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i
TACIT COLLUSION IN A DYNAMIC DUOPOLY WITH
INDIVISIBLE PRODUCTION AND CUMULATIVE
CAPACITY CONSTRAINTS
Glenn Loury
No.
55*7
May 1990
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
TACIT COLLUSION IN A DYNAMIC DUOPOLY WITH
INDIVISIBLE PRODUCTION AND CUMULATIVE
CAPACITY CONSTRAINTS
Glenn Loury
No.
55;«7
May 1990
A Dynamic Duopoly with Indivisible
Production and Cumulative Capacity Constraints
Tacit Collusion in
by
Glenn C. Loury*
J. F.
of Government
Harvard University**
Kennedy School
June 1990
Abstract
This paper studies a dynamic, quantity setting duopoly game characterized as follows:
Each firm produces an indivisible output over a potentially infinite horizon, facing the constraint that its cumulative production cannot exceed an initially given bound. The environment is otherwise stationary; the remaining productive capacities of the firms at any moment are common knowledge; the firms choose production plans contingent on these capacities which are mutual best responses in every contingency. The resulting Markov Perfect
Equilibria are analyzed using a two—dimensional backward induction, and compared with
the equilibria which emerge when precommitment to time paths of output is possible. It is
shown that the ability to precommit can be disadvantageous; that collusion in Markov
Equilibrium is facilitated by the symmetrical placement of the firms; and that having
greater capacity confers basic strategic advantage on a firm by enabling it to credibly
threaten future production. The model solves an open problem in the theory of exhaustible
resource economics by imposing subgame perfection in a resource oligopoly with independent stocks. It also formalizes the intuition that, when indivisibilities are important, tacit
coordination of plans so as to avoid destructive competition is facilitated by establishing a
convention of "taking turns" — that is, a self—enforcing norm of mutual, alternate
forebearance.
*
The author thanks
helpful discussions;
Ariel Halperin, Eric Maskin, Robert Rosenthal and John Vickers for
and the Bradley Foundation, the Olin Foundation, and the Center for
Energy Policy Research, MIT,
**
for research support.
This paper was written while the author was
Department of Economics, MIT.
in residence as a Visiting Scholar at the
I.
Introduction
This paper studies tacit collusion in a dynamic model of duopoly.
indivisibility in production at
any moment., and an intertemporal constraint on the
cumulative amount of output which
is
to
accomodate
its
essential, limited
production
We
is
whether to produce
opponent's action.
The
possible over the horizon of the game.
On
explores the strategic implications of these two conditions:
indivisibility, a firm deciding
Firms face an
On
at a point in
analysis
the one hand, due to the
time has limited flexibility
the other hand, due (say) to the presence of an
and non-reproducible factor of production, each firm's capacity
progressively diminished by decisions
it
for future
has taken to produce the past.
study the Markov Perfect Equilibria of the resulting dynamic game when each
firm's strategy specifies a rate of production at
acities for future
any moment
production remaining as of that moment.
as a function of the joint cap-
An
equilibrium
is
a pair of such
state—contingent rules which are optimal for each firm, given the other's strategy, from
every conceivable state. In a sense our
game
is
a "closed—loop", "multiple stock" version of
the "oil"igopoly problem encountered in the exhaustible resources literature.
our knowledge, the
first
with multiple stocks.
the context of
This
is,
to
treatment of perfect equilibrium in a depletable resource model
Previous work has succeeded in imposing subgame perfection only in
"common
The assumed
1
pool" situations, in which there
is
a single limiting resource. 2
indivisibility in production further distinguishes the present analysis
from existing work in resource economics. Our firms decide whether or not to produce at
each moment, but are unable to vary the non-zero rate at which output
regulate an "on-off control variable, facing a constraint on the total
'
variable can be "on" over the horizon of the game.
'See
Loury (1986), and the literature cited
is
supplied.
amount
They
of time this
In effect, their output decisions are
there.
An early contribution was the study of competition in the exploitation of a common fishery
by Levhari and Mirman (1980). Related analyzes include Reinganum and Stokey (1985V
and the recent, more general, work of Sundaram (1989) and Dutta and Sundaram (1989).
2
qualitative choices about participation.
Dilemma
in
The game
which the capacity to play "tough"
Dilemma
—
analogous to a dynamic Prisoner's
in the future diminishes
We focus
frequency of having played "tough" in the past. 3
study of the Prisoner's
is
on the
the extent to which, in a
with an increased
classic question in the
non—cooperation
equilibrium,
the players can avoid dissipative competition through mutual forebearance.
Characterization of closed—loop equilibrium in
several state variables
is
non—zero sum
a very formidable, largely open, problem.
rare outside of the linear—quadratic context, though a
number
differential
games with
Explicit solutions are
of economically interesting
examples have been identified with a special structure permitting profitable study. 4 In
this
paper we
make
progress by simplifying the problem in two ways:
As noted, we
the set of actions from which the players can choose at any instant; and,
actions, once chosen,
must remain
fixed for
These assumptions permit explicit solutions
Our
ability to solve the
model
some
is
be obtained.
exploited to pose a
number
of interesting
questions for which general answers are unavailable in the existing literature on
duopoly:
Are firms more, or
less,
they can, and do, commit at the
tion in
which such commitment
tween the
the other?
size
is
and distribution of
How
dynamic
able to avoid mutually destructive competition
initial
when
date to specific future actions, relative to a situa-
not possible?
What
is
the relationship in equilibrium be-
payoffs, on the one hand,
will the distribution of
that
(possibly quite short) interval of time.
for perfect equilibria to
explicitly
we assume
limit
and capacities
to produce,
remaining capacity evolve over time?
How
on
are
A closely related model is the continuous time Prisoner's Dilemma game of Rosenthal and
Spady (1989). There Markov Perfect Equilibria in strategies defined on a two dimensional
3
state space of cumulative profits are studied.
A good discussion of the differential game approach to dynamic oligopoly problems is in
Fudenberg and Tirole (1986). Two linear—quadratic analyses are Fershtman and Kamien
(1987), who study a deterministic, one-state-variable model of duopolistic pricing, and
Hansen, Epple and Roberds (1985), who solve a stochastic two-state-variable model of
resource depletion, though without imposing sugbgame perfection. Other models with a
structure permitting profitable study of subgame perfect equilibrium are the memoryless
patent race of Reinganum (1981), the racing model of Harris and Vickers (1985), the
capacity investment model of Spence (1979), extended in Fudenberg and Tirole (1983), and
the duopoly pricing model with menu costs of Halperin (1990).
4
these considerations affected by the rate of discount and the elasticity of market
Among
the results obtained are: (1) state—contingent strategies imply equilibrium
payoffs at least as great,
(2)
symmetry
mine
and often greater, than those possible with open—loop
the interest rate, and to the elasticity of
is
demand;
that,
at the initial date to future actions
when such commitment
to their
mutual advantage
if
is
inversely related to total capacity, to
(4) the ability to credibly threaten future
more than
Result (1) implies that the ability to
can be disadvantageous! Result (2) suggests
not possible, firms are better able to maintain high prices
neither of
vides theoretical support for the
them enjoys a
common
large
market share! Result
(4) pro-
intuition that a firm with a substantial share of
Arabia in OPEC!) can disproportionately influence the course of play.
total capacity (Saudi
To
is
endowments consistent
a basic strategic advantage which allows the appropriation of a
proportionate share of the industy's total payoff.
commit
the extent of
strategies are employed; (3)
with attaining tacit collusion in perfect equilibrium
production
strategies;
of capacities facilitates collusion in closed—loop equilibrium, but can under-
when open-loop
it
demand?
obtain these results
we employ the two
simplifications mentioned to transform a
complete information, continuous time game of dynamic duopoly into a more tractable
crete time analogue.
Markov
strategies
is
This conversion
is
defined in section
described in section
III.
A
II.
dis-
Perfect equilibrium in
two-dimensional backward induction
is
em-
ployed in section IV to solve for explicit equilibria, which are compared to the open-loop
Nash
equilibria of the
ior in the discrete
same game. In section
model
V we
consider the limit of equilibrium behav-
as period length shrinks, raising
issues discussed briefly in that
some technical and conceptual
and the concluding section of the paper. 5
Because of the indivisibility in production, coordination of play is closely linked with the
idea of "precedence" in this model. Our discrete time equilibria exploit this, and the related notion of "alternation" heavily. In continuous time, when players are free to change
their actions at any instant, it is nolonger clear how to define who produced "last period".
Thus, it is not clear how to interpret the limit, as period length goes to zero, of the discrete
time equilibria analyzed in section IV. See the treatment of extensive form games in continuous time by Simon and Stinchcombe (1989) for a fuller discussion of these issues.
5
H.
The Model
We
A
consider the following model of competition between two independent firms,
and B, facing an indivisibility in produc ion, with given limitations on their cumulative
capacities to produce.
Time
is
continuous over an infinite horizon,
A
flow rates of production of firms
that
2
a b
that
q =(q ,q )e{0,l}
;
Q
e{0,l,2}, Vr>0.
feasible
to
A
either a firm
b
is
2
is
of)
a
inactive, or
and q
it is
b
is
At date r the
respectively, are such
producing at a unitary
the industry's production rate at
r;
obviously
a number R. (j=A,B)
denoting
production for that firm over the course of the game;
Capacity
the vector of initial capacities/stocks.
active; a firm exits the
is
is
firm's initial capacity (or stock)
cumulative (resource cost
Re(R a ,R r )gIR
a firm
a
Q =q +q
any moment.
rate at
is,
and B, denoted by q
re[0,oo).
game when
its
falls at
a unitary rate
capacity has fallen to zero. There
is
if
no way
add to stocks during the play of the game. The firms' stocks are common knowledge.
The demand
ically.
There
is
side of the
market
is
passively modelled; buyers do not behave strateg-
an inverse demand function, P(-), which
only on the total rate of flow of output of the two firms.
P(l)=l>e=P(2)>0. (For completeness,
dustry flow revenue
inelastic.
If e>l/2,
falls as
let
is
time invariant and dependent
Monetary units
P(0) be any finite number.)
the rate of production increases, so
then we will refer to
demand
as being
we say
elastic.
are such that
If
that
There
is
e<l/2 then
demand
common, uniform
their respective discounted flows of
rate r>0.
The
is
no other cost of
production beyond the opportunity cost implied by the depletion of capacity.
ceipts are discounted at the
in-
Future re-
firms' payoffs are simply
revenue received over the infinite horizon as a result of
their joint paths of supply.
We
study the equibbria of the following discrete time analogue of the continuous
time set—up introduced above: Assume that the firms cannot change their flow rate of
output except at particular, exogenously given and uniformly spaced moments in time
that
is,
except at the beginning of a "period."
tain their production at the
new
—
After any change they are bound to main-
rate until the next such
moment. These moments when
adjustments are permitted are
—
length of a "period"
discrete time
We
ize.
common knowledge
the interval between dates
dynamic game
is
which emerges
Suppose then that T(h)E{0,
Associate with each
be taken.
t",
Monetary flows over
tions,
and may be regarded
Suppose that
at dates
V» and V R
where:
tel
intervals
as equivalent
length h.
],
the
Vc =
and
lump sum equivalent
initial state, s
and b =max{beI|b-h<R
is
actions can
will often refer to
is
t
"date
[h-t, h-(t+l)).
worth
6(h)
.
at the beginning of
Then
dollars at
M-P{Q
S
l
tel
each period.
their respective payoffs are
l
'
)],
of the flow of one dollar over an interval of
Unique payoffs are implied by every path of firms' actions via
h,
we
evolve over the course of the game.
that
A
„-rh
Given periods of length
Define the
We
meaning the interval
lump sums paid
tGl
for /?(h)=[
allowed.
[r,r+h), reT(li), are constant given our assump-
t
!
t",
when new
the set of dates
is
the firms take the actions q
f
Va
A =../3Sto.P(Q)];
t
(2.1)
is
the
whose equilibria we character-
h,
a dollar received on date
;
h>0 be
a change of action
reT(h) the integer t=r/h.
Define the discount factor 8(h)se
Let
firms.
in these equilibria in the limit, as h-»0.
h, 2h, 3h,...}
meaning the instant r=h-t, and to "period
date zero.
when
defined for each value of
also study the behavior
two
to the
inessential.)
The
(h,R)=(a
R }. (We
consider
now how
(2.1).
the firms' productive capacities
Let I be the set of positive integers, and IeIu{0}.
