Dynamic Price Competition and Tacit Collusion II

Secret Price Cuts
Price Rigidities
Reputation for Friendly Behavior
Dynamic Price Competition and Tacit Collusion II
Takuya Irie
April 29, 2017
Takuya Irie Dynamic Price Competition and Tacit Collusion II

Secret Price Cuts
Price Rigidities
Outline
Secret Price Cuts
A Simple Example
Price Competition
Price Rigidities
Markov Perfect Equilibrium
Maskin and Tirole (2001)
The Dynamic Programing Equations
Proﬁts Are Bounded Away from Zero
Folk Thorems
Concluding Remarks

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
Secret Price Cuts
▶ Consider the case in which its rival’s prices are not observable.
▶ In this case, a ﬁrm must rely on the observation of its own
realized market share or demand to detect any price cutting.
▶ However, when demand is random, a low market share may be
due to a price cut or to a slack in demand.
▶ Thus, if demand is random, price cuts are hard to detect,
which leads to hinder collusion.

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
A Simple Example
▶ Two firms charge pm as long as their profit has been high in
the past.
▶ Punishment Phase: If a firm observes a low demand (due to a
price cut or a low demand) or if it itself has undercut pm in the
last period, it charges a low price for some periods of time T.
▶ Collusive Phase: The firms charge pm after the punishment
phase is completed, until the next deviation or slump in
demand.
▶ This model predicts periodic price wars.
Note: Price wars are triggered by a recession, contrary to
Application 3 (Fluctuating Demand).

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
▶ Under imperfect information, the fully collusive outcome
cannot be sustained.
▶ It could be sustained only if the firms kept on colluding, but a
firm that is confident that its rival will continue cooperating
has every incentive to undercut.
▶ Thus, full collusion is inconsistent with the deterrence of price
cuts.

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
Price Competition
Assumptions
▶ Two ﬁrms chooses price every period.
▶ The goods are perfect substitutes and MC = c.
▶ The demand is split in halves if in a tie.
▶ Demand is stochastic:
q =
{
0 w.p. α
D(p) w.p. 1 − α.
(1)
▶ pm and Πm: the monopoly price and the monopoly proﬁts in
the high-demand state
▶ The demand shock is i.i.d. over time.

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
▶ A firm that does not sell at some date is unable to observe
whether the absence of demand is due to the realization of
the low-demand state or to its rival’s lower price.
▶ It is always common knowledge that at least one firm makes
no profit (Check!).
▶ In the infinitely repeated version of the game, we look at an
equilibrium with the following strategies:

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
Strategies
▶ Collusive Phase (CP): Both firms charge pm until one firm
makes a zero profit.
▶ Punishment Phase (PP): Both firms charge c for exactly T
periods, where T can a priori be finite or infinite.
The game begins in the CP. The occurrence of a zero profit
triggers a PP. At the end (if any) of PP, the firms revert to the CP
and charge pm as long as they both make positive profits.

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
Optimal Length of the Punishment Phase
▶ Look for a length of the PP such that the expected present
discounted value of profits of each firm is maximal subject to
the constraint that the associated strategies form an
equilibrium.
▶ Let V + (resp. V −) denote the present discounted value of a
firm’s profit from date t on, assuming that at date t the game
is in the CP (resp. starts from the PP).
▶ Then we have
V +
= (1 − α)(Πm
/2 + δV +
) + α(δV −
), (2)
V −
= δT
V +
. (3)
▶ Incentive constraint:
V +
≥ (1 − α)(Πm
+ δV −
) + α(δV −
) (4)

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
▶ By some computations, we have
1 ≤ 2(1 − α)δ + (2α − 1)δT+1
. (5)
▶ Because the game starts in the CP, the highest proﬁt for the
ﬁrms is obtained by solving the following program:
max
T
V +
(
=
(1 − α)Πm/2
1 − (1 − α)δ − αδT+1
)
subject to inequality 5.

Secret Price Cuts
Price Rigidities
A Simple Example
Price Competition
▶ The RHS of inequality 5 is increasing (resp. decreasing) w.r.t.
T if α < 1
2 (resp. α ≥ 1
2).
▶ Since V + is decreasing w.r.t. T and inequality 5 is not
satisfied for T = 0, if α ≥ 1
2, there exists no T satisfying
inequality 5.
▶ If α < 1
2, it suffices to choose the smallest T that satisfies the
incentive constraint.
▶ We thus obtain a finite optimal length of punishment, which
implies periodic price wars.

