Lecture Notes: Industrial Organization
Joe Chen
6.6
75
Secret price cuts
As stated earlier, a firm weights two opposite incentives when it ponders price cutting: future
losses and current gains. The highest level of collusion (monopoly price) is sustainable with
the severest level of punishment (eternal reversion to pricing at marginal cost). When price
choices are perfectly observable, it makes sense to resort to extreme punishments because
such punishments never occur in equilibrium and therefore are costless to firms (they are
“just” threats). In some situations, maximal punishments need not be optimal.
Suppose firms initially coordinate on the monopoly-price perfect equilibrium, and some
firm deviates by undercutting the price in the first period. By the trigger strategy with
maximal punishment, firms charge the marginal cost forever after the deviation. But the
firms, who expect no profits from period 2 on, have every incentive to renegotiate to avoid
the punishment phase and reach another equilibrium anew. The possibility of renegotiating
undermines the strength of punishments and, therefore, adds incentive to undercutting.
This opens the door for the discussion of renegotiation equilibrium in supergames. We
will not get into that here.
Instead, we consider a situation where the demand is stochastic and firms cannot tell a
secret price undercutting from a slump in the demand. Under such an uncertainty, mistakes
are unavoidable and maximal punishments need not be optimal.
Under imperfect information, the fully collusive outcome is not sustainable. The collusive outcome could be sustained only if firms kept on colluding (charging the monopoly
price) even when making small profits, because even under collusion small profits can occur
as a result of low demand. However, a firm that is confident that its rivals will continue
cooperating even when their profits are low has every incentive to undercut secretly. Thus,
fully collusive outcome is inconsistent with preventing (deterring) secret price undercutting.
Lecture Notes: Industrial Organization
Joe Chen
6.6.1
76
Setup — price game
Porter (1983) and Green and Porter (1984) propose a supergame model that formulizes the
issue of secret price cutting. In their model, Green and Porter assume quantity competition.
Here, we will go through a version of the model where firms compete in prices. The essence
is the same.
Consider a market of two firms choosing prices in every period. The goods are perfect
substitutes and are produced at constant marginal cost c. Consumers all buy from the low
price firm, and the demand is split in halves if both firms charge the same price. In each
period, there are two possible states of nature. With probability α, there is no demand (the
“low-demand state”); with probability 1 − α, the demand is D(p) > 0 (the “high-demandstate”) Denote pm ≡ arg maxp (p − c)D(p), and Πm = (pm − c)D(pm ). Note that a firm
that does not sell at time t is not able to observe whether the absence of demand is due to
the realization of the low-demand state or to rival’s secret price cut.
Let’s look for an equilibrium with the following trigger strategies:
• At the beginning, both firms charge pm ;
• Both firms continue to charging pm until one or both of them makes zero profit and
the game goes to the punishment phase. Call the phase that both firms charging
pm the collusive phase;
• In the punishment phase, both firms charge the marginal cost c for exactly T periods
(T can, a priori, be finite or infinite), and revert to the collusive phase.
Note that at least one firm makes zero profit, whenever it happens, is common knowledge, even though both firms does not observe its rival’s profits. Note also that the punishment phase is unavoidable in equilibrium; in other words, the punishment phase is on
the equilibrium path.
Lecture Notes: Industrial Organization
Joe Chen
6.6.2
77
Trigger strategy equilibrium
To look for an equilibrium given the above trigger strategies is to look for a T 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 + denote the expected present discounted value of a firm’s profit from t on, given
that at t − 1, the game is in the collusive phase. Let V − denote the payoff of a firm’s profit
from t on, given that at t, the game starts the punishment phase. Then:
⎧
⎨ V + = (1 − α)( Πm + δV + ) + αδV −
2
⎩ V − = δT V +.
Or,
⎧
⎪
⎨V + =
⎪
⎩V − =
(1−α)Πm
2[1−(1−α)δ−αδ T +1 ]
(1−α)δ T Πm
.
2[1−(1−α)δ−αδ T +1 ]
(*)
And, the incentive constraint is:
V + ≥ (1 − α)(Πm + δV − ) + αδV − ;
or (using the definition of V + ),
δ(V + − V − ) ≥ Πm /2.
(**)
Note that, on the one hand, V − must be sufficiently lower than V + to prevent undercutting.
This implies T has to be long enough. On the other hand, because punishments are costly
and occur with positive probability, T should be chosen as small as possible given that the
constraint is satisfied. Substituting the equations of (*) to equation (**) yields:
2(1 − α)δ + (2α − 1)δ T +1 ≥ 1.
