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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
/,
QUOTAS AND THE STABILITY OF IMPLICIT COLLUSION
by
Julio J. Rotemberg
and Garth Saloner"
WP#1 778-86
May 1986
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
'"quotas and the stability of
implicit collusion
by
Julio J. Rotemberg
and Garth Saloner-WP#1 778-86
May 1986
"Sloan School of Management and Department of Economics, M.l.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
:?
-^o,
^-
i
AUB 2 7
Ills'"
1986
|
!o
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
will
ensue
if
collusion that can be sustained
Our
strategic variables.
variables
Since a quota reduces the
punish a deviating domestic firm, the amount
ability of the foreign firms to
We study both the case
The more powerful the
any firm deviates.
threat, the more collusion that can be sustained.
In such
by the threat that
a setting the firms in an industry sustain collusive prices
more competitive pricing
.
in
is
This
of
correspondingly diminished.
which sales and the case
results are strongest
when
which prices are the
in
prices are the strategic
quotas always make monopolization of the domestic market more
:
difficult in that case.
This
is
in
sharp contrast
to the results in static
imperfect competition models in which quotas tend to make
domestic firm to act as a monopolist at home.
It
is
it
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
difficult for the firms to sustain collusive
make coDusion 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
developed by Abreu (1982) involve an outcome
This
in
.
can,
willing to tolerate the ensuing losses,
it
if
it
makes
result,
is
it
is
which the domestic firm earns
because even with a very large
zero profits
of the style
tariff,
the foreign firm
charge a price so low that
impossible for the domestic firm to earn profits at home.
tariffs
do not affect the abUity of the duopoly
to maintain
As a
monopolistic outcomes.
There the maximum punishment the foreign firm
Contrast this with a quota.
can
still
inflict
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,
it
as pointed out
has a larger incentive to deviate from the
by Davidson (1984), trade restrictions
country to
share of the domestic market.
Thus the domestic
favor and thereby giving
also
make
more
it
In this case of maximal punishments,
firm's incentive to abide
to lose
if
it
deviates.
therefore, our results have the
In his classic paper he
showed that a single domestic producer who faced
a competitive foreign
a single domestic
producer and
than with a quota.
tariff
by
by altering the market
opposite implication of those of Bhagwati (1965).
would act more competitively with a
it
the domestic country have a larger
let
the collusive scheme can at least be partiaUy restored
its
since the domestic
This general intuition must be qualified somewhat by the
less costly to the foreign
shares in
This generally
a single foreign
market
When we consider
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
In that case,
to sustain.
then, the consequences of tariffs and quotas are similar.
As mentioned previously, our results are strongest for the case
prices are the strategic variables.
When
quantities are the strategic variables
we
we analyze the case
in
in
which
which
find that only small quotas break the
discipline of the oligopoly while large quotas,
if
anything, strengthen the
incentives to go along with the monopolistic outcome.
Thus our
case parallel closely those of Davidson (1984) for tariffs.
results for this
He uses quantities
as the strategic variables in a model very similar to ours and shows that small
promote collusion, while the opposite
tariffs
true for large tariffs.
is
After considering the effects on monopolization in the country that imposes
the quota
we study the foreign repercussions
we used
that the arguments
We
of quotas.
notice immediately
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
a
capacity constrained in this way.
If
be sold in the domestic market, the presence of a
quota requires that the domestic firm
a limited capacity to
is
Suppose that
limit.
sell
more
at
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.
in the
Thus, for the reasons discussed above, more competition results
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.
In that case,
time
and
in either
Suppose, for example, that the
are not perfectly correlated across
market taken alone,
unable to sustain the monopoly level of prices in
and Saloner (1906), when demand
is
outweigh the punishment which
meted out
is
demand.
states.
sufficiently high,
As Bernheim and Whinston (1986) show,
sustain higher profits
all
in later,
in
the firms
As
in
Rotemberg
(which
is
may
the incentive to cheat
"more normal", periods.
such settings the firms can
when they share two markets with imperfectly
In the simplest case,
may be
also the one that
correlated
we study), the
markets are of equal size 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
be able to sustain monopoly profits each period.
Imposing a quota on one of the
markets, however, means that the firms lose the countervailing force that
smooths the incentive to cheat when demand
Instead, the foreign market
is
high
in the
foreign market.
transformed into a market with fluctuating demand
is
and monopoly profits may no longer be sustainable there.
Thus the
market or
imposition of a quota can increase competition in the foreign
domestic market.
in the
The natural question
is
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
The presence
that 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 HI.
use the analysis developed
in
increased competition abroad
show that
in
is
if
the domestic firm
demand
In Section IV,
is
we
in
We
capacity constrained.
that this can occur even
not capacity constrained in Section V.
show that increased competition can result
market.
and
these sections to show that quotas can lead to
the presence of fluctuating
the domestic firm
II
In Section VI
when
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
at
home
We consider an
There are two countries, domestic and foreign.
oligopolistic industry with one domestic
the simplest case
restriction
in
is
we assume
both countries.
Marginal cost
is
buy the good
so that consumers can only
slightly contradictory assumption
P
=
a
where P
is
constant and equal to c
is
own country
in their
it
made mainly
2
.
for tractability
Thus
the foreign
Yet, and this
does at home.
by the two firms are viewed as perfect substitutes
is
This
sales abroad.
However, the markets are segmented
firm can charge a different price abroad than
Demand
start with
Foreign firms do not face any transportation costs from
shipping the good to the domestic country.
sold
makes no
that the domestic firm
relaxed in Section IV.
To
and one foreign firm.
,
the goods
the domestic market.
in
given by:
-
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
two firms quote the same price they share the market
assumptions about the way
gets half the market.
feasible
in
which this sharing
The second
is
the presence of quotas the foreign firm
case.
Thus the
There are two possible
done.
