,t)
=
t).
(22)
''a process Y is progressively measurable if, as a mapping from fJ x 3?+ to 9?, its restriction to the time set [0, i\ is
measurable with respect to the product sigma-field generated by Tt and the Borel sigma-field of [0, (]. The progressive
sigma-field is the sigma-field on fJ x 3?.^. generated by all the progressively measurable processes. A subset of n x S+
is progressively measurable if it is an element of the progressive sigma-field. A good reference for these is Dellacherie
and Meyer (1978).
THE EXIT GAME
5
In words, T{S\(t>)
exit time of its
14
the
is
time after S that the strategy
first
<f){S)
able to
is
and
is,
we need the two
Proposition 3.1 Suppose
> S
time with 4>(S)
<i>{TiS;<P))
=
4'{S)
and T{S\<t>) precisely correct, however, we need to show
at each optional time
tell
That
T{S;(t>).
properties
instructs the firm to exit prior to the
opponent.
To make our interpretation of
that one
(f>
we need
to
</>(5)
and
T{S;<t>) are optional times.
It is
for these
regularity conditions.
that
^ and S
E.
(f)
and T{S;(p)
a.s.,
show
whether one should continue or exit according to
5*
is
also
is
Then
(t>{S)
is
T{S;<t>)
> S
a.s.
an optional time.
an optional time with
an optional
Moreover,
T{S;4>)a.s.
Proof.
To prove
or 0(5) A
<
/"(
6
that 0(5)
as
Borel measurable in
5
t.
is
an optional time
is
it
suffices to
prove that 0(5) A
an optional time. By the right-continuity of
By
the composition of two mappings,
it
is
t
we know
<f){Lj,t) in t,
follows that 0(a;,
/"(-measurable,
S{(jj)
0(<)
At) At E
is
Tf
Next note that
5(w)) A
<^(u;,
By
the fact that {5(u;)
optional time.
The
i
=
0(w,5(u;) A0l{5(u;)<t}('^)
=
0(a;, 5(a;)
><}£/"(
assertion that 0(5)
Next we want to show that
T{S;<t>)
A
is
defined in (21).
stochastic interval [5, oo)
that T{S;4>)
is
is
By
opponent and
t.
In words,
itself still
discussed by Perry and
necessity of this,
t
for all
'l{5(u/)>(}(t^)
a.s. is
os-
G
t
.?^(
as
0(5)
is
an
obvious.
an optional time. Observe that
:
(u,t) e
/I is
An[S{u),oo)},
a progressive
also a progressive set, then Dellacherie
<
from the hypothesis that 0(<)
last assertion of the
right-continuous in
=
+
set,
and the
fact that the
and Meyer (1978, IV. 50) shows
an optional time.
Note that the
<j>{t)
> 5
is
'A(w.5'(w)A0l{5(u-)>t}('^)
an optional time, we know 0(5) A
the hypothesis that
Finally, the last assertion follows
its
is
Ol{S(u;)<(}('^)
= mf{t >
T{S{uy,4>)
where
5
as
A
+
it
>
it
says that a firm will indeed exit at 0(T(5;0))
remain
at 5.
This
is
'^We thanks David Kreps
is,
one should remain
for pointing this
out to
us.
=
I
t.
6
T{S;(p))
if
$
is
both
related to the kind of intertemporal consistency
Reny (1990) and Simon and Stinchcombe
That
right-continuous in
above proposition follows from the hypothesis that
suffices to consider the following
1/2.
is
(1988).'-^
example. Let 0(<)
in the
=
1
To understand the
for all
industry from time
(
G
[0,
1/2]
and
to time 1/2 but
TEE EXIT GAME
3
/
At time 1/2, one
T{0;(f>).
him
instructs
not sure what to do.
is
Before
we turn our
equilibrium,
for firm
i
when
in order. Let
is
profit function is
7r,j
7r,2,
that
/ji
and
<
/21
and
/3,i
and
/3,2,
/jj
him, however, to exit
tell
be the unique optimal exit barrier
I'j
and the
exit cost
<
assumed
'22-
That
it
to denote inf{<
> S Xt <
:
1*^} for
span. For example,
has no chance to be a monopoly. Given the ordering on
7r,i
firm
its life
Proposition 2.2 shows that
1
<
/*j
We
/t^.
a stronger than firm 2 in that firm
is
will use <^_, to
form exit game
is
n-ij
>
denote firm
i's
a pair of strategies
tt
^[|T(O;0)AT(O;0_.)
SUp,e$
let
G $ X
$
,-rt^^^^x,)dt
A
/;_,, hi-j,
=
and
Note that these
2.
But
/92j-
Vij
this
is
not
we use T'AS)
be the functions
replaced by
/J
barriers
/9,j.
0, solves:
(/>_,-,
e-^^(°--*)/3.2l{T(O;0)<T(O;^_.)}
+ ^"'"^^°''*~''^il(^T(O;.^_i))l{T(O;fli)>T(O;0_.)}
i
>
I's exit
Nash equiUbrium of the extensive
so that given
-
/?ij
assume further
For convenience,
replaced by Kij and
opponent's strategy.
((t>ii4'2)
and
;r2j
/* 's.
any optional time 5. Also,
defined in (13), (17), and (20) respectively, with
We
in the single firm
the optimal exit barrier
is
l'^
perfect
the
earlier,
is,
Then
is /3,j.
depicts one possible relative positions of
1
subgame
I22 is
assumptions can be insured by the hypothesis that
necessary. Figure
'
1,2.