Secret Price Cuts
Price Rigidities
Price Rigidities
▶ We assumed that past price does not affect their profits; i.e.
prices are adjusted continuously.
▶ However, changing one’s price every day or every minute
would often be prohibitively expensive, so prices are likely to
exhibit short-run rigidities.
▶ In addition, on the demand side, past prices may affect the
firms’ current goodwill through consumers’ learning about the
good or switching costs.
▶ On the supply side, past prices affect current inventories.

Secret Price Cuts
Price Rigidities
A Simple Example (Asynchronous Timing)
Assumptions
▶ Two firms produce perfect substitutes.
▶ At odd (resp. even) periods, firm 1 (resp. firm 2) chooses its
price.
▶ pi,t+1 = pi.t, where pi,t is a price chosen by firm i at date t.
▶ In period t + 2, firm i may choose a new price, which again
will be locked in for two periods.
Then, firm i’s objective is to maximize
∞∑
t=0
δt
Πi
(pi,t, pj,t). (6)

Secret Price Cuts
Price Rigidities
▶ We look for a perfect equilibrium in which the firms’ price
choices are simple in that they depend only “payoff-relevant
information.”
▶ For example, p2,2k affects firm 1’s profit at date 2k + 1 and
will be termed payoff-relevant.
▶ We write this as p1,2k+1 = R1(p2,2k); i.e. firm 1’s strategy is
conditioned as little information as is consistent with
rationality.
▶ Similarly, p2,2k+2 = R2(p1,2k+1).
▶ These reaction functions are called Markov reaction functions
and a perfect equilibrium in which the firms use Markov
strategies is called a Markov perfect equilibrium (MPE).

Secret Price Cuts
Price Rigidities
By “one-shot deviation principle,” it suffices to check that for any
current price p2 = p2,2k at time 2k + 1, firm 1’s reaction
p1 = p1,2k+1 maximizes
Π1
(p1, p2)+δΠ1
(p1, R2(p1))+δ2
Π1
(R1(R2(p1)), R2(p1))+· · · (7)
and firm 2 behaves similarly.

Secret Price Cuts
Price Rigidities
▶ Consider a dynamic game in which, in each period t, player i’s
payoﬀ πi
t depends only on the vector of the player’s actions
at, that period, and on the current (payoﬀ-relevant) “state of
the system” θt ∈ Θt; i.e. πi
t = gi
t(at, θt).
▶ Assume that
1. player i’s possible actions Ai
t depend only on θt; i.e.
Ai
t = Ai
t(θt);
2. θt is determined by the previous period’s actions at−1 and
state θt−1;
3. each player maximizes E(
∑
t δt−1
πi
t).
▶ In period t, the history of the game, ht, is the sequence of
previous actions and states:
ht = (θ1, a1, θ2, a2 . . . , θt−1, at−1, θt).

Secret Price Cuts
Price Rigidities
▶ Since the only aspect of the history that directly affects player
i’s payoffs and action sets starting in period t is the state θt,
it is natural to consider strategies that depend only on the
state θt rather than on the whole history ht.
▶ In this kind of games, we can easily derive a MPE (a fortiori a
subgame perfect equilibrium) using Markov strategies.
Note: By contrast, in an arbitrary dynamic game, we must first
derive the set of states in order to discuss Markov strategies
(maybe so hard.....).

Secret Price Cuts
Price Rigidities
Merits of Markov Perfect Equilibrium
1. MPE is successful in eliminating or reducing a large
multiplicity of equilibria in dynamics games.
2. MPE is successful in enhancing the predictive power.
3. By preventing non-payoﬀ-relevant variables, MPE has allowed
researchers to identify the impact of state variables on
outcomes.
4. Markov strategies reduce the number of parameters to be
estimated in dynamic econometric models.

Secret Price Cuts
Price Rigidities
We look for the necessary and sufficient conditions that correspond
to a symmetric equilibrium; i.e. R1 = R2 = R.
Assumptions
▶ There is a finite number of possible prices ph.
▶ R : ph → pk is a reaction function, where ph is the price to
which one of the firm is currently committed and pk is the
price chosen by the other firm.
▶ αhk ≥ 0: the transitional probability that the firm reacts to
price ph by charging price pk (∴
∑
k αhk = 1)
▶ Π(pk, ph): the instantaneous profit of the firm when its price
is pk and the price of its competitor is ph
Then, R will be interpreted as a Markov chain.

