4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Dynamic oligopoly
Secret price cuts and deliveries
Introduction
Secret price
cuts
4820–3
Secret
deliveries
Optimal
punishments
Geir B. Asheim
Discussion
of cartel
theory
Department of Economics, University of Oslo
ECON4820
Spring 2010
Last modified: 2010.02.02
Motivation
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Up to now: Models of cartels (for which competition is
restrained by the threat of retaliation) where
Perfect monitoring of past actions (prices or quantities)
Introduction
Outline
Secret price
cuts
Secret
deliveries
Optimal
punishments
Discussion
of cartel
theory
Threats are never called and hence
price wars and/or fierce quantity competition
will never be observed according to such models
Are there models that (in the spirit of Stigler, A model of
Oligopoly, J Pol Econ 72 (1964) 44–61)
model hidden actions
(secret price cuts and/or secret deliveries)
still allow for collusion through the threat of retaliation
leads to threats being undertaken in equilibrium
Outline
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Do secret price cuts undermine a price cartel?
Tirole 6.7.1
Introduction
Outline
Secret price
cuts
Secret
deliveries
Optimal
punishments
Discussion
of cartel
theory
Do secret deliveries undermine a quantity cartel?
Green, Porter, Non-cooperative collusion under imperfect
price information, Ecma 52 (1984) 87–100
Abreu, Pearce, Stachetti, Optimal cartel equilibria with
imperfect monitoring, J Econ Theory 39 (1986) 251–269
Discussion of collusion under the threat of retaliation
Problems of prediction and credibility
Secret price cuts (Tirole 6.7.1)
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Analysis
Results
Secret
deliveries
Optimal
punishments
Discussion
of cartel
theory
Homogeneous good market; two firms competing in price;
the usual assumptions on the demand function D(·).
If one firms secretly undercuts,
then no demand for the other firm.
Secret only if there is a probability α > 0
that demand = 0 for any price.
If one firm observes no demand,
then this is mutually known:
Either the other firm has undercut
Or demand has fallen to zero
Therefore, the event that one firm observes zero demand
can be used as a trigger of punishment.
Strategy profile
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Analysis
Results
Secret
deliveries
Optimal
punishments
The firms agree to set p1 = p2 = p m
until one firm observes 0 demand.
Then firms set p1 = p2 = c in T − 1 periods
as punishments.
Average discounted payoff if firms follow the agreement:
Πm
v = (1 − α) (1 − δ) 2 + δv + αδ T v
m
(1 − δ)v + α(δ − δ T )v = (1 − α)(1 − δ) Π2
Discussion
of cartel
theory
m
m
(1 − α)(1 − δ) Π2
(1 − α) Π2
v=
=
1+A
(1 − δ) + α(δ − δ T )
T
T −1 ).
where A ≡ α δ−δ
1−δ = α(δ + · · · + δ
Subgame-perfect equilibrium?
4820–3
Dynamic
oligopoly II
v = (1 − α) (1 −
+ δv + αδ T v
v ≥ (1 − α) (1 − δ)Πm + δ T v + αδ T v
Geir B.
Asheim
Introduction
m
(1 − δ) Π2 + δv ≥ (1 − δ)Πm + δ T v
T
Πm
≤ δ−δ v
(1 − δ)
2
Secret price
cuts
Analysis
Results
Secret
deliveries
Optimal
punishments
Discussion
of cartel
theory
m
δ) Π2
Short run gain
Long run loss
m
Πm
2
T
δ−δ
Π
(1 − α) m
(1
−
α)
2
≤ (δ + · · · + δ T −1 )
= 1−δ δ−δT Π2
1+A
1 + α 1−δ
(1 − α)(δ − δ T )
1≤
(1 − δ) + α(δ − δ T )
Does an SPE exist?
4820–3
Dynamic
oligopoly II
(1 − α)(δ − δ T )
1≤
(1 − δ) + α(δ − δ T )
Geir B.
Asheim
Introduction
Secret price
cuts
Analysis
Results
Two questions:
What is the lowest δ which allows collusion to be disciplined
through the threat of retaliation? Let T = ∞.
1≤
Secret
deliveries
Optimal
punishments
(1−α)δ
1−(1−α)δ
≤ (1 − α)δ
1
1
δ ≥ 2(1−α)
⇒ α< 2
1
2
Discussion
of cartel
theory
Is the severest punishment (T = ∞) optimal if δ >
1
2(1−α)
Because punishment occurs in equilibrium,
punishment must not be made more severe than necessary!
Secret deliveries
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Secret
deliveries
Analysis
Results
Optimal
punishments
Discussion
of cartel
theory
Homogeneous good; n firms competing in quantities;
the usual assumptions on the demand function D(·).
If one firm makes a secret delivery,
then the other firms observe a lower price.
Secret only if there is a demand uncertainty:
pt = θt P(q1t + q2t )
θt ∼ F (θ)
F (θ) : probability for θt ≤ θ
Both firms observe the market price pt . Therefore,
the event that the firms observe a price smaller than
a threshold p̃ can be used as a trigger of punishment.
Strategy profile
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Secret
deliveries
Analysis
Results
Optimal
punishments
Discussion
of cartel
theory
The firms agree to produce a small quantity q1 = q2 = q̃
until pt < p̃ is observed.