Note that the action taken by firm
is
1,
both the monopoly case and the duopoly case are higher than those of firm
last
for
=
</)(l/2)
eliminates this possibility.
t
there axe always j firms in the industry during
optimal exit barrier for firm 2 when
and
in
=
attention to the existence of a Nash equilibrium and a
some notation
problem when the
(f>{t)
1/2, but 4>{T(0;<p))
His strategy at that time,
to remain in the industry while his strategies after time 1/2
immediately! The right-continuity of
for
=
immediately after time 1/2. This specification implies that T{0;4))
exit
1
15
i,
or the exit time of firm
i,
in
a Nash equilibrium
((/>i,</>2)
an optional time
T{0;
That
is,
on the
set {T(0](f>i)
A
>
set
{T(0;
<?!),)
4>i)l{T(0;4>,)<T(0;<t>.,)) +^'ll{T(0;*,)>T(0;rf._.)}-
<
T{Q;4>^i)}, where firm
7(0; (^_,)}, where the opponent exits
Nash equilibrium
{(f)i,4>2)
is
i
exits first,
first,
firm
i
it
exits at T{0;4>,); while
behaves
a subgame perfect equilibrium^^
if
like
for
on the
a monopoly.
any optional time 5,
T{S;^t) solves
fT(S;0)AT(S;<*_.)
sup
IP E[
:$
e-^^'-'K,2{Xt)dt
J/s
-
e-^(^(^^^)-^)/3.2l{T(5;0)<T(5;0_.)}
+ e-^('^(^'^-'^-'Ka{Xr^s-,4>.,))hns-S>ns;4>-.)}\^s],
(24)
where E[-|^5] denotes the expectation conditional on TsOur
definition of a
subgame
perfect equilibrium
is
a direct generalization of Selten (1975) to continuous time.
THE EXIT GAME
3
A subgame
equilibrium
16
perfect equilibrium {4>i,4>2)
((t>\,d>'2),
lies in
4>'-
in
0,.
Rather
it
any subgame. This, however,
differ
any other subgame perfect
- T{S; 4>',)l {ns;4>',)<T{S;4>'_,))
"••'•
on a
set of (u;,<)
it
(25)
does not require the
focuses on the uniqueness of the exit times taken by the
the right sense of uniqueness.
is
the implied exit action taken by firm
which
for
time S. This definition of uniqueness seems rather weak as
uniqueness of the mappings
two firms
if
we have
T{S;<p,)l{T(S;4>,)<ns;4>-,)}
for all optional
said to be unique
is
i
fi is
significance of
There can be two strategies
in equilibrium.
whose projection onto
The
(^,
of a strictly positive probability.
</>,
and
But
they can imply the same exit times of the two firms in the subgame starting from an optional time
S
as long as (25)
true.
is
The optimizations
<jf)
of (23)
and (24) look formidable
G $. The following proposition shows that (23)
the optimization
is
performed by looking
Proposition 3.2 Let 5 G T.
T
= T(S\0)
SUP.,T
equivalent to a
mappings
much simpler problem
in
which
for optional times.
T
For every r G
Thus, (24) for a given
a.s.
is
as they are looking for complicated
S
is
> S
with t
a.s.,
there exists a
<i>
E
^
so that
equivalent to
- e-(-^)A2l{r<T(5;*_.)}
Elfs^^^''"-'^ e-^i'-^K,2iXt)dt
'•>^
(26)
+ e-^(^(^^*-)-^'i;.l(XT(S;*_.,)l{.>T(S;^_.)}|/-S
Proof.
Let r G
T
with r
> S
a.s.
It
suffices to
show that there
exists
.
<f>
£ ^ so that T{S;
4>)
=
t
Define
a.s.
(f>iuj,t)
=
l[O,rH]('^,0l"(w)
+
tl(^(^),oo)(w,0-
Then
A = {(w,0
It is
known
n X
3?+
:
<i>{u,t)
that the stochastic interval [r(a;),oo)
Meyer (1978, IV.62). Then
To
G
characterize the
sidering satisfies the
that the longer
its
<f)
e ^.
subgame
It is
is
perfect equilibrium,
monotone property studied
opponent stays
progressive measurable; see Dellacherie and
in
we
first
Huang and
in the industry, the earlier
3.1 For any optional time S, firm
in firm j
's
time from S on, for
i
=
[t(uj),oo).
also easily verified that T{S;<t>)
Lemma
exit
= t,te^+} =
1,2
i 's
j
/
i.
r F-a.s.
note that the exit
I
game we
are con-
Li (1990, Proposition 4) in the sense
a firm
will exit as
optimal exit time from
and
=
S on
is
the best response.
monotone decreasing
THE EXIT GAME
3
Consequently, firm
exit
z's
time
is
17
optimal exit time in response to an upper (lower) bound of
i's
optimal exit time
T'2(S)
is
optimal exit time of firm
it is
of firm
=
T(S\(f>-,)
if
and
oo,
than
later
its
monopoly
Proposition 3.3 Let
T'i{S)
is
if
{<pi,<p2)
G $X
$
6€ a
T{S;4)-,)
=
Therefore, the
S.
opponent starting from any
its
and cannot be
exit time
certain to remain a duopoly throughout
opponent's
Note that starting from any time 5, firm
exit time.
i's
response to any feasible strategy of
in
i
5 cannot be
optional time
time when
bound
a lower (upper)
its
earlier
than
its exit
span.
its life
subgame perfect equilibrium. Then for
all
optional time
S,
T'iiS)
<
+
T{S;(l>,)l{T{S;<i,.)<T{S;4,.,)}
7''l('5')l{T(5;0,)>T{5;0_.)}
Next we consider the optimal strategy of firm
suggested in the single firm problem. Suppose firm
than or equal to a
critical level
Ij
I's
best response
Theorem
2.1
must
form. At any optional time
[ui, /*2]
either
(-oo,
firm j
and firm j
to consider the case
=
fi2{x)
-E,
=
where we
for y
A 2 < X <
z
V
[e-^^("-)(/.2(u.)
