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QUOTAS AED TEE STABILITY
OF IMPLICIT COLLUSIOJL-"
By
Julio J. Rotemberg
and Garth Salcmer*
Number 419
l'iay
1986
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 021 39
4 1986
QUOTAS ABD THE STABILITY
OP IMPLICIT COLLUSION
By
Julio J. Rotemberg
and Garth Salcmer*
May 1986
Number 419
*Sloan School of Management and Department of Economics, M.I.T. respectively.
We would like to thank Robert Gibbons and Paul Krugman for helpful comments
and the National Science Foundation (grants SES-8209266 and IST-8510162
respectively) for financial support.
QUOTAS AND THE STABILITY OF IMPLICIT COLLUSION
by
Rotemberg
and Garth Saloner*
Julio J.
May
1986
*Sloan School of Management and Department of Economics, M.I.T.,
respectively.
We would like to thank Robert Gibbons and Paul Krugman for
helpful comments and the National Science Foundation (grants SES-8209266 and
IST-8510162 respectively) for financial support.
I
Introduction
This paper shows that the imposition of an import quota by one country can
protection can reduce the
lead to increased international competitiveness:
price in the country that imposes the quota, the foreign country, or both.
somewhat paradoxical result emerges from
a
model of implicit collusion.
threat,
will
ensue
if
any firm deviates
the more collusion that can be sustained.
ability of the foreign firms to
punish
collusion that can be sustained
We study both the case
in
is
The more powerful the
.
Since a quota reduces the
a deviating domestic firm,
the amount of
correspondingly diminished.
which sales and the case
which prices are the
in
Our results are strongest when prices are the
strategic variables.
In such
by the threat that
a setting the firms in an industry sustain collusive prices
more competitive pricing
This
strategic
quotas always make monopolization of the domestic market more
variables:
difficult in that case.
This
is
in
sharp contrast to the results
imperfect competition models in which quotas tend to make
domestic firm to act as a monopolist at home
.
It is
it
in static
easier for the
also different from the
impact of tariffs which, in a dynamic setting like the one studied here, do not
necessarily make
Whether
it
tariffs
more
make
difficult for the firms to sustain collusive
collusion
more
difficult to sustain or not
outcomes.
depends on
the severity of the punishments that the firms can reasonably be expected to
inflict
on their cheating rivals.
The maximal punishments
of the style
developed by Abreu (1982) involve an outcome in which the domestic firm earns
because even with a very large
zero profits
.
can,
willing to tolerate the
it
if
it is
makes
result,
it
This
is
tariff,
ensuing losses, charge
the foreign firm
a price so low that
impossible for the domestic firm to earn profits at home.
tariffs do not affect the ability of the
duopoly to maintain
As a
monopolistic outcomes.
Contrast this with
can
still
inflict
There the maximum punishment the foreign firm
a quota.
on the domestic firm
is
to sell its entire quota.
Thus
yields positive profits for the domestic firm.
firm faces a lower punishment,
monopolistic outcome.
fact that,
This generally
since the domestic
has a larger incentive to deviate from the
it
This general intuition must be qualified somewhat by the
as pointed out
less costly to the foreign
by Davidson (1984), trade restrictions
country to
let the
make
it
domestic country have a larger
Thus the domestic
share of the domestic market.
also
firm's incentive to abide
by
the collusive scheme can at least be partially restored by altering the market
shares in
its
favor and thereby giving
it
more
to lose
if
it
deviates.
In this case of maximal punishments, therefore, our results have the
opposite implication of those of Bhagwati (19G5)
showed that
a single domestic
producer who faced
would act more competitively with a
a single domestic
In his classic paper he
.
producer and
tariff
a single
a competitive foreign
than with a quota.
market
When we consider
foreign producer, the opposite result
emerges.
If
the punishments are not as grim as this, but instead involve reverting
to the equilibrium
from the one-shot game in each period, then large tariffs
(which are anologous to small quotas) make collusion more difficult to sustain.
In that case, then,
the consequences of tariffs and quotas are similar.
As mentioned previously, our results are strongest
prices are the strategic variables
.
When
quantities are the strategic variables
for the case in which
we analyze the case
we find that only
discipline of the oligopoly while large quotas,
if
in
which
small quotas break the
anything, strengthen the
incentives to go along with the monopolistic outcome
.
Thus our results
case parallel closely those of Davidson (1984) for tariffs.
for this
He uses quantities
as the strategic variables in a model ve~y similar to ours and shows that small
promote collusion, while the opposite
tariffs
true for large tariffs.
is
After considering the effects on monopolization
the quota
we study the foreign repercussions
we used
that the arguments
of
the country that imposes
in
We notice immediately
quotas.
show that more competition emerges
to
country
in the
that imposes the quota also imply that quotas lead to increased competition
abroad
the the domestic firm
if
is
We model the capacity
capacity constrained.
constraint in a simple way, assuming that the firm can produce at constant
marginal cost up to some
quota
is
limit
and cannot exceed that
imposed when the domestic firm
the monopoly output
is
to
be sold
in the
quota requires that the domestic firm
sell
is
Suppose that
limit.
a
capacity constrained in this way.
If
domestic market, the presence of a
more
at
a limited capacity to supply the foreign market.
home.
Therefore,
it
has only
In that case its capacity
constraint has the same effect abroad as the quota does on the imports of the
foreign firm.
Thus, for the reasons discussed above, more competition results
in the foreign market.
It is
possible for quotas to lead to increased competition abroad even in
the absence of capacity constraints, however
levels of
demand fluctuate over
the two markets.
time
and
Suppose, for example, that the
.
are not perfectly correlated across
In that case, in either market taken alone,
unable to sustain the monopoly level of prices in
and Saloner (1986)
,
when demand
outweigh the punishment which
is
is
demand.
meted out
in later,
in
.
As
in
may be
Rotemberg
may
the incentive to cheat
"more normal", periods.
such settings the firms can
when they share two markets with imperfectly correlated
In the simplest case,
markets are
states
sufficiently high,
As Bernheim and Whinston (1986) show,
sustain higher profits
all
the firms
of equal size
(which
is
also the one that
we study), the
and have perfectly negatively correlated demand.
In
.
.
that case the two markets taken together are perfectly stable over time and
hence, since the firms' incentive to cheat doesn't fluctuate over time, they may
Imposing a quota on one of the
be able to sustain monopoly profits each period.
markets, however, means that the firms lose the countervailing force that
smooths the incentive to cheat when demand
Instead, the foreign market
high in the foreign market.
is
transformed into
is
a
market with fluctuating demand
and monopoly profits may no longer be sustainable there.
Thus the imposition
market or
of a
quota can increase competition
in the domestic market.
The natural question
is
in the foreign
whether the separate
analyses discussed above can be combined to show that the imposition of a quota
can contemporaneously increase competition in both the domestic and foreign
markets
.
We show that
The presence
this is indeed the case
of implicit collusion is not the only reason
result in increased competition in both markets
model developed
in Dixit
.
why quotas may
Indeed we demonstrate that the
and Kyle (1985) can be modified
to
show that potential
entry can have the same effect.
The paper
is
organized as follows
:
We show that quotas can increase
competition at home with prices as the strategic variables in Section
with quantities as the strategic variables in Section
III.
II
and
In Section IV,
we
use the analysis developed in these sections to show that quotas can lead to
increased competition abroad
show that
in the
the domestic firm
presence
is
if
the domestic firm
of fluctuating
this can occur
not capacity constrained in Section V.
in
We
capacity constrained.
demand that
show that increased competition can result
market.
is
even when
In Section VI
we
both the domestic and the foreign
Section VII concludes the paper with a discussion of the effect of
potential entry
.
II
Quotas and price competition
home
at
We consider an
There are two countries, domestic and foreign.
oligopolistic industry with one domestic
the simplest case
restriction
is
Marginal cost
consumers can only buy the good
by the two firms are viewed
P
is
is
c
it
.