The
first is that
that any possible market division
and that market shares are also
the market.
is
.
implicitly
agreed upon.
may be forced
the
to
each
is
However,
in
have less than half of
latter assumption is 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
the market.
If
4
Then,
2
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)^/4b < 6(a-c)^/4b(l-6)
where the RHS
of this equation represents the future profits that are given
by cheating and
long as
6
6
is
the rate at which future profits are discounted.
equals at least 1/2, the monopoly price
We now consider the
effect of a quota.
allowed to import be E(a-c)/b so that
sold
Similarly,
of
if
e
is
1/2,
it
demanded
can
is
sell
So as
sustainable.
Let the amount the foreign firm
is
scaled by the amount that would be
A quota where
under perfect competition.
firm to supply the amount
it
is
up
e
is
1
would allow the foreign
at a price equal to
the amount
demanded
marginal cost.
at the
monopoly price
(a+c)/2, and so on.
Notice as an aside that any quota which
is
binding
equilibrium, i.e. which reduces the amount Imported,
measures
of domestic welfare.
This
is
at the original
raises the standard
so because, even
if
the price remains at
(a+c)/2, the domestic firm having higher sales,
now earns higher
profits.
c
Domestic welfare
is
only increased further
the national identity of firms
if
the price actually falls
however, we focus mainly
a slippery concept,
is
Since
.
on the competition-enhancing affects of quotas.
We begin studying the equilibrium with quotas by analyzing the
punishments for deviating from the
standard practice
(1971),
in
understanding.
implicitly collusive
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 these may be preferable
even when they are lower than the maximal punishments.
the firm can
place
is
sell
of the firm to signal,
to post prices other
signalling
is
if
the owners of
the firm forward, the only equilibrium until the sale takes
the one-shot Nash equilibrium.
managers
First,
7
Second,
it
is
in the interest of the
once punishments begin, that they are unwUling
than those of the one-shot Nash equilibrium.
credible
maximal punishments as
in
If
this
Abreu (1982) become
Q
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 of prices)
show that
this equilibrium is the
.
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)
(a+c)/2
=
and
(iii)
e)
2
it
is
[a+c-E (a-c) ]/2.
is
is
(a-c)(2E-E^)^^^/2
-
(2)
charged by both firms with probability zero
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 of Krishna (1986).
A higher quota
highest and lowest price charged.
Kreps and Scheinkman (1984) show that the
entire distribution of prices
is
(a
stochastically dominated
higher
e)
lowers both the
by that with
a lower
quota.
There
comment.
c)
2
is
one more feature of this punishment equilibrium that deserves
At this equilibrium the domestic firm earns a present value of (a-
2
What must be noted
(1-e) /4b(l-6).
is
that the domestic firm can never earn
less than this present discounted value of profits at
The reason
one involving maximal punishments.
firm can gviarantee for itself that
is
for this
2
it
is
that the domestic
2
wUl earn (a-c) (1-e) /4b per period by
posting a price equal to [a + c-E (a-c) ]/2b.
the domestic firm
any equUibrum; even the
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
(a + c)/2 while the foreign firm,
higher price.
which
is
firm continues to be
subject to a quota, naturally prefers a
Yet we concentrate on the question of whether the duopoly can
We do
sustain the "monopoly" price of (a + c)/2.
First,
this for two related reasons.
higher prices are more difficult to sustain so that
sustain (a+c)/2 they cannot sustain any higher price.
whether the introduction
in
if
the firms cannot
Second, we are interested
of a quota lowers equilibrium prices
from their free
trade level of (a+c)/2.
The
firm
is
sustainability of the price (a + c)/2
expected to seU
allowed to
sell its
have no incentive
deviate since
is
expected
it
at this price.
entire quota,
to deviate.
earns more
foreign firm
the foreign firm
In this case,
game.
Similarly,
2
Instead,
/2b.
if
to sell
y(a-c)/b (0<y<l)
1/2,
in
which case
it
sells its entire
1/2,
in
which case
it
sells
Consider
2
c)
/2b.
(£
first the
It will
which the
in the collusive
arrangement.
its profits
it
in
which
t
is
-
quota, and the case in which
it
By
+
E
-
6e(2e
-
2
E
)^/2
exceeds
deviating, the foreign firm earns E(a-
e(2e
-
E^)^/^]/2(l-6)
(3)
or:
V >
sells
(a-c)/2b.
former case.
E
are y(a-
smaller than or equal to
thus choose to deviate unless:
y)/2 < 6[p
-
in
entire quota at a price essentially identical to
its
We analyze separately the case
(a+c)/2.
the foreign firm
deviates by undercutting the price slightly,
it
demand or
either total
if
the domestic firm never deviates but the foreign
Then, by going along with the collusive arrangement,
c)
is
the domestic firm will always
we consider the intermediate case
So,
supposed
is
for instance,
E(a-c)/b, at this price the foreign firm will
in the static
to sell nothing,
firm always deviates.
If,
depends on the amount the foreign
1
.
(4)
11
Note that for small
e,
must essentially equal
y
that the price will roughly equal (a+c)/2
punished
9
Thus
at the
is
Note also that the higher
is
the firms' discount factor,
it
is
monopoly price
On
that price.
is
being
sell essentially its
y/2
if
it
goes along
cheats
it
sells
it
can
2
the other hand
-
If it
while
of (a + c)/2,
Thus the domestic
y/2 < 6[E/2
allowed to
knows
firm
6,
entire
the less
willing to sell without cheating.
consider the domestic firm.
cheating.
goes along or
it
the foreign firm
Now
it
deviates unless
.
capacity.
whether
The foreign
e.
sell
(a-c)(l/2
-
vi)/b
(a-c)/2b at
2
earns only (a-c) (1-e) /4b per period after
it
firm
is
deterred from price-undercutting
if:
e^/4]/(l-6)
-
or:
y < 6e
-
6e^/2.