-
T{y)
J/,
X,
>
i^{x,Ui,lj){f,2{ui)
=
inf{i
y,2 G
5?.
u, is
>
demand
is
lower
:
Thus
Xt
u,
(27)
=
+
will exit
i
when Xt
u,
>
/_,.
enters the set
Otherwise, Xt
becomes a monopoly and
will follows
Harrison (1985, Chapter 3), firm
-
<
T(l,)
\e-^^('')v,^{l,)\T{ui)
i^{x,lj,U,)fi2{lj)
+
>
T(/,)]
i){x,lj,Ui)Vii{lj),
y} for any scalar,
must maximize
9i;.(x,u,/j)
=
i's
firm j plays (27) can be written as:
+ E,
0,2)
i
a barrier policy of the following
— oo,/j] where
By
/3.-2)|T(u,)
r(/,)]
is
in
determined as follows.
UjJi2]-
when
+
(
and firm
when Xs = x G
[e-^^(^)/.2(/,)|r(r.,)
fi2{x)
recall that
- E,
reaches
it
will exit
payoff under the barrier policy depicted above
Viix,Ui,l,)
present, firm
is still
unique optimal strategy thereafter. The value
It suffices
does not exit until the
i,
a unique solution to (24) which
S when
Ij] first
firm plays a simple strategy
any optional time S. The arguments identical to that
from above or from below before
will reach the set
its
is
^
U.S.
= iRi{s>t:X,<lj}.
solve (24) for
show that there
j, j
T,\{S)
every subgame, namely,
in
<f>j{t)
Firm
when the other
i
<
Ci
(28). Direct
L /„ / ^
hi{u,lj),
computation
yields:
(28)
THE EXIT GAME
3
where Ci
18
a strictly positive term and
is
=
/i.(x,y)
^
x,y £
for
and
for
=
f
+
7 r(^.2(2)
Jy
-
rA2)(e''*'''-'''
Using this expression,
1,2.
+
e-'''^'-^^)dz
vn{y),
(30)
straightforward to verify the following
is
it
+
0,2
proposition:
Proposition 3.4 Let j /
i
and suppose
unique optimal exit time in response to
4>-,it)
=
>
inf{5
t
X, <
:
Then firm
£ ^+.
Ij} 'it
is
4>j
r(0l{X.,„€A.}+J;'(0l{X.(.,€^;}'
=
where r{t)
inf{s
>
^^
f
'
and
is
/ii(-, •)
When
t
:
>
inf{2/
/j
:
<
/i.(y,/,)
Aj
[u,, /i],
=
>
I,
:
i
finds itself playing a
demand
demand
falls
to the level
lower than
is
demand
trades off this potential future gains with the current losses.
for firm
as
i
it
may become
will suffer losses longer
i
if
the
and
The
Xs >
loss to
set
monopoly
Ai
=
1*2,
firm
i
exits
when
the
demand
[ui,l'2]
characterizes the situation
the larger the set Ai (a smaller u,
In particular,
is
U2
first
=
/ji
and A2 =
is
falls
T'^ or u,
if /j
['111^22]-
=
00.
>
/*2,
Now
when
if /j
<
1*2
Firm
2's
falls
and
u^
in the industry, firm
and firm j
demand
=
2
and
So firm
exits.
turns out to be a
when the demand
rises to
profit
becomes so dismal and firm
below
/'j.
firm
i
Continuing on, firm
/*i,
in
t
=
i
exits.
will suffer
By Lemma
3.1,
exits as a duopoly, or equivalently,
firm j will only exit after firm
Ij.
i
the future.
exits as a duopoly.
Now
take j
in
=
I
and
/^
cannot
should exit
it
=
respond to firm
i
/jj
I's
for this
strategy
I22.
then firm j exits before firm
take j
i
demand
optimal exit time
below
knows that
(/j, Ui), it
has no chance to become a monopolist and
i
below
when the demand
the other hand,
monopolist at
firm
time the demand
to exit immediately
On
).
Thus
sustain even as a monopolist.
Then
/^j;
the other hand, a rising
balance,
the more likely that firm
Ij ),
optimally the
G
be balanced out by the prospect of becoming a monopoly
the tougher the opponent (a smaller
case.