Yet, and this
does at home.
for tractability
Thus the foreign
,
the goods
as perfect substitutes in the domestic market.
given by:
= a -
where P
constant and equal to
own country 2
in their
made mainly
slightly contradictory assumption is
Demand
This
However, the markets are segmented
firm can charge a different price abroad than
sold
sales abroad.
Foreign firms do not face any transportation costs from
shipping the good to the domestic country.
so that
is
To start with
firm.
we assume that the domestic firm makes no
relaxed in Section IV.
in both countries.
and one foreign
bQ
(1)
the industry price and
Q
is
the sum of the amounts sold
by the
domestic and foreign firms
In this section price
is
This means that
if
one
supplies the entire market.
If
the strategic variable.
firm quotes a price lower than the other,
it
There are two possible
two firms quote the same price they share the market.
assumptions about the way in which this sharing
gets half the market.
feasible
The second
is
the presence of quotas the foreign firm
case.
Thus the
latter
done
.
The
first is that
that any possible market division
and that market shares are also
the market.
is
assumption
implicitly
agreed upon.
may be forced
is
the
to
each
is
However, in
have less than half of
the only one that makes sense in this
We
start
by analyzing equilibrium under free trade.
This equilibrium
3
the standard duopoly equilibrium in a repeated setting.
is
We assume that the
firms try to sustain the monopoly price (a + c)/2 and that they each serve half
9
A
Then,
the market.
If
if
either firm deviates,
market so that
neither firm deviates, each earns (a-c) /4b per period.
it
undercuts the price slightly and captures the entire
However, after
earns twice this amount.
it
a deviation the firms
are assumed to revert forever to the noncooperative equilibrium for the
corresponding one-period game which has a price equal
This punishment which gives each firm profits of zero
possible punishment that can be conceived
industry
5
It is
.
if
to the
is
marginal cost
c.
also the strongest
the firms are free to leave the
then impossible to make them earn negative profits.
With this punishment, each firm will be deterred from deviating as long as:
(a-c)
2
2
/4b < 6(a-c) /4b(l-6)
where the R.HS
of this
by cheating and
long as
6
6
equation represents the future profits that are given up
the rate at which future profits are discounted.
is
equals at least 1/2, the monopoly price
We now consider the
effect of a quota.
allowed to import be c(a-c)/b so that
sold
under perfect competition.
firm to supply the amount
Similarly,
if
of (a+c)/2,
z
is
1/2
,
it
A
sell
is
sustainable.
is
Let the amount the foreign firm
scaled
quota where
demanded
can
it
So as
is
by the amount that would be
z
is
1
would allow the foreign
at a price equal to marginal cost.
the amount demanded at the monopoly price
and so on.
Notice as an aside that any quota which
is
binding at the original
equilibrium, i.e. which reduces the amount imported, raises the standard
measures
of domestic welfare.
This
is
so because,
even
if
the price remains at
.
8
the domestic firn
(a+c)/2,
Domestic welfare
is
having higher sales, now earns higher profits.
only increased further
the national identity of firms
if
the price actuaUy falls
.
Since
however, we focus mainly
a slippery concept,
is
g
on the competition-enhancing affects of quotas.
We begin studying the equilibrium with quotas by analyzing the
punishments for deviating from the implicitly collusive understanding.
standard practice
(1971),
in
As
is
models of repeated oligopoly (see for example, Friedman
Rotemberg and Saloner (1986) and Brock and Scheinkman (1986)) we assume
that firms revert to the
Nash equilibrium
in the
deviates from the collusive understanding.
who focuses instead on
While, as
one-shot game
This
if
in contrast to
is
any firm
Abreu
(1982)
the maximal feasible punishment for any deviating firm.
we show below, our results are not very
sensitive to our assumption of
one-shot Nash punishments, there are three reasons
why
even when they are lower than the maximal punishments.
these
may be preferable
First,
if
the owners of
the firm can sell the firm forward, the only equilibrium until the sale takes
place
is
the one -shot Nash equilibrium.
managers
to
of the firm to signal,
7
Second,
it
is in
the interest of the
once punishments begin, that they are
post prices other than those of the one-shot Nash equilibrium.
signalling
is
credible
unwillin g
If this
maximal punishments as in Abreu (1982) become
c
infeasible.
Third, the one-shot Nash equilibrium
easier to characterize
is
than the maximal punishments
The
static
one -shot game to which firms revert when they are punishing each
other has no pure strategy equilibrium.
Kreps and Scheinkman (1983) analyze the
mixed strategy equilibrium, and compute the expected profits of the firm with
larger capacity.
They
also exhibit one
such mixed strategy equilibrium (for
which they compute the entire distribution
show that
this equilibrium is the
of prices)
.
It is
straightforward to
unique mixed strategy equilibrium and thus
to
.
also
..
compute the expected profits
We now
list
of the foreign firm at this equilibrium.
The arguments behind
the salient features of this equilibrium.
these assertions are contained in the appendix:
The highest price charged by both firms
(i)
The lowest price charged
(ii)
o =
(a+c)/2
and
it
(iii)
e)
2
(a-c)(2E-E
-
1/2
2
)
[a+c-E (a-c) /2
is
]
is
/2
(2)
charged by both firms with probability zero
is
In equilibrium,
the domestic firm has expected profits of (a-c)
/4b per period while those of the foreign firm equal
e
2
(1-
(o-c) (a-c)/b
Notice that this static equilibrium has the features of the differentiated
products model
Krishna (198G)
of
A higher quota
.
is
higher
e)
lowers both the
Kreps and Scheinkman (1984) show that the
highest and lowest price charged.
entire distribution of prices
(a
stochastically dominated
by that with
a lower
quota.
There
comment.
c)
2
(1-e)
2
is
one more feature of this punishment equilibrium that deserves
At this equilibrium the domestic firm earns a present value
What must be noted
/4b(l-6).
is
that the domestic firm can never earn
less than this present discounted value of profits at
one involving maximal punishments
firm can guarantee for itself that
The reason
.
it
will
posting a price equal to [a+c-E (a-c) ]/2b.
the domestic firm
is
(a-
of
any equilibrum; even the
for this
is
that the domestic
2
2
earn (a-c) (l-£) /4b per period by
This
what gives quotas their
limit
on the punishability of
ability to
break the monopoly
price
With the value of the punishments in hand we now turn to the analysis of
the repeated game.
The price preferred by the domestic firm continues
(a+c)/2 while the foreign firm, which
higher price.
is
to
be
subject to a quota, naturally prefers a
Yet we concentrate on the question of whether the duopoly can
10
We do
sustain the "monopoly" price of (a + c)/2.
this for two related reasons.
higher prices are more difficult to sustain so that
First,
sustain (a+c)/2 they cannot sustain any higher price.
whether the introduction
in
if
the firms cannot
Second, we are interested
quota lowers equilibrium prices from their free
of a
trade level of (a+c)/2.
The
firm
is
sustainability of the price
expected to
allowed to
sell its
sell at this price.
entire quota,
have no incentive to deviate.
deviate since
is
If,
depends on the amount the foreign
for instance,
sell
In this case, the domestic firm will always
Similarly,
2
supposed
Instead,
/2b.
if
to sell
which case
it
sells its entire
in
which case
it
sells
Consider
c)
2
/2b.
(e
profits are y(ait
in
which
z
is
-
z
it
+
z
-
6e(2e
-
e(2e
1/2
2
z
)
exceeds
(a-c)/2b.
By
deviating, the foreign firm earns c(a-
-
1/2
2
e
)
]/2(1-6)
(3)
or:
V >
sells
smaller than or equal to
thus choose to deviate unless:
v)/2 < 6[y
-
its
quota, and the case in which
former case.
first the
It will
which the
entire quota at a price essentially identical to
its
1/2, in
1/2,
in
y(a-c)/b (0<y<l) in the collusive arrangement.