(5)
Equation (5), which
y
must be relatively small
since higher levels of y
is
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
To
sustainable.
(5)
contradict one another the monopoly price
see under what conditions this occurs
is
not
we combine the conditions
and obtain:
e{
1
-
6
-
6[(2e
-
e^)^'^^
-
E/2]l < 0.
(6)
For
small
E
enough we can neglect the term
second order) and the condition
clearly violated.
is
saw that the foreign firm must be allowed
But the domestic firm always requires that
strictly smaller than
e
for y equal to
;
in
e
it
square brackets (which
When
e
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
increasing in
is
up
e
entire quota
when
.62 for (6)
cheats
to
the term in brackets
1
z
until
it
cheats
is
square
only
this the foreign firm cannot sell its
e
,
6
must exceed about
y)/2 < 6[y
-
E
exceeds 1/2 so that when the foreign firm
e
Then the foreign
earns (a-c) /4b.
-
Yet this analysis
in
be satisfied.
consider the case in which
(1/2
The term
positive.
For this maximal applicable
.
2
it
is
reaches .553.
e
beyond
to 1/2 since
relevant for
Now
and
between
E
+
e(2e
-
firm will cheat unless:
e^)^''^]/2(1-6)
or:
y > (l-6)/2
+
-
6e
6£(2e
-
t'^)^^^
(7)
.
which must now be satisfied together with
(5)
for monopolization to be feasible.
So we substitute (7) (holding with equality) in (5) and obtain:
(l-6)/2 < 6e(2£
For
demanded
E
-
£^)^/^
equal one, that
is
-
6e^/2.
when
at the competitive price,
(8)
the foreign firm can sell the entire quantity
(8)
requires that
6
exceed 1/2 as
it
did
13
under free trade.
Since the
(and thus of
of 6/(1-6)
6)
RHS
of (8) is strictly increasing in t,
required to make (8) hold,
the level
when
falls strictly
e
rises.
To summarize
the results of this section, more restrictive quotas (starting
quota which allows the foreign firm to
at a
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
one which satisfies
strictly smaller than the
When these equations hold with
with equality.
For lower values of
sustainable.
equality, monopoly
However,
the price falls.
t,
(or (8))
(6)
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
deviations by the foreign firm lead this firm to earn zero
if
profits from then on (which for low
e
is
a
much harsher punishment than
maximal punishment) the foreign firm wUl require that
so as to refrain from deviating.
1/2
and any positive
This
is
y
briefly
compare the results
analyzing tariffs in a similar setting.
of the foreign firm are higher,
domestic firm.
6
equal to
e.
in this setting is in
direct contrast to the conclusions of the static analysis reported
We now
the
equal at least (1-6)e
inconsistent with (5) for
The increased competition brought about by quotas
(1985).
For
by the
The repeated game
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
has been analyzed by Bernheim and Whinston (1986), who consider optimal
punishments
firms
in
the style of
Abreu (1982).
Then, both the domestic and foreign
earn zero profits when they are being punished
still
.
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 (1986) show that the equilibrium that Involves the highest
profits for the duopoly as a whole
now has
a price higher than (a + c)/2.
This
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
maximal punishments the classic results of Bhagwati
(1965) about competition between a foreign and a domestic firm, are precisely
reversed.
A
quota, because
punished effectively, makes
consequence.
it
makes
it
impossible for the domestic firm to be
difficult to collude,
while a tariff has no such
This raises the intriguing possibility that this
governments seem
However,
it
it
to prefer quantitative restrictions to tariffs
is
the reason
12
must be pointed out that the robustness of the monopoly outcome
with respect to a tariff
is
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 they deserve to
be qualified somewhat further.
The analysis hinges
critically
on the presence
15
Since this means that only the foreign firm can
of only one domestic firm.
punish the domestic firm,
of a quota.
punishment
this
ability is curtailed in the
presence
there were more domestic firms in the market, they could
If
revert to a price of
c
if
any
of
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
only one domestic firm
in this section of
Ill
may be
a
good one.
Quotas and quantity competition at home
In this section
we assume
This has two consequences.
competitor
amount.
when
First,
deviates.
neither firm eliminates the sales of
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
that quantities are the 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 to less sustainabUity of
now only holds
very
little.
for
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.
2
earns (a-c) /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-
2
c)/8b which would give
profits of 9(a-c) /64b in the period of the deviation.
it
From then on, the firms would revert
would
will
to
Cournot-Nash equilibrium so that each
2
(a-c)/3b and earn profits of (a-c) /9b per period.
sell
be deterred from cheating
[9/64
-
Now
6
-
1/9] (a-c)^/b(l-6)
So suppose that
level of sales,
it
is
to the
If
e
is
above 3/8, so that the quota
monopolizing level of output,
has no effect.
it
lower than this level but higher than the Nash equilibrium
that
i.e.
,
exceed 9/17.
consider a quota of E(a-c)/b.
exceeds the best response
each firm
if:
l/8](a-c)^/b < 6[l/8
which requires only that
So,
z
is
between 3/8 and
1/3.
Then
the quota does not
affect the foreign firm's ability to punish the domestic firm from deviating
(since each firm produces (a-c)/3b
in
makes the foreign firm's own deviation
cannot now exceed E(a-c)/b.
Cournot-Nash equilibrium.
it
done
in the
So such a quota has no effect when the firms can
absence of a quota.
is
For suppose that
selling (a-c) /4b,
constrained,
somewhat lower.
since
it
6
is
it
when
it
now earns
will still not
This means that
With
z
not only wUling to go along with
is
less
when
it
deviates than
may be possible
if
In essence,
it
its
does
if
it
sales are
to sustain the
output by having the foreign firm produce less than half of the
domestic firm produce more than half.
may
just below 9/17 so that
be induced to cheat even
it
actually
this cannot be
not sustainable in the absence of a quota.
between 3/8 and 1/3 the foreign firm
isn't
Furthermore,
possible to sustain this monopolistic outcome even
the collusive output
it
less profitable since this deviation
sustain the monopolistic outcome without a quota.
make
However,
monopoly
total,
and the
the "excess" unwillingness to
cheat on the part of the foreign firm can be used to free up the constraint on
17
the domestic firm.