Then firms produce the Cournot quantity q1 = q2 = q C
in T − 1 periods as punishments.
If an SPE, it is because an increase in quantity, although
leading to a short run gain, incr. the prob. of a price war.
Whereas reducing the quantity, although reducing the prob.
of a price war, is out-weighted by a short run loss.
What is the probability that pt < p̃ is observed, given that
all firms abide by their tacit agreement?
p̃ p̃ Pr θt P(nq̃) < p̃ = Pr θt < P(nq̃)
= F P(nq̃)
≡ α̃
Write
Π̃ = E (θP(nq̃) − c)q̃
ΠC = E (θP(nq C ) − c)q C
Payoffs
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Secret
deliveries
Analysis
Results
Optimal
punishments
Discussion
of cartel
theory
Average discounted payoff if firms follow the agreement:
v = (1 − δ)Π̃ + (1 − α̃)δv + α̃ (1 − δ)(δ + · · · + δ T −1 )ΠC + δ T v
v = (1 − δ)Π̃ + (1 − α̃)δv + α̃ (δ − δ T )ΠC + δ T v
v = (1 − δ)Π̃ + δv − α̃(δ − δ T )(v − ΠC )
(1 − δ)v + α̃(δ − δ T )v = (1 − δ)Π̃ + α̃(δ − δ T )ΠC
(1 − δ)Π̃ + α̃(δ − δ T )ΠC
Π̃ + AΠC
C
v=
+
=
=
Π
1+A
(1 − δ) + α̃(δ − δ T )
T
T −1 ).
where A ≡ α̃ δ−δ
1−δ = α̃(δ + · · · + δ
Π̃−ΠC
1+A
Subgame-perfect equilibrium?
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
v = (1 − δ)Π̃ + δv − α̃(δ − δ T )(v − ΠC )
First-order condition:
0 = (1 −
Secret
deliveries
Analysis
Results
Optimal
punishments
Discussion
of cartel
theory
∂ Π̃ δ) ∂q
i q̃
∂ Π̃ ∂qi q̃
∂ Π̃ ∂qi q̃
−
∂ α̃ ∂qi q̃ (δ
=
=
Marginal short run gain
∂ α̃ δ−δ T
∂qi q̃ 1−δ
∂ α̃ ∂qi q̃ (δ
− δ T )(v − ΠC )
(v − ΠC )
C
+ · · · + δ T −1 ) Π̃−Π
1+A
Marginal long run loss
Note that it is not certain that the punishment Π̃ − ΠC
will be imposed; it might be imposed in any case.
This is reflected by A > 0.
Discussion of this equilibrium
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Secret
deliveries
Analysis
Results
Optimal
punishments
Discussion
of cartel
theory
In equilibrium, all firms produce q̃. Still, with probability α̃, price
below p̃ will be observed. The players know that this is caused
by low demand. Even so, they implement the punishment phase.
Why?
It is individually rational for the firms to participate in the
punishment, given that all others do so. The punishment is
implemented as an SPE. They understand that if they refrained
from punishing because no-one has deviated, the discipline of
the firms to keep the cartel agreement would disappear.
In what cases, are secret deliveries
– rather than secret price cuts – appropriate? OPEC?
Optimal punishments
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Secret
deliveries
Optimal
punishments
Discussion
of cartel
theory
In the case of secret deliveries, Cournot reversion,
with return to cartel solution after T periods, is arbitrary.
Note that it is profitable to let q̃ > q m .
Loss of first-order, gain that less discipline is needed (reduced
probability for price war, reduced duration).
Abreu, Pearce, Stachetti (1986) find that the optimal symmetric
punishment has a stick-and-carrot form. (Optimal in the sense
that it maximizes expected discounted payoff).
Note that symmetric punishments are not restrictive
as the deviator (if any) cannot be identified.
Optimal punishments
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Secret
deliveries
Optimal
punishments
Discussion
of cartel
theory
Quantity
competition
Perfect
monitoring
Hidden
actions
Cournot
reversion
Friedman
RES (1971)
Green-Porter
Ecma (1984)
Opt. sym.
punishment
Abreu
JET (1986)
APS
JET (1986)
Price
competition
Perfect
monitoring
Hidden
actions
no
problem
Tirole
6.7.1.1
Bertrand
reversion
Discussion of collusion under the threat of retaliation
Problem I: Prediction
4820–3
Dynamic
oligopoly II
The folk theorem yields many equilibria.
Geir B.
Asheim
Introduction
Secret price
cuts
Secret
deliveries
Optimal
punishments
Discussion
of cartel
theory
How to choose between all these equilibria?
On the basis of subgame perfectness
we cannot argue that one equilibrium
is more reasonable than another.
But: Multiple equilibria
⇓
Coordination
⇓
Efficient equilibrium (?)
Discussion of collusion under the threat of retaliation
Problem II: Credibility
4820–3
Dynamic
oligopoly II
Geir B.
Asheim
Introduction
Secret price
cuts
Secret
deliveries
Optimal
punishments
Discussion
of cartel
theory
Coordination before stage 1
⇓
Coordination before later stages
⇓
Punishments that harm all players are not credible
⇓
Undermine the credibility of equilibria
that are based on such punishments.
What equilibria are renegotation-proof ?
Can renegotiation-proofness improve the ability to predict?
Are these insights important for competition policy?