We analyze separately the case
(a+c)/2.
the foreign firm
deviates by undercutting the price slightly,
it
demand or
either total
if
we consider the intermediate case
So,
Then, by going along with the collusive arrangement,
c)
is
nothing, the domestic firm never deviates but the foreign
firm always deviates.
is
the foreign firm
e(a-c)/b, at this price the foreign firm will
earns more in the static game.
it
expected to
foreign firm
(a + c)/2
.
(4)
11
Note that for small
z,
must essentially equal
y
that the price will roughly equal (a + c)/2 whether
q
punished
Thus
is
Note also that the higher
is
the firms' discount factor,
at the
it
is
monopoly price
On
cheating.
sell
2
-
v/2
e
if
it
it
goes along
cheats
it
it
firm
sells
it
can
2
hand
Thus the domestic
-
If
(a+c)/2, while
of
the other
V/2 < 6[e/2
being
6,
the less
willing to sell without cheating.
consider the domestic firm.
that price.
is
essentially its entire
it
the foreign firm
Now
allowed to
goes along or
deviates unless
.
capacity.
it
The foreign firm knows
z.
sell
(a-c)(l/2
p)/b
-
(a-c)/2b at
2
earns only (a-c) (1-e) /4b per period after
is
deterred from price-undercutting
if:
/4]/(1-6)
or:
y < 6z
-
Equation
li
bz
2
/2.
(5),
(5)
which
is
must be relatively small
since higher levels of
y
if
valid also
when punishments
the domestic firm
is
to
are maximal,
shows that
be deterred from cheating
make cheating more attractive without increasing
its
cost to the firm.
If
(4)
and
(5)
contradict one another the monopoly price
To see under what conditions
sustainable.
this occurs
is
not
we combine the conditions
and obtain:
e{
1 -
6
-
6[(2e
-
z
2 1/2
)
-
e/2]} < 0.
(6)
12
For
small
t
enough we can neglect the term
second order) and the condition
But the domestic firm always requires that
than
t
for y equal to
;
When
clearly violated.
is
saw that the foreign firm must be allowed
strictly smaller
square brackets (which
in
z
it
z
small
is
is
of
enough we
to sell essentially its entire quota.
y
be smaller than
6e
which
is
has too strong an incentive to
undercut the monopoly price
For
brackets
up
z
entire quota
when
(6)
Now
cheats
z
until
cheats
consider the case in which
-
square
only
this the foreign firm cannot sell its
z
,
6
must exceed about
y)/2 < 6[v
t
-
exceeds 1/2 so that when the foreign firm
z
Then the foreign
earns (a-c) /4b.
(1/2
is
in
to be satisfied.
2
it
Yet this analysis
For this maximal applicable
.
The term
positive.
is
reaches .553.
z
beyond
since
to 1/2
it
the term in brackets
1
increasing in
is
relevant for
.62 for
and
between
£
+
e(2e
-
1/2
2
e
)
firm will cheat unless:
]/2(1-6)
or:
y > (l-6)/2
+
6e(2e
-
5e
-
2 1/2
e
(7)
.
)
which must now be satisfied together with
So we substitute (7)
(l-6)/2 < 6e(2e
For
demanded
e
(5)
for monopolization to be feasible.
(holding with equality) in (5) and obtain:
-
1/2
2
e
)
equal one, that
is
-
6e
2
/2.
(8)
when the foreign
at the competitive price,
(8)
firm can sell the entire quantity
requires that
6
exceed 1/2 as
it
did
.
.
13
under free trade.
of 6/(1-6)
Since the
(and thus of
6)
RHS
of
strictly increasing in z,
is
(8)
required to make
(8)
hold,
falls
the level
when
strictly
t
rises
To summarize the results
at a
more restrictive quotas (starting
section,
of this
quota which allows the foreign firm to
sell
demanded
the entire amount
at
the competitive price) monotonically reduce the ability to monopolize the
Note that, for a given
market.
domestic market
is
6,
the quota that gives the minimum price in the
strictly smaller than the one
When these equations hold with
with equality.
For lower values of
sustainable.
which
(or (8))
(6)
equality, monopoly
However,
the price falls.
z,
satisfies
is
just
the quota
if
is
reduced significantly more, the price starts rising again as the prices charged
even in the one-shot game
rise.
For
equal to zero, the monopoly price
t
is
reestablished.
It
must be pointed out again that the increased competition brought about
by quotas
is
not sensitive to the use of one -shot Nash punishments.
instance even
if
deviations
by the foreign firm lead
profits from then on (which for low
maximal punishment) the foreign firm
so as to refrain from deviating.
1/2
and any positive
is
z
will
This
is
a
this firm to
require that y equal at least (1-6)e
inconsistent with (5) for
We now briefly compare the results
analyzing tariffs in a similar setting.
of the foreign firm are higher,
domestic firm.
by the
The repeated game
has been analyzed by
6
equal to
z
in this setting is in
direct contrast to the conclusions of the static analysis reported
.
earn zero
much harsher punishment than the
The increased competition brought about by quotas
(1985)
For
in
A
to those
by Krishna
one would obtain when
tariff simply implies that the costs
tariff rate,
than the costs of the
which the two firms have different costs
Bemheim and Whinston
(1986),
who consider
optimal
.
14
punishments
firms
still
in
the style of
Abreu (1982).
Then, both the domestic and foreign
earn zero profits when they are being punished
This means that
.
the incentives to participate in the implicit agreement that requires that the
price (a + c)/2 be charged,
do not change as a result of the tariff.
Moreover,
Bernheim and Whinston (198G) show that the equilibrium that involves the highest
profits for the duopoly as a whole
now has
higher than
a price
This
(a + c)/2.
occurs because the profit maximizing price from the point of view of the foreign
firm
is
now higher.
So a tariff has the potential for increasing the domestic
price above the monopoly price.
Thus,
in the case of
(1965) about competition
reversed.
A
maximal punishments the classic results of Bhagwati
between
quota, because
punished effectively, makes
consequence.
it
foreign and a domestic firm, are precisely
makes
it
impossible for the domestic firm to be
while a tariff has no such
difficult to collude,
This raises the intriguing possibility that this
governments seem
However,
it
a
it
to prefer quantitative restrictions to tariffs
must be pointed out that the robustness
with respect to a tariff
is
is
the reason
12
of the
monopoly outcome
sensitive to the use of maximal punishments
.
If
instead, reversions to the Bertrand outcome are used, a tariff which raises the
foreign firm's costs barely below the monopoly price of (a + c)/2 makes the
monopoly price unsustainable.
The reason
for this
is
that to maintain this
price the domestic firm must give a sizeable fraction of the market (1-6) to the
foreign firm.
Thus
if
the foreign firm's costs are near (a + c)/2 the domestic
firm will actually earn more during the period of punishment (when
price barely below the foreign firm's costs) than
when
it
it
charges a
goes along with the
monopoly price
The results
of this section are sufficiently striking that
be qualified somewhat further.
The analysis hinges
critically
they deserve to
on the presence
15
Since this means that only the foreign firm can
one domestic firm.
of only
punish the domestic firm, this punishment ability
of a quota.
there were more domestic firms
If
revert to a price of
c
if
any
of
curtailed in the presence
is
in
the market,
they could
all
them deviated and thus keep the oligopoly
the monopoly outcome even in the presence of a quota.
On
at
the other hand,
if
the
domestic firms are capable of concerted cheating, the approximation considered
in this section of only
Ill
one domestic firm may be a good one.
Quotas and quantity competition at home
we assume that quantities are the
In this section
This has two consequences.
when
competitor
amount.
neither firm eliminates the sales of
First,
deviates.
These are maintained
in quantities,
which
is
prices are the strategic variables.
The
its
at the implicitly colluding
Second, punishments are characterized by reversion to the single period
Nash equilibrium
when
it
strategic variables.
a
much milder form
of
punishment than
13
results of this section are directly comparable to Davidson's (1984)
analysis of tariffs, in which he also assumes that quantities are the strategic
variables.
collusion
to import
We show that the result that quotas lead
now only holds
very
little.
for
to less sustainability of
very restrictive quotas that allow the foreign firm
For larger quotas, we find that collusion
anything, enhanced by the quota.