Thus
for a quota to promote competition
equilibrium (a-c)/3b,
detail.
We
t
must be below the Nash
it
We now analyze such
must be less than 1/3.
on the part of the foreign
first look for the lowest level of sales
firm which makes
willing to go along with an equilibrium
it
Then we analyze whether
(a+c)/2.
Suppose that the foreign firm
and that the firms attempt
the domestic firm
by the foreign
this price given these sales
p(a-c)/b in equilibrium
to sell
of output.
derive the minimum level of y for which the foreign firm
By cooperating
deviate.
2
sells Its
It
output
The domestic firm
earns (a-c) y/2b.
requirement that
deviates
sells
-
We
y)/b.
This
y.
is
We now
not want to
(a + c)/2 so that
at a price of
it
monopoly output
show below that the
be smaller than 1/3 Implies that when the foreign firm
z
E(a-c)/b.
Thus when
it
deviates
2
it
Note that the profits to the foreign firm when
p)/b.
in
it
will
sells that portion of the
not sold by the foreign firm, or (a-c) (1/2
is
willing to go along with
is
monopoly level
to sustain the
whose price
firm.
supposed
is
a quota in
because the higher
is
y the
earns (a-c) e(1/2
it
-
+
e
deviates are increasing
lower are the domestic firm's sales
when they both go along with the proposed equilibrium output.
After
game.
its
deviation the market reverts to the equilibrium in the one-shot
Since the quota
is
less than the
Nash equilibrium value
foreign firm sells E(a-c)/b while the domestic firm sells
this level of output,
E)/2b.
Hence the foreign firm
(a-c)^{E[l/2
i.e.
or (a-c) (l-E)/2b.
if:
+
y
-
e]
-
will
its
Thus the foreign
-
e
the
best response to
2
firm earns (a-c) e(1-
be deterred from deviating
y/2!/b < 6(a-c)^[y
of (a-c) /3b,
if:
(1-e) ]/2b(l-6)
y >
E
-
6e^/[1
2e(1-6)]^^.
-
Note that for values of
for this
is
when
that
near zero, the foreign firm
e
the implicit agreement allows
(9)
is
deviating unless
it
that for sufficiently small
is
already selling
all
as
allowed to
is
that
The reason
price does not differ appreciably from
small,
Then
(a+c)/2, even during the punishment phase.
much
deviate unless
to sell nearly its entire quota.
it
the quota
firm's interest to sell as
will
it
can and
it
always
is
in the
foreign
cannot be deterred from
essentiaUy
sell
it
its
This means
entire quota.
e
the foreign firm has no punishment power since
it
is
allowed to bring in.
Given
surprising that we establish next that for sufficiently small
this,
e,
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
best response to this,
c)
2
(a-c) (l-y)/2b,
2
(l-y) /4b in the period in which
would
sell
The domestic firm would
selling (a-c)vi/b as above.
is
it
if
deviated.
it
It
would then earn
From then on the foreign firm
deviated.
2
2
(1-e) /4b per period.
So the domestic firm would be deterred from deviating only
-
v)^/4
-
(1/4
-
increase
is
y,
easy to see that
when
u falls,
as long as y
is
if:
p/2) <
6[l/4
It is
(a-
(a-c)E/b so that, by an argument similar to that above, the domestic
firm would earn (a-c)
(1
sell its
-
y/2
-
(1
-
E)^/4]/(l-6).
in spite of the fact that profits
the inequality in (10)
smaller than 6/(1-6)
is
more
(which
is
(10)
from deviating
likely to hold the smaller
true by assumption).
The
19
reason for this
Is
that the principal effect of a decrease in y
that the
is
domestic firm earns more by going along with the collusive agreement.
make the domestic firm as
firm sell only as
much
as
willing to go along as possible
required to make
is
(9)
we
Thus
the foreign
let
hold with equality.
Substituting this expression for y in (10), multiplying by (1-6) 1-2e (1-6)
[
collecting terms,
E^(l-26^)
-
we divide
E^(l-6)[4(l-6^)
+
+
by
e
this expression
2
by
(after dividing
(11)
range of
t.
and
26]
e'^(1-6)[4(1-6)
obtain (1-26 )<0, which requires
of
2
]
the inequality (10) becomes:
+
If
to
2
e
This means that
6
46(1-6)
2
+
6^] < 0.
and take the
limit as e
if
6
exceeds .71
it
is
goes to zero we
Furthermore, the LHS
smaller than about .71.
decreases monotonically with
)
(11)
e
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
smallest discount factor consistent with collusion
under
free trade (9/17), one can demonstrate numerically that the monopolistic outcome
is
not sustainable for
e
equal to .01 but
These are such low values
of
e
is
sustainable for
e
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
competition abroad.
Sections
and
II
to
III
we adapt the arguments presented
In this section
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
If
(1).
both firms have infinite capacity, there
a monopolistic equilibrium in which (a + c)/2
as
6
identical so that their
is
charged
in
both markets as long
equals at least 1/2 when prices are the strategic variables and
least 9/17
when
becomes
So marginal cost equals
infinite.
in
is
equals at
that the domestic firm actually has limited capacity given
If
k equals
c
for instance,
2,
that this capacity
is
by
untU the firm produces k(a-c)/b and then
the domestic firm can supply the
The
quantity demanded at the competitive price in both markets.
section
6
quantities are strategic variables.