<
0}
Xs
/_,
Falling
On
On
a bad news.
is
to reach
falls
a monopolist sooner.
reach Ui, the prospect of the potential future
much
<
But, remaining in the industry gives firm
/'j.
i
too
/'j,
So staying
Ij.
the opportunity to become a monopolist
if
/i.(y,/,)
duopoly gajne at S with a demand
i
Similarly,
=
otherwise,
incur losses as the current
means firm
/,
defined in (30).
firm
good news
(31)
{-oo,lj],
if inf{t/
0},
oo,
1
the opponent will not exit until the
will
=
e Ail) Aj}, A,
A',
i's
I2
=
i
/22i ''^at is,
does, and firm
firm 2 exits
i
will exit as
when
the
a
demand
3
THE EXIT GAME
below
falls
is
l22-
Then
19
=
Ui
and Ai =
oo
demand
not to exit until the
falls
Firm
0.
below
Combining the above, we have thus
never exits as a duopoly and
1
its
unique strategy
/jj.
identified a
subgame
perfect equilibrium in which the
"strong" firm (firm 1) always exits as a monopolist and the "weak" firm (firm 2) exits as a duopoly
since
<
/*j
I21
and
Proposition 3.5
>
I22
'12-
Put, for all
i
G R+,
Mt) =
is
{<P\i4>2)
a
n2{t).
subgame perfect Nash equilibrium,}*
The expected discounted future
profits in the
equilibrium for the two firms, or equilibrium payoffs, are, respectively,
^
^^
=
V2{x)
where
Vij
is
and
tt
t
is
'
(33)
replaced by nij
f3
gij{x,y)
time
^
I22;
t;22(x),
defined in (4) with
This equilibrium
ifi<
\ uii(x)
=
and
and where
Pij,
- e{x,y)f„{y).
fij{x)
(34)
a "stationary equilibrium" in that the strategies for the two firms at any
is
a "copy" of their strategies at time
In this case, one easily verifies that T{S;
0.
4>i)
=
<f)i{S)
with probability one.
Proposition 3.5 and Proposition 3.3 also gives a trivial sufficient condition for the uniqueness
of the
subgame
subgame
perfect equilibrium:
perfect equilibrium.
survive as a monopolist
is
if /J2
When
1^2
<
<
^21'
'21
1
demand below which
in
1
1
is
the unique
firm 2 cannot
cannot survive as a duopoly. So
any subgame as firm 2
is
much
too weaker than
and thus we have a unique subgame perfect equilibrium.
However, other equilibria
l*j
*^^ level of
even higher than that at which firm
naturally, firm 2 always exits earlier than firm
firm
'
^^^ equilibrium of Proposition 3.5
in this case. If this
is
may
arise
if /21
<
/i2-
when demand
the case,
is
See Figure
between
/21
a duopoly and both can survive as a monopolist. Thus there
We now
One
identify candidates of other equilibria.
easily verifies that 0,
g
4>,
for
t
=
1,2.
1
for the
and
/j2,
assumed order of
barriers
neither firm can survive as
may be more than one
equilibrium.
3
THE EXIT GAME
20
Suppose that firm 2 does not
subgame, namely,
4>2{t)
Proposition 3.4 firm
—
exit until the
^2i(0' which
is
lower than or equal to
an upper bound of firm
is
optimal exit time
I's
demand
response to firm
in
By
equilibrium exit time.
2's
strategy
2's
every
in
/jj
is
r(01{.v.,„6^,}+Tr,(01{x.,.,e^.}
where Ai =
that ui
>
subgame
^2 = (-oOi'al'
[tii,/j2]'
As a consequence of
/ji-
=
^^'^ ''(O
Lemma
inf{s
as a duopoly in
equal to
is
response.
its
Since
response here
is
is
A-i
:
AiU
X, £
time of firm
bound
Then
infinity.
its
t
3.1, this exit
perfect equilibrium exit times. This lower
Suppose that ui above
>
(35)
1
Theorem
time
is
reaches
h (>
The
^22
show
2.1
we suppose that
an empty set and firm
bound on
('21 ''12]-
Arguments
is
T'j'2(5) as
does not exit
1
subgame
its
its
the unique
perfect
subgame
similar to those used to prove
that, in response to firm I's exit time defined in (35), firm 2's optimal exit
a stationary two-barrier policy at an optional time 5 as a duopoly: exits when demajid
A3 before
^21) ^'^d
=
reaches A\, where Ai
it
is
a duopoly,
it
its exit
becomes a new upper bound of
3.1 since (35)
is
a lower
(-00,
When
U
/2]
when the demand
similar to that for (31).
is
=
the
bound of
firm
I's
decision.
its
^'^^
below
demand
is
/ji-
below
in the future
Furthermore, this best response exit time
subgame
subgame
[u2»'22]
falls
always trades off the potential of being a monopoly
against the current duopoly losses in
for firm 2
and A3
[ui,/i2] a^ in (35),
"2 being two critical numbers; otherwise, exits
interpretation of this two-barrier policy
and firm 2
We
G
u\
a lower bound on
is
equilibrium exit times at any subgame, the equilibrium of Proposition 3.5
perfect equilibrium. So
also verify
a tighter lower bound than
a lower
is
One can
A2}.
perfect equilibrium exit times by
perfect equilibrium exit times at
Lemma
any subgame.
repeat this procedure to generate tighter and tighter upper bounds on firm 2's and tighter
and tighter lower bounds on firm
I's
subgame
has a fixed point other than the equilibrium of Proposition 3.5, this fixed point
perfect equilibrium
and provides the exact
respectively, for firm I's
and firm
2's
largest lower
subgame
If this
procedure
is itself
a subgame
perfect equilibrium exit times.
bound and the exact
least
upper bound,
perfect equilibrium exit times. This fact
is
recorded
in the following proposition:
Theorem
that
/ii(
3.1 Suppose that there exist uj,
«;,/•)
^.