Thus our
Davidson's (1984) since he shows that
is,
if
results here are similar to
small tariffs
(which correspond to large
quotas) enhance collusion while large tariffs do the opposite.
Again, the firms are assumed to be able to monopolize the domestic market
in the
absence of quotas.
earns (a-c) 2 /8b.
If
By going
along with selling (a-c)/4b each, each firm
either firm were to deviate
it
would choose
to sell 3 (a-
16
c)/8b which would give
woulci revert to
From then on, the firms
2
/64b
in
the period of the deviation.
Cournot-Nash equilibrium so that each
2
would
will
profits of 9(a-c)
it
sell
(a-c)/3b and earn profits of (a-c) /9b per period.
be deterred from cheating
[9/64
-
2
l/8](a-c) /b <
which requires only that
Now
6
6
So,
if:
[1/8
-
1/9] (a-c)
2
/b(l-6)
,
exceed 9/17.
consider a quota of E(a-c)/b.
If e
is
above 3/8, so that the quota
exceeds the best response to the monopolizing level of output,
So suppose that
it
is
t
is
between 3/8 and 1/3.
affect the foreign firm's ability to
Then the quota does not
punish the domestic firm from deviating
(since each firm produces (a-c) /3b
in
Cournot-Nash equilibrium.
makes the foreign firm's own deviation less profitable since
cannot now exceed t(a-c)/b.
it
done in the absence of a quota.
the collusive output
is
For suppose that
(a-c) /4b,
isn't constrained,
somewhat lower.
since
it
6
is
it
when
it
now earns
will still
it
may
be
With
e
not only willing to go along with
is
less
when
may be
it
deviates than
if
it
its
does
if
it
sales are
possible to sustain the monopoly
output by having the foreign firm produce less than half of the
domestic firm produce more than half.
actually
just below 9/17 so that
not be induced to cheat even
This means that
it
firms can
this cannot
not sustainable in the absence of a quota.
between 3/8 and 1/3 the foreign firm
selling
when the
Furthermore,
possible to sustain this monopolistic outcome even
However,
this deviation
So such a quota has no effect
sustain the monopolistic outcome without a quota.
make
has no effect.
it
lower than this level but higher than the Nash equilibrium
that
level of sales, i.e.
each firm
In essence, the "excess"
total,
and the
unwillin gness
cheat on the part of the foreign firm can be used to free up the constraint on
to
17
the domestic firm.
for a quota to promote competition
Thus
equilibrium (a-c)/3b,
detail.
t
must be less than
must be below the Nash
it
We now analyze such
1/3.
quota
a
in
first look for the lowest level of sales on the part of the foreign
We
firm which makes
willing to go along with an equilibrium
it
Then we analyze whether the domestic
(a + c)/2.
by the foreign
this price given these sales
Suppose that the foreign firm
and that the firms attempt
to sustain the
is
willing to go along with
is
firm.
supposed
is
firm
whose price
y(a-c)/b in equilibrium
to sell
monopoly level
of output.
We now
derive the minimum level of y for which the foreign firm will not want to
By cooperating
deviate.
2
output at a price of
sells its
it
The domestic firm
earns (a-c) y/2b.
sells that
not sold by the foreign firm, or (a-c) (1/2
requirement that
deviates
it
sells
-
We
y)/b.
E(a-c)/b.
Thus when
it
deviates
2
it
in y.
This
because the higher
when they both go along with
game.
its
is
y the
earns (a-c) e(1/2
it
this level of output,
is
i.e.
U[l/2
if:
deviates are increasing
the proposed equilibrium output.
less than the
or (a-c) (l-c)/2b
Hence the foreign firm
2
+
t
lower are the domestic firm's sales
Nash equilibrium value
foreign firm sells E(a-c)/b while the domestic firm sells
(a-c)
-
deviation the market reverts to the equilibrium in the one -shot
Since the quota
E)/2b.
show below that the
be smaller than 1/3 implies that when the foreign firm
z
Note that the profits to the foreign firm when
After
it
portion of the monopoly output
y)/b.
is
(a + c)/2 so that
+
v
-
e]
-
will
.
its
Thus the foreign
-
e
(a-c)/3b, the
best response to
2
firm earns (a-c) e(1-
be deterred from deviating
2
y/2}/b < 6(a-c) [y
of
if:
(1-e) ]/2b(l-6)
18
y > t
6e
-
2
2 E (l-6)]
-
/[1
Note that for values of
when
near zero, the foreign firm
to sell
it
the quota
is
nearly
firm's interest to sell as
deviating unless
it
much
allowed to
is
that for sufficiently small
already selling
all
as
that
can and
Then
it
The reason
it
always in the foreign
is
cannot be deterred from
essentially
sell
deviate unless
price does not differ appreciably from
small,
it
will
entire quota.
its
even during the punishment phase.
(a + c)/2,
is
(9)
.
t
the implicit agreement allows
for this is that
14
its
This means
entire quota.
z
the foreign firm has no punishment power since
it
is
Given
allowed to bring in.
surprising that we establish next that for sufficiently small
this,
z
,
it
is
it
not
the equilibrium
cannot involve the monopolization of the domestic market.
Suppose that the
Consider the domestic firm's incentive to deviate.
foreign firm
selling
is
best response to this,
c)
2
(1-y)
would
2
(a-c)y/b as above.
The domestic firm would
(a-c) (l-y)/2b,
deviated.
/4b in the period in which
sell
it
if
it
It
would then earn
From then on the foreign firm
deviated.
2
(1-c)
2
/4b per period.
So the domestic firm would be deterred from deviating only
-
2
y)
/4
-
(1/4
-
increase
is
y,
if:
y/2) <
6[l/4
It is
(a-
(a-c)c/b so that, by an argument similar to that above, the domestic
firm would earn (a-c)
(1
sell its
-
y/2
2
-
(1
-
/4]/(l-6).
(10)
easy to see that in spite of the fact that profits from deviating
when
y falls,
as long as y
is
the inequality in (10)
smaller than 6/(1-6)
is
more
(which
is
likely to hold the smaller
true by assumption).
The
.
19
reason for this
that the principal effect of a decrease in y
is
is
that the
domestic firm earns more by going along with the collusive agreement.
make the domestic firm
firm sell only as
much
as willing to go along as possible
as
required to make
is
Substituting this expression for
2
(1-26
2
3
-
(l-6)[4(l-6
E
)
+
we divide
If
4
e
2
+
)
+
by
z
this expression
2
by
(after dividing
range of
z
.
the foreign
hold with equality.
(9)
multiplying by (1-6) [1-2e (1-5)
2
]
and
26]
(1-6)[4(1-6)
obtain (1-26 )<0, which requires
of (11)
(10),
let
to
the inequality (10) becomes:
collecting terms,
e
y in
we
Thus
2
z
This means that
6
2
46(1-6)
2
+
6
< 0.
]
and take the
limit as
if
6
exceeds .71
it
is
z
goes to zero we
Furthermore, the LHS
smaller than about .71.
decreases monotonically with
)
(11)
z
in the relevant
always possible to sustain
the monopoly price
If,
instead,
6
between 9/17 and
is
.71,
the monopolistic outcome
sustainable with free trade but not with a sufficiently small quota.
is
In this
case of linear demand, however, the quota must be very small for this result to
hold.
If
we take the
free trade (9/17)
is
,
smallest discount factor consistent with collusion
under
one can demonstrate numerically that the monopolistic outcome
not sustainable for
z
equal to .01 but
These are such low values
of
z
is
sustainable for
z
equal to .015.
that even the noncooperative solution looks
essentially identical to the monopolistic solution.
IV Quotas and competition abroad: Constant demand
We now turn
to the implications of the existence of a
quota at home for
.
20
competition abroad.