Now assume
k(a-c)/b.
is
idea of this
sufficient to maintain the monopolistic outcome
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
6
that market
6
Now consider
is
6
about
is
about
.
.65.
If
the effect of a quota.
important to notice
is
is
sustainable in the world as a
instead k
at
This
is
is
left
over to
that the domestic firm
is
firm sells (a-c)/2b at
the foreign country.
sell in
is
it
is
never willing
punishing the foreign
selling
it
abroad
This means that the domestic firm
sold at home.
e
we can use
(6)
6
that sustains the monopoly outcome abroad.
value of
6
strictly below
to a value of
is
.65 in the
is
Letting k-1/2 be
minimum
For k barely above 1/2 no
sufficient to maintain that outcome (in contrast
absence of a quota).
For k equal 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.
Hence
Here the monopoly outcome only broke down for tiny quotas.
domestic firm has a capacity of (1/2 +e) where
e
is
negligible,
outcome can be maintained with free trade as long as
other hand, once the domestic market
firm
6
less than
(or (8) for k bigger than 1) to discover the
value of
1
if
effectively
is
capacity constrained abroad with a capacity of (k-1/2) (a-c)/b.
denoted by
to sell
so because the marginal revenue from selling a unit
home always exceeds the marginal revenue from
(a-c)/2b
equal to one, the
is
The domestic
more that (k-1/2) (a-c)/b units abroad, even when
firm for deviating.
equal to k/2
e
62
home and has (k-1/2) (a-c)/b units
What
with
equivalent to twice the domestic market) to obtain the
is
for which the monopoly outcome
value of
critical
(6)
in
So first
to foreign firms.
Then we can use
barely above 1/2.
This value of
whole.
becomes closed
unable to meet demand
is
is
is
willing to sell at most
e
is
abroad.
6
if
the
the monopoly
exceeds 9/17.
On
the
closed to the foreign firm, the domestic
This means that for
e
small
below .71, the monopoly outcome won't be sustainable abroad.
enough
if
V Quotas and
competition abroad: Fluctuating
demand
This section considers a somewhat different model
in the
country tends to increase competition
collusion
is
The
other country.
idea
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.
which a quota
in
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
Rotemberg and Saloner (1986), and negatively correlated across markets as
Bernheim and Whinston (1986).
To
simplify the analysis
we assume
in
in
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
we assume
is
high
in
one market
it
is
low in the other.
that there are N/2 firms in each market while
do not restrict attention to linear demand curves.
symmetric, the monopoly price in the high state, p
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
15
we
23
That
free trade.
is
we consider
a situation in which
all
N
firms sell in both
markets and derive the condition under which no single firm wants to deviate
from charging p
in the
country
in
which demand
is
high and p
in the other.
Prices are again the strategic variable so that any deviation triggers a
permanent reversion
which price equals marginal
to the static equilibrium in
cost.
Let
be the total monopoly profits
IT
those in the market with low demand.
former
in the
market with high demand and
As long as p exceeds marginal
the
cost,
bigger than the latter since the monopoly always has the option of
is
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.
(tt"
it
Thus each
+
TT^)(1
1
< 6/(1
-
is
high and
now and
firm
1/N) < 6(tt"
is
+
in all
In the other.
it
future periods
On
its
thus earns
tt
in
the other hand,
by
It
share (1/N) of
deterred from deviating as long as:
tiVn(1
-
6)
(12)
or:
N
-
-
(13)
6).
We assume that either the number
factor big
enough that
of firms
Is
small
enough or the discount
(12) 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.
fewer (N/2) active firms
in the
First,
it
into the domestic
means that there are
domestic market while N firms continue to
sell
in the
As a
foreign market.
the conditions under which monopoly will be
result,
As
sustainable will be different depending on which market has high demand.
Rotemberg and Saloner (1986) show,
in this setting
sustain collusion
when demand
monopolization
is
possible,
monopolization
is
possible in the high
Consider
/N abroad.
The
Thus
low.
is
always easier to
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
It
is
it
high. Suppose that the
is
firm that undercuts this price earns
losses from the ensuing competition are smaller for foreign
countries since domestic countries
may
also lose
any profits they make
So we concentrate on the incentive to deviate of foreign firms.
they lose the expected value of profits
in the future.
If
a foreign firm will deviate
•tt"(1
-
1/N) > 6(tt"
Note that the
of
RHS
+
(it
is
+tt)/2.
if:
ttV2N(1-6).
(14)
of (14) equals half the
RHS
of (12).
The
costs in terms
foregone future profits from deviating are half what they would be
firms had access to both markets.
of (14)
and
is
(14) to hold simultaneously.
in the
it
is
In particular,
if
is
bigger than
it
is
When
it
is
ti
,
the
if
in
the profits
(14)
is
impossible to sustain monopoly abroad
high state of demand so the quota increases competition.
It is
LHS
(12) holds as an equality
just sustainable with free trade) then
(14) holds,
foreign
if
possible for (12)
high and low demand states are different, the inequality
always satisfied.
in the
Yet, insofar as
bigger than half of the LHS of (12) so that
(so that monopolization
home.
at
they deviate
Since demand
independently distributed over time, these expacted profits equal
So,
instead of
it
easy to show that when (12) holds
it
Is
always possible to sustain
25
monopoly abroad
the low state of demand.
in
(12)
ensures that deviations
Yet,
when demand
would prevail
if
is
low,
will
This results from the fact that
when demand
not take place even
constant.
is
expected future losses from deviating exceed those that
demand were expected
This
to stay low forever.
from the fact that even when (14) holds,
it
is
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
one used
to the
refrain from undercutting p
-
Ti^(l
1/N)
=
6(Tr^
+
s
and the ensuing profits
to derive
(14),
s
it
Then by an
.
a foreign firm will just
if
TiV2N(l-6)
or:
TT^
=
so that
IT
ttV{2(N-1)(1-6)/6
s
equals n
1
when
-
1]}
(15)
(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
is
also true
when
linear.
briefly to an analysis of the domestic market.