(h,RAb
(h,R
R )),
as follows:
suppress explicit dependence of
initial state is a pair of integers
a
s
=max{aeI|a-h<R A },
on h and
R
when
denoting the "number of plays"
which each firm can make over the course of the game, where a "play" involves supplying
output at the constant unitary rate for a period of duration
by neglecting the
fact that the initial stocks
may
denote the state space. In this way, for every
h,
and where we "round—off"
not be evenly divisible by h.
initial
vector of capacities
ReR
2
Let
SeI«I
and every
.
h>0, we assign an
—
initial state
eS, representing this association by:
s
s.
is
the state of the
actions during period
t,
then the state
is
apparent that
IQ.
Markov
if
al
game
at date
date t+1
and
t,
satisfies:
if
is
q.
s.
,
-R.
It
the vector of firms'
—
s,
q,
Perfect Equilibrium
obvious that, for any given history of play prior to date
It is
=
,
(h,R)~h
s
action paths subsequent to date
t,
t,
the set of feasible
and the firms' conditional valuations of these paths,
depend on that history only through the state of the game
"state of the game", as defined here,
is a
at
t,
s
=(a,,b,
"payoff relevant state" in the sense,
Maskin and Tirole (1989): Conditional valuations are independent of the
stationarity
assumed
big enough;
all
Feasibility
in (2.1).
histories prior to
That
is,
e.g.,
of
the
history, given the
depends only on whether remaining stocks are
which reach the state
t
).
s,
imply the same
set of feasible
subsequent actions.
We
employ the concept
game, with h
from S into
MPE.
Markov Perfect Equilibrium (MPE)
fixed, in the analysis to follow.
[0,1],
reporting for each
during a period when the state
tegy
of
is
seS.
This restriction involves
Cim
In general then, strategies are functions
the probability of participation in the market
However, we
shall restrict attention to
less loss of generality
since pure strategy equilibria always exist in this model,
cient here.
If,
at
some node
s,
A
getting "stuck" there for a period
with pure strategies.
—
may
appear to be the case,
and mixed strategies are
ineffi-
risk
a source of lost industry profits which does not exist
Because we are primarily concerned with understanding when a rev-
Given a pair of pure Markov strategies on
during the play of the
point
than
pure stra-
and B play "0" with positive probability, they
enue maximizing outcome occurs in equilibrium, we
tice
for the discrete
restrict attention to
S,
game may be envisioned
pure strategies.
the resulting motion through the lat-
as exemplified
by Figure
1.
At each
seS four continuations are possible, corresponding to the four elements qe{0,l}
These outcomes
result in
motion of the state from
s
to
some successor node, s'=s—q.
2
.
If,
at
node
s,
both firms stay out of the market
remains where
it
began.
With Markov
(i.e.,
q=(0,0)), then s'=s, and the state
amounts
strategies this
to remaining at
s
forever,
an obvious impossibility in an equilibrium unless s=(0,0). The three other continuations
correspond to one or both firms being in the market, and result in motion of the state
either "due South",
"due West", or "Southwest", as depicted by the arrows in the Figure.
Eventually the state reaches one or both of the axes, representing the exit of one or both
firms.
At
this point, the firm
with remaining capacity produces one unit in each period
until the state reaches the origin.
/ /
/
I
,
Figure
<
1
—
<
s
i
1
1
i
i
/
,
'
1
1
.
I
I
^
H
->
I
1
1
Thus, with each profile of pure Markov strategies and each
initial state
may
be
associated a unique path through the lattice and, via (2.1), unique payoffs for the firms.
Given the period length, these payoffs are a function of the
and the parameters
e
and
6.
Every path terminates
initial
in a finite
node, the strategy pair,
number
of steps.
Current
period payoffs (divided by /3(h)) at each node depend on the actions taken there as follows:
B
1
1
Table
1
1
We now
E
A E{<r
precede formally, introducing some notation and definitions. Define:
A :S-*{0,l}|CT A (0,b)=0,VbeI},
E
A B
B e{<7
(a,0)=0,VaeI}, and £eE *E
b :S^{0,1} aB
|
A
pure strategy profile (or simply, a profile)
profile
generates a path {s
creE
s
(3.1)
Q
s
=
s,
=
s
t
We
denote this sequence
s'e{s,(s,i7)},
in (2.1)
and that
with
V R (cr)(s),
cr(s.(s,<7)),
Definition 3.1:
Definition 3.2:
path
A
t-l
is
s,
we may
since
ular initial state.
it
We
say that point
are not credible.
s'
is
under a
s
if
seS, every
a successor of
se{s,(s', cr)}.
V
s
under a
Identifying
.
(<r)(s)
if
q,
and
way, as the respective sum in
in the obvious
<reX,
ctgS
a
is
Nash Equilibrium from sGS
A
V A (flr)(s)
>
V A (<rA ,&B )(s)
VB (a)(s)
>
V B (a A ,c B )(s) VatfZ 8
o*gS
for every
is
a
Va eS
A
,
if:
and
.
Markov Perfect Equilibrium
if it is
a
Nash
SGS.
all
MPE's
in pure strategies.
Nash Equilibrium
is
weaker
requires strategies to be best responses to each other only from a partic-
In a
NE
neither firm gains by deviating from its strategy at any node
it
anticipates the response to the deviation to be as
reported by the strategy of the other firm
beyond
state
defined inductively as follows:
define the value function for each firm,
along the subsequent path, given that
strained,
From any
{s,(s,o-)}.
Let S cS be the set of
MPE
cre£.
tGl
a predecessor of
strategy profile
Equilibrium from
than
~ ^ s t-l)'
strategy profile
A
an element
and
seS given profile
at state
(2.1) along the
,s,,...,s.,... },
{s,(s,<7)}.
s'
is
feasibility,
MPE
by the
Yet these responses are not themselves con-
definition.
Therefore,
NE may
depend on threats which
requires that promised responses to deviations
with subsequent equilibrium play.
must be consistent
We now
we
In the sequel
space.
a
formalize the idea that a profile
is
MPE
an
on some subset of the state
construct equilibna by finding particular strategies that are best
responses to each other from initial states in a given region of
extending the resulting
S,
We say
profile to the entire state space so as to constitute a full equilibrium.
W
states
is
closed to deviations from
W, and
stay in
by a\a- the
Given a
closed to deviations
VseW, Va 6E
Let S|
cr'eE
by replacing
A
A
,
Vcr
B
and a' sa on
(a)
W
t
aeE
(b)
g
is
is
a
a
NE
from
implies
W
is
a-,
{0,1}
,
a
W. Denote
j=A,B.
we say
that
W
cr'
eSi
,
v
.
set in question.
closed to deviations from
It is
a.
W
MPE on
a;
B
w
then
)}cW.
if:
and
VseW.
s,
MPE
on
W.
Obviously,
aeE,
if
Restricted equilibria on a given set
thought of as equivalence classes of strategy
if
generated by
2
-»
{s (s,a\a
t
restricted
denote the class of restricted
W
and
A )}cW,
closed to deviations from
is
a with
and a function a:W
{s (s,a\a
:
strategy profile
W cS
agreement on the
seW
the following condition holds:
if
B eE
strategy under
j's
WcS
set of states
from a
A
Definition 3.4:
paths from
if all
one firm's deviation frcm such a path also yields a point in
profile obtained
Definition 3.3:
is
if
a
a profile
that a set of
profiles, in
which membership
also obvious that
Theorem
if
aeE
then
w
and only
if
MPE
establishes that every restricted
1
can be extended to the entire state space in a manner consistent with
thus be
defined by
is
aeE.
may
,
Markov
Perfection.
Consider the following simple example which illustrates the definitions. For nel
define
is
any
S
set
contains.)
n{s=(a,b)eS| a+b<n}.
Obviously,
S
containing the origin, and containing
Now, consider the
strategy profile
whenever he has a positive stock.
A
is
all
closed to deviations from any
points "southwest" of any point
o=(a.,(7
waits until
creE
B
R)
defined as follows:
B
it
plays
has exhausted before playing any-
(as
10
Tl
thing.
n'=Max{nGl \6
Let
Then,
>e}.
cannot gain from deviating; and
A
if
—
we have
n'>l,
e>6
will deviate only if
*
ae£,
that
kn
n
.
But
for
it is
2<n<n'+l: B
easy to see that
*
a£E
.
For max{a,b} large enough, A
To
establish that a profile
will deviate.
a ccnstitutes an
seS, neither firm can increase his payoff
the path
{s,(s,<7)},
given that
a
is
MPE
it
suffices to
show
that, for every
by a single deviation from a at some point along
followed by both firms after the deviation.
That
is,
only "one—shot" deviations need be considered. This follows from the obvious fact that,
a sequence of deviations
which
is
desirable,
raises the firm's payoff.
it
must contain a "one—shot" deviation
Moreover, since no firm
will deviate at
alone produces, only two forms of deviation are relevant:
prefers to stay out of the market, or
o-(s)e{(l,0),(0,l)}
Either
at
if
some node
a node where he
ct(s)=(1,1)
and someone
and the firm not producing prefers
to be active.
At the node
determine
if
s,
A, and given the putative equilibrium profile
for the firm
a profitable deviation
is
possible
we need
compare two quantities:
to
a, to
(i)
Pe+SVj. (a)(s'), the current revenue flow when q=(l,l) plus the discounted continuation
value of moving from
s
to s'=s—(1,1); and
(ii)
6V Ja)(s'
'),
the discounted continuation
value occasioned by the actions q=(0,l) which lead to the node
<r(s)=(l,0),
or
if (ii)
A
a at
s.
If
and a(s)=(l,l), then
A
deviates.
never deviates from
exceeds
(i)
erize the desirability of deviations for firm
If
a profile
a provides no incentive
actions at a given node, assuming
Vi(u)(s) and V-
R
Definition 3.5:
must
(cr)(s)
Given
Bellman Inequalities
it is
at
s
if
(i)
exceeds
(ii)
and
If
<r(s)=(0,l),
The analogous conditions charact-
B.
for either firm to deviate
from
its specified
followed thereafter, then the firms' value functions
satisfy standard
strictly positive
quantity
s"=s—(0,1).
s?S,
dynamic programming conditions. Thus:
we say
cr(s)^(0,0), and:
that a strategy profile
aeS
satisfies the
11
(a)
(3.2)
o(s)=(l,l) *
and
Then
1:
cr£S|
It is
Markov
>
^(^0,1)),
V B (a)(s) = 0e+«VB (a)(s-{l,l))
>
SV
a(s)=(0,l)=> /36+iV
(c)
o(b)=(1,0)=» /3c+*V
(a)(s-(l
A
B
(3e+6V
(a)(s-(l
>
>
B
(a)(s-{l,0)).
A (a)(s-(0,l)) = V A (a)(s).
l))<«V (ff)(s-{l 0)) = V (a)(s).
B
l))<«V
B
>
the foregoing definitions and discussion with the following useful result:
WcS
let
(a)
W
closed to deviations from
(b)
<7
(c)
a
Given
W
A (a)(s-(l,l))
(b)
We summarize
Lemma
V A (a)(s) =
If
.
is
aeS
the profile
A (a,0)=cr B (0,b)=l,
satisfies
a
W:
a;
any non—zero points
and
(c)
on
all
of
S,
then
two—dimensional backward
(a,0)
and (0,b)e W;
V. and V R
at the nodes
ctgS
.
induction, to characterize
Perfect Equilibria in pure strategies for this game.
For arbitrary values of
on
the Bellman Inequalities at every strictly positive se W;
satisfies (b)
possible, using a
at
satisfy the following conditions
This
may
all
be seen as follows:
s—(1,1), s—(1,0) and s—(0,1), there
is
always some pair of actions q=cr(s) which permit the Bellman Inequalities to be satisfied
at
s.
That
one of them
over,
is,
for
fails,
strictly positive
s
in
and ceE,
which case one of the inequalities in (3.2)(b) or
V A (o-)(n,0)=V B (a)(0,n)=/?(l-^n )/(l-^)
fore, starting at
either both inequalities in (3.2)(a) hold or
the point
(1,1)
for all
at
given the values already assigned at to
s,
Bellman Inequalities are
s,
satisfied at each step.
values for the functions V.(a)(-) and
nodes
s
—(1,1),
MPE on
W
a
MPE
at
It is
V R (c-)(-)
s—(1,0) and s—(0,1). Employing
a subset
satisfying (b) of
Lemma
1.
MoreThere-
and working systematically "backward", up and to the
right in the lattice, one constructs a pure strategy
a
a
(3.2)(c) holds.
by successively assigning values
s'<s, in such a
way
to
that the
possible to do this so that at each node
have already been determined
this induction,
we prove
at the
that a restricted
of the state space can always be extended to the entire lattice in such
12
a
way
that the resulting profile constitutes a
Theorem
Proof:
we
and
ctg£|
W)
equilibrium on
there exists
a*eS
S.
such that:
o*(s)=a(s), VseW.
See the apppendix.