Sections
and
II
In this section
to
III
we adapt the arguments presented
in
argue that under certain circumstances they can be
interpreted as providing conditions under which competition increases abroad.
We assume that the domestic and foreign markets are
demand curves are given by
a monopolistic equilibrium in
as
6
equals at least 1/2
least 9/17
when
becomes
which
both firms have infinite capacity, there
(a + c)/2 is
when prices are
charged
in
is
both markets as long
the strategic variables and
6
equals at
quantities are strategic variables.
Now assume
k(a-c)/b.
If
(1).
identical so that their
that the domestic firm actually has limited capacity given by
So marginal cost equals
infinite.
If
k equals
c until
the firm produces k(a-c)/b and then
for instance,
2,
the domestic firm can supply the
quantity demanded at the competitive price in both markets
section is that this capacity
is
The idea
.
of this
sufficient to maintain the monopolistic outcome
in both countries with free trade.
However, with
a
quota that forbids
all
imports into the domestic country, the domestic firm must devote a larger
fraction of its capacity to producing for the protected domestic market.
then effectively has a very small capacity for selling abroad.
It
This has the
same effect abroad as would a quota imposed by the foreign country.
We
is
first treat the case in
which prices are the strategic variables.
equal to 1/2 (the benchmark case) and k
is
less than 2,
If
6
the two firms cannot
maintain the monopolistic outcome in both markets even in the absence of a
quota.
This results because the domestic firm does not have sufficient
punishment power
outcome can
to
at best
keep the foreign firm from cheating.
be sustained for values of
6
Thus the monopoly
that are greater than that
required to sustain monopoly under free trade without capacity constraints
We consider how large the
"critical" value of
6
must be for different
values of k. Note first that k must be at least equal to 1/2 for the problem to
21
be interesting since otherwise the domestic country
own market when
its
suppose that k
is
(the world market
value of
that market becomes closed to foreign firms.
value of
critical
Now
6
is
6
about
is
.65.
If
consider the effect of a quota.
important to notice
is
is
at
instead k
is
equal to one, the
is
is
left
The domestic
over
This
firm sells (a-c)/2b at
to sell in the foreign
that the domestic firm
is
it
is
never willing to
selling
abroad
it
z
we can use
(6)
6
that sustains the monopoly outcome abroad.
value of
6
strictly
to a value of
is
.65 in the
1
is
Letting k-1/2 be
.
minimum
For k barely above 1/2 no
sufficient to maintain that outcome
absence
of a quota).
For k equal
(in contrast
to 1,
a value of
.65
required 'when only .62 was required under free trade.
We now turn
to the case in
which quantities are the strategic variables.
Here the monopoly outcome only broke down for tiny quotas
domestic firm has a capacity of (1/2 +e) where
z
is
other hand, once the domestic market
firm
is
is
willing to sell at most
below
.71,
z
is
abroad.
Hence
.
negligible,
outcome can be maintained with free trade as long as
6
less than
(or (8) for k bigger than 1) to discover the
value of
below
if
effectively
is
capacity constrained abroad with a capacity of (k-1/2) (a-c)/b
denoted by
sell
punishing the foreign
This means that the domestic firm
sold at home.
country.
so because the marginal revenue from selling a unit
is
home always exceeds the marginal revenue from
(a-c)/2b
equal to k/2
z
sustainable in the world as a
more that (k-1/2) (a-c)/b units abroad, even when
firm for deviating.
So first
about .62.
home and has (k-1/2) (a-c)/b units
What
with
(6)
in
equivalent to twice the domestic market) to obtain the
is
This value of
whole.
Then we can use
barely above 1/2.
for which the monopoly outcome
6
unable to meet demand
is
6
if
the
the monopoly
exceeds 9/17.
On
the
closed to the foreign firm, the domestic
This means that for
z
small
the monopoly outcome won't be sustainable abroad.
enough
if
22
V Quotas and competition abroad: Fluctuating demand
This section considers
country tends
collusion
is
somewhat different model
in
which
quota
a
The idea
country.
to increase competition in the other
in
one market, they are small
In this setting,
one
that
is
in the
other market and
the overall incentive to cheat (taking the two
markets together) are fairly constant over time.
If
one looks at either market
Then
separately, however, the incentive to cheat fluctuates over time.
in
in
maintained under free trade only because whenever the net benefits
from cheating are big
viceversa.
a
if,
as
Rotemberg and Saloner (1986), future punishments are largely independent
the current period's incentive to cheat, then those punishments
insufficient to deter cheating
high.
It
if
the current incentive to cheat
may be
is
particularly
thus becomes more difficult to prevent them from cheating.
imposition of a quota has the effect of separating the two markets,
makes
implicit collusion
Our
specific model
more
of
it
Since the
thus
difficult.
assumes that demand
is
random
in
both countries, as in
Rotemberg and Saloner (1986), and negatively correlated across markets
Bernheim and Whinston (1986).
as in
To simplify the analysis we assume that there are
only two states of demand (high and low) whose realizations are independently
distributed over time in any one country and perfectly negatively correlated
across markets.
So when demand
In this section
is
high in one market
we assume that there are N/2 firms
do not restrict attention to linear demand curves.
symmetric, the monopoly price in the high state, p
it
in
is
low in the other.
each market while we
Since the countries are
,
is
the same in both
countries and similarly for the monopoly price in the low state, p
the analysis by ensuring that these two prices p
.
We
start
and p are sustainable with
That
free trade.
is
we consider
situation in which
a
all
N firms
sell
markets and derive the condition under which no single firm wants
from charging p
in the
country
in
which demand
is
high and p
in
both
to deviate
in
the other.
Prices are again the strategic variable so that any deviation triggers a
permanent reversion
to the static
equilibrium in which price equals marginal
cost.
Let
be the total monopoly profits in the market with high demand and
it
those in the market with low demand.
former
As long as p exceeds marginal
even when demand
charging p
is
A
high.
firm that deviates from the monopoly
outcome undercuts the price charged by the others slightly.
the market in which
deviating
it
+
demand
gives up both
these profits.
U
the
cost,
bigger than the latter since the monopoly always has the option of
is
(tt
v
is
now and
Thus each firm
TrVl
"
high and
1/N) < 6(n
U
is
in the other.
On
thus earns
it
in
the other hand, by
in all future periods its share
(1/N) of
deterred from deviating as long as:
1
+
it
It
tr
)/N(l
-
(12)
6)
or:
N
-
1 < 6/(1
-
6).
(13)
We assume that either the number
factor big
enough that
(12)
of firms is small
enough or the discount
holds so that with free trade the two markets can be
completely monopolized.
We now consider what happens when no imports are allowed
country.
This drastic quota has two effects.
First,
fewer (N/2) active firms in the domestic market while
into the domestic
it
means that there are
N
firms continue to sell
24
in the
As
foreign market.
the conditions under which monopoly will be
a result,
sustainable will be different depending on which market has high demand.
Rotemberg and Saloner (1986) show,
in this
sustain collusion
when demand
monopolization
is
possible,
monopolization
is
possible in the high
Consider
/N abroad.
.
Thus
low.
always easier to
is
it
whether complete
to see
we study the conditions under which such
Any
demand
state.
country when demand
first the foreign
firms try to charge p
tt
is
setting
Suppose that the
high.
is
firm that undercuts this price earns
The losses from the ensuing competition are smaller
may
countries since domestic countries
also lose
they lose the expected value of profits in the future
.
Since
If
U
(1
-
1/N)
>
6(tt
U
Note that the E.HS
of
1
+
of
tt
they deviate
(tt
is
and
(14)
)/2N(1-6)
(14)
(14)
.
equals half the R.HS of (12).
The costs
Yet, insofar as
than half of the LHS
to hold simultaneously.
of
(12)
In particular,
tt
so that
if
(12)
is
bigger than
it
is
in the
tt
When
(14) holds,
it
is
terms
foreign
if
,
the
if
the profits
is
impossible to sustain monopoly abroad
high state of demand so the quota increases competition.
