Summary of Tirole (1988) Chapter 6.

1. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Dynamic Price Competition and Tacit Collusion I
Takuya Irie
April 29, 2017
Takuya Irie Dynamic Price Competition and Tacit Collusion I

2. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Outline
Introduction
Conventional Wisdom
Collusion
Detection Lags
Asymmetries
Static Approaches to Dynamic Price Competition
Kinked Demand Curve
Discussion
Supergames
The Theory
Applications
Takuya Irie Dynamic Price Competition and Tacit Collusion I

3. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Introduction
▶ Although in chapter 5 we analyzed one-shot competition, in
practice firms are likely to interact repeatedly.
▶ Since a firm must take into account not only current profits
but also the possibility of a price war, repeated interaction
may upset the Bertrand outcome.
▶ As a result, firms would recognize their interdependence and,
therefore, might be able to sustain the monopoly price
without explicit collusion (i.e., in the noncooperative manner).
▶ This is called “tacit collusion.”
Takuya Irie Dynamic Price Competition and Tacit Collusion I

4. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Collusion
Detection Lags
Asymmetries
Collusion
Assumptions
▶ Two firms with marginal cost c
▶ q = D(p): demand function
▶ Π(p) = (p − c)D(p): industry profit when the lowest price
charged is p
Start from a situation in which firms charge the monopoly price
pm ≡ arg max Π(p) and each makes profit Πm/2, where
Πm ≡ Π(pm).
Takuya Irie Dynamic Price Competition and Tacit Collusion I

5. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Collusion
Detection Lags
Asymmetries
▶ Suppose that a firm is trying to deviate from pm.
▶ If the firm raises its price above pm,
1. its rival will stay put at pm
;
2. the firm’s profit will be zero.
▶ If the firm cuts its price under pm,
1. its rival will match the price;
2. the firm’s profit will be Π(p)/2 ≤ Πm
/2.
▶ Therefore, deviating from pm is not profitable under this
conjecture.
▶ However, there are factors that may prevent collusion.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

6. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Collusion
Detection Lags
Asymmetries
Detection Lags
▶ Prices may remain somewhat hidden; for example
manufacturers may sell to a small number of big buyers.
▶ In this case, since the retaliation is delayed, it is less costly to
a price-cutting firm.
▶ Therefore, tacit collusion is harder to sustain.
Important
Information lags make the dynamic interaction less relevant.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

7. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Collusion
Detection Lags
Asymmetries
▶ A similar point can be made about the existence of some large
sales situation, such as the arrival of a big order from a large
buyer.
▶ In such a case, one would predict that collusion may break
down.
▶ ∵ The short-run private gain from undercutting is large
relative to the long-term losses associated with a subsequent
price war.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

8. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Collusion
Detection Lags
Asymmetries
Factors that eliminate the threat to collusion
Trade associations, resale-price maintenance, rule-of-thumb
pricing, and basing-point pricing
basing-point pricing system: a price system in which the buyer
pays a base price plus a set shipping price depending on the
distance from a specific location.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

9. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Collusion
Detection Lags
Asymmetries
Asymmetries
▶ Heterogeneity in both costs and products make coordination
difficult.
▶ For example, a lower-cost firms would like to coordinate on a
lower price than the higher-cost firms.
▶ Under asymmetric costs, there is no “focal” price on which to
coordinate.
An example of “focal” point
When you are waiting for your friends in Shibuya, it is natural for
you to wait in front of Hachiko. (Personally, I like to wait in front
of TSUTAYA. )
Takuya Irie Dynamic Price Competition and Tacit Collusion I

10. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Kinked Demand Curve
Discussion
Kinked Demand Curve
Assumptions
▶ Two firms, i = 1, 2, with marginal cost c
▶ q = D(p): demand function
▶ Π(p) = (p − c)D(p): industry profit when the lowest price
charged is p
▶ pf : “focal” price
We can think of pf as the current market price or the steady state
price. Then, we can obtain the same conjecture as the previous
slide.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

11. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Kinked Demand Curve
Discussion
▶ The reaction function for firm j is as follows:
Rj(pi) =
{
pi if pi ≤ pf
pf if pi > pf .
(1)
▶ The residual demand function for firm i is as follows:
˜Di(pi) =
{
D(pi)
2 if pi ≤ pf
0 if pi > pf .
(2)
▶ See Figure 6.1 and 6.2.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

12. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Kinked Demand Curve
Discussion
With such beliefs about its rival’s reaction, firm i maximizes
(pi − c) ˜Di(pi); (3)
i.e.,
max
pi
(pi − c)
D(pi)
2
subject to pi ≤ pf
. (4)
Takuya Irie Dynamic Price Competition and Tacit Collusion I

13. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Kinked Demand Curve
Discussion
An additional assumption
Π(p) = (p − c)D(p) is increasing to the left of pm and decreasing
to its right; i.e., it is quasi-concave.
Then, the optimal price for firm i is as follows:
p∗
i =
{
pf if pf ≤ pm
pm if pf > pm.
(5)
Therefore, if pf ∈ [c, pm], a situation in which both firms charge pf
is an “equilibrium” as long as each firm expects its rival to react as
described above (this issue taken up below).
Takuya Irie Dynamic Price Competition and Tacit Collusion I

14. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Kinked Demand Curve
Discussion
Criticisms
▶ This story is too successful in explaining tacit collusion.
▶ We have no indication how firms end up at a given focal price.
▶ The focal price may change when costs change.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

15. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Kinked Demand Curve
Discussion
▶ Consider the third criticism above.
▶ Suppose that for the initial marginal cost c the focal price is
pf (c).
▶ If c′ > c, the price remains at pf (c).
▶ If c′′ < c, the price goes down to the new focal price pf (c′′).
Result
Upward rigidity but not downward rigidity
▶ See Figure 6.2 and the black board.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

16. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
Kinked Demand Curve
Discussion
Discussion
▶ In a static game, each player’s strategy is independent of the
other players’ strategies.
▶ A situation in which both firms just charge pf without the
beliefs above is not a Nash equilibrium.
▶ In this case, the Nash equilibrium leads to the Bertrand
outcome (i.e., p∗
i = c).
▶ In order to formalize an equilibrium in which both firms
charge pf , we need to introduce a dynamic-game approach.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

17. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
The Theory
Change the assumption
▶ We replicate the basic Bertrand game T + 1 times, where T
can be finite of infinite.
▶ Πi(pit, pjt): firm i’s profit at time t when it charges pit and
and the other firm charges pjt.
Definition
This game is called a repeated game, or a supergame.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

18. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
▶ Each firm maximizes the present discounted value of its profit;
i.e.,
max
pit
T∑
t=0
δt
Πi
(pit, pjt), (6)
where δ is the discount factor.
▶ At each date t, the firms choose their prices (p1t, p2t)
simultaneously.
▶ The price strategy depends on the history
Ht ≡ (p10, p20; . . . ; p1,t−1, p2,t−1). (7)
Takuya Irie Dynamic Price Competition and Tacit Collusion I

19. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Definition
A strategy profile is called a subgame perfect equilibrium if for any
history ht, at date t, firm i’s strategy from date t on maximizes the
present discounted value of profits given firm j’s strategy from
that date on.
▶ First we assume that the horizon is finite: T < ∞.
▶ Then, the equilibrium of (T + 1)-period price game leads to
the Bertrand outcome repeated T + 1 times (Prof. Matsui
would say “check!”).
Takuya Irie Dynamic Price Competition and Tacit Collusion I

20. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
▶ Consider the horizon is infinite: T = ∞.
▶ It is easy to check that the Bertrand equilibrium repeated
infinitely is an equilibrium of this game.
▶ However, this equilibrium is no longer the only equilibrium.
Definition
The following strategy is called trigger strategy :
pit(Ht) =
{
pm if Ht = (pm, pm; . . . ; pm, pm) or t = 0
c otherwise.
(8)
Takuya Irie Dynamic Price Competition and Tacit Collusion I

21. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
If
Πm
2
(1 + δ + δ2
+ . . . ) ≥ Πm
, (9)
which follows if δ ≥ 1
2, then these trigger strategies are equilibrium
ones because if a firm deviates from pm, its present discounted
value of profits will decrease.
Intuition
If a firm undercuts the monopoly price, it gets almost monopoly
profit during the period of deviation but it destroys collusion in the
later periods.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

22. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Folk theorem
Any pair of profits (Π1, Π2) such that
Π1
> 0, Π2
> 0, and Π1
+ Π2
≤ Πm
(10)
is a per-period equilibrium payoff, which is amount to
(1 − δ)
∞∑
t=0
δt
Πi
(pit, pjt), (11)
for δ sufficiently close to 1.
Since aggregate payoff cannot exceed Πm and equilibrium profits
cannot be negative, when δ is close to 1, everything is an
equilibrium.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

23. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
▶ This supergame theory is also too successful in explaining
tacit collusion.
▶ We have no indication how a “focal equilibrium” is chosen.
▶ However, in the literature we often introduce the following
assumptions:
1. In a symmetric game the focal equilibrium is symmetric.
2. The focal equilibrium must be Pareto optimal.
▶ In this case, per-period payoff will be Π1 = Π2 = Πm/2 when
δ ≥ 1/2.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

24. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Application1: Market Concentration
Assumption
n firms with the same marginal cost charge pm and share the
market equally.
▶ The per-period and per-firm profit is Πm/n.
▶ Thus, the cost of being punished for undercutting decreases
w.r.t. n.
▶ The short-run gain from undercutting pm slightly is
Πm
(1 − 1/n) − ε. (12)
Takuya Irie Dynamic Price Competition and Tacit Collusion I

25. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
▶ On the other hand, the long-run loss is
Πm
n
(δ + δ2
+ δ3
+ · · · ). (13)
▶ For collusion to be sustainable,
Πm
n
(δ + δ2
+ δ3
+ · · · ) > Πm
(1 − 1/n) − ε (14)
⇔ δ >
n − 1
n
(15)
must hold (let ε = 0).
▶ Therefore, market concentration facilitates tacit collusion.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

26. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Application2: Long Information Lags and Infrequent
Interaction
Punishment might be delayed for the following two reasons:
▶ Information lags
▶ Infrequent interaction
Infrequent interaction means that δ is low. That’s all.
Consider the first reason.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

27. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Assumptions
▶ Duopoly model
▶ Prices are observed two periods after they are chosen.
▶ A firm’s profit and demand are observed by this firm at least
two periods later.
Then, the monopoly price is sustainable in equilibrium iff
Πm
2
(1 + δ + δ2
+ · · · ) > Πm
(1 + δ) (16)
⇔ δ >
1
√
2
, (17)
which implies that information lags are also a cause of breakdown
of collusion.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

28. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Application3: Fluctuating Demand
Assumptions
▶ Demand is stochastic:
q =
{
D1(p) w.p. 1
2
D2(p) w.p. 1
2.
(18)
▶ D1(p) < D2(p) for all p.
▶ The demand shock is i.i.d. over time.
▶ The two firms learn the current state of demand before
choosing their price simultaneously.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

29. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Look for a pair of prices {p1, p2} such that
1. both firms charge price ps when the state of demand is s;
2. {p1, p2} is sustainable in equilibrium;
3. the expected present discounted profit of each firm along the
equilibrium path
V =
∞∑
t=0
δt
(
1
2
D1(p1)
2
(p1 − c) +
1
2
D2(p2)
2
(p2 − c)
)
(19)
=
(
1
2
D1(p1)
2 (p1 − c) + 1
2
D2(p2)
2 (p2 − c)
)
1 − δ
(20)
is not Pareto dominated by the other equilibrium payoffs.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

30. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
▶ Let Πs(p) = (p − c)Ds(p) be the profit in state s.
▶ Assume that the two firms take the following trigger strategy:
for each i,
pit(Ht) =
{
pm
s if Ht = (pm
s , pm
s ; . . . ; pm
s , pm
s ) or t = 0
c otherwise,
(21)
where pm
s ≡ arg max Πs(p).
▶ Let Πm
s ≡ Πs(pm
s ) be the monopoly profit in state s.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

31. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
▶ If the monopoly profit can always be sustained, then
V =
(Πm
1 + Πm
2 )/4
1 − δ
. (22)
▶ If a firm deviates,
1. the future loss: δV
2. an extra gain: Πm
s /2
▶ For collusion to be sustainable for all s, since Πm
2 > Πm
1 ,
Πm
2
2
≤ δV (23)
must hold.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

32. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Then, we get
(23) ⇔ δ ≥ δo ≡
2Πm
2
3Πm
2 + Πm
1
. (24)
Note that δ0 ∈ (1
2, 2
3).
Intuition
▶ Since the future loss is an average of high and low profit, it is
smaller than it would be if the high demand were to persist
with certainty in the future.
▶ Therefore, when δ ∈ [1
2, δ0), while in the case of a
deterministic demand full collusion is sustainable, in this case
it cannot be sustained in the high-demand state.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

33. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
▶ Consider the case that δ ∈ [1
2, δ0).
▶ Choose p1 and p2 as follows:
max
p1,p2
(
1
2
Π1(p1)
2 + 1
2
Π2(p2)
2
)
1 − δ
(25)
subject to
Π1(p1)
2
≤ δ
(
1
2
Π1(p1)
2 + 1
2
Π2(p2)
2
)
1 − δ
(26)
Π2(p2)
2
≤ δ
(
1
2
Π1(p1)
2 + 1
2
Π2(p2)
2
)
1 − δ
. (27)
Takuya Irie Dynamic Price Competition and Tacit Collusion I

34. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
▶ This maximization problem is equivalent to:
max
p1,p2
{Π1(p1) + Π2(p2)} (28)
subject to Π1(p1) ≤
δ
2 − 3δ
Π2(p2) (29)
Π2(p2) ≤
δ
2 − 3δ
Π1(p1). (30)
▶ Then, we get p1 = pm
1 , and p2 is chosen so that
Π2(p2) =
δ
2 − 3δ
Π1(pm
1 )1. (31)
▶ Note that p2 < pm
2 .
1
See footnotes 17 and 18.
Takuya Irie Dynamic Price Competition and Tacit Collusion I

35. Introduction
Conventional Wisdom
Static Approaches to Dynamic Price Competition
Supergames
The Theory
Applications
Thus, for δ ∈ [1
2, δ0), some collusion is sustainable:
▶ In the low state of demand, firms charge the monopoly price;
▶ In the high state of demand, firms charge below the monopoly
price.
Conclusion
Although in the usual sense the price may be higher during booms,
this model implies that there exists a price war during booms.
Takuya Irie Dynamic Price Competition and Tacit Collusion I