Theorem
turn
WcS
Given
1:
full
now
1 is
an important tool which
to a substantive analysis of
solve the
model
MFE
shall
We
be used repeatedly in the sequel.
strategy profiles.
The argument
explicitly to find sta.te—contingent plans of action
is
constructive;
which are mutually
consistent with revenue maximization by both firms from every conceivable state.
We
begin by defining tacit collusion and describing open—loop equilibrium in the model.
IV.
On
the Possibility of Collusion in
Markov
Perfect Equilibria
A. Tacit Collusion
In this simple setting, with only
*r o possible actions,
t
In order to maximize the
takes a very primitive form.
arrange to avoid supplying at the same time
It is
must
of their payoffs the firms
that reduces discounted industry revenues.
obvious that the industry revenue maximizing output at any strictly positive node
s=(a,b) depends only on the total (a+b).
behavior at seS.
the
if
sum
the notion of tacit collusion
MPE
under
a,
Regarding
ueS
achieves collusion at s
M(a),
is
MPE
With
a from the
demand
,
we
call it
initial
it is
node
a achieves collusion.
a collusive path.
s
iff
is
"large", then
we say
If
that
a path satisfies
Industry revenues are maxim-
a generates a collusive path from
s
.
obvious that Q*(s)h1, s^(0,0), since supplying two units
instead of one lowers both current and future revenue.
and a+b
be this optimal collusive
cr»(s)+(7o(s)=Q*(s). The region of collusion
if
the set of nodes at which
inelastic
Q*(s)e{0,l,2}
tacit collusion (or coordination, or cooperation),
{s,(s,a)}cM(a), seS and ce£
ized in the
Let
With
elastic
demand,
if
s=(a,b)
Q*(s)=2: Because of discounting, the opportunity cost of sup-
plying another unit today vanishes as total capacity grows large, while the marginal rev-
13
enue of two instead of one unit of output
is
/?(2e— 1)>0.
Q*(a,b)=l, a+b<n; and Q*(a,b)=2, a-i-b>n. The
n"
(4.1)
That
n
is,
is
= Max{neT| ^j
>
2e+f(^y-
)}
So
critical
if
n
e>l/2 then,
is
= Max{neT|
for
some n>l,
given as follows:
n
l
6
>
2e-l}.
the largest total stock at which the opportunity cost of an incremental unit
of production, given that one unit
is
to be supplied in each subsequent period,
is
no
less
than the marginal revenue of raising output from one to two units. With inelastic demand,
collusion in equilibrium requires the avoidance of simultaneous production everywhere.
With
elastic
demand, such coordination
B. Collusion in
is
only relevant
when the
total stock
Markov
consider the latter briefly.
is
so
committed, then
ize
the
sum
(a,b)eS
2:
it
may
There
we mean
that firms can bind themselves
If
one firm believes
has a corresponding path which maximizes
in these paths.
Such an equilibrium achieves
avoids simultaneous production
of the firms' payoffs.
tacit collusion
Lemma
if it
play
same game, we
its payoff.
to be compatible with the available capacities of the firms,
Nash Equilibria
the node (a,b)
By "open—loop"
time paths of actions which are known to both.
With paths constrained
sider the
n.
Perfect Equilibria with
the outcomes achieved in the open-loop (precommitment) equilibria of the
the other
than
Open Loop Equilibrium
In order to contrast the possibilities for collusion in
at the initial date to
is less
The following
when
to
do so
is
we
tacit collusion
necessary to
result characterizes the region in
con-
from
maximS
where
be achieved using open-loop strategies.
exists
an open—loop Nash Equilibrium from the
which avoids simultaneous production everywhere along
strictly positive
its
path
if
node
and only
if:
14
Proof:
from
+
jMax{a,b}
(4.2)
In an
its last
in these
Min{a,b}
< n' e
open—loop Nash Equilibrium neither firm wishes
date of positive supply to
two dates
be
for either firm
its first
such a
0,
librium without simultaneous production
is
n
Max{nel|
6 >_e}.
to shifts a unit of output
date of zero supply. Letting the difference
shift
pays
impossible
e>6
if
e>6
if
.
for
So an open—loop equi-
some
firm, along every
path where one or the other, but not both of them, supplies on successive dates.
Consider
the problem of finding a sequence of a
of:
difference in order
between the
0(a,b).
between the
last "0"
Thinking of
and the
"i's"
last "1"
first
"1";
and b
and the
is
first "0";
and
Denote
n.inimal.
such that the larger
"O's"
(2) the difference in order
this
minimal difference by
A
sequence as the tLne path of supply for firm
this
in
equilibrium from initial node (a,b) which avoids simultaneous production,
neither firm will deviate
0(a,b)
is
realized
0(a,b)<n'.
if
by placing the
The reader can
lesser in
number
verify that the
two
of the
an open-loop
we
see that
minimal difference
digits in the
middle of the
sequence and surrounding them as evenly as possible on either side by the other
Assuming that a>b, and that a
is
even, this
is
illustrated in Figure
1,
l,...l, 0, C,
a/2
digit.
2:
>|
|<
1,
(1) the
0,...0, 1,
1,
1....1
a/2
b
Figure 2
Examination of
0(a,b)
this
= Min{nel|n
Figure shows that this disposition cannot be improved upon, and that
>
^Max(a,b)+Min(a,b)}.
With open—loop
d
strategies each firm takes the other's
path as given, and so anti-
cipates no consequence of an increase in cuvrent supply besides a reduction of its
acity for future production.
own
So the "shadow price" on a marginal unit of capacity
present value of revenue from the last unit sold.
There
exist
is
cap-
the
endowments from which each
15
firms' initial
shadow
price on marginal capacity
output, along every collusive path.
whatever the elasticity of demand, total
C.
its
e,
for
marginal revenue at zero
which
collusion in
open—loop equilibrium. (Note: 2n'<n.)
size
and
symmetry
relative
maximal industry revenues
show that something
in
Markov
Our method
Perfect Equilibrium.
effective at achieving coordination because,
game contingent upon
an open—loop equilibrium.
is
constructive;
is
we
Intuitively, closed-loop strategies are
by making the entire future course of the
current actions, they can impose on the firms a cost of increased out-
strate formally in Proposition
1
that
when
implying a vanishingly small shadow price
symmetric and the periods
key concept in
and Markov
We
total production capacity is arbitrarily large,
for
the marginal unit,
if
capacities are relatively
in a non-cooperative, perfect equilibrium.
this construction is the notion of
strategies, colluding to
"turn—taking". With {0,1}
maintain a high price amounts to specifying
regions of the state space within which one firm has precedence, and the other
to forebear.
demon-
short, the firms can always coordinate their actions tacitly so as
maximize discounted industry revenue
actions
true regarding
shall exhibit a
put far greater than the foregone discounted revenue from the last unit sold.
A
So,
of capacities can
close to the opposite of the foregoing
particular equilibrium profile to establish this point.
to
both
(4.2) fails,
Precedence and Alternation with Markov Strategies
We will
more
than
less
From such endowments,
firms produce on the initial date in every
militate against achieving
is
Along a collusive path, one firm produces
until a
node
is
is
expected
reached which signals
the other's "turn", the other then produces until the state passes out of
its
region of
precedence, and so on. In equilibrium the firm without the turn must prefer forebearance
to "playing out of turn."
A
special case of this turn-taking partition
might be called "pure alternation,"
where the firms share the market by producing on alternate dates. To motivate the
alternating strategy profile, consider a
one—shot game with payoffs
as given in
Table
1.
16
This
game
return of
has (1,1)
e.
as its strictly
dominant Nash Equilibrium, providing each firm with a
Suppose that the firms were to encounter
of periods, with
common
6 being their
librium would imply substantial losses.
game over an
this
discount factor.
For small
e
Both firms would be much
infinite
sequence
the one-shot equi-
better-off
if
they could
coordinate their actions so as to avoid "head to head" competition, yet each prefers to
supply in the current period. Resolution of this Prisoner's
that the firms reach
A
will
agreement
tacit
as to
who
last period stays
out
this period".
If
them
is
is:
not too low
be better to wait until tomorrow for one unit of revenue, and the implied oppor-
It is
units of revenue every
e
easy to see that such an alternating convention, supported by the threatened
punishment of reverting to "(1,1) forever
1
librium of the infinitely repeated
game
"competing forever". That
and only
(
in such a situation
the discount factor
tunity to wait yet again the day after tomorrow, than to claim
day.
requires
and when.
will supply,
natural understanding which might develop between
"Whoever produced
it
some
Dilemma—like problem
43 )
is, if
«
This condition
is
<-
if
'
if
deviation occurs,
and only
if
a subgame perfect equi-
"waiting to alternate"
2
if:
is
5-(l+6 +^+...)
=
S/(l-6
2
)
is
better than
> e/{l-8).
Or,
lh-
important in the analysis of Markov Equilibrium to follow. Note that
holds with inelastic
demand
if
it
the periods are short enough, but never with elastic demand.
Return now to the dynamic game with cumulative capacity constraints which we
have been considering. Formally, define a profile
for all strictly positive
seS:
<reE
a(s)+cr(s— u(s))=(l,l).
as
an alternating strategy
The path
{s,(s,a)}
profile
if,
therefore involves
the firms supplying the market alternately in successive periods for as long as both have
positive stocks.
supplies
There are exactly two such
when the
profiles, distinguished
stocks of both are positive and their
without loss of generality that firm
A
sum
is
by which of the firms
an even integer.
We
assume
produces under the alternating strategy profile
17
when the sum
Notice
of positive stocks
how the reduction
even, and denote the corresponding profile by
is
of the continuous
game
to
one
in discrete
aeE.
time with a countable
state space, induced by the assumption that actions cannot be instantly changed,
being
is
heavily exploited in defining this form of precedence.
Figure 3
b+
s
t
(*>)}
s=(a,b)
^_i
{s (s,fr)}
l*~
t
i
i
i
i
—
1
i
i
,
1
1
•
«-
-•
a
(a-b,0)
Take an arbitrary point
a,
which we
call
the
alternating path
several alternating paths.
a,
ahead" to s—(1,1).
from
s,
denoted
VseS.
By
A
deviation from
definition of
a,
to deviation
if
itself
<r<e
under
These paths have a
{s,(s, cr)}.
to (0,0).
stair-
Figure 3 illustrates
a means either remaining at
s—(l,l)=[s— a(s)— <r(s— a(s))]=s
a always leads to a node belonging to
from
itself.)
A
useful
way
to state this fact
is
2
S, are closed to
s,
or
"jumping
(s,o-)e{s,(s,cr)}, for
alternating path
its
any alternating
that, along
path, the consequence of deviation from "waiting one's turn"
For
s
(Indeed, only an alternating strategy profile generates paths which are closed
s.
It is
nodes to
set of all successor
Notice that these paths, regarded as subsets of
s>>(0,0), so a deviation from
through
and consider the
and include nodes along the axes leading
step appearance,
deviations from
scS,
is
always to "lose one's turn."
easy to see that the turn—taking convention represented by
it
must be that a (a,l)=l, VcreS
A
a-1
a payoff of at least /3[e+£(l-<S
can get from playing "0."
Since
5r
)/(l-<5)]
A (a,l)=0
:
By playing
"1" at
a
(a,l)
is
A
not an
MPE.
guarantees
which exceeds /36{l-^)/{l-S), the best
when
a
is
even,
it
follows that
a£E
.
it
18
Nevertheless, there
is
some
non-trivial subset of nodes, implicitly defined as follows,
which the alternating profile a
For a
Definition 4.1:
3(e)
=
a restr icted
is
{s6S|Vs'6{s (s,a)},
aeE, g,i y
2:
est subset of
Proof:
3(e)
ations from
if
^c£|
if
W
WcJ(t). That
then
3(e)
Therefore
1, if
we can show
node s£.3(e).
is
closed to deviations from
firm
If
"waits
j
it
receives
it is
turn" at
its
fie
by definition, precisely the
s
its
c)
if
of receiving in perpetuity
,
,
iff
itself, it
V.(3-)(s)><5/?(j^).
erty of a "locally," at a node
partial converse of
when
it
does
Theorem
2,
s
and
for this general rule only
its
V.(a)(s).
must be that
Thus,
a does not pay.
A
firm "waits
d
its
come does not exceed the value
,
the revenue loss to the other firm caused by
Moreover, the argument
is
This continuation yields the payoff
nodes where deviation from
the value of that turn,
follows
are satisfied at every
This result leads to the following general rule about turn—taking:
turn" for one period
closed to devi-
payoff there
waiting to have the market to
set of
is
immediately, but loses the oppor-
V(b-)(s)=6-[0+5V.(a)(s-(l,l))]. So the deviation pays
.
the larg-
The Theorem then
a.
that Bellman Inequalities (3.2)(b or
(3e+8V -(o)(s—(1,1))- But, because
out of turn
is
the countable union of alternating paths, each of which
tunity to have the market to itself nexl period.
is,
3(e)
is
deviates by playing "out of turn"
3(e)
is,
on which a mutual agreement to "take turns" could be self—enforcing.
strictly positive
If it
Moreover,
ap')=Q, then Vfa)(a')<6-0§=$, j=A,B}.