Secret Price Cuts
Price Rigidities
▶ Let Vh be the discounted value pf the profit of a firm that
chooses its price when the other firm has chosen ph in the
preceding period, and Wh be that of the second firm.
▶ Then, we have
Vh = max
pk
[Π(pk, ph) + δWk]. (8)
▶ This yields the following set of equations:

Secret Price Cuts
Price Rigidities
Vh =
∑
k
αhk[Π(pk, ph) + δWk], (9)
Wk =
∑
l
αkl[Π(pk, ph) + δVl], (10)
[Vh − Π(pk, ph) − δWk]αhk = 0, (11)
Vh ≥ Π(pk, ph) + δWk, (12)
∑
k
αhk ≥ 1, (13)
αhk ≥ 0. (14)
To check that strategies form an equilibrium, it suﬃces to compute
Vh and Wh for all h and check that these equations are satisﬁed.

Secret Price Cuts
Price Rigidities
▶ Show that in a symmetric equilibrium the average per period
must exceed Π(pm)/2 for δ close to 1; i.e. profits in a MPE
cannot be close to the competitive profit.
▶ The price grid is assumed discrete.
▶ Let V (p) (resp. W(p)) denote the present discounted value of
profits of the firm whose turn it is (resp. is not) to choose a
price.

Secret Price Cuts
Price Rigidities
▶ Consider the case in which a firm chooses pm + k, where k is
“small.”
▶ Then, the other firm’s present discounted value of profit is
V (p) = max( max
p<pm+k
[Π(p) + δW(p)],
Π(pm + k)
2
+ δW(pm
+ k), max
p>pm+k
δW(p))
(15)
▶ Let p∗ be the smallest price that solves the RHS of 15.
▶ Then, a firm’s reaction to pm + k is not lower than p∗; i.e.
R(pm + k) ≥ p∗.

Secret Price Cuts
Price Rigidities
Case a: p∗
≥ pm
▶ In this case, starting from any price, each ﬁrm’s payoﬀ when it
plays is at least
δ2
[Π(pm
− k) + δW(pm
− k)]. (16)
∵ it could raise its price to pm + k, and undercut to pm − k
after its rival’s reaction.
▶ Similarly, we have
W(pm
− k) ≥ δ3
[Π(pm
− k) + δW(pm
− k)]. (17)

Secret Price Cuts
Price Rigidities
▶ Thus, each firm’s intertemporal profit is as least
(
δ2
1 + δ + δ2 + δ3 + δ4
)
Π(pm − k)
1 − δ
. (18)
▶ For δ close to 1, this profit is as least
1
4
(
Π(pm − k)
1 − δ
)
, (19)
which amounts to
Π(pm − k)
4
(20)
close to
Π(pm)
4
(21)
for a fine price grid.

Secret Price Cuts
Price Rigidities
Case b: p∗
< pm
▶ In this case, we have
Π(p∗
) + δW(p∗
) ≥ Π(pm
) + δW(pm
). (22)
▶ On the other hand,
W(pm
) ≥ δ
Π(p∗)
2
+ δ2
W(p∗
). (23)
▶ Then we have
(1 − δ)W(pm
) ≥
δ
1 + δ
(
Π(pm
) −
Π(p∗)
2
)
. (24)
▶ Since Π(pm) − Π(p∗)
2 ≥ Π(pm)
2 and δ
1+δ ≃ 1
2, the average profit
per period and per firm exceeds one-forth of the monopoly
profit.

Secret Price Cuts
Price Rigidities
Folk Thorems
Concluding Remarks
▶ Asymmetries in information are likely to induce firms to raise
their prices in a situation of repeated price interaction.
▶ In a repeated price game with asymmetric information about
marginal cost or demand, each firm sacrifices short-run profit
by raising its price in order to build a reputation for changing
high prices.

Secret Price Cuts
Price Rigidities
Folk Thorems
Concluding Remarks
Infinitely Repeated Games of Complete Information
Assumptions
▶ Infinitely repeated game of the following n-person “static”
game
▶ Ai: action space for player i
▶ Πi(ai, a−i): payoff function for player i
▶ The set of pure strategies is finite.
▶ We do not distinguish between pure and mixed strategies
(think of Ai as the set of mixed strategies).
The static game is called the “constituent game.”