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Joe Chen
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When the game starts in the collusive phase, the highest profit for the firms is obtained
by solving the following problem:
max V + =
T
(1 − α)Πm
£
¤
2 1 − (1 − α)δ − αδ T +1
s.t. 2(1 − α)δ + (2α − 1)δ T +1 ≥ 1.
Observe first, V + is decreasing in T (again, this implies we need to find T as small as
possible). Following conclusions can be drawn:
• For all T , 2(1 − α)δ + (2α − 1)δ T +1 < 1, if α ∈ [1/2, 1]. This is because when α = 1/2,
2(1 − α)δ + (2α − 1)δ T +1 = δ < 1, and 2(1 − α)δ + (2α − 1)δ T +1 is decreasing in α.
When α is “high”, no equilibrium with the above trigger strategies is sustainable;
• Assume 2(1 − α)δ ≥ 1 ⇔ (1 − α)δ ≥ 1/2, the above trigger strategy equilibrium
is guaranteed using maximal punishments (T = ∞). Note that this generalizes the
result for the deterministic demand case which corresponds to α = 0.
• To maximize V + subject to the incentive constraint, given α and δ, it suffices to
choose the smallest T (finite) that satisfied the incentive constraint.
As an example, if α = 1/4, the incentive constraint requires 3δ − δ T +1 ≥ 2. This is
possible only when δ ≥ 2/3. If δ = 0.7, the smallest T is around 5.46.
Now we have price wars that are “involuntary” in that they are triggered, not by a price
cut, but by an unobservable slump in demand. Note also that price wars are triggered by
recessions, contrary to the Rotemberg-Saloner model.
Lecture Notes: Industrial Organization
Joe Chen
6.7
79
Price rigidities
In the supergame framework, we assume that firms always choose prices simultaneously (the
synchronicity assumption). Two features of this setup are important: First, a firm’s current
profit is not affected by its rival’s previous price choices when it chooses its own price (the
game is a repeated game, not a fullly fledged dynamic one); second, the only reason a firm
conditions its pricing behavior on previous price choices is that the other firms do so. Hence,
the firms’ strategies are bootstrap strategies in that: The achievement of collusion stems
from a subtle self-fulfilling expectation. There are no “real” business strategies, such as
trying to regain losing market shares.
In reality, past price choices affect current profits in many scenarios. This raises the
discussions that price reactions are not bootstrap reactions but are “real” attempts to
regain market shares. Past price choices may affect current profits through:
• Price rigidity: the existence of menu cost;
• On the demand side, consumers may face costs of product learning, or switching costs,
or both;
• On the supply side, past prices affect current workload, if orders take time to be filled.
In this subsection, we introduce the concept of Markov perfect equilibrium (MPE)
through the examination of a model of asynchronous pricing. Let’s consider two firms
producing perfect substitutes. At odd (respectively, even) periods, firm 1 (respectively
2) chooses its price. At any period t, a price pit chosen by firm i lasts for two periods:
pit = pi,t+1 . In period t + 2, firm i chooses a new price, which again will be locked in
for another two periods. The assumption that firm 1 (respectively, firm 2) chooses a price
at odd (respectively, even) periods and the assumtpion that the prices are lock-in for two
periods, are not important; the key is the lock-in of prices for some periods.
We look for an equilibrium in which the firms’ price choices are “simple” in that they
depend only on the “payoff-relevant” information. More precisely, at date 2k + 1, firm 2
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Joe Chen
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is still committed to the price p2k it chose at period 2k. Note that this price affects firm
1’s profit at period 2k + 1 and therefore, it is “payoff-relevant”. Consider a strategy of the
form: p1,2k+1 = M1 (p2,2k ). That is, firm 1’s strategy is conditioned on as little information
as is consistent with profit maximizing. Similarly for firm 2, p2,2k+2 = M2 (p1,2k+1 ). Mi (·) is
called a Markov reaction function. A Markov perfect equilibrium is a perfect equilibrium
in which firms use Markov strategies. For price p2,2k at time 2k + 1, firm 1’s reaction must
maximize its objective function given that the firms will react according to M1 (·) and M2 (·)
in the future. Mathematically, at period 2k + 1, denote p2,2k as p2 , and p1,2k+1 as p1 , firm
1’s profit from period 2k + 1 on is:
V 1 (p2 ) = max Π1 (p1 , p2 ) + δΠ1 (p1 , M2 (p1 )) + δ 2 Π1 (M1 (M2 (p1 )), M2 (p1 )) + · · ·
p1
In equilibrium, p1 = M1 (p2 ) must maximize the profit expression for all p2 . Firm 2 behaves
similarly.