S
h.
Lemma
from
MPE:
above and 0<f<l define the region of alternation, 3(e), by:
as
t
Theorem
from
it
having played
used the alternating prop-
immediate successor. So we have the following
relevant whenever precedence changes hands.
19
Corollary
1:
Suppose aeE
.
Then
a(s)
+
cr(s— cr(s))
=
and a-(s)=0 imply that
(1,1),
V.(a)(s)a/?(|^), for je{A,B} and seS.
The next
result provides
n
For s=(a,b)eS define n(s)E|a—b|, and k(s)=min{a,b}. Let
the larger firm and
functions on
s
the smaller firm at
i(s)
2k ~2
<|>(n,k)
2:
0(e)={seS
s,
i,je{A,B}.
e 6
n
-
<5
[Vkel, k<k(s):
(}>(n,k)
+ (l-^"2 ).^),
<|>(n(s)+l,k)>e
if
j(s)
denote
(Explicit dependence of these
be suppressed.) Define the function
will often
(4.4)
Corollary
also introduce nota-
Recall that n'EMax{n6l| S >e}.
tion which will be used extensively in the sequel.
Definition 4.2:
We
an explicit description of .0(e).
for
as follows:
nel and kel.
a-, \(s')=0,
and
<j)(n(s),k)>e
if
s
aw Js)=0}. Moreover, when
(4.3)
holds then
0(e)={seS|Vt>O, n(s (s,a))<n'}.
t
See the appendix.
Proof:
With
indivisible production forebearance,
by one firm or the
other,
is
essential if
they are to share the market and maintain a high price. Theorem 2 and Corollary
that such restraint
is
consistent with the firms' interests only
if
0(e)
is
too large for a firm to be willing to "wait
consists of nodes
generated by
able (in
a.
from which
Thus,
MPE), meaning
if
the
its
turn" to receive
this criterion is satisfied at
initial
node
s
€
it.
is
Now
greater
the region
0(e) we say that
alternation
is
implement-
that a tacit agreement to share the market through simple "turn-
how the implementability
demand
Any
every point along the path
taking" could be implemented by a state—contingent, self-enforcing rule.
If
imply
the "contingent right to
produce" has been allocated in such a way that payoffs never exceed f3{jzr)return
1
of this
norm depends on
Corollary 2 shows
the size and distribution of capacitities.
sufficiently inelastic, in that (4.3) holds, then
0(e) contains any node
1
20
from which alternation never causes the stocks to
differ
by more than n' without regard
,
the interest rate or longer the period length, the less elastic
to hold.
With
symmetry. The higher
So, for £<yr~T, impleir.entability depends only on
to total capacity.
demand
elastic
(4.4) implies a
demand must be
for this result
trade—off between the aggregate
size of cap-
acity and the inequality of its distribution, in determining whether alternation
mentable. For e>T^-c,
smaller
is
n,
n
only
<j)(n,k)>e
(i.e.,
n<n') and k
the larger can k be while maintaining
total capacity, the
more equally must
it
be held
Another implication of Theorem 2
limit as
6 >e
if
of the set
h-»0
h- 3>{t),
is
Thus, the greater
(j)(n,k)>e.
the Proposition below.
1
proof examines the
Its
a to a
to extend
1:
For any fixed
£6(0,1), r>0,
which achieves collusion whenever
and the
initial
Let
total capacity
R~, e,
and
r
V
R~eR.+R r
_
|
R.-Rg
|
of the paper.
an
MPE
exists
enough
R=(R ,R R ), where
.
such that {s
(s
t
whenever h<E,
S.
*
x>0. K>0, and aeS
exists
,
on
Formally:
be given, and consider the vector of capacities
There
MPE
full
relevant, given that the periods are short
holdings are sufficiently symmetric.
R A +R B =K.
Proof:
it is
and
the
is
implementable.
This method foreshadows limit arguments developed more fully in section
Proposition
imple-
not too large; the
is
for alternation to be
and uses Theorem
is
o
(h,R),cr)}cM(a)
<x, and Q*(s (h,R))=l.
See the appendix.
The proof
is
based upon the observation that, when the periods are short and the
firms equally placed private incentives to conform with the alternating profile nearly
,
coincide with "social" incentives to
max mize
i
industry revenues
:
If
each firm has kxR"/2h
C
units,
then (4.4) implies that neither firm deviates from a
other hand, coordination
(4.1).
But, as h-tO and
is
if
required to maximize industry revenues
<5->l,
r)1,
i
e<(Tqrr)(l + ^
these condition both require that R~
<
1
if
e<-Jl+6
)•
On
2k—
),
the
by
-ln(2e-l)/r. Thus,
the notions of "alternation" and "coordination" are intrinsically linked in this model.
21
MPE Extension of the Alternating Convention
D.
We now
is
complete the program initiated in Theorem 2 by exhibiting a profile which
The extension
a perfect equilibrium extension of a to the entire lattice.
The
the following general form on S\ 3(e):
while the smaller firm supplies nothing
3(e) within a given number
reaches
only on the smaller firm's endowment.
and only
if
of periods.
if,
by
The
The argument
Definition 4.3: For strictly positive seS, and
so doing, the equilibrium path
m(s) e
Theorem
3:
+cd,
With any
is
if
of periods depends
based on a backward induction
We
III.
On
l<m(s)<m,/\, otherwise
in
require the following:
0<e<l, the "waiting time" m(s)
if
is
(j)(l,k(s))>e,
given by:
and
<Kl,k(s))<e.
ueX which
arbitrary sequence {m, :k>l}cl associate the profile
a as follows:
extends
3(e)-,
ct(s)eo-(s).
On S\3(e),
cr.,. \(s)=l;
a-,
\(s)=0
if
For each e£(0,l) there exists a unique sequence
a-, \(s)=l.
{m?:k>l} such that the associated
Proof:
number
critical
m(s) = min{meI|V;c, Kk<k(s): <J)(n(s)+l-m,k)>e},
(4.5)
profile takes
larger firm supplies one unit at every node,
two dimensions, as suggested oy the discussion of section
profile
a*,
determined in this way,
is
MPE.
an
See the appendix.
Comparing
(4.4)
and
time that the smaller firm
(4.5), in light of Corollary 2,
[i(sj]
we have
associated this strategy for
and only
if
one
may
see that
would have to wait before reaching 3(e)
produce, given that the larger firm
if
a
[j(s)]
i
supplies constantly.
on S\
you can get there within m,
.25(e):
"Wait
m(s)
if it
is
the
does not
With every sequence {m,
for
3(e) when holding k
periods; otherwise, produce."
be interpreted as the parameters of an "optimal waiting policy" for
i,
So
}
units
{mf} may
given the assumed
22
behavior of
ing" by
The Theorem
j.
together with constant production by
i
on
asserts that, given alternation
j
"optimal wait-
.25(f), this
on S\ .2* (e) are mutual best responses.
This waiting strategy extends the set of nodes at which simultaneous production
avoided (and thus collusion
is
achieved) beyond the region of alternation,
-29(e),
ing interpretation of the
MPE
a*
is
the
the larger firm exploits
t
its
by giving
An
precedence (and therefore, enhanced payoff) to the firm with greater capacity.
is
interest-
position so as to
garner a greater than proportionate shar^ of the collusive profits by credibly threatening to
"produce
all
the time" on S\.25 (e), thereby persuading the smaller firm that
interest to forebear
node
To
\.
.<
We
and
has the dominant play
j
will wait as long as
it
noted above that
(or, credible
in its
outcome consider a
see the basis of this
such that k(s)=l, and n(s)=n>n'-l.
s
such a node. Thus
this,
whenever m(s)<mf
,
it is
a-, _\(s)=l,
threat) of "1" at
s;
i
WeE,
at
knows
can have the market to itself within n' periods. This
reasoning, applied recursively, generates the full sequence of optimal waiting times, {mf}.
When demand
We
found.
Corollary
3:
inelastic in the sense of (4.3) these waiting times
e<Tj-r then the sequence
If
We use
.25(e)
a nested induction to construct
and k(s)=l, then
waits for .25(e)
it is
m
,
earning @6
proof of Corollary
2,
firm
j
^
'.
obvious that
So
i
vi
k>l, determining
...,m?
1
k(s)=k.
} as
i's
i(s)
aits if
a*
is
The proof
rests
on the
fact that
given by:
is
{^=1} on
first
s,
since,
point reached in
i(s)
best play
S\ 25(e).
earning
m(s)<n'— lEm?,
if
optimal for
i's
3.
)>f}, kel.
either produces at
and only
given in (4.6), k>2, and consider
a*
Theorem
~1
best response to
produces under a at the
Inductively, assume that following
{m?,
{mf },
m* = Max{m6l|<5m -( T^)-(l+<52k
Proof:
can be explicitly
give the derivation in the text to illustrate the conceptual basis of
(4.6)
seS\
is
/3e,
or
by the
.29(e).
whenever k(s)<k— 1,
from seS\
m(s)<mf implies m(s—(l,l))<mf
If
.25(e)
,
for
where
when e<yq—
i-
23
Corollary 2 implies that m(s)=n(s)— N, for Ne{n',n'— 1}, and one easily verifies from (4.6)
m£e{m£_, m?_,— 1},
that
"1" at
If it
k>2, so
,
s,
continuing optimally from s-(l,l),
^(e)
waits for
better.
earns:
it
^'
6
Therefore, by induction,
On
the other hand,
does better to play "1" at
a*
s
and then "0"
earned
arrives at
after
s'
s—(1,1),
two
-(1—8
for
this
true
is
plays "0" at
i
periods, but
if it
s
for fixed
for
which
m(s)>mw
•>,
we note
Finally,
s
less
if
has exhausted
If
So a*
s,
coordination
is
i(s)
plays "1"
If it
two periods, having
at the successor node, it also
along the way.
6j3e
implies
Clearly
i
if it
plays "0".
do
will
m
Therefore, induction on
cfi
optimal for
is
at
i
any
s
i
if
m(s)<mw
\,
in
n
,,
>
For s£S\3(e),
which case the ensuing path
forebearing until this region
m(s)>mf/
satisfies:
The
is
reached, and alternating
0<t<k(s)—l, then both firms produce until
but
i
withdraws
seM(o*) and Q*(s)=l only
achieved at any node,
it is
i
m(s—t-(l,l))=mf/ \., l<t<k(s)— 1, the
if
at the
node s—t-
(1,1)
and
(s,o*)}cM(o*). That
{s
t
is, if
\.
while the induction hypothesis on
establishes that
equilibrium path begins with both producing, but
waits for &(e).
after
s')
coordinated play as capacities decline.
m(s—t-(l,l))>m.w
capacity. If
its
k,
is
when k(s)=k and mf<m(s)<mf+m-l,
a* obeys a kind of "inverse entropy" law:
and only
leads directly to .2>(f), with firm
play thereafter.
2
)/(l-<5 )].
l<m(s)<m£/
The induction hypothesis on k
plays "1" at
MPE
that the
achieved at
is
plays
i
completing the proof.
market never evolves toward
coordination
2k_2
.(l-<5
such that
now having earned only
followed by induction on
k,
s
and then "1"
can do no better than the latter course
m,
all
+2
firm
Simple algebra shows waiting
some node (say
arrives at
it
no worse than the former course
i
)/(l—6).
s)
when k(s)=k and m(s)=mf+m.
i(s)
the former course dominates the latter.
implies
/3-[e+^
earns:
optimal from
is
Suppose
s.
alternatively,
If,
fie.
at
it
if
k(s)=k and m(s)=m?+l, the same algebra now shows
if
m>2, and consider the best play
at
m(s—(l,l))=m(s)<mf <m?_ .. Hence,
achieved at every subsequent node.
24
E.
The Limits
We
complete
to Collusion in Perfect Equilibrium
this discussion of ta
collusion by showing that
;it
possible to
it is
achieve the industry's revenue maximurr in an
MPE
nodes than M(o*) when demand
At the same time, our constructive methods
show why collusion
distributed.
is
For the
is
inelastic.
not possible in any equilibrium
rest of this section
we assume
nodes seS from which there exists some
Any
MPE
that
ctgE
reached
is
t
wb en
where
that V-(<7)(s,)<^/?(t—4)
is
great at
s
j
is
at
s
{s (s,cr)}cM(a).