Secret Price Cuts
Price Rigidities
Folk Thorems
Concluding Remarks
Definitions
▶ A payoff vector Π = (Πi, Π−i) is individually rational if
Πi > Πi∗ for all i, where Πi∗ = mina−i maxai Πi(ai, a−i).
(Πi∗ is called reservation utility.)
▶ It is feasible if there exists feasible strategies a = (ai, a−i)
such that Πi = Πi(a) for all i.
Payoff:
V i
=
∞∑
t=0
δt
Πi
(ai(t), a−i(t)) (25)
(ai(t) is the action chosen by i at t.)
Average payoff:
vi
= (1 − δ)V i
(26)

Secret Price Cuts
Price Rigidities
Folk Thorems
Concluding Remarks
Folk Theorem
Let aN = (aN
i , aN
−i) be a Nash equilibrium of the constituent
game, ΠiN = Πi(aN
i , aN
−i), and v = (vi, v−i) such that v is feasible
and vi > ΠiN for all i. Then there exists δ0 < 1 such that v is an
equilibrium payoﬀ vector for all δ ≥ δ0.
Proof: For simplicity, suppose that there exists pure strategies
a = (ai, a−i) such that vi = Πi(ai, a−i) for all i. Consider the
following strategies. Each player plays ai as long as all players have
stuck to strategies a before. If someone has deviated in the past
period, the player plays aN
i . Then, by deviating today, a player
gains at most a bounded amount; on the other hand, he looses the
gain from future cooperation:
(vi
− ΠiN
)(δ + δ2
+ · · · ), (27)
which tends to inﬁnity as δ tends to 1. 2

Secret Price Cuts
Price Rigidities
Folk Thorems
Concluding Remarks
Finitely Repeated Games of Complete Information with
Multiple Equilibria in the Constituent Game
▶ Although collusion cannot be sustained in the ﬁnitely repeated
game in which there exists only one Nash equilibrium (e.g.
Bertrand price game), when the constituent game has several
Nash equilibrium, it is possible to obtain equilibria other than
the Nash equilibrium in the repeated version of the game.
▶ See the example in 6.7.3.2.

Secret Price Cuts
Price Rigidities
Folk Thorems
Concluding Remarks
Finitely Repeated Games of Incomplete Information
Assumptions
▶ Finitely repeated game in which player i is “sane” w.p. 1 − α
and “crazy” w.p. α (Crazy player’s strategies are at the
disposal of the modeler.)
▶ If player i is sane, the action space is Ai and the payoff is
Πi(ai, a−i).
Theorem (Fudenberg and Maskin (1986))
Any individually rational and feasible payoff vector of the
constituent game (for sane player) can be sustained as a perfect
Bayesian Equilibrium of the finitely repeated game for arbitrary
small probability α, as long as the horizon is sufficiently large and
the discount factor is sufficiently close to 1.

Secret Price Cuts
Price Rigidities
Folk Thorems
Concluding Remarks
Sketch of the proof: Let aN = (aN
i , aN
−i) be a Nash equilibrium of
the constituent game when players are sane w.p. 1. Let ΠiN be
the corresponding payoﬀs and vi = Πi(ai, a−i). Assume that the
crazy player i plays ai as long as all players have done so in the
past and aN
i if anyone has deviated in the past. For a sane player,
the one-period payoﬀ from deviating is bounded above. On the
other hand, the loss of future cooperation with the crazy players is
αn−1(vi − ΠiN )T if T is the horizon and δ = 1. Hence, for T ≥ T0
it cannot be optimal for player i to deviate from ai even if he is
sane, where T0 is the value at which the gain and the loss are
equalized. 2

Secret Price Cuts
Price Rigidities
Folk Thorems
Concluding Remarks
Concluding Remarks
▶ Although we have seen three approaches to price collusion
(supergames, price rigidities and reputation), these
“theoretical heterogeneity” is needed here.
▶ Since the proliferation of theories is mirrored by an equally
rich array of behavioral patterns actually observed under
oligopoly, these approaches may be complements rather than
competitors.

Dynamic Price Competition and Tacit Collusion II

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