If
domestic
firms were to deviate from the collusive arrangement they could be punished in
both markets.
oncG.
On
In this case,
the other hand
deviating firms would deviate in both markets at
we have already derived the net benefits from deviating
abroad under the assumption that punishments would only be meted out abroad.
To
complete the analysis we thus look at the net benefits from deviating in the
domestic market alone under the assumption that firms will only be punished in
Then, the
the domestic market for doing so.
the
sum
of these
two net benefits.
Suppose that the domestic market
high, a deviating firm would earn
then lose
It
it
is
instead of
would thus be deterred from deviating
tt"(1
2/N) <
-
The RHS
meted out only
the share
fully monopolized.
share of total expected profits in
its
total net benefits to deviating are
6(tt"
of (16)
in
is
+
its
its
When demand
share (2/N) of
if:
RHS
On
While the punishment
of (12).
one market, the share of each firm
LHS
would
(16)
the same as the
substantially smaller than the
It
.
own market from then on.
ttVn(1-6).
we previously considered.
n
is
in this
the other hand the
of (12)
market
LHS
is
is
double
of (16) is
since deviations are taking place in
one market alone whUe, in addition, 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
(12) is satisfied (even with equality)
if
strict inequality.
In this case
showing that
monopoly
is
This applies to the case
in
then (16)
is
satisfied as a
which demand
is
high at home.
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 (16) but an even smaller LHS.
So the net benefits to deviating are
negative here as well.
In
summary, when
holds, monopolization
(12)
is
always possible in the
protected domestic market. Indeed, since protected domestic firms strictly
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
in a
richer model, to make monopolization feasible abroad as well.
the market unevenly as
Section
becomes easier for the domestic firms has the
we must allow the
this
II.
feasible at
is
set of firms that
done
charge the lowest price to
split
Bernheim and Whinston (1986) and as we do
in
Bernheim and Whinston (1986) show that the possibility
in
such
of
arbitrary divisions aUows "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 willing to deviate.
potentially achievable
when demand
does not work, however, when
there
is
N
is
In this
fluctuates relatively
equal to
2
home when
it
little.
deviates abroad.
Then
Its
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
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
we consider the
In this section
competition at home (as in Sections
Sections IV and V)
fluctuating
possibility that a quota will increase
and
11
The model we use
.
demand model
of Section
as well as abroad (as in
111)
to illustrate this possibility is the
V except
to permit competition to
that,
increase at home, we consider a strictly positive quota.
analysis
we
one firm
in
restrict attention to the case in which
each country, and
6
is
Also,
N equals
to simplify the
so that there
2,
equal to 1/2, so that 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 wUl 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 of the losses
home) plus the net profits from
at
deviating abroad.
Suppose that the foreign firm has a share
its
net profits from cheating
when demand
those
when demand
notice
fall
is
that,
as
low are (1-3o/2)it
is
s
is
1
high are (l-3a/2)Ti -otT/2.
net profits from cheating
those
when demand
a of the foreign market.
when demand
is
s
high are (3a/2-l/2)iT -(l-a)Ti /2.
we increase
a,
-ott
/2 while
Similarly the domestic firm's
low are (3a/2-l/2)iT
1
is
s
1
1
s
-(l-a)-!! /2
and
The important thing
to
the foreign firm's net profits from deviating
by the same amount 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
Then
there
make both firms
We therefore now concentrate only on the
will
sum
Thus,
of each
always be a
willing to go along.
total incentive to deviate.
In
29
the foreign country this
s
So unless
state.
abroad
is
it
is
equal to
home
at
We can then write the quota
enough
small
Is )/2
-it
1
it
in the
low state and
foreign firm
is
t
is
given by P=a -bQ (where
in state
Suppose that
less than 1/2.
is
as equal to
i
decreased incentive to deviate
is
(a -c)
e
[e (1-6)
+
/2
6e
-
violated which
6z {2t -t
This
(a -c)/b
at the
)^
to maintain
expression
s
1
abroad.
)
in
monopoly
(17) with
i
at
if
the quota
monopoly price the
(4)
the
we can obtain the
]/h
y
one firm's
(17)
home.
replaced by
1
This
6
is
the same as that (6) be
is
equal to 1/2 in Section
in the
is
At least one of these net incentives
i
two markets taken together
given by
when demand
plus the expression in (17) with
II.
domestic market.
in the
consider the net incentive to deviate
we try
where
1)
given by:
is
for the case in which
thus a net incentive to deviate
Now
(tt -tt
we proved
equals either u or
i
Once again by varying
Note that the condition that (17) be positive
by
positive
the other's increased incentive so we can
focus on the total incentive to deviate.
if
is
high
in the
-it
Then we can obtain from
allowed to seU y (a -c)/b.
domestic firm's net profits from deviating.
is
si )/2
the total incentive to deviate
foreign firm's net profits from deviating while from (5)
There
(it
one of the two states.
in
Suppose demand
is
(tt
is
s
(tt
1
-it
)
plus the
high abroad.
It
is
replaced by u when demand
is
positive so that
it
given
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
is
so negative that the monopoly outcome can be maintained abroad even in the high
state.
That this
is
unlikely for small quotas should be apparent since, then.
the static game gives about as
many
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
home.
at
a price p at
home
The amount demanded
(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 seU
let
sells the difference
e
at
is
(a -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:
E^(a'-c)(p-c)/b
-
(p-c)[(a'-p)-E(a^-c)]/b
+
(a'-c)^(l-E V/4b.