)+a(s
From
,)=(1,1).
this
way on
<7.(s,)=0.
But
means that the
this
Ignoring
If
1
we
the difference in
at
s
in
larger firm
would not
any revenue maximizing
in the proof of
may
MPE.
Theorem
can then use Theorems
1
and
2,
is
So
a
be determined by an induc-
A
3.
limit
is
the degree of disparity in initial holdings consistent with an
MPE
Corollary
t
must have begun with the smaller firm waiting. There
argument similar to that used
We
the set of
there will be no pt'th which maximizes industry revenue and keeps
tive
collusive path.
M*
between the firms' "regions of pre-
on how long a firm would be willing to wait, which
struct an
Denote by
t
cr(s
have been willing to forebear in order to arrive
limit
c^tt-t-
the firm for which
the larger firm's payoff below this bound.
any such path arriving
set of
capacities are very unequally
satisfying:
t
holdings
when
collusive path {s,(s, cr)} occasionally passes
cedence", so that a date
know
from a considerably larger
ascertainable in
MPE generating
and the notion of alternation, to con-
generating a collusive path from any node within these limits.
Markov
strategies momentarily, let us study those time paths of output
which avoid simultaneous production everywhere without either firm's implied payoff be-
coming "too large". Call the path {q.:0<t<t} admissible from
Vt; (2)
q +q.+...+q
=
s;
and any intermediate date
and
t,
(3) the
value of the sequence {q
0<t<t, does not exceed
/?(t-4).
s
if
(1) q.e{(0,l),(l,0)},
:t<t'<t}, for either firm
Consider the recursion:
a
25
(4.7)
(i)
n
=n', and
(ii)
c
=
Max{cel|
k
HV
.
k
+
)
)
where
kel,
^-V^
for
-
V k-1>
k=l,
Vo
with
~=
2,
...,
and
< /?({=£)},
cW
,
S
+
+c,— 1,
,
/3(|=f
1+C,
rC,
^fe
~=
=n,
n,
#££
)•
Then we have:
Lemma
3:
Moreover,
An
admissible path from
for kel,
^-n^j+l
c <
<
s
exists iff n(s)<n,
where {n,},
%,
/
kel, solves (4.7).
c+1, where cEMax{ceI| f{l-S)/{l-f
+1 )>e}>l.
Consider the problem of finding the longest possible sequence of "l's" and "O's",
Proof:
given that exactly k
"O's"
are to be used in the sequence, such that no "tail" of the
sequence ever has a present value exceeding /?(t—r), where a "1" earns the value
earns nothing, and the discount factor
this sequence, kel.
s
if
and only
n(s)<n,
if
tion problem:
With
c,
the
is
\.
/
The
number
is
follow
n,
+k
be the maximal number of "l's"
k— 1
to k.
So,
c,
maximal
its
as the
num-
=(k+n, )—{k— 1+n, _,), which
is (i).
length, given
k.
So
(ii)
and
(iii)
from the Principle of Optimality and the definition of present value.
c =
(ii)
and
Max{cel|
(iii):
c
<5
/?(^^)<V k </?(^),
(l-<5)/(l-(5
c+1
)>e}
kel.
<
< c,
Substitution into
c
Min{cel|
(5
(l-<5)/(l-<5
(ii)
c
yields:
)<e}
= c,
Since tx^>£, c>1; while the definitions above imply c<c+2, and hence c<c, <c+l.
over,
in
the Bellman equation for this optimiza-
is
which may be added to the sequence
of "l's"
a "0"
obvious that an admissible path exists from
it is
recursion (4.7)
the present value of a sequence at
From
(4.8)
Let
6.
{h, } so defined,
ber of allowable "O's" increases from
V,
is
/?,
if
f{l-5)/{l-f
+1
)>e>f +1 (l-6)/{l-f
Define N"={seS|k(s)=0 or n(s)>n,
given in (4.7).
Notice that
if
/
x—c,
n(s)>hw s—c,
/
\
+1
),
\},
/
then
c,
where
then, by
=c, Vk>l.
{n, }
Lemma
and
3
{c, },
k>l.
More-
d
k>l, are as
and Corollary
1,
no
col
26
lusive
MPE
path
is
possible in which the smaller firm produces at
maximum
from serT attaining an industry revenue
n(s)— [h,
path
at
\-Ci
/
/
J periods before producing. Once
some node s'erT
for
involves firm
it
is
to collusion in
sgS, or
(4.7):
If
(b)
If
(c)
in the
Appendix we prove that
if
l<t<k— 1. Thus, although
If
n(s)=h
k(s)
^
k
=0 and
then *
j(s)
^+m k ^
path in
profile
then seM*.
IT,
a on
rT
as follows:
3j (b) =1;
m k(s) +h k(s) >n(s)>h k(s) ^ k(s)
k(s)=0, or
n(s)>n
k(s)
an admissible path exists from
to reach the only admissible
s
c,_. units supplied by
provides a powerful tool for assessing the limits
it
Given any sequence {m,:k>l}cl define the
4:
(a)
For
would wait from
if i(s)
Theorem
an "open—loop" concept,
MPE.
MPE
which n(5')=h,/>,—c,/>,, the ensuing path, necessarily
the larger firm, followed by one unit from the smaller firm, for
admissibility
path
waiting at least
i(s)
has produced on a collusive
must follow the pattern of production implied by
admissible,
MPE
Thus, any
s.
then
,
<r
j(s)
=l and
=0;
&.
(s)
then a(s)=(l,l).
*
Then:
Proof:
(i)
3
{m k :k>l}
which o^Eivr; moreover,
(ii)
{seS|n(s)<h
k
/
s+niw JcM*.
See the Appendix.
The kinship
MPE
for
of
Theorems
and
3
Theorem
4 should be apparent.
3 constructs the
a on ^(e)cS, showing that constant supply by
a* from the restricted equilibrium
the larger firm, and optimal waiting by the smaller one, are mutual best responses on
S\.2>(e).
Theorem
the "boundary" of
4 constructs the restricted
rT
MPE
j
by specifying behavior on
together with optimal waiting by
responses, given this specified boundary behavior.
some ae£
N
to coincide with the unique admissible path in that set,
showing that constant supply by
there exists
a on
such that
the fact that, for any s^N,
o=u on
we can choose
~o
rT.
a
The second
is
We
are mutual best
a restricted
part of
so that the path
nating play on S\rT, and hence (s (s,ff)}cM(ff).
t
Since
i
MPE
Theorem
{s,(s,o}}
and then
4
is
on
IT,
based on
implies alter-
conjecture, but have as yet not proven,
27
d completely characterizes
that
Theorem
plies
some
3 together
show
clearly
why
dynamic duopoly with
low price on his
on each firm's
first
/
\+m,
^ then s£M*.
,
Lemmas
from that node. So
coordination via non-cooperative play
possibility of coordination turns
when they
more
is
and
difficult to
use state—contingent
With open-loop
indivisible production.
2
strategies, the
on the ability to find a path for which the discounted value
last
supply date
is
no
less
than the discounted value of the
With Markov play what matters
date of forebearance.
is
whether the
produce can be allocated so that neither firm has to wait when the value of contin-
right to
on a prorated per period
dination.
Comparing
Theorem
al
maximum
firms precommit to paths of actions than
of the high price
n(s)>n,
if
that the existence of an admissible path from a given node im-
attains an industry revenues
rules, in this
uing,
in the sense that
MPE
when
achieve
4(ii) states
M*
(4.2)
and
induced by a
basis, exceeds the fall in price
(4.7) reveals that the
former
is
the
loss of coor-
more severe
constraint.
4 suggests that the concepts of "alternation", "forebearance", and "optim-
waiting" are sufficient to fully describe
actions in this setting.
As the proof
how
of the
the firms might tacitly coordinate their
Theorem
reveals, the
extreme admissible path
implied by (4.7) constitutes a "boundary" separating the "interior" of rT from the rest of
S.
From any
serT not
do
so.
seS\rT the firms agree to alternate until reaching this boundary; from any
on the boundary, the smaller firm waits
for the
boundary
if it is in its interest
to
Play along the boundary gives the larger firm sole access to the market roughly the
fraction
1— e
firm knows
it
of the time [see (4.8)].
This tacit agreement
to be in the other's interest to
is
self—enforcing, because each
conform from any node
in the state space.
V. Depicting Discrete Equilibrium Dynamics in a Continuous State Space
Consider a continuous formulation of this game, without the assumption that output rates are fixed for intervals of length
h.
The
state space
is all
capacity pairs,
2
[R
,
.
o
Each firm's (Markov) strategy maps capacities into production
The
profile
rates:
q-:R
i=A,B.
-»{0,l},
q=(q»,q-n) partitions the state space into regions of production,
q.
(1),
and
28
forebearance,
(0), for
q-
each firm, and induces a vector
the system over time: -r— =—q(R
),
R
gram shows motion "due South" on
"Southwest" on q.
(l)fiq
qT
i
0)f1q
For
R
R=(R.,R R ),
(1),
the associated phase dia-
"due West" on
qT
(l)nq^
and
(0),
Payoffs from any node are given in the obvious way, as
(!)•
R
g-ven.
determining the evolution of
field
An
the discounted integral of instantaneous revenue flows.
equilibrium consists of a stra-
tegy pair from which neither firm can affect a unilateral improvement in
payoff from
its
any node, where strategies are restricted so that the differential equation describing the
evolution of the system always has a solution.
q.
(1)
length h.
let
{a :h>0} be a family
For h fixed, a
the path {s.(s (h,R ),a
R r+h-R r = -h
By analogy with
R
,
)}.
-
MPE
of
ah
a unique initital state
sets
1
l
{o-
(s
h-^0
-
ah
(
'
S
o(
h
:h>0} defines a vector
h
the limit exists.
,
to evolve over time in
(h,R )«h
s
c7h
=
K( S o( R o)' )) -h
the continuous case,
r
R
-R
o
(h,R
'
R r))'
field
Vr-h-t,
on K
cr*
a family of
MPE,
{cr*
defined in
Theorem
3 treats
h parametrically, so
:h>0}, as envisioned above. But
alternating profile, and
{a
tering" for infinitesmal
h.
(s
tel.
given by:
the motion over time of
would mimic closely the path of capacities and the payoffs generated by a
MPE
determined,
r ))},
The implied phase diagram, showing
The
is
So,
g = -^(R ),-lim{a
if
imply that the
of the discrete game, indexed by the period
cause the vector of capacities,
will
Given
the following manner:
is
restriction will
have a relatively simple topological structure.
Now,
as
Any such
(h,R))} has no limit, as
The continuous approach
a*
h-»0:
is
it
,
(R
:
r>0),
for "small" h.
actually defines
an extension of a
,
the
Alternation leads to "chat-
outlined above could not produce
behavior in equilibrium like that which emerges in the equilibria we have studied for the
discrete
game, with h "small". This
is
not necessarily a criticism of the discrete approach;
one might as readily conclude that the continuous formulation
it fails
is
flawed, to the extent that
to capture the strategic possibilities inherent in a convention of
"turn—taking".
29
Moreover, this "chattering" in the limit need not prevent a phase diagrammatic depiction
of equilibrium dynamics;
With
we can simply
lim{a
define: (j,^) e
mind, note that the equilibrium a* defined
this definition in
into three mutually exclusive
partitions the state space S
and
regions: a region of alternation, 3>; a region of forebearance
region of simultaneous production,
Jr ={seS|0<m(s)<m?/
"small"
and,
Section
IV (and
[-(0,1)], if
if
s
Q
|
explicit
m(s)>mw
(h,R)G
(h.RJe,? as
s
3
when
Abusing notation, write Re
explicit
3
[«?,
dependence of 3) on
(h,R))=A
if
state space K
2
&]
if
3E>0 such
limiting dynamics implied by the
= (3U3r U<^')
Re3
if
RG^
if
Moreover, w(x,y)
=
x
e~
+
_2ry/
-e
w(x,y)
<
-ln(e), e<^;
MPE
2ry
+ ^(l^T ') >
w*(y); and,
w(x.y)
(5.1)
w*(y)=i.ln[(l+e-
(5.2)
w*(y)
=
Clearly,
Q
(h,R)6
The
2
in
,
a*, for
& as h-»0;
results in
which these
that, Vh<E,
s
Q
(h,R)6^ \&,
=
x
lim {h-m*,
K
h-.0
(\.j>\\}lV n tt "
e,Vye[0,y];
Reo^if w(x,y) > w*(y).
+ j-ln^ +
(
e_2~) e
^
e<£;
Max{0, l.ln[s(z)^(z)l£^]
}|
'
a* on the continuous
(family)
2ry
)/26],
&\.
then h-+0=>h-n(s (h,R))-»x and
are as described above, where:
rx
<
e.
formulae cannot be found in the discrete case.
