4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Dynamic oligopoly — Tacit collusion
4820–2
Context
Theory of
repeated
games
Geir B. Asheim
Application
of repeated
games
Department of Economics, University of Oslo
Price
rigidities
ECON4820
Spring 2010
Empirical
studies
Motivation
4820–2
Dynamic
oligopoly I
Static price competition → Aggressive behavior
Geir B.
Asheim
Introduction
Outline
Static quantity competition
→ Somewhat less aggressive behavior
Context
Theory of
repeated
games
Application
of repeated
games
Price
rigidities
Empirical
studies
But correct prediction?
Is competition restrained by the lack of available capacity?
Alternative:
Competition is restrained by the threat of retaliation:
If one firm undercuts, then an aggressive response
from competitors.
Short run gain must be compared to long run loss.
Outline
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Context
Given number of firms
Collusion among firms vs. cooperation among individuals
Introduction
Outline
Context
Theory of
repeated
games
Application
of repeated
games
Theory of repeated games
Finitely repeated games
Infinitely repeated games
Application of repeated games
Price
rigidities
Price competition
Empirical
studies
Quantity competition
Price rigidities
Empirical analysis
Context
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Oligopoly
Collusion vs.
cooperation
Theory of
repeated
games
Application
of repeated
games
Price
rigidities
Empirical
studies
Given number of firms
Collusion among firms vs. cooperation among individuals
Why a given number of firms?
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Oligopoly
Collusion vs.
cooperation
Theory of
repeated
games
Application
of repeated
games
Durable investments
Technological know-how
Barriers to entry
Price
rigidities
Empirical
studies
Collusion among firms vs.
cooperation among individuals
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Oligopoly
Collusion vs.
cooperation
Collusion among firms
Efficiency loss due to higher prices to consumers
→ Usually illegal
Overt collusion supported by third-party enforcement
is not possible
Tacit collusion supported through repeated interaction
(deviation triggers future punishment)
Theory of
repeated
games
Application
of repeated
games
Price
rigidities
Empirical
studies
Cooperation among individuals
Efficiency gain, e.g., due to lower transaction costs
→ Promoted
Implemented by many mechanisms:
Supported through repeated interaction
Social preferences; e.g., reciprocity
May be supported by social evolution
Theory of repeated games
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Theory of
repeated
games
Finitely
repeated
games
Infinitely
repeated
games
Application
of repeated
games
Price
rigidities
Empirical
studies
The analysis of dynamic oligopoly is based
on the theory of repeated games.
Let G = N, (Ai ), (ui ) be a normal form game,
where Ai is the set of actions for player i
and ui is the payoff function for player i. Two classes:
Finitely repeated games: G is played finitely many times.
Infinitely repeated games: G is played infinitely many times.
Equilibrium concept:
Subgame-perfect equilibrium
Finitely repeated games
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Theory of
repeated
games
Finitely
repeated
games
Infinitely
repeated
games
Application
of repeated
games
Price
rigidities
Empirical
studies
Assume that G has a unique Nash equilibrium. Examples:
Prisoners’ dilemma
Bertrand model
with homogenous products and linear and identical costs
Cournot model
with homogenous products and linear demand and costs
Then: A unique SPE of the finitely repeated game, in which the
unique Nash equilibrium of G is played in every stage. Why?
However, Kreps, Milgrom, Roberts, Wilson (aka “Gang of Four”)
Rational cooperation in finitely repeated prisoners’ dilemma, J
Econ Theory 27 (1982) 245–252 show how firms cooperate to
sustain reputation for friendly behavior; Tirole 6.5. Relevant?
Note that if G has multiple Nash equilibria, then a large set of SPEa;
Benoit, Krishna, Finitely repeated games. Emca 53 (1985) 905–922
Infinitely repeated games
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Theory of
repeated
games
Finitely
repeated
games
Infinitely
repeated
games
Application
of repeated
games
Price
rigidities
Empirical
studies
Tacit collusion requires infinitely repeated interaction.
A δ-discounted infinitely repeated game of G
is an extensive game where G is played infinitely many times,
where players can observes the actions of previous rounds
and where the payoff function for player i is given by
ui∗ ((at ))
∞
= (1 − δ)
δ t−1 ui (at )
t=0
Important papers:
Characterizing the set of SPEa for a given δ: Abreu, Extremal
equilibria of oligopolistic supergames, J Econ Theory 39 (1986)
191–225; Abreu, On the theory of infinitely repeated games with
discounding, Emca 56 (1988) 383–396
Characterizing the set of SPEa when δ → 1: Fudenberg, Maskin
(1986) The folk theorem in repeated games with discounting and
with incomplete information, Emca 54 (1986) 533–554
Application of repeated games
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Price competition
Introduction
Context
Theory of
repeated
games
Application
of repeated
games
Price
competition
Quantity
competition
Price
rigidities
Empirical
studies
Quantity competition
Is it true that collusion is facilitated by (Chamberlin, 1929)
Fewer firms
Shorter detection lags
Symmetric firms
What other factors facilitate collusion?
Price competition (1)
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Theory of
repeated
games
Firms 1 and 2 have constant unit cost c. In each stage, firms
choose prices simultaneously. The chosen prices determine
quantities in this stage and becomes observable for both firms
before the next stage. Infinite number of stages. Payoff is
profits discounted at constant discount factor δ ∈ (0, 1).
Consider the following paths, where p ∈ (c, p m ]:
(pt ) = (p, p), (p, p), . . .
Application
of repeated
games
(nt ) = (c, c), (c, c), . . .