=
0,
^
hi(u2J\) =
i
^
I
0,
/J,
uj
*'"'^ ^2
"""•
'21
<
^2
<
"t
^
'1
-
0)
<
'22;
^^^^
and h2(u\,l2) =
>
\n{{y>i;:h2iy,ll)<0},
if
00,
otherwise
inf{y
/J
,
:
My,'r) <
'12
<
"2 *"^'*
THE EXIT GAME
3
21
and h,{x,y) are defined
in (30) with the
A2 = (-oo,/2] U
T{t\4>\)
<
T^.it) andT^^it)
a "copy" of
reach U2,
it is
and firm
lower than
if
{<t>'i,4>'2)
is
T{t-<t>2)
3.1
<
^r™
/21
1
decreases to
1^2
and firm
immediately and firm
On
1
1 is
1
continues to
from above, so firm
the
demand
decrease to
the
demand
increases to reach u\ before
be
will
will
exits
1
firm
cis
Xq <
I2,
firm 2 exits immediately and firm
There are two interesting features of
demand has been
the
weak firm
increasing. This
(firm 2) exits as the
if
Xq G
is
between
to reach uj before
it
decreases to reach
when the demand has been
The equations
/j
and
that determine uj,
3.5.
<
(t>i{t)
is
/J
1
(24).
Xq G
If
the
plays
its
1
if
(/J,U2)i
demand
rises to
a long time. Thus firm 2
to reach
demand
/j,
which
reaches Uj from
monopoly
in the future
continue until
(/2,Ui), firm
and firm
^22'
Xq G
if
1
as the prospect of being a
/2
Xo >
if
exits
when
monopoly
is
continues as a monopoly.
monopoly strategy henceforth.
First, either firm
When demand
may
it
is
may
exit
between
reaches
exit as the
when
the
and
U2,
/j
down
to
/j;
or
demand goes up
Second, the strong firm exits before the weak firm does
I2.
I2,
First,
monopoly. Second,
demand drops
up to reach U2 before
drifts
happens when the demand
/J,
and Uj ^^^ from the
The
is
first
between
/J
and
^2-
order conditions for the
existence of a solution to these equations with the
desired order implies the existence of a stationary
one in Proposition
(t>i{t)
take cases, and
[uJ,/!], as firm 2 will
Uj, the strong firm (firm 1)
declining. This
two firms' optimization problems of
the
Fifth, if
two occasions.
in
demand
when the demand
if
this equilibrium.
happens
profit.
the prospect of being a
decreases to
it
as a
not exit until either the
exits immediately.
1
/Jj
in the industry for
gloomy. Otherwise, firm 2 exits when the demand reaches
if
We
a monopoly throughout. Third,
the other hand,
knows that firm 2
/j?
["n'l]) a"<^
another subgame perfect equilibrium, then
gloomy and firm 2 becomes a monopoly. Fourth,
Finally,
X^ G ^i U ^2)1 ^1 =
:
a "stationary equilibrium" in the sense that every
is
becomes a monopoly.
/J2'
t
T2(t) a.s.
a bad news for firm 2 as firm
1
>
inf{5
^"""^
te^+,
both stay on with the former making a negative
1
below or reaches
is
=
Then
y.
which the relative positions of the barriers are depicted.
in
[u2,/22]i firm 2 exits
firm 2 and firm
f^ for x <
can thus describe the equilibrium by looking at 0,(0).
when the demand
firm 2 exits
exits
1
<
Theorem
of
We
4>i{Q).
also refer to Figure
Xq G
Moreover,
[u2,^22]-
The equilibrium
is
r(<)
subgame perfect equilibrium, where
a
= —
=T^x{t)l{X.^,,eAr}+r(t)l{x^,„€A,},
<t>2it)
is
understanding that J^
subgame
perfect equilibrium in addition to the
Otherwise, Ai becomes an empty set and A2 becomes {—00,122]- Then one
concludes that the lower bound on firm
starting from an optional time
As a consequence, there
S
is
I's
Ti*i(5)
exists a unique
subgame
perfect equilibrium exit time at any
and the upper bound
subgame
for firm 2's exit
perfect equilibrium.
time
subgame
is
T22(5).
3
THE EXIT GAME
The
22
following example demonstrates
how one
uses
Theorem
3.1
to construct scenarios that
imply either a unique or multiple subgame perfect equilibria.
Example
3.1 Let
be increasing step
assume
We
that
/i
<
l3,j
=
for all i,j and
functions with
Choose
2.
Let
a,2
>
0, 6,2
>
b,\
/*!, /21, /i2.
a,_,,
and 1°
6,j
0,
and
x°,
<
i°2
f'^''
^J -
^'2-
^^^^
and x?i so that
/jj
through the following steps:
,
<
/21
<
lli, /ji
<
^ii- «"<^ '12
"
'21
<
^21
~
'ii-
a,
a*
/<
>
0.
choose the parameters,
1.
>
a,i
can be shown that
r,'(x)
'
^
= e"-^-e--4 >
'
I
and for ^ <
(or a'
>
<
°'
'{">"'
0,
j/z <
(37)
^
'
0,
a,),
^w.)-,(-.)){:°; ;::»:
Thus, for
e
>
(38)
sufficiently small, the following hold,
'?('2*i-'ti+0>r/(-(/t2-/2*i-20),
(39)
- '21 - ^(0) ^ ^(/l2-/2'i)-^(0)
- '21 - 20 - ^(0)
v(-{lh - '2*1)) - ^(0)'
(40)
and
Vil'u
^('l2
since r){l^i-lii)
I21
—
as determined in step
III
Choose
> riilU-^21) > ^(-('12-^21))
e
and
bi2
''22
_
vilh
r/(/-2
622
^21
by (40).