Define w(x,y)=lim{h-m(s (h,R))}, and w*(y) =
°
The
2:
s
[B].
h^O
Proposition
3
;
^=-(1,1),
j(s
.
,
,
by the smaller firm, 3f and a
xe|R.—Rg|, yEMin{R.,R R } and zEMax{R.,R R };
h-k(s (h,R))-»y.
Theorem
The dynamics generated by
>.}.
as h-»0;
h->0, for
in
VReR
collectively exhaustive
their proofs) can be used to solve for the regions of K
various motions obtain, even
Denote
Drop the
<tf.
and o^fseS
37=-^),
h, satisfy:
^=-(1,0)
>.}
(h,R))}, as h->0,
(s
e>
l,
£> 2' and:
]
30
for
u
(y)=[^=^-^—
The argument
Proof:
(4.6),
>
is
that, Vh<K:
<j)[n(s
o
e>l, w*(y)=0, y<y<y, for ye(0,y).
A
a straightforward consequence of taking limits in (4.4), (4.5) and
except for the derivation of (5.2), which
3K>0 such
When
0<y<y=-ln(2f.-l)/2r.
yJ
4+(2e-l)e
By
Appendix.
in the
is
Corollary
2,
Re
3>
if
(h,R))+L,k)>e, Vkel, k<k(s (h,R)). Then (4.4) implies:
cj>(n(s
(h,R))+l,k(s (h,R))]^e"
rx
-e~
o
2ry
2ry
+ ^(l^e~ ),
h -
as
0.
Moreover, from (4.5) one sees that m(s (h,R)) must "solve": <K n—m+l,k)«e. Equating
the limit above with
ting h^O in
(5.1).
substituting
e,
(4.6) gives:
The derivation
e
s
^
of (5.2)
is
*
(x—w)
for
)(l-^
x,
and solving
more co-nplex
since
no
when OrriS equation
(A. 2) in the proof of
It
follows from (5.1) that,
when
if
z
r
1
e
--In -
— x+y >
r
L
as
—It^
-
y+ e
h-»0,
which
is
{mf }
is
explicit representation of
Theorem
3
must be
c<^, a coordination failure occurs
r
h
w, gives w(x,y). Let-
_2ry
k_1 i
-rw
*-(l+e
2-e
)
),
available
a* for "small"
for
utilized.
under the
MPE
~ry
that a coordination failure must occur in any
l
.
The comparable
,
N
limit in (4.2) implies
J
open—loop equilibrium
if
z
>
1
--In
fe-
2ry
l
*—
•-
e
.
-•
This gives a clear measure of the superiority of closed-loop play for achieving coordination.
A
comparable measure can be derived from
contingent play vanishes as
tion 2,
is
firms' payoffs for various
endowment
and
state-
(5.2) converge
It is
easy to
configurations using the results of Proposi-
and so to verify our assertion that the larger firm garners a greater than propor-
depict the limit phase diagrams for ihe
The power
MPE
in (5.1)
continuous in the elasticity of demand.
tionate share of industry earnings from nodes
4B
which shows that the advantage of
Observe that the expressions
e-»l.
as e-»l/2, so equilibrium behavior
compute
(5.2),
of this limit
a defined in Theorem
argument
4.
is
R£& where
MPE
made
w(x,y)»w*(y). Figures
a* in the respective cases
clear
by
its
For there the discrete game
e<jr
4A and
and e>^.
application to the restricted
is
very difficult to analyze,
A
30
Figure
4
/
/
^45
F
ieur e
4 B
v
-
3o
b -
y
*.-i.Uic^^iV|
F i°;u
Q
re
4 c
/
/
/
/
31
but
its
continuous analogue
employed to
see that
is
transparent.
Lemma
and equations
(h,R)erT, for h sufficiently small,
s
extreme admissible path defining the boundary of
(by (4,8)) c,e{c, c+1}, where c=Max{et-:r|c<
line
3,
rT
—
with slope (relative to the larger firm's axis)
-r^-,
and
2>-ln(e)/r +
if
can be
(4.8),
(—)-y.
The
"chatters" with infinitesmal h, since
But
-}.
(4.7)
as
h-»0
boundary becomes a
this
along which the larger firm produces
on average the fraction 1— e of the time. (See Figure 4C.) The proof of Theorem 4 shows
that one can extend
a
The
alternate turns.
way
in such a
that, "above" this line, the firms agree to take
a requires that "below"
profile
Optimal waiting
constantly and the smaller firm waits optimally.
boundary of
cient use of an option to go to the
Let us see
how
decide
how
a,
may
be derived.
From equation
problem facing the smaller firm
produce the fraction
and nothing
for
e
of the time.
duration 1— e
any node
im{h-m,
,,
/
(A. 3) in the
s
for
each time
j3
it
effi-
a there.
-q\\}> an<^
hence
Appendix one can
which n(s)>h,
move
,
\
to
is
to the ex-
produces, but only gets to
(Because of discounting, earning
/?
for duration
e,
better than earning @e continuously, so waiting might
is
For each unit of time that the firm produces, the wait increases by roughly
units of time, since the
firms produce has slope
boundary of
1.
before beginning to wait, then
(5.3)
n(y,t,w)
=
N
is
its
—
has slope
Passing to the limit,
the wait for the boundary of rT
initially
while the path along which both
the smaller firm has y units of stock,
if
w, and
if
the firm produces for
t
|.{[l-^r
rt
]
+ ^-^[e-rt/'W ^/ 6]},
w(y) = Max{w>0| n(y,0,w)
>
Max
0<t<y
y>0, w>0, and 0<t<y.
II(y,t,w)}.
if
units of time
earnings are:
Accordingly:
(5.4)
equivalent to the
long to earn @e by producing each period, before waiting to
treme admissible path where the firm can earn
pay.)
at
is
produces
taking the payoff implied by
rT,
the limit of the waiting times, w(y)El
the behavior implied by
verify that the
this line the larger firms
32
One may
if
and only
from
verify
(5.3) that
d"U/dwdt>0,
w<w(y). Notice as well that
if
is
II
n(y,0,w)>fl(y,t,w), Vte[0,y],
so that
not generally a concave function of
This fact has the implication that the equilibrium path under a
the initial conditions, since the
y.
[E.g, this continuity
is
argmax
in (5.4)
crucial to the derivation of (5.2).]
endowment would cause
iod.
A number
from
(5.3)
and
Proposition3:
may
be desirable for
some node, and yet a
slight de-
by the following:
(5.4), as exemplified
Assume
e<l/3.
1-e
Then w(y)E--__
In
the inequality
is
if
-ry/<h
.
L
1-e
z < —ln(e)/r +
ry
(1— e)y/e
Some
+
MPE
extension of a
w(y), for h sufficiently
reversed, then xhe low price obtains at every date until the
smaller firm has exhausted, under any
Proof:
it
and
of interesting results along these lines can be derived with remarkable ease
attains the high price continuously
If
Thus,
w
strategy to switch to producing for a protracted per-
its
r
small.
not be continuous in
need not be a continuous function of
the smaller firm to wait to reach the boundary of N" from
crease in
may
t.
MPE
extension of
a,
when h
is
small enough.
Using subscripts to denote partial derivatives, (5.3) implies:
r-
1
n
w)
=
(y,t,w)
=
(y,t
t
2
r" n
l
-rt
^T
-w
r
e
rt 6
).[(^)e- / + e- y/
r
2
rt e
£
r t -w
)-[(^) e- / -e- y/
+e
e
-e r
e
rt
(
t
];
while,
(
].
tt
Thus:
-1
[r
2
n +r~ n
t
e<l/3. So
n,
VI.
]
=
r
e
(
t_w
maximum
rt/e
2
1
)-[(^)(- -= ^)e~
<0 3 n,,>0, and n,,<0
attains an interior
for
tt
3
II
-2e~
ry/e
]
> 0,
0<t<y, Vw>0, since
>0. In particular, the function n(y,t,w) never
with respect to
t
on the interval
any y>0 there exists w(y) such that n(y,0,w)>n(y,y,w)
[0,y].
iff
Thus, since
IT.
>0,
y<w.
Conclusion
In the model of dynamic duopoly studied here firms regulate an "on-off production
'
'
33
non—replenishable
process, depleting their respective supplies of an
a constant rate whenever their process
If
only one of
"on".
When
them produces an any moment the
moment, the remaining capacity
In
investigated.
hi
to
both are producing the price
price
is
Two
paths of production which are mutual best responses.
In
any
at
notions of equilibrium are
commit
firms
low.
is
Each of them knows,
high.
produce of the other.
open—loop Nash Equilibrium the
factor of production at
at the initial
Markov
date to time
Perfect Equilibrium, the
firms specify state—contingent plans of actions which are mutual best responses in every
The capacity
contingency.
of the firms, in these alternative strategic settings, to achieve
coordination of their actions via non-cooperative play has been systematically investigated.
By assuming
firms can change their actions only on exogenously given, equally
spaced moments in time, we convert the problem to a discrete time
game which, due
to the
dichotomous production structure, can be explicitly solved. The possibility
for coordin-
ation to be attained via non-cooperative play, in either strategic setting,
characterized
(in
Lemmas
2
and
3)
by two
relatively simple integer
characterizations allow us to derive the
ability to
precommit undermines the
strategies are
The
more
programming problems. These
main comparative
results of the paper:
possibility of coordination;
effective at coordinating play
is
when
that the
and that state-contingent
firms are symmetrically placed.
analysis emphasizes a difference between discrete and continuous time
dynamic
games recently stressed by Simon and Stinchcombe (1989). The extensive form analyzed
the discrete game, though restricted to
Markov
strategies, reveals a rich structure largely
hidden from view in a continuous time treatment. The idea of "precedence"
ing their play contingent on
"who
did
imal coordination in this environment.
what
Of
last
period"
— is
— firms' mak-
fundamental to achieving max-
course, this finding
is
related to the
"on—off
process of production which has been assumed here, but the point would seem to be of
general import.
The
results in section
V
in
more
on the limit behavior of our Markov Equilibria
suggest the possibility of profitably using this discrete approach in other settings where
continuous time formulations have been reflexively adopted.
'
34
Appendix
Proof of Theorem
1:
k
The
Define S ={(a,b)eS|a+b<k}, kel.
closed to deviations
E =f
Now,
|{^i|gK
v k>l};
I
se(S
from any
k+1
\S
k
),
^E.
S.
clearly,
one of the alternatives
are nested, cover
}
,
-,
For fixed k>l,
|gi
|
k
Therefore E,
ugE.
profile
{S
sets
cS
for
and hence
,, k>l,
•
any
ctgE,
,,
IS
holds at
(3. 2)(a)-{c)
s.
Moreover,
the profile
se(S
k+1
<7*eE
is
WcS
and aGS,
\S )n\V, set
,
o*(s)=o(s).
(2)
At each seS
k+1
\(S
k
2,
UW),
...:
Now
with the Bellman Inequalities there.
1
implies that
a*eE,
such k and
a on S\S
s,
.
w
|W
<7eEi
At each node
(1)
o* con-
*
implies
aeE,
l
,
s
K
,
Thus
k>l.
nW
*
*
Lemma
for every
assign a value to
*
sistent
and
the following inductive procedure constructing
For successive values of k=l,
well defined.
k
w
K
for
the continuation payoffs in (3.2)(a)—(c) do not depend on values taken by
Therefore, given
and are
S,
,
at each step
,
Therefore u*e£
k.
q
.
Proof of Corollary 2:
Let seS be strictly positive, with n=n(s), k=k(s), i=i(s) and j=j(s). By the
definition of the alternating profile we have that:
2
2k -2
n+1
2k-2
(i) V.(a)(s) = ^.[(l-^
-(l-8
it a-.(8)=0; and,
)/(l-« ) + 8
) /(1-5)},
V.(a)(s)
(ii)
=
8P(l-6
2k
)/(l-P),
a (s)=0.
if
i
Now, by Definition
a,
RHS
se^(e)
of
if:
(i)
(1)
cr.(s
exceeds the
RHS
if,
for every
of
if
Given
n>l.
for
if
n<n'
.
a strictly positive successor of
s
under
any n>0, k>l. Therefore, when a-(s)=0 then
Kk<k(s) and n=n(s); and,
a.(s)=0 then se3{e)
if:
(1)
(2)
[RHS (i)]<#3(^j),
for l<k<k(s).