It is
LHS
holds as an equality
high and low demand states are different, the inequality in (14)
always satisfied.
in
possible for (12)
(so that monopolization is just sustainable with free trade) then
in the
)/2.
+tt
foregone future profits from deviating are half what they would be
(14) is bigger
home.
at
if:
firms had access to both markets.
of
for foreign
demand
independently distributed over time, these expacted profits equal
So, a foreign firm will deviate
instead of
it
any profits they make
So we concentrate on the incentive to deviate of foreign firms.
tt
As
easy to show that when (12) holds
it
is
always possible to sustain
.
monopoly abroad
in the
low state of demand.
(12)
ensures that deviations
Yet
when demand
would prevail
if
is
low,
will
This results from the fact that
not take place even
when demand
constant.
is
expected future losses from deviating exceed those that
demand were expected
to stay
from the fact that even when (14) holds,
it
is
This
low forever.
still
possible to sustain
This can be seen as follows:
substantial profits in the high state.
turn results
in
Let the
sustainable price (the highest price that can be charged without inducing any
firm to deviate) in the high state be p
argument analogous
S
1/N)
-
(1
S
=
6(tt
s
it
tt
Then by an
.
will just
if
1
+
and the ensuing profits
one used to derive (14), a foreign firm
to the
refrain from undercutting p
tt
s
)/2N(1-6)
or:
t
s
so that
1
=
tt
s
it
/[2(N-1)(1-6)/6
equals
1
tt
when
-
(15)
1]}
(12)
holds as an equality and exceeds
when
it
it
holds
as a strict inequality
So far, we have shown that under plausible conditions the foreign market
becomes more competitive while
still
retaining substantial profits.
This result
does not hinge on the fact that prices are used as the strategic variables. As
in
Rotemberg and Saloner (1986),
sustain
when demand
is
high.
it
As we showed there, this
quantities are the strategic variables
We now turn
only requires that collusion be difficult to
when demand
is
also true
when
is linear.
briefly to an analysis of the domestic market.
If
domestic
firms were to deviate from the collusive arrangement they could be punished in
26
In this case,
both markets.
On
once.
deviating firms would deviate
both markets
in
at
the other hand we have already derived the net benefits from deviating
abroad under the assumption that punishments would only be meted out abroad.
complete the analysis we thus look at the net benefits from deviating
domestic market alone under the assumption that firms
Then, the
the domestic market for doing so.
will
To
the
in
only be punished in
total net benefits to deviating are
the sum of these two net benefits.
Suppose that the domestic market
high, a deviating firm would earn
then lose
It
its
is
instead of
it
share of total expected profits in
would thus be deterred from deviating
U
tt
(1
-
fully monopolized.
2/N) <
The RHS
meted out only
6(tt
of (16)
in
u
is
+
share (2/N) of
its
its
LHS
of
in addition,
would
on.
(16)
the same as the
substantially smaller than the
It
.
is
if:
itVnCI-S).
the share we previously considered.
ir
own market from then
RHS
of
On
While the punishment
(12).
one market, the share of each firm
one market alone while,
When demand
in this
the other hand the
(12)
market
is
LHS
(16) is
of
is
double
since deviations are taking place in
the net benefit
is
smaller because there
are fewer firms active in the market and so the firm's share was larger to start
with.
So
if
(12) is satisfied
strict inequality.
In this case
showing that
monopoly
is
(even with equality) then (16)
is
satisfied as a
which demand
is
high at home.
This applies to the case
in
demand
is
low in the foreign market and our previous analysis
there
is
a net loss
from deviating there applies.
sustainable in the absence of a quota
home market, then
When demand
it
is
is
when demand
Thus
is
if
high in the
also sustainable with the quota.
low at home the corresponding condition at home has the same
.
27
RHS
as
but an even smaller LHS
(16)
.
So the net benefits to deviating are
negative here as well.
summary, when
In
holds, monopolization
(12)
always possible
the
in
since protected domestic firms strictly
Indeed,
protected domestic market.
is
prefer to go along with monopolization than to deviate when (12) holds as an
we can conclude that protection may make monopolization
equality,
home when
The
was not feasible with free trade
fact that collusion
potential,
To do
it
this
in a richer
we must
II.
becomes easier for the domestic firms has the
model, to make monopolization feasible abroad as well.
allow the set of firms that charge the lowest price to split
the market unevenly as
Section
feasible at
is
done
Bernheim and Whinston (1986) and as we do
in
Bernheim and Whinston (1986) show that the possibility
in
such
of
arbitrary divisions allows "punishment power" to be shifted from a market in
which there
is
more than enough
monopoly outcomes.
to
one in which there isn't enough to sustain
In our example, the domestic firms can offer the foreign
Domestic firms would
firms a share higher than 1/N of the foreign market.
tolerate this for fear of losing their domestic profits
if
they deviate abroad.
Foreign firms would earn more when they go along with the collusive arrangement
and thus would be less
potentially achievable
willing to deviate
when demand
does not work, however,
there
is
when N
is
.
In this
fluctuates relatively
equal to
2
home when
it
little.
deviates abroad.
Then
is
This approach
so that as in Sections
only one firm in the domestic market.
fear retaliation at
way monopolization
II
this firm does not
and
need
Its incentives to deviate
abroad are the same as those of foreign firms and giving the foreign firm a
disproportionately large market share only increases the domestic firm's
incentive to deviate abroad.
III
to
.
VI Quotas and Competition in Both Markets
competition at home (as in Sections
Sections IV and V)
fluctuating
possibility that a quota will increase
we consider the
In this section
II
and
The model we use
.
demand model
of Section
III)
as well as abroad
to illustrate this possibility is the
V except
to permit competition to
that,
increase at home, we consider a strictly positive quota.
analysis
we
restrict attention to the case in which
one firm in each country, and
6
is
(as in
equal to 1/2,
N
Also,
equals
so that
to simplify the
so that there
2,
monopoly
is
is
just
sustainable in both markets with free trade.
Since any firm that deviates in one market will be punished in both, any
deviating firm will undercut the implicitly agreed upon price in both countries.
So the incentive to deviate in any one country can be thought of as the sum of
the net profits from deviating at home (i.e. the benefits from deviating at home
minus the present value
home) plus the net profits from
of the losses at
deviating abroad
Suppose that the foreign firm has
its
net profits from cheating
when demand
those
is
when demand
notice
fall
is
that,
as
when demand
s
1
when demand
Similarly the domestic firm's
s
high are (3a/2-l/2)ir -(l-a)ii /2
we increase
by the same amount
a,
s
low are (3a/2-l/2)ir -(l-a)ir /2 and
1
is
s
(1-3cx/2)tt -an /2 while
1
is
Then
market.
1
low are
is
high are (l-3a/2)Tt -air/2.
net profits from cheating
those
a share a of the foreign
.
The important thing
the foreign firm's net profits from deviating
Thus,
as the domestic profits from deviating increase.
as long as in each state the total net profits from deviating (the
firm's net profits from deviating) are zero,
redistribution of shares that will
to
sum
each
of
there will always be a
make both firms
We therefore now concentrate only on the
willing to go along.
total incentive to deviate
.
In
29
the foreign country this
s
So unless
state.
abroad
is
ti
is
(tt
home
at
We can then write the quota
enough
small
1
equal to
the low state and
in
-ti
tt
(
si )/2
the total incentive to deviate
t
in
-tt
is
the high
positive
one of the two states.
in
Suppose demand
is
Is )/2
foreign firm
is
1
t
given by P=a -bQ (where
is
in
state
Suppose that
less than 1/2.
is
allowed to
sell
y (a
as equal to
i
z
(a
i
equals either u or
-c)/b where
at the
if
the quota
monopoly price the
Then we can obtain from
-c)/b.
1)
(4)
the
foreign firm's net profits from deviating while from (5) we can obtain the
Once again by varying
domestic firm's net profits from deviating.
decreased incentive
to deviate is the other's
focus on the total incentive to deviate.