(
<
-
{/jj
-
I'u
+
r/(-(/t2-/2',
hi
and
<
rj{l'2x
611
611 by (39)
and
(38), also noticing
thatl^^-l^ <
I.
that satisfies (39), (40)
Note that bn >
^V (37)
>
/2i)/2-
Then, set
- '/(O)
-20)-r?(0)'
- '21 - '7(0)
- /21 - 2f) - r/(0)
&21 by (37).
Furthermore,
T?(-(/u-'2i))-^(0)
b,j
so that
(41)
(42)
GAME OF AN INCUMBENT VERSUS A POTENTIAL ENTRANT
4
3.
Let
=
I22
+
'12
where
<5'
6
>
so chosen that
is
4.
we choose
Finally,
S follows
and 1°
aij
from (43) since limiio'22 —
d,j
=
a,j/b,j, for i,j
=
/21
+
e
Let
for
all
l^
and u^ =
l'^^
=
-
^
'12-
to satisfy
=
/^
where
'^
r,(-(/*2-/2i))-'7(0)"
621
The existence of such a
23
-lln(l +
x°
cf.,)
1,2.
f-
Then
/iiCuJ,/^)
=
by (41) and h2{ul,l^)
=
by (42). Also,
y G (/t2,'22]'
>
h2{yJ\2)
by (44)- Therefore,
It is
much
1^2
e'"'^'(-&22(r/{-(/;2
=
and
00
l^
=
/12.
-
This can be done since
1
^(0))
+
h2,(i){l\2
1,
>
i?(/2i
hr{i;2,l2i)
will
~ ^n) >
-
l2x)
So, (Ti,T2) in (36) with above u'
we simply
bu
and firm
-
easier to construct the case in which there
example, after completion of step
h,(y,i;^)
/;i))
is
-
^(0)))
and
I' is
>
0,
an equilibrium.
a unique equilibrium.
In the above
set
r;(-(/t2-/^i))-r/(0)-
^(~('i2
~
^21))-
= e^-'H-buivi-ilu -
Then, for any y G
'2*1))
-
^(0))
+
['2i''r2]
bniriil'2^
not exit before firm 2's longest possible exit time T2\.
-
I'u)
-
r?(0)))
>
0,
The uniqueness of
the
equilibrium follows from Theorem 3.1.
Game
4
In this section,
of an Incumbent Versus a Potential Entrant
we
investigate the situation where firm
option to exit, while firm 2
exit.
This
is
thus a
game
is
1 is initially
in the
market and has a single
not in the market at the beginning and has options to enter and then
of an incumbent versus a potential entrant, henceforth abbreviated as
simply the entry game.
Before we proceed formally,
we note the
following. First, in
any subgame perfect equilibrium,
once the two firms are in the industry in the same time, therehence, they must be playing a subgame
perfect equilibrium in the exit
game
enters, then afterwards, firm 2
must follow
discussed in Section
its
3.
Second,
if
firm
1
exits before firm 2
unique optimal single firm entry and exit decisions
GAME OF AN INCUMBENT VERSUS A POTENTIAL ENTRANT
4
characterized in Section
For simplicity, we assume that there exists a unique subgame perfect
2.
By
equilibrium for the exit game.
Given
in Proposition 3.5.
game with a
entry
4)\
time with
firm
1
^ X
:
focus on
??+ H^
>
<l>\{t)
a.s.
t
subgame
strategy before firm
time
t,
if <f>2{t)
=
at
firm
t.
1
We
1
if
and the two observations noted above,
perfect equilibria,
and on firm
firm
In words,
exits,
I's
if
where
entry decision before firm
2's
time
t
1
=
4>2{^) is
t.
Similarly, let
</)2
fi
:
analyzing the
t
is
(f>i{t)
X Jf+
an optional time with ^^i^) >
assume that
4>i
(t>\
and
4>2
in Section 3.
are right-continuous in
and
t
I's
exits.
firm 2 has not entered and firm
if <f>\{t)
the one
is
our attention on firm
restrict
strategy before firm 2 enters, where
at
and only
we can
in
an optional
1
has not exited,
i-i-
3?+ be firm 2's
a.s. for all
has not exited and firm 2 has not entered, firm 2 enters immediately
condition stipulated for
4>2,
K+ be
immediately
will exit
the analysis of Section 3, this unique equilibrium
this hypothesis
exit decision before firm 2 enters
Let
24
z G
if
3?.
If
and only
satisfy the measurability
Denote by $i and $2 the space of aU the possible
and
4>\
respectively.
Define
T{S;(f>\)
= mf{t> 5:0?(O =
O
TiS;<t>'2)
= ini{t>S:4>lit) =
t}
and
for ail
A
e
(p\
<^i
subgame
and
<p2
G $2- By Proposition
perfect equilibrium of the entry
r(5;(^i) solves, for
all
T{S;4>\) and T{S;(f)2) are optional times.