6.7.1
The kinked-demand story revisited
Let D(p) = 1 − p, and firms are producing at marginal cost c = 0. The price grid is discrete:
ph = h/6, where h = 0, 1, ..., 6. Note that p0 = 0 is the competitive price, and p3 = 1/2
is the monopoly price. Consider a symmetric reaction function M1 (·) = M2 (·) = M (·) as
follows:
p
Π(p) = 36p(1 − p)
M (p)
p6 = 1
0
p3
p5 = 5/6
5
p3
p4 = 2/3
8
p3
p3 = 1/2
9
p3
p2 = 1/3
8
p1 = 1/6
5
p0 = 0
0
p1
⎧
⎨ p1 with prob. α
⎩ p with prob. 1 − α
3
p3
According to M (·), starting from p3 , if a firm raises its price, its rival does not follow suit.
If a firm undercuts to p2 , its rival reacts with a price war. At p1 , the firms engage in a
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Joe Chen
81
war of attrition. Both firms want the price to go back to p3 ; however, each of them wants
the other firm to move first, because the relenting firm loses market share in the short run.
The outcome is a typical mixed strategy behavior in which firms raise price with positive
probability.
The pair of strategies (M (·), M (·)) forms an MPE when the discount factor is close
enough to one. To check, we need to verify that no firm would deviate from M (·):
• No price cutting at p3 . At p3 , undercutting to p2 results in: 8 + δ · 0 + δ 2 · 0 + δ 3 · V (p3 );
pricing at p3 results in: V (p3 ) = 4.5(1 + δ + δ 2 + ...) = 4.5/(1 − δ). So, as long as δ is
close enough to one, undercutting is not profitable;
• At p2 , continue the price war. Charging p1 results in: 5 + δ · W (p1 ), where W (p1 )
is the payoff when: a firm chose p1 in the previous period, it is now the other firm’s
term to choose price, and firms use (M (·), M (·)). Note that at p1 , firms are indifferent
between staying p1 and raising the price to p3 . Hence, 2.5 + δW (p1 ) = δV (p3 ) or,
W (p1 ) = V (p3 ) − 2.5/δ. So, the payoff of charging p1 is: 2.5 + δV (p3 ). This is larger
than the pay off of charging p3 which is δV (p3 ).
• At p1 , firms play a mixed strategy with the probability of playing p1 defined as:
4.5
δ
− δ}
| 1{z
raising price to p3
or,
= 2.5 + δ {α[2.5 + δV (p1 )] + (1 − α)[5 + δV (p3 )]};
|
{z
}
continuing p1
α=
5+δ
.
5δ + 9δ 2
Note that V (p1 ) = δV (p3 ).
Let p3 be the focal price, this equilibrium is the same as the static kinked-demand-curve
story, but now the reactions are “real” and fully rational.
Lecture Notes: Industrial Organization
Joe Chen
6.7.2
82
Edgeworth cycle
There also exist equilibria in which the price never settles. For instance, consider the
following strategy:
p
Π(p) = 36p(1 − p)
M (p)
p6 = 1
0
p4
p5 = 5/6
5
p4
p4 = 2/3
8
p3
p3 = 1/2
9
p2
p2 = 1/3
8
p1
p1 = 1/6
5
p0 = 0
0
p
⎧0
⎨ p0 with prob. β
⎩ p with prob. 1 − β
5
Lecture Notes: Industrial Organization
Joe Chen
6.7.3
83
Some final thoughts
Despite multiple equilibria, it can be shown that in every MPE, profits are always bounded
away from the competitive profit (which is 0). Both supergames and Markov games suggest
some collusion is always possible as long as the discount factor is close enough to one. In
Markov games, unlike supergames, the current profit is determined by pervious actions.
When firms compete in prices, the response to a rival’s previous action would be to regain
its losing market share. Based on this reasoning, suppose one introduces demand fluctuation
into the model, one would expect price adjustments more sluggish during booms than during
recessions (more Green-Porter alike).
Some sort of collusion is possible because of the “fear” of retaliation (triggering a price
war). However, the motives for retaliation are very different with the two different setups.
In supergames, the price war is a purely self-fulling phenomenon. A firm charges a low price
because it expects the other firms to do so (bootstrapping). In Markov games with price
rigidities, the reaction of one firm to a price cut by another firm is motivated by its desire
to regain a market share that has been and continues to be eroded by its rivals aggressive
pricing strategy.
In an intertemporal setup, it is also possible that firms can sustain collusion through
nonphysical factors such as reputation. We stop here.