Condition (2)
[RHS (ii)]<#?(^)
for l<k<k(s)
is
for
and
redundant in each
(4.4), condition (1) is equivalent to: (J)(n(s),k)>e, l<k<k(s), if cr.(s)=0:
and, <J>(n(s)+l,k)>e, l<k<k(s),
only
for
(ii),
n=n(s)-l; and, (2) [RHS (ii)]<£,5(t4)
case
s'
}=0, and V.(o-)(s')<^3) when 3j(-s'j=0. Notice that
[RHS (i)]<$(yEj)
When
l<k<k(s)-l.
se .25(e)
when
V.(a-)(s')<£(jEj)
the
4.1,
if
cr.(s)=0.
But
if
(4.3) holds
then
(j)(n,k)>e,
V k>l,
if
and
This completes the proof.
Proof of Proposition 1:
Suppose e<l/2. Let R=(R.,Rr>) be some vector of
initial stocks
such that
35
R A +R B =R.
Given any h>0, Corollary
|R
(i)
A -RB
(i)
imply there exists an
the definition of a,
<
o
-ln(e)/r,
(ii)
MPE
a coinciding with a on
we conclude
that
and
R=(R A ,R B ),
let
R A +R B =R~,
with
s=s (h,R).
It
o
follows
from
and define |R
that
C4. 4)
5(h)
in (A.l) as
for
h
2k ( s
H
h-+0,
MPE
e~
2 together
£^(f), from
s
|ex,
satisfying
Min{R
ir-,
•
a=a on
Fix
A ,R B }Ey.
(s)=0
if
3){t).
h>0
and only
if:
e-\l+5(h)]-6{h)
>
s
^[l
+ ^h)]-l
s,
n(-),
and
k(-), to see that
if:
2ry
> (2e-l)/(2e~
rx
-l).
as:
(iv)
will
a -R b
using the definitions of
sufficiently small
(iii)
it
Since
-2>(e).
sG^(e) when
-^h)^
As long
and
for
}{t>)
(A.l)
Take the limit
s(h,R)e .25(c)
1
which vanishes
,a)}cM(a).
{s.^s
Suppose e>l/2, and continue to denote by a an
Let
Theorems
gaurantees :hat (4.3) holds.
profile
if:
and
n(s (h))<n'(h)— 1 (neglecting rounding error,
assures that
small h), while condition
|
(h,R)e 3(i), VR">0,
s
< ln(^)/r.
h
(ii)
Condition
2 implies
be possible to select
the analysis leading to (4.1)
implies h-n(s (h,R))
-»
R<
-ln(2e-l)/r,
x>0 and y>0
we know
—ln(2e— l)/r,
such that x+2y=R~, and
Q*(s )=1
that
as
h-»0.
Thus
(iii)
n(s)>n(s)+2k(s)sR~.
iff:
(iv) is satisfied
Now, by
holds.
whenever
But
R
is
(4.1)
below
the threshold at which collusion becomes desirable. For such R" it is always possible to
divide the total endowment such that (iii) holds. Thus, for h is small enough we can find
R=(R A ,R B ), R A +R B =R~,
such that {s
(s
t
Q
(h,R),a)}cM(a) whenever Q*(s (h,R))=l.
Q
Proof of Theorem 3:
We have to show that the profile t* is well defined, and that neither firm will
deviate from it at any node seS\ 3){i). This amounts to showing two things: (1) that
there exists a unique sequence of integers {m?} such that on S\J?(e) the strategy "wait
3)(e) from
s
if
you can
get there within
mw
\
periods"
is
a best response for
i(s)
for
to the
strategy "always produce" for j(s); and, (2) the strategy "always produce" is a best
response for j(s) to the strategy given in (1) for i(s). The proof of Corollary 3 establishes
(1) in the case e<8/(l+6), when an explicit formula for {mf } can be found. The nested
induction employed there may be extended to the general case e€(0,l) as follows.
Consider S(k)={seS\i5(e)|k(s)<k}, k>l. Given any integers {m,, ..., m, }, the
.
36
Theorem, and thus each firm's value function, is uniquely determined
node in S(k). Therefore, the following induction uniquely defines a sequence of
integers {mf}, k>l, and so, by the definition in the Theorem, a profile o*:
profile defined in the
at every
m*
(A. 2)
m*k
e
=
Max
{m(s)}
s.t.
m
k(s)=k, and £
(
s
)
+1
if
2k
-[—o
0e+8V.,Jo*)(s-(l,l)),
>
]
1[s)
i-r
seS
mt =0
where
n'-l; and for k>2,
the inequality always
This definition
fails.
is
meaningful because at each
k>2 the profile a* and values V-(cr*)(-), i= A, B, are already defined on S(k— l)U^(e);
and, k(s)=k implies s—(l,l)eS(k— l)U^(e).
ment that waiting
ever
i(s)
m(s)>mf
if
integers
mf
are defined by the require-
to reach an alternating path is dominated for i(s) by producing whento show is that waiting for one period at s is optimal for
What we have
.
and only
The
m(s)<mf Vs6S\^(e). For
if
,
this
we use the
following nested induction.
i(s) will not deviate from a* on S(k— 1), k>2; and, (b) the inequality
where k(s)=k and m(s)=m>l. Then that inequality also fails at s',
where k(s')=k and m(s')=m+l. By (b), waiting to reach -2>(e) from s' is worse for i(s)
than waiting for one period, producing for one period, and then following o* While, by
(a), producing for one period and then following a* is at least as good as producing for
one period, waiting for one period, and then following a*. Thus, producing and then following o* dominates waiting to reach i?(e) from s'. So either, for all seS(k) with
k(s)=k, producing for one period and then following a* is better than waiting to reach
@(e), whence rn^EO; or, for some m?>0, the inequality in (A. 2) holds iff m(s)<mf For
Suppose: (a)
in (A. 2) fails at
s,
.
.
m
follows a* on S(k) if it does so on S(k— 1),
i(s) follows a* on S(l), then imply that the
"optimal waiting strategy" defined in (A. 2) is firm i's best response on all of S.
fixed k, this induction on
k>2.
The induction on
We now
k,
implies
and the
i(s)
fact that
from constant production on
a*. We do so by induction on k. Clearly
j(s) will not deviate from o* at any node where i(s) is waiting, so we need only consider
s tor which m(s)>mw y We argued in the text that j(s) follows
a* on S(l). Assume
show that the
larger firm will not deviate
S(k), k>l, given that the smaller firm follows
this is so
on S(k— 1), k>2. Take any seS(k) such that k(s)=k and m(s)>mf. Let
the node reached
and
let
if
j(s)
follows
a*, let
m=m(s'). There are three
"
s
cases:
(1)
be the node reached
m>mf_,
;
(2)
if
j(s)
deviates from
m<m?;_, and
;
s'
(3)
be
a*,
m=m?_,
be playing "1"
at the successor to node s, whether j(s) follows a* or not. But then j(s) cannot gain
by deviating. For, given discounting, firm j does better to follow a* at s and then
deviate at s', than to deviate at s and then follow o* at s", since both paths terminate after two periods at the same node. Yet, by the induction hypothesis, the payoff from
the former course is a lower bound on the value of following a* at s, and the payoff to the
latter course is the best that can happen if firm j deviates at s.
Case (2): In this case i(s) will be waiting to reach @{e) from the successor to s,
whether j(s) follows o* or not. So j(s) does better under a* from s" than it would
do under the pure alternating profile. Moreover, since s£3(e), it follows that:
Case
(1):
By
the definitions, m(s'
V j(s) (0(s"')>V
So,
were
j(s)
to play "0" at
s, its
')=m+l. So
in this case i(s)
will
(a)(s"')>«±3j(5)
value there would exceed
6(3(1— e)/(l— S).
But then the
37
reasoning of Theorem 2 and Corollary 1 implies firm j will not wait to "take its turn"
under such circumstances. Hence, j follows a* at s.
Case (3): In this case if j(s) follows a* it goes to the node s' where i(s) begins
waiting to reach -S>(c), while by deviatirg it goes to s" where i(s) is not waiting.
know from Case (2) that if i(s) were to begin waiting at s" then it would not pay for
firm j to deviate at s. Thus, we can dispose of this case if we can show that firm j is
worse off at s" when firm i produces there than when firm i begins waiting there. Let
t' denote the first date at which firm i(s) waits along the path {s,(s' ',o*)}, 0<t'<k—1.
We
t'=0, we are in Case
If
A
(2).
m(s
proof
critical fact for this
(s",a*))
<
is
the following:
m(s"), l<t'<k-l.
t/
The waiting time
does not rise as one
to reach 3){e)
moves along the path
{s (s'
',&*)}
t
To
of n for
which
2, m(s)=Min{meI|Vk, l<k<k(s), <}>(n(sV-m+l,k)>e}.
2 in the text implies that, as k(s) falls, the critical value
by Corollary
see this notice that,
The discussion following Corollary
,(s",cr*))<k(s")=k-l, by the
k(s
But n(s,,(s",<7*))=n(s"), and
4>(n,k)>e, l<k<k(s), does not decline.
Thus, m(s ,(s",a*))<m(s")-
definitions.
t
t
With
this observation in hand,
we complete the proof by showing
that firm
j
is
worse off under a* at s" if firm i produces there, than if it does not. Given that firm
is about to begin waiting for .0(e), it is obviously better for firm j if the wait is longer.
Define V*(t)=V./\(o )(s"), given that t'=t;let s=s.(s",a), and m=m(s,). We need
J|c
to
show that V*(0)>V*(t), l<t<k— 1. Frcm the
rt
,
=
V*(t)
where s,E3(e)
favors
j,
is
/?e(i=|
the
first
,=m
+ fitffi
node reached
+
-V
S
is
which firm
.=m.
assumed. Biu m,
>
pe{(^
have:
t
0<t<k-l,
),
implies
§
t _,
H
+ p{s\^l
1
=s. —(1,1). Therefore:
1+m t
t+1
({^
))
1+t+m
+
Now
because s,6^(e), and
{V
&
j
is
)-^V
(&)(s
j(s)
waiting at
t
s\,
we
-(l,l))}
(5^)(s
j(s)
t
t
J
0<t<k-l.
have: V.(a)(s.)>0t+6V.(cr)(s.—(1,1))-
i
v
t
j
j
t
Substituting this inequality into the expression above yields the following:
V*(t)-V*(t+1)
So V*(0)>V*(t), l<t<k-l.
Proof of Theorem
(i)
We
n, >ri,
We
p5 {l-e)(l-6
conclude that
j
1+m l
)
>
0,
will not deviate
0<t<k-l.
from a* in Case
(3).
4:
begin by noting that N"
_-, k>l.
l
>
is
closed to deviation
Thus we must show that
i
no greater than the value implied
is
1+m t
)}
we
Since a longer wait by
waits.
j
f+1
H^j-
(a)(s
j(s)
t
t
v*(t)-v*(t+i)
at
)
,<m,, we know that V*(t+1)
and since m,
above when m,
)
o*
1+t+m,
t
r
,
,
definition of the profile
from
a, in
view of the fact that
neither firm can gain by deviating at any node
i
1
1
1
38
Neither will deviate when
seN".
from waiting
will not deviate
The
o-(s)=(l,l).
Consider
{m k }cl,
when
will not deviate
i(s)
},
nodes seN
for
a does not exceed
the successor node under
some {m,
at
a-
\—c,
/
/
2 implies
that, for
is
when
n(s)=0, and neither firm will deviate
/
j(s)
since its payoff at
\,
So what must be shown
<5/?(t—r).
structure of the argument is similar to that used for Theorem 3.
firm i(s) We assert the existence of some sequence of integers
first
yet to be determined, nich that
"a., x(s)=0 iff n
A
nested induction of the sort used in the proof of
behavior of firm
must be
i
—c,<n(s)
Theorem
for
to produce, then
i
Let N" E{seN|k(s)<k}, kel.
establishes that the optimal
3
k—
k
of the "optimal waiting" form:
optimal
it is
iff
<n ^ s ^- ii
c
+
k( s )~ k(s)
k(s) ™k(s)"
h,/ „\-Cw >><n(s)," seN.
a best response to "a-/ \(s)=l
h,
Theorem
of
which n(s)=n,
is
with
The proof
alone produces.
it
some seN \N
I.e., if for
it is
also optimal for
to produce
i
k—
k
s'eN \N
for which n(s')>n(s). The argument turns on the fact that, due to discounting, it could never pay i to wait in order to reach a node at which i will produce,
when it could produce and reach a node from which it might choose to wait. So we need to
at
show that the length of
true for k=l, since
i
m^n'—c.>0.
(4.7))
gives
would wait a
Now
—
\N
N
waiting by
,
total of n' periods to play its
and consider
takes
i
to a
it
at least
is
Suppose, inductively, that {in,,
best play on
i's
optimal wait on N"
i's
m,
.