The foreign firm gains nothing by cheating but
the static game
e
(a
-c)(p
its
cost from reversion to
is:
-
a]/b
so that the overall benefits from deviating are:
{p-c)(t\a^-c)
whose minimum
is
-
(a^-p))/b
(a^-c)^(l-E^)^/4b
+
reached for a p equal
+
E^a^-c) (o-c)/b
[a +c-£ (a -c)]/2.
There, this
expression equals:
{-E^/2
+
e[1
-
(2e-e^)^^^
+
E^]!(a-c)^/2b
(18)
31
which
is
maximum
negligible for small
e
.
The
cost from deviating at home,
fact that
0,
(18)
small.
is
is
small
means that the
maximum
Yet,
profits in the
high state abroad must satisfy:
(tt^
-
TrV2
-
so that for small
to
*
it
<
impossible to make
is
s
it
equal to
in
which
Tn
it
as would be required
monopolize the foreign market.
VII
.
Conclusions
We have shown
in a variety of
models
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.
Indeed, any action by the government which makes
when coUusion breaks down more
profitable
Thus Government imposed minimum
makes collusion
itself
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
to increase competition
The
the industry.
by encouraging the entry
we consider
setting
new (domestic) firms
very closely related
is
to that
into
developed
They analyze an industry with an established foreign
and Kyle (1985).
in Dixit
of
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
in the
it
must compete with the foreign firm
domestic market
(while
market), the domestic government provides
its
fixed costs.
competes
both markets.
in the
foreign
with sufficient revenues to cover
then finds entry attractive, enters, and increases
It
Since the domestic market was monopolized in any event, the
competition abroad.
domestic country
it
it
in
it
is
made better
off since
consumer surplus
is
unaffected and
rents have been shifted towards the domestic country.
This analysis can easUy 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 weU.
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 of K when
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
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.
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
fvirther
order
in
is
own market
to
is
that the domestic country
encourage
it
to enter.
that the domestic firm
in
may
may have
However, what the
not need complete
which case domestic welfare can be improved even
by allowing some foreign competition.
In contrast to
our analysis
is
this result holds
whether the
a quota or a tariff.
All that is
in Section II,
instrument used by the domestic country
required
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 coUusion or of potential entry are
important, quotas can lead to greater competitiveness.
APPENDIX
appendix we consider the one- shot game
In this
and
strategic variables
which one
in
made
establish the claims
s =
[a
+
namely that:
in the text,
The lowest price charged
(ii)
(a+c)/2
=
and
(Hi)
-
is:
this lowest price is
charged by both firms with probability zero
the foreign firm earns e(o-c) (a-c)/b.
In equilibrium,
of these in turn:
The highest price charged by both firms
the domestic firm,
it
will
never choose
to
charges
its
highest price,
But the price that maximizes
which
(11)
is
its
than
Hence the domestic firm knows that
be undercut with probability one.
when
it
is
undercut for sure
is
s,
therefore the highest price charged by the domestic firm.
The lowest price charged
is
if
it
e
-
(a-c) (2e-e2)1/2/2
and
First,
it
.
(o)
This price
cannot be the case
its
In
which the domestic firm would be willing
o
(i.e.
has a number of interesting
in
an equilibrium that the foreign
firm always charges a price strictly above a.
would always make more than
(iii)
(o-c) (a-c)/b.
could thereby be sure to capture the entire market
undercut the foreign firm)
properties.
(a+c)/2
o =
Consider the lowest price
charge
will
profits
equilibrium, the foreign firm earns
to
it
sell less
charge a price higher than the
highest price charged by the domestic firm.
it
s
is
given that the foreign firm can only
First note that,
when
is:
{a-c){2t-t^)^'^^/2
We discuss each
(i)
We
E(a-c) ]/2
-
c
which prices are the
of the firms is capacity constrained.
The highest price charged by both firms
(i)
in
profits at s
If
it
did so the domestic firm
by charging a price between
o
and
35
the lowest price charged by the foreign firm, which contradicts the fact that
is
highest price.
its
domestic firm,
mean
unless
would benefit from undercutting
at o so that
it
would be unwilling
the lowest price charged
is
charged by the
it
with positive
This would imply that the domestic firm couldn't expect to capture
demand
of
a is the lowest price
if
cannot be charged with positive probability since that would
that the foreign firm
probability.
all
it
Secondly,
s
charge
to
by the domestic
On
o.
the other hand
the foreign firm (who
firm,
seeks only to undercut the domestic firm) would never charge a price of
who charges
must be lowest price charged by the domestic firm
it
o.
So o
with zero
probability.
Since the foreign firm must make equal expected profits at
prices
it
charges
we can compute
in equilibrium
its
profits
when
being able to
sell its
entire quota so these profits equal
If
its
it
the domestic firm
charges
is
a.
sure to
When
sell
it
does this
the entire
£
is
is
assured of
(o-c) (a-c)/b.
demand given by
profits from charging a price p are given by:
(p-c)(a-p)/b
=
(a-c)^/4b
2
-
[p
-
2
which must equal (a-c) (1-e) /4b
=
(a+c)/2
-
(a-c)(2E-£^)^^^/2.
the
expected profits simply by
its
computing
all
(a+c)/2]^/b
at a price of o.
(Al)
So o equals
17
(A2)
(1),
FOOTNOTES
proven by Mookerjee and Ray (1986) and Bernheim and Whinston (1986)
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
is
it
presented, for
instance, in Rotemberg and Saloner (1986).
4
The
first of
demand,
if
these assumptions
5
it
The second assumption
punishment
6
is
is
large.
is
the division of the spoils that
able to resist a larger temptation to deviate
Thus
collusion
what leads Abreu (1982)
The abUity
considered
7
is
easiest to collude.