1
(a -c)
2
1
[E (l-6)
+
6e
i2
/2
-
1
1
1/2
l2
)
given by:
is
]/b.
Note that the condition that (17) be positive
violated which
There
thus a net incentive
is
Now
if
we proved for the case
we try
monopoly
to maintain
s
1
(it
-ti
abroad.
which
to deviate in the
6
(17)
is
the same as that (G) be
is
equal to 1/2 in Section
II.
domestic market.
consider the net incentive to deviate in the two markets taken together
expression in (17) with
by
in
one firm's
increased incentive so we can
This
6£ (2£ -E
y
)
i
at
home.
replaced by
1
This
given by
when demand
plus the expression in (17) with
At least one
is
i
is
s
(tt
1
-tt
)
plus the
high abroad.
replaced by u when demand
of these net incentives is positive so that
it
given
It is
is
low
is
impossible to sustain monopoly at home.
Now we
consider whether
it
is
possible to have an outcome different from
the monopoly outcome at home such that the net incentive to deviate at home
so negative that the
state.
That
monopoly outcome can be maintained abroad even
this is unlikely for small quotas should
in the
be apparent since, then,
is
high
:
30
many
the static game gives about as
profits as the monopoly outcome.
This
suggests that there are no prices between the monopoly price and those that
prevail under static competition for which the costs of reverting to competition
are high.
We now consider
and study the overall profits from deviating
p at home
The amount demanded
home.
at
a price
(a-p)/b and, since the distribution of this demand across firms
this price is
immaterial,
We consider
more formally.
this issue
we
the foreign firm sell
let
sells the difference
e
(a
at
is
-c)/b while the domestic firm
between the quantity demanded and the quota.
The domestic
firm's net profits from deviating are then given by:
i
E
1
(a
The foreign
the static game
t
1
1
-c)(p-c)/b
-
1
(p-c)[(a -p)-E(a -c)]/b
+
(a -c)
by cheating but
firm gains nothing
2
its
(l-E
2
I
)
/4b
.
cost from reversion to
is:
(a -c) [p
-
o]/b
so that the overall benefits from deviating are
1
1
a
(p-c)(E (a -c)
whose minimum
is
-
(a
1
-p))/b
reached for
+
(a -c)
2
(l-c
2
1
)
/4b
1
+
E
1
(a -c) (o-c)/b
a p equal [a +c-t (a -c)]/2.
There, this
expression equals:
l-z
2
/2
+
e[1
-
(2e-e
2 1/2
)
+
2
E
2
]}(a-c) /2b
(18)
31
which
is
maximum
negligible for small
t
The
.
cost from deviating at home,
fact that
<t>
,
(18)
small.
is
means that the
small
is
Yet,
maximum
profits in the
high state abroad must satisfy:
S
(tt
-
1
•n
)/2
-
so that for small
to
<t>
$
it
<
impossible to make
is
s
tt
m
equal to
it
as
would be required
monopolize the foreign market.
VII.
Conclusions
We have shown
in a variety of
models
in
which
implicit collusion is
important that quotas have the potential of increasing competition, not just in
the domestic market, but also abroad.
those obtained
when there
is
These results stand
a single domestic firm
in
sharp contrast to
and goods from abroad are
supplied competitively (Bhagwati (1965)) as well as those obtained when domestic
and foreign firms act as
The
in a
one -shot game.
intuition underlying our seemingly paradoxical results are not
restricted to quotas.
competition
less viable.
when
Indeed, any action by the government which makes
collusion breaks
down more
profitable makes collusion itself
Thus Government imposed minimum price
floors could
have precisely
the same effect.
One natural question
to
ask
is
whether our results depend
critically
on the
existence of implicit collusion or whether they can also be generated in other
models of strategic interaction.
would be modified
in the
In this conclusion
One particular concern
is
how our
results
presence of entry.
we sketch an argument that shows that quotas
also tend
.
32
to
increase competition by encouraging the entry of new (domestic) firms into
The setting we consider
the industry.
to that
developed
They analyze an industry with an established foreign
and Kyle (1985).
in Dixit
very closely related
is
firm and a domestic potential entrant.
They show
that
it
possible that
is
if
the domestic country prohibits imports that there will be more competition in
the foreign market.
This occurs because the potential domestic firm may find
unprofitable to enter
if
By protecting
it
it
must compete with the foreign firm
in the domestic
market
(while
market), the domestic government provides
its
fixed costs.
It
is
competes
both markets
in the foreign
with sufficient revenues to cover
then finds entry attractive, enters, and increases
Since the domestic market was monopolized in any event, the
competition abroad.
domestic country
it
it
in
it
made better
off since
consumer surplus
is
unaffected and
rents have been shifted towards the domestic country.
This analysis can easily be extended to include the case of a nonzero
Moreover,
quota.
it
not only does competition increase abroad, but
in that case,
Consider the case where the firms
increases in the domestic country as well.
compete with outputs as their strategic variables
if
entry occurs.
Suppose
further that the domestic potential entrant will find entry worthwhile
earns
its
K
is
greater than
than the domestic monopoly revenues.
quota which
is
its
Now
domestic Cournot revenues and less
let
the domestic government impose a
such that the domestic firm earns revenues
best-response to the quota.
to enter
it
competitive revenues abroad plus some amount, K, of revenues in the
domestic market, where
its
if
and does
so.
Once
it
of
K when
In that case the potential entrant
is
it
produces
willing
has entered the firms produce their Cournot
outputs in the foreign market, and in the domestic market the foreign firm
its
quota while the domestic firm
increases in both countries.
1G
sells its
best-response
to that.
sells
Competition
33
The
protect
Dixit
its
and Kyle (1985) insight
domestic firm
above analysis points out,
insulation in
its
order
in
is
own market
to
that the domestic country
is
encourage
it
to enter.
that the domestic firm
in
may have
However, what the
may not need complete
which case domestic welfare can be improved even
further by allowing some foreign competition.
In contrast to our analysis in Section
instrument used by the domestic country
required
is
is
II,
a
this result holds
quota or a
whether the
tariff.
All that is
that the domestic government increase the domestic profits of the
domestic potential entrant to the point where entry
tariff raises the costs of the foreign firm,
is
they make
attractive.
it
Since a
"less aggressive" in
the domestic market which in turn makes that market more attractive to the
potential entrant.
Thus when considerations
to
of implicit collusion or of potential
important, quotas can lead to greater competitiveness.
entry are
.
34
APPENDIX
appendix we consider the one-shot game
In this
strategic variables and in which one of the firms
establish the claims
=
[a
(a + c)/2
=
and
(a-c)(2s:-0
-
In equilibrium,
1/2
is:
/2
charged by both firms with probability zero
the foreign firm earns
We discuss each
(i)
will
is
never choose
to
it
charges
its
highest price,
But the price that maximizes
which
(ii)
is
its
The lowest price charged
will
profits
is
than
Hence the domestic frrm knows that
be undercut with probability one.
when
if
the foreign firm earns
it
it
is
undercut for sure
z
-
(a-c) (2s-e2) 1/2/2
(o-c) (a-c) /b
(a)
and
is
s,
First,
it
.
This price
(iii)
In
.
which the domestic firm would be willing
could thereby be sure to capture the entire market
undercut the foreign firm)
properties.
