Lecture Notes: Industrial Organization
Joe Chen
6
64
Oligopoly Pricing — Dynamic
The simple Bertrand price game is one-shot in nature. As mentioned previously, the
introduction of repeated interaction makes firms to take into account the possibility of
price wars and long-run losses when they decide whether to undercut a given price. Thus,
the time dimension itself can work to soften the intensive price competition in oligopolistic
markets. In an oligopolistic setting, the threat of a vigorous price war can be sufficient
to deter the temptation to cut prices. Hence, firms might be able to collude in a purely
noncooperative manner. This opens a door to our discussion of tacit collusion. We will
consider a variety of theories that explain tacit collusion. Before we dive into different ideas,
we stop a little bit to talk about factors facilitating or hindering collusion (conventional
wisdom), and static approaches to dynamic price competition.
6.1
Collusion hindering (facilitating) factors
Conventional wisdom suggests several factors may hinder or facilitate collusion. The point
is that there are (always) two opposite forces working behind tacit collusion: the temptation
to undercut and making current (short-run) profits; the retaliations (price wars) and future
losses associated with undercutting.
• Secrecy: detection lags and few buyers (lumpiness);
• Multimarket contact;
• The existence of large sales;
• Decreasing returns to scale;
• Asymmetry: difficult to coordinate;
• Market concentration (number of firms).
We will see later on how to “model” some of these intuitions.
Lecture Notes: Industrial Organization
Joe Chen
6.2
65
Static approaches to dynamic price competition
Because dynamic pricing behavior is difficult to analyze and the tool to do so is developed
only “recently”, there is a considerable literature that attempts to formulize the dynamic aspects in a static content. We briefly mention the kinked demand theory and the conjectural
variation approach.
6.2.1
Kinked demand theory
This theory tries to explain the absence of frequent price cutting in oligopolistic markets.
Let there be two firms , i = 1, 2, with unit cost c. The demand is: q = D(p). Let pf denote
the “focal price”, or, the steady-state (long-run) price. Each firms reasons as follows: If
p ≥ pf , the rival will not follow suit; if, instead, p < pf , the rival will match. Thus, firms:
max (pi − c)D(pi )/2
s.t. pi ≤ pf .
When pf ∈ [c, pm ], pf is an equilibrium. This equilibrium results in a demand curve with a
kink.
6.2.2
Conjectural variation approach
This approach proposes a conjectural variation function to answer the question whether
firms are colluding.
ej (ai ). Hence, firm i:
Suppose firm i believes that firm j reacts according to R
ej (ai )).
max Πi (ai , R
ai
ej are differentiable, we have:
Let ai = qi . If Πi and R
ej (qi ))qi − Ci (qi ).
max P (qi + R
qi
FOC yields:
ej0 )qi = P − Ci0 + qi P 0 + qi P 0 R
ej0 = 0.
P − Ci0 + P 0 (1 + R
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Joe Chen
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e0 = 0, corresponds to Cournot competition, and R
e0 =
Hence, zero conjectural variation: R
j
j
−1 corresponds to competitive outcome. As to the collusive outcome, it requires positive
ej 0 > 0.
conjectural variation, i.e., R
The traditional criticisms of industry models using conjectural variations are its lack
of a theoretical framework and that each firm’s conjecture about the output response of
the other firms would not be confirmed if such a firm actually altered its output level from
the equilibrium. As a result, there has been considerable interest in oligopoly models with
“consistent” conjectural variations. A conjectural variation is consistent if it is equivalent to
the optimal response of the other firms at the equilibrium defined by that conjecture. Perry
(1982) showed that when the number of firms is fixed, competitive behavior is consistent
when marginal costs are constant; when marginal costs are rising, the consistent conjectural
variation will be between competitive and Cournot behavior. Finally, when free entry is
allowed and redefine consistency to account for such, the only competitive behavior will be
consistent.
Lecture Notes: Industrial