Each individual firm
This
not restrictive since, with constant
the firms can sustain any collusive outcome, they can also sustain
the monopoly outcome.
makes
is
is
easier the larger
to focus
if
the
the punishment.
is
on the maximal possible punishments.
of tariffs to shift rents from foreign to domestic firms is
in a static
In other words,
model by Brander and Spencer (1985).
following a deviation, the owner of the deviating firm signs
a 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
will
date at which ownership passes.
involved
in a finite-horizon
.
The contract
receive the profits that accrue until the
Then
the current owner and the rival are
game which has as
its
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
37
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
e)
2
at the
is
this loss.
this loss,
below
o.
To obtain
It
must then be compensated
a lower
10
firm earn less than
it
in later
this price v
it
periods for taking
we assume
that,
after taking
2
-
[
(a-c)/b] {e/2
+
[
Then
6(2e-e^)/(1-6) ]^^^| so that, to
order the foreign firm earns £(a + c)/2 even when
Footnote 7 establishes that for
e
small,
\i
it
is
being punished.
must be essentially equal
for the foreign firm not to cheat even with maximal punishments.
valid in this case as well,
monopoly
is
does
must charge a price
the domestic firm earns the entire monopoly profits (a-c) /4b.
V must equal at least (a+c)/2
first
bound on
as
that,
2
one- shot Nash equilibrium for at least one period
V which
is
earn at least (a-c) (1-
to
To make the foreign
/4b(l-6) at any equilibrum.
for this
unsustainable for small
e
to
e
Since (5)
is
even with
maximal punishments.
11
This conclusion, which
is
also derived
by Mookerjee and Ray (198G)
for a
slightly different game,
is
somewhat striking since the domestic firm makes
there
is
static competition
positive profits
if
by simply charging
less than the foreign firm's marginal cost inclusive of tariff.
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 period)
a price just
In
so low that its losses
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.
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
seUs the entire quota when
market, the domestic firm
amount
this
it
greater.
Then by
suffices to
RHS
When
deviates.
To show
show that
this inequality
is
e
it
that this latter amount
less than
15
if
e
to
larger than
is
Suppose
(l/4 + vi/2).
This means that
of the inequality.
impossible
response
it
were
would also have to be greater than 1/4 plus
expression whose maximum, reached when
is
the firms are monopolizing the
selling (a-c) (l/2-y)/b so that the best
(a-c) (l/2 + y)/2b.
is
the quota
half the
is
it
e
is
e
must exceed 1/2 minus an
1/3 and
6
is
This
equals 1/18.
1,
less than 1/3.
is
Bernhein and Whinston (1986) consider the more general case of imperfect
They show
correlation.
that the sustainability of collusion
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
They
Brander and Spencer (1981).
is
committed to selling
its
pre-entry
output once entry occurs. In that setting, the incumbent can deter entry by
producing a large enough output.
to prefer to
produce
Imposing a
a lower output
tariff
on the incumbent may lead
even though that
will result in
However, since the incumbent
stUl has the option of detering entry,
chooses instead to allow entry
it
price
when
it
does
so.
So,
must be the case that
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
39
would have been
if
however, depends
Since there
is
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.
post-entry.
17
This result,
to
respond optimally
This leads to the Cournot outcome
in
its
to the
both markets
In that case the above analysis applies.
Note that this expression
for s (the highest price
is
strictly declining in
as are the espressions
2
2
charged by the domestic firnm) and (a-c) (1-e) /b, the
So our analysis of competition
expected profits of the domestic firm.
consistent with that of Krishna (1985).
differentiated products.
£
is
She analsyzes a duopoly producing
She shows that,
in the single
shot static game,
reducing a 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.
REFERENCES
Dilip, "Repeated Games with Discounting: A General Theory and an
Application to Oligopoly", Princeton University, December 1982, mimeo.
Abreu,
Bernheim, B. Douglas, and Michael D. Whinston, "Multimarket Contact and
Collusive Behavior," Harvard University, April 1986, mimeo.
Bhagwati, Jagdish, "On the Equivalence of Tariffs and Quotas" in R.E. Baldwin et
eds. Trade, Growth and the Balance of Payments: Essays in Honor of Gottfried
Haberler .Noth Holand, Amsterdam, 19G5.
al.
Brander, James and Barbara Spencer, "Tariff Protection and Imperfect
Competition", in Monopolistic Competition and International Trade edited by
Henryk Kierzkowski, Oxford: Oxford University Press, 1984.
^
"Tariffs and Extraction of Monopoly Rents
of Economics
14 (1981), 371-389.
under Potential Entry," Canadian Journal
.
Brock, WUliam, A. and Jose, A. Scheinkman: "Price Setting Supergames with
Capacity Constraints" Review of Economic Studies 52, July 1985, 371-82.
,
Davidson, Carl, "Cartel Stability and Tariff Policy," Journal of International
Economics 17 (1984), 219-237.
.
Avinash K. and Albert S. Kyle, "The Use of Protection and Subsidies for
Entry Promotion and Deterrence," American Economic Review 75 (March 1985), 139Dixit,
152.
Friedman, James W.
"A Non- Cooperative Equilibrium for Supergames", Review of
Economic Studies 28, 1971, 1-12.
:
.
Helpman, Elhanan, "Increasing Returns, Imperfect Markets and Trade Theory,"
Foerder Institute for Economic Research Discussion Paper 18-82, 1982
Kreps, David M., and Jose A. Scheinkman, "Quantity Precommitment and Bertrand
Competition Yield Cournot Outcomes," Bell Journal of Economics 14 (Autumn 1983),
326-337.
Krishna, Kala, "Trade Restrictions as Facilitating Practices," Harvard
University, October 1984, mimeo.
Mookherjee, Dilip and Debraj Ray, "Dynamic Price Games with Learning-By-Doing,"
Stanford University, March 1986, mimeo.
Rotemberg, Julio J. and Garth Saloner, "A Supergame-Theoretic Model
Wars During Booms," American Economic Review forthcoming.
,
of Price-
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