Price
competition
Quantity
competition
Price
rigidities
Empirical
studies
Result
The simple strategy profile σ((pt ), (nt ), (nt )) is a
subgame-perfect equilibrium if and only if δ ≥ 12 .
Price competition (2)
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Theory of
repeated
games
Application
of repeated
games
Price
competition
Quantity
competition
Price
rigidities
Empirical
studies
Proof.
Enough to consider unilateral one-period deviations. Why?
Is a unilateral one-period deviation from (p, p) profitable?
(1 − δ)(p − c)D(p)
Supremum of payoff with deviation
Not profitable if 1 − δ ≤
1
2
≤
1
2 (p
− c)D(p)
Payoff without deviation
or, equivalently, δ ≥ 12 .
Is a unilateral one-period deviation from (c, c) profitable?
Payoff with deviation is non-positive.
Payoff without deviation equals zero.
What if there are more than 2 firms? δ ≥ n−1
n
Few firms and short detection lags facilitate collusion.
Price competition – Asymmetric costs
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Firms 1 and 2 have constant unit cost c1 and c2 , where c1 < c2 .
Consider the following paths, where p1m = arg maxp (p − c1 )D(p):
(pt ) = (p1m , p1m ), (p1m , p1m ), . . .
(nt ) = (c2 − , c2 ), (c2 − , c2 ), . . .
Introduction
Context
Theory of
repeated
games
Application
of repeated
games
Price
competition
Quantity
competition
Price
rigidities
Empirical
studies
Since firm 1 will have positive profit, even after collusion has
broken down, we obtain:
Result
The simple strategy profile σ((pt ), (nt ), (nt )) is a SPE
if and only if δ ≥ δ where 12 < δ < 1.
Hence, asymmetric costs can prevent collusion. But what if the
firms agree on asymmetric market shares? Tirole, Exercise 6.5
Which firm gets the higher market share?
Price competition – Fluctuating demand
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Low-demand D1 (p) with prob. 12 and high-demand D2 (p) with
prob. 12 . Assume that D1 (p) < D2 (p) for all p with D2 (p) > 0.
m
Write Πm
k = maxp (p − c)Dk (p) and pk = arg maxp (p − c)Dk (p)
Consider the following paths:
(pt ) = (pkm , pkm ), (pkm , pkm ), . . .
Context
Theory of
repeated
games
Application
of repeated
games
Price
competition
Quantity
competition
Price
rigidities
Empirical
studies
(nt ) = (c, c), (c, c), . . .
Since the short run gain is higher when demand is high:
Result
The simple strategy profile σ((pt ), (nt ), (nt )) is a SPE
2Πm
1
2
2
if and only if δ ≥ δ where 2 < δ = 3Πm +Π
m < 3.
2
1
1
2 < δ < δ? Rotemberg, Saloner, A supergame-theoretic model of
business cycles and price wars during booms, Am Econ Rev 76 (1986) 390–407
What if
Quantity competition (1)
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Theory of
repeated
games
Firms 1 and 2 have constant unit cost c < 1. In each stage,
firms choose quantities simultaneously, and P = 1 − q1 − q2 .
The chosen quantities in this stage and becomes observable for
both firms before the next stage. Infinite number of stages.
Payoff is profits discounted at constant discount factor δ ∈ (0, 1).
Consider the following paths:
1−c
1−c 1−c
(qt ) = ( 1−c
4 , 4 ), ( 4 , 4 ), . . .
Application
of repeated
games
1−c
1−c 1−c
(nt ) = ( 1−c
3 , 3 ), ( 3 , 3 ), . . .
Price
competition
Quantity
competition
Price
rigidities
Empirical
studies
Result
The simple strategy profile σ((qt ), (nt ), (nt )) is a
9
.
subgame-perfect equilibrium if and only if δ ≥ 17
Quantity competition (2)
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Theory of
repeated
games
Application
of repeated
games
Price
competition
Quantity
competition
Price
rigidities
Empirical
studies
Proof.
Enough to consider unilateral one-period deviations.
1−c
Is a unilateral one-period deviation from ( 1−c
4 , 4 ) profitable?
9
1 1
2
1
(1 − δ) 64 − 8 (1 − c) ≤ δ 8 − 9 (1 − c)2
Short run gain
Long run loss
1
1
Not profitable if (1 − δ) 64
≤ δ 72
or, equivalently, δ ≥
9
17 .
1−c
Is a unilateral one-period deviation from ( 1−c
3 , 3 ) profitable?
No, Nash equilibrium in the stage game.
(n+1)2 −4n
(n+1)2 −16n2 /(n+1)2
What if there are more than 2 firms? δ ≥
Few firms and short detection lags facilitate collusion.
Price rigidities
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
Introduction
Context
Theory of
repeated
games
Application
of repeated
games
Price
rigidities
Empirical
studies
How to formalize the kinked-demand-curve story?
→ Tirole 6.4 & 6.7.2.
→ Covered later.
What is the effect of a “meet-the-competition” clause?
What is the effect of a “most-favored-nation” clause?
Empirical studies of collusion
4820–2
Dynamic
oligopoly I
Geir B.
Asheim
The railroad cartel
Porter, Bell J Econ 1983
Ellison, RAND J Econ 1994
Introduction
Context
Theory of
repeated
games
Application
of repeated
games
Price
rigidities
Collusion among petrol stations
Slade, Rev Econ Stud 1992
Collusion in the soft-drink market: prices and advertising
Gasmi et al., J Econ & Manag Strat 1992
Empirical
studies
Collusion in procurement auctions
Porter, Zona, J Pol Econ 1993 (road construction)
Pesendorfer, Rev Econ Stud 2000 (school milk)