This
s/
for
obviously
is
one unit, so (consulting
a
,}cl, k>2, are such that
\N
optimal behavior at seN
his
node
...,
c,
which n(s/)=n\
—c,
,
which c
after
causes the following pattern of play to ensue: i produces for one period, j produces for
periods, i produces for one period, j produces for c,
know
c
periods, etc.
lc_i
2
We
that
v ;(^)( s
be the
sum
ic
)
=<^v
k-l'
for
V k-1
given in
of firms' capacities at
(i)
s/.
k
k—
be optimal at seN" \N"
^Ww,K
i
(ii)
s."
if
The
T k E2k + n k-V=n/ + k +( c 1 +---+ c k--l)
Let
Therefore firm
is
k
)/(i-6)
i's
payoff at
- sv _
k r
—
on
well defined
and only
N~
,
>
Pe+6-V.(~a)(s-il,l))
1
equality in
,
^
= Max {/3e(«
+
l ~°
(ii)
l<
and
W
=c
=0.
s/
is
given by:
we conclude
that waiting
if:
i
\
where m(s)sn(s)—h, +c,
at
4J )-
w k e VjC&k^o = p(i-s
Assuming, inductively, that V(a)(s)
will
(
t
m(s)+c, ,+...+Ci
k_1
<k
The
inequality in
.
^-W,K
6
(ii)
states: "i
},
l
prefers to wait
uses the induction hypothesis to infer that, conditional on
i
—
producing at s, i's optimal Kjhavior on N"
takes the form: continue to produce until,
with k—t (possibly zero) units left, it pays to wait for the node s/_., and associated
payoff
W,
__.
."
We need
to
show that the inequality
in (ii) holds
when m(s)=c,
,
so as to
c
39
conclude that
waits at serT
i
Lemma
In the proof of
(iii)
Substituting
U
1
\N
and only
if
we noted
3
k
+ |=|
/?[!
into
(iii)
k
Wk
>
]
^-
+
k
and rearranging, we see that
(ii)
k
k
for
P{^)>V k > P(^fr)
.hat
> fll
n(s)<h,+m,,
if
k>l. So by
,
(i):
k>2.
],
(ii)
hold
will
if,
for l<t<k:
k
k
l
some m,el.
k
l
-[1-6+6-6
> e(l-6 ) + 6
.[i-6+6e-8
].
Recalling the definition of T, and using a bit of algebra we have that (ii) holds if:
6
(iv)
]
Ck
(v)
.(i-^^).[l-(5
(5
c
Now by
By Lemma
may
8
(4.7)(ii),
T
^f)]
k+Ck
>[e-6
t
"].(l-^
c
k_ ]
(
e-[l-t
W k =/?(l-£2t )/(l-<$2 )+£2t W k_
,
and
t
W
fc
> [RHS
]
k>l;
)
if
so, 6
k
k
we
,+...+c, _.>t+l, then
c,
of (v)], and
_t <j9(l-^(
l<t<k-l.
))
{l-6+6e)>e, k>2.
holds.
(ii)
2
2
(4.8), 6 /(l+6+6 )<e<6/(l+6),
l<m<t. Then by
m =l,
k_
So,
"t )<-r.
-j_T
f
>
(v)]
Vk_1 <(^)
and
]>e
c
Obviously,
[LHS of
conclude that
c
.(
-[l-6(l-6)V
k>l.
3, c, >1,
Suppose
k
Ck - 1+ '" +Ck -t
2
^)/(l-<5
).
So
and
holds
(ii)
c
k
if,
Also,
<2.
for
l<t<k:
p
k
5
But
p
^
[j^ ~ 0~V(l-£)W k _
]>e,
c
l<m(s)<c
that
k
The
.
integer
best play on N*
i's
It
m k is
remains to show that firm
which n(s)>hw N+m,
closely parallels that
n,
/
and
employed
three cases, distinguished
a-, for
{m,}
cr(s)=(l,l).
by the behavior
K
s
,
might wait.
a
of
If
it
holds}—
k
>0.
will
i
(ii)
We
holds
conclude
seF
for
sketch the argument here, as the reasoning
Theorem
at the nodes
s'
j
the fact that <r(s)=(l,l) implies V-(<r)(s)
is
If a-(s' ')=()
3.
and
<x(s')=(l,l) then
will produce,
then
it
j^^-
inductively defined in this manner.
in this part of the proof of
if j does or does not follow a- at s.
that involves waiting to reach a node where
definition of h\
We
if
Wk </?(^), firm
cannot gain by deviating from a at
j(s)
reached
reach a node from which
which holds
Thus we have proven that
m k EMax{m(s)| (ii)
given by:
given by
is
}>e,
=l, Vk>l. Since
k
wait at least one period to reach an alternating path.
for
2
t
[j^
t
t>8 /(l+6), then
if
6
if
2
(4.8) implies that
2k
k
which holds
when
j
it
Again, there are
s'
',
respectively
will not deviate, since
could produce and
will not deviate, since
too large for
j
by the
to forebear at
j
s", where
in order to reach
i
forebears.
There remains the case in which a-(s')—0,
but a(s' ')=(1,1), so by deviating j avoids reaching a node where i would have begun to
wait. Argument analogous to that used for Theorem 3 disposes of this case as well.
The point, as in the proof of Theorem 3, is that j never gains by delaying i's wait.
Bear
in
mind that
i
is
waiting to reach the unique admissible path in rT at some node
s/
40
for
—c,
which n(s/)=n,
Vi,
)/(l—6)+6
were to begin waiting
i
—c, +m,
k(s)=k and n(s)=n,
P(l—8
If
.
then the payoff to
s— (1,1), then the implied payoff
latter payoff
from a
at
is less
s, it
any t>l. And yet,
at
j
would rather follow a
it
(4.7), <T^ V,
at
s
will not deviate
j
MPE
a restricted
on
however,
If,
n~ 1
which
for
i
by
)/(l-6)+^
mV
than
at
s'
than deviate and reach
s'
s'
'
is:
i
were to begin at
k_1
_•</?( t—r)- Thus,
begin waiting at
i
s
as a consequence of this waiting
j
^e+^(l-^
is:
s
strictly prefer that
beginning to wait there. So
is
to
from a node
s/
ai defined in (4.7).
than the former since, by
would
conclude that a
V, _,
for
,
for
if
deviates
j
'—t
'
The
.
-
(1,1), for
with
i
from a under any contingency, and we may
N~.
*
We
(ii)
now show
that there
is
sequence {m,
So we
will
{seS|n(s)<h,
that
some ueS such
}
we know
that
that
/
\+m,
a=a on
/
N".
JcM*.
By
{s,(s,o)}cM(a), for
have proven the Theorem
if
aeEiTT,
Theorem
definition of the profile
all
we can show
Since
seN
for
implies
a and the
which n(s)<n,
that, for every
1
/
\+mw
seS\N", there
\.
some
is
*
which generates a collusive path from
ctgE
a which has the property that
it
reached, after which
a.
Note that
it
follows
We
do so by constructing an extension of
generates an alternating path from
(s,ar)}n^0,VseS\K
{s
s.
since
until
s
N~
is
For seS\N~,
(a,0)eN~ and (0,b)eN~, Va,b>0.
t
let
s
that
at
s.
be the
first
n(s)<h,/-N
When
node reached from
If
s
Then
along {s,(s,&-)} such that seN".
k(s)>0 and a(s)=a(s) we
will say that
a
is
it
must be
a
"compatible" with
at s, then redefine a so that firm A
odd, instead of even. With this redefinition in
a and a are not compatible
produces when the
sum
of stocks
is
mind
we can
state: VseS\K there exists an alternating profile a which is compatible with
the sense that they both have the same firm producing at the first node reached from
under a that
lies in
N~.
With a
^(s)E{s
so defined, let
(s,a)}llN"
and
let
a[s]
a
in
s
be a
t
strategy profile defined on y£(s) so that-
Vs'£^(s)\N~. In words, for given seS\N",
and the compatible alternating path from
N",
and follows a on N\
that
<r[s]
It is
Clearly
it
a[s]
s,
is
s.
c-[s](s')=a(s'),
the profile defined on the union of N"
it generates alternation outside of
such that
^(s)
obvious that
generates a collusive path from
showed that
j[s](s')=a(s'), Vs'cN", and
We
is
closed to deviation from
now show
that
cr[s]eE
cr[s],
and
WcV
only be conerned with points s'6 ^(s)\N\ The proof of Theorem
pays to deviate from alternation only when the payoff at the node reached
we need
without deviating exceeds 0{j^)- Neither firm's payoff exceeds this bound at
s.
At any
2
6
41
predecessor of
and successor of
s
along the path {s.(s,a)}, each firm's payoff consists of
s
receiving /? on alternate dates until the node s is reached, and hence may be written:
2
2
2t
2t
2t
2t
)/(l-<5 )+ <5
.( cr)(i)<^( l—
)/(l— <? )+ <5 /?(^^)</?(^|), in view of the fact that
/?( l-<?
V
So neither firm
e<8/(l+6).
*
cr[s]eSi
w
By Theorem
qy
_
1,
3
3,
anywhere on ^(s). Hence
<r[s]
_
*
aeS such that
Derivation of Equation (5.2):
We seek w*(y)Elim{h-mw
in
from
will deviate
/,
ct[s]e<7
From
t-,\\}.
on
<s6(s),
and so seM*.
the implicit definition of
{m£ }
MPE
equation (A. 2), and the properties of the
that m(s)=mf / \ if and only
we conclude
(A.3)
m
^<5
(
s
)
1-62k(s)l
+1
>
.
a* established in the proof of
the following condition holds:
given
Theorem
+ ^+^-tuM.MTC*?
ihxW^
Kt<k
}
1-6'
1-6
>•
if
n
>/M *(s)+2.
12"
1-6
L
i(s) does better tc wait for 3(e) from s than to produce (and to
continue producing until waiting becomes optimal), but that this would not be so if the
wait were one period longer. In view of the fact that h-k(s (h,R))-*y, as h-»0, we have that
(A.3) states that firm
h-m(s)-»w*(y) as h-k(s)-»y in (A.3). in other words, (5.2) is obtained from (A.3) by
w*(y) with lim{h-m(s)}, and y with lim{h-k(s)}.
The key to the derivation is the observation that h-t->0 as h-»0. Assuming this is
true for the moment, the limit of the RES of the first inequality in (A. 3) can be evaluated
using calculus. Introduce the infinitesmais dysh-t and dw»h-[m(s)—m(s—t (1,1))] for hxO;
let h-m(s)-*w* and h-k(s)-*y as h-»0. In the text we noted that, by taking limits in (4.4)
letting h->0 in the latter, identifying
•
—ln[pd-(e—^)e
the waiting time defined in (4.5) satisfies: h-m(s)-»H
ry
]Ew(x,y), as h-k(s)-y
and h-n(s)-tx. Therefore, simple differentiation yields the following:
(i)
dw
s
w(x,y)-w(x,y-dy)
s
dv.
'
r
.
-gjxj) dy
2(2e-l)e
-l+(2e-l)e
2ry
where u(y)E[l-(2e-l)e
]/[l+(2£-l)e
and taking hsO, (A.3) becomes:
,..,
(«)
e
-rw*
l-^~
:' ry
Now, using the
].
2 ry
,
-[-21— ]~fdy J+
r
1
2ry-
•
e
-rdy
*-e
5r7
dy
= [l-u(y)].dy,
definition of 6(h)
-r(w*-dw)
l-e~
K
'•[—
r
and
/3(h)
2r ( y " dy
-^
)
J.
Note that (ii) has the intuitive interpretation that firm i, with capacity y, is indifferent
between waiting w* units of time to share the market, or producing for a short interval
and waiting thereafter. Letting dy->0 in (ii), and using (i) yields the following:
(iii)
It is
easy to see that
(iii)
iw
2ry-2iyi
= e-e--rw
.g.(i-e-^). u (y)+e-^].
implies (5.2).
42
Notice that u(0)=(l-e)/e<l; u'(y)<0; and u(y)=0, for yE-ln(2e-l)/2r.
—
So there
r
^. For y>y the value of w*(y)
a unique ye(0,y) satisfying: [2e—u(y)]/[2—u(y)] = e
implied by (iii) is negative, so the first inequality in (A. 3) fails when h is sufficiently
small, and hence w*(y)=0. Finally, we must verify the presumption that h- 1-»0 as h-»0,
upon which the above argument is based. This presumption is valid if, for hsO and firm
indifferent between producing and waiting, producing for any positive interval leaves it at a
point where waiting is preferred. This is Equivalent to the requirement:
is
i
<9[w*(y)—w(x,y)]/5y
But by
(iii),
w*(y)
is
<
decreasing, while by
whenever w*(y)=w(x,y).
(i),
w(x,y)
is
increasing in y.
o
43
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ll\l
023
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