3.1,
game
is
composed of
(</>5,</>2)
6 ^1 x $2 so that
optional time S,
sup E[
e-^('-^)7rii(X,)(f5
/
-
e-^(^-^)/3n l{T<r(S;0|)}
T>S
where
vi
sup
defined in (32), and T(S;<f)2) solves
is
E
[e-^(^-^)(r2(Xr)
-
a22)l{r<T(5;<)}
+
^''^^*^^*'^"^^^2l(^T(5;^J))l{r>T(5;<)}l-^5]
,
T>S
where
V2
is
defined in (33) and
1)21
is
defined in (19) with x and
a
replaced by
7r2i
and 021,
respectively.
The
following proposition reports a stationary
The proof
subgame
of this proposition, being similar to that of
firm 2 sets a critical level u^ such that
it
perfect equilibrium for the entry game.
Theorem
enters the market the
3.1,
first
is
omitted. In this equilibrium
time demand
is
above u^ before
4
GAME OF AN INCUMBENT VERSUS A POTENTIAL ENTRANT
firm
exits
1
otherwise
it
and then plays the unique subgame perfect equilibrium of the
behaves optimally as a monopolist after firm
exits.
1
as to balance the marginal benefit of being a duopoly presently
a future monopoly. Firm
1
firm 2's entry; otherwise,
it
sets a critical level
plays the unique
such that
/*
subgame
it
The
25
game
exit
value Uj
is
thereafter;
determined so
and the marginal benefit of being
exits
demand
if
is
below
perfect equilibrium of the exit
l\
before
game
after
firm 2 enters.
We
will use uji
and U22 to denote the entry barrier
for firm 2 as
a monopoly and as a duopoly,
respectively, as described in (18) of Proposition 2.1. In addition, define
T|i(i)
=
'in{{s
>t:Xs>
u^j}.
Proposition 4.1 Put
where
l\
G
('ii5'i2)i ^2
+
4>\{t)
=
r(i)l{x.,„6^,}
4>lit)
=
T|i(r(0)l{A:.,.,eA.}
^n(^(0)l{X.,„€^.}'
+
(45)
r(Ol{x.,„e^.},
S ("22''^) ^^^ unique solutions to hi{u2J\)
=
(46)
and ^2(^2, m) — ^ ^^^^
hiiuj)
=
^|"(7r„(z) + r-/J„)(e"'("-^)-e-'"("-^))d2 +
h2{u,l)
=
7(- r(T22(^)-ra22)(e"-(^-"-e-''*(^-'))ciz
/3ii
+
t;i(u),
•''22
+
(^e-C--') + ^e-*('--'))
+
(a22
/322))
+
t)2l(/),
and
T{t)
with Ai
— — 00, /J], and A^ —
(
= mf{s >t:XseAi\J
\u\^-\-oo).
Then
{4>i,<f>2) is
Three interesting observations can be made.
existence of a potential entrant
potential entrant
demand
demand
l\
This
to exit.
is
before
it
levels in the future,
Second, the entrant sets
^22-
force the
First, in the equilibrium, l\
incumbent to
its
increases to reach u^.
makes firm
due to the opportunity of
profit in the future.
1
exit earlier than
its
The
1
time.
potential of
if it
>
/Jj.
when
Thus the
there
is
no
This occurs when
becoming a duopolist
incurrence of current losses.
exits higher than its
becoming a monopolist
its
life
less tolerant to the
entry level U2 before firm
Therefore the entrant sacrifices
monopoly
subgame perfect equilibrium.
and hence the incumbent has a shorter economic
decreases to reach
at higher
may
a
A2],
duopoly entry
level
waits long enough for firm
1
current duopoly profits in exchange of the potential
5
GENERALIZATIONS
Finally, there axe
two possible results of the incumbent versus entrant game: "peace" or "war".
By "peace", we mean
market by
as a
5
26
that the entrant enters after the incumbent exits and both firms share the
different time segment. If the entrant enters before the
duopoly
in
incumbent
exits,
then they battle
the market.
Generalizations
In contrast to Dixit (1989), the single firm entry-exit
re-entry once a firm exits.
Our approach can be
Suppose that the cost of entry and re-entry
simplicity,
assume that
tt is
bounded,
when
the optimal expected profits
the current
the firm
is
tt
is
is
the
same equal
demand
to a.
The
does not allow
cost of exit
and continuous. Let v(x;
x,
is
bounded and
r
and the firm
>
market, the problem the firm faces
in the
in Section 2
generalized to count for re-entry, however.
strictly increasing,
of the majket, respectively. Given that
When
problem analyzed
0,
is
I)
in the
is
For
is (3.
and v{x; 0) be
market and out
v{x;I) and v{x;0) are bounded.
to find an optimal exit time that
solves:
v{x; I)
=
sup
E
(47)
Jo
When
outside the market, the firm seeks an optimal entry time that solves:
v{x;0) = sup
Note that (47)
is
E
[e-'"^(t;(XT; /)
-
a)l
(48)
.
equivalent to
[n{E\e-^'^{fiXT)-viXT;0) +
TeT
(i)
i-
Assume
to begin that v{x\ I) and v{x;0) are continuous in x.
Assume
(1990) shows that there exist solutions to (47) and (48).
Theorem
1
of
Huang and
Li
that these solutions are barrier
policies.
Let v{x,l;I) and v{x,u;0) denote the expected future profits,
spectively,
Letting i
under a barrier policy with the
=
exit
and entry
barriers,
/
v{x,l;I)
=
f{x)-9{xj)f{l) + e{x,l){v(l,u;0)-(i)
v{x,u;0)
=
e{x,u){v{u,i,I)-a)
u in (49) and i
v{l,u;0)
=
=
/
in (50),
we can
for
i
<
and outside the market
in
and
u.
for
————
Then,
x >
/,
and
,
u.