(a+c)/2
o -
Consider the lowest price
charge
it
sell less
therefore the highest price charged by the domestic firm.
equilibrium,
to
s
charge a price higher than the
highest price charged by the domestic firm.
when
(o-c) (a-c)/b
given that the foreign firm can only
First note that,
it
z
of these in turn:
The highest price charged by both firms
the domestic firm,
is:
(a-c) ]/2
z
this lowest price is
(iii)
We
namely that:
in the text,
The lowest price charged
(ii)
o
-
c
+
which prices are the
capacity constrained.
is
The highest price charged by both firms
(i)
s
made
in
a
(i.e.
has a number of interesting
cannot be the case In an equilibrium that the foreign
firm always charges a price strictly above
would always make more than
its
profits at
a.
s
If
it
did so the domestic firm
by charging
a price
between
o
and
the lowest price charged by the foreign firm, which contradicts the fact that
is
highest price.
its
domestic firm,
Secondly,
a
if
the lowest price charged by the
is
cannot be charged with positive probability since that would
it
mean that the foreign firm would benefit from undercutting
demand
of
unless o
it
at a so that
it
would be unwilling
charge
to
On
a.
the lowest price charged by the domestic firm,
is
with positive"
This would imply that the domestic firm couldn't expect to capture
probability.
all
s
the other hand
the foreign firm (who
seeks only to undercut the domestic firm) would never charge a price of
who charges
must be lowest price charged by the domestic firm
o.
So
with zero
it
probability.
Since the foreign firm must make equal expected profits at
prices
it
charges in equilibrium we can compute
computing
its profits
being able to
If
its profits
sell its
when
charges
it
from charging
=
(a-c)
is
a price
2
/4b
-
sure to
[p
-
does this
(a-c)(2£-£
1/2
2
)
sell
the entire
z
is
is
assured of
(o-c) (a-c)/b
.
demand given by
2
-
2
(a+c)/2
it
p are given by:
which must equal (a-c) (1-e) 2 /4b
a =
When
/2.
the
expected profits simply by
entire quota so these profits equal
the domestic firm
(p-c)(a-p)/b
a.
its
all
(a+c)/2] /b
at a price of o.
(Al)
So
a
equals
17
(A2)
:
(1),
a
3d
FOOTNOTES
proven by Mookerjee and Ray (1986) and Bernheim and Whins ton (198G)
1
This
2
The notion
is
of
segmented markets
is
discussed, for the case of static
imperfectly competitive models, in Helpman (1982).
3
For the specific demand and cost functions used here
instance,
4
The
in
if
assumptions
5
it
6
The
is
they can also sustain
the division of the spoils that
easiest to collude.
punishment
is
not restrictive since, with constant
The second assumption
Each individual firm
This
is
the firms can sustain any collusive outcome,
the monopoly outcome.
makes
is
large.
is
able to resist a larger temptation to deviate
Thus
collusion
what leads Abreu (1982)
easier the larger
is
to focus
ability of tariffs to shift rents
a
the
on the maximal possible punishments.
from foreign to domestic firms
In other words, following a deviation,
if
the punishment.
is
considered in a static model by Brander and Spencer (1985)
7
presented, for
Rotemberg and Saloner (198G).
first of these
demand,
is
it
is
.
the owner of the deviating firm signs
contract to pass ownership of the firm to a third party for a nominal fee
(which can even be less than the scrap value of the plant)
specifies that the current
owner
date at which ownership passes.
will
.
The contract
receive the profits that accrue until the
Then the current owner and the
involved in a finite -horizon game which has as
its
rival are
unique equilibrium the
outcome from the one -shot game.
8
Such signalling might be credible, for example,
managers who are "old-fashioned"
in the
if
there are some types of
sense that they have never conceived of
maximal punishments, and who therefore revert to the one-shot Nash equilibrium
.
5
I
in
any reversionary period.
9
This result does not depend on the use of one-shot Nash punishments and can
The reason
be derived also with maximal punishments.
mentioned above, the domestic firm can be sure
z
)
is
this loss.
this loss,
below
To obtain
It
a
must then be compensated
at least
(a-c)
as
2
(1-
it
in later
must charge
a price
periods for taking
lower bound on this price v we assume that, after taking
2
at least
(a+c)/2
-
[
(a-c)/b]
U/2
+
[
6(2e-e )/(l-6)
order the foreign firm earns E(a+c)/2 even when
Footnote
7
does
it
Then
the domestic firm earns the entire monopoly profits (a-c) /4b.
v must equal
10
o.
that,
is
firm earn less than
one-shot Nash equilibrium for at least one period
v which
first
earn
To make the foreign
/4b(l-6) at any equilibrum.
at the
to
for this
establishes that for
for the foreign firm not to cheat
valid in this case as well,
z
small,
y
it
is
1/2
]
is
so that,
to
being punished.
must be essentially equal
even with maximal punishments.
monopoly
]
unsustainable for small
z
to
z
Since (5)
is
even with
maximal p uni shments
11
This conclusion, which
is
also derived
by Mookerjee and Ray (1986)
for a
slightly different game,
is
somewhat striking since the domestic firm makes
there
is
static competition
positive profits
less
if
by simply charging
than the foreign firm's marginal cost inclusive of
that yields zero profits for the domestic firm
is
The equilibrium
constructed as follows:
the first period the domestic firm charges a price, u,
(in that
tariff.
a price just
In
so low that its losses
period) are exactly equal to the present value of the profits from the
one-shot game from the second period on.
slightly higher than u.
The foreign firm charges
The foreign firm charges
this price
a price just
every period until
the domestic firm charges u for one period; thereafter both firms revert to the
one-shot outcome each period.
38
For evidence on this fact and some alternative explanations see Deardorff
12
(1986)
.
This hinges critically on the fact that we do not use Abreu's (1982) severe
13
These are however
punishments.
difficult to characterize in this case.
This inequality can be used to demonstrate that the foreign firm always
14
sells
when
the entire quota
market, the domestic firm
amount
this
is
it
the quota
it
greater.
Then by
To show that
.
show that
suffices to
this inequality
is
z
it
less than
15
impossible
if
z
is
(l/4+y/2)
would also have
expression whose maximum, reached when
z
is
amount
this latter
This means that
half the E.HS of the inequality.
is
the firms are monopolizing the
(a-c) (1/2-p) /b so that the best response to
selling
(a-c) (1/2+y) /2b
is
When
deviates.
z
to
Suppose
.
it
were
be greater than 1/4 plus
must exceed 1/2 minus an
1/3 and
6
is
This
equals 1/18.
1,
less than 1/3.
Bernhein and Whinston (1986) consider the more general case
They show
correlation.
larger than
is
that the sustainability of collusion
of imperfect
a monotonically
is
decreasing function of the degree of correlation across markets, so that we are
considering the extreme that makes collusion most sustainable.
16
This result
is
in contrast to that of
consider a model in which the incumbent
Brander and Spencer (1981)
is
committed to selling
its
They
.
pre-entry
output once entry occurs. In that setting, the incumbent can deter entry by
producing
to
a large
enough output.
prefer to produce
a
Imposing a
still
chooses instead
it
when
it
to allow
does so.
on the incumbent may lead
lower output even though that
However, since the incumbent
price
tariff
entry
will result in
has the option of detering entry,
must be the case that
So, in their analysis,
any
it
entry.
if
it
receives a higher
tariff big
enough
to
induce
the domestic firm to enter necessarily raises the domestic price above what
it
it
2
would have been
if
however, depends
Since there
is
J
the domestic firm was deterred from entering.
critically on the
absence
of
perfecteness of their equilibrium.
no logical link between the incumbent's pre-entry output and
post-entry output, the incumbent should be expected
entrant's post-entry output.
17
Note that this expression
for
s
respond optimally
to the
This leads to the Cournot outcome in both markets
is
(the highest price charged
expected profits
strictly declining in
as are the espressions
2
(1-e)
So our analysis of competition
of the domestic firm.
differentiated products.
t
by the domestic firnm) and (a-c)
consistent with that of Krishna (1985).
a
to
its
In that case the above analysis applies.
post-entry.
reducing
This result,
She analsyzes
She shows that,
a
2
/b,
the
is
duopoly producing
in the single shot static
game,
quota raises domestic prices as the domestic firm need not be so
concerned with foreign competition.
to the case in
Our analysis shows
which the duopoly produces
a
that her result extends
homogeneous good.
"
40
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(
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,
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"
'
.
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Date Due
Lib-26-67
3
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