(49)
(50)
solve linear equations (49)
l-O{l,u)0{u,l)
re-
.
and (50) to obtain:
(52)
CONCLUDING REMARKS
6
27
Thus, the expected future profits are given by (49) and (50) under a barrier policy with any pair
The
of parameters (l,u).
search for the values of the optimal barriers
that maximize v{x,u\0) and v(xj;l).
—ou
where C\ and C2 are
=
It
=
and
——— =
C2/i2(/, u),
ol
-i(a +
and 7 are given by
(53)
Jl
/3)+ r(7r(0)
7
+
7-/3)(e-'"("-^)-e<''("-^))(iz,
(54)
Jl
(14)
-
(16). Therefore, the
/ii(r,u*)
as
and u
-{a + fi)- r(n{z)-ra)ie^'^'-'^-e-^'^'-'^)dz,
=
h2{l,u)
It is
/
can be shown that
Ci/ii(/,u),
7
a,, a,,
reduced to finding
strictly positive terms,
hi{l,u)
and
is
=
0,
and
optimal barriers must solve:
/i2(r,ti*)
=
0.
(55)
then straightforward to verify that v{x;I) and v{x;0) are indeed continuous functions of x
we have assumed
earlier.
Finally,
we can use the
principle of
dynamic programming
to verify
that there do not exist non-barrier policies that strictly dominate the optimal barrier policy. This
justifies
our searching for an optimal barrier policy.
Finally, differentiating hi{l,u) with respect to u gives
^Ml^ = (e-(''-') - e-<'*("-'))(m - x(u)).
ou
So, hi{l,u)
and
h{l,-)
>
is
for
/
<
u
<
increasing for
7r~^(ra) since x(-)
/
<
u
< x~^(ra). Thus, by
u*
Similarly,
is strictly
> ;r~Vra) >
increzising
and
/ii(/,/)
=
')~^{ot
+
/3)
>
(55)
n~'^iO).
we can show that
Given the above solution to a single firm's problem, we can then apply the technique developed
in Section 3 to analyze a duopolistic entry-exit
6
game,
albeit
more complicated.
Concluding remarks
We have
analyzed optimal entry-exit decisions for a single firm and
both in continuous time with Brownian motion uncertainty.
for firms
Under very general hypotheses, we
have demonstrated the existence and uniqueness of an optimal policy
policy
is
behaving strategically,
for
a single firm. The optimal
a barrier policy and the entry and exit barriers are solutions to algebraic equations.
7
REFERENCES
28
In a duopolistic context,
subgame
perfect equilibria.
we have worked
One
equilibrium
directly in continuous time
always the one
is
a monopoly throughout and thus the weak firm
may
exist,
is
in
which the strong firm behaves
always a duopoly
in its lifetime.
In these other equilibria, both the strong firm
however.
and shown ways to look
for
like
Other equilibria
and the weak firm may
exit
even when the demand has been increasing on the average.
The method used
decisions. Indeed,
many
our analysis can potentially be useful to
in
any situation
in
other optimal economic
which dichotomous choices are made over time under uncertainty
can be formulated as an optimal timing problem and our technique
may
apply.
References
7
1.
Chung, K.L., Lectures from Markov Processes
to
Brownian Motion, Springer- Verlag, New
York, 1982.
2.
3.
C, and P. Meyer, Probabilities and Potential A: General Theory of Process,
North-Holland Publishing Company, New York, 1978.
Dellacherie,
Dixit,
"Entry and Exit Decisions under Uncertainty," Journal of Political Economy
A.,
97:620-638, 1989.
4.
Fudenberg, D. and
J. Tirole,
"A Theory
of Exit in Duopoly," Econometrica, 56:
943-960,
1986.
RAND
5.
Ghemawat,
6.
Harrison, J. M., Brownian Motion and Buffered Flow, John Wiley
7.
P.,
and B. NalebufF, "Exit,"
Journal of Economics, 16: 184-194, 1985.
k
Sons,
New
York, 1985.
Huang, C. and L. Li, "Continuous Time Stopping Games with Monotone Reward Structures,"
Mathematics of Operations Research, 15: 496-507, 1990.
8.
Merton, R.C., Continuous Time Finance, Basil Blackwell, Cambridge, Massachusetts, 1990.
9.
Perry, M. and P. Reny, "Non- Cooperative Bargaining with (Virtually)
published manuscript, University of Western Ontario, 1990.
No Procedure,"
"Reexamination of the Perfectness Concept of Equilibrium Point
Games," /nternat/onai Journal of Game Theory, 4:25-55, 1975.
un-
Extensive
10.
Selten, R.,
11.
Simon, L., "Basic Timing Games," unpublished manuscript, University of Call fomi a at Berkeley, May, 1987.
12.
Simon,
L.
and M. Stinchcombe, "Extensive Form Games
in
in
Continuous Time: Pure Strate(To appear in Economet-
gies," discussion paper, University of California at Berkeley, 1988.
rica.)
13.
Wilson, R.,
"Two
Surveys: Auctions and Entry Deterrence," Technical Report No.
ford Institute for Theoretical Economics, 1990.
3,
Stan-
Q:It
ut
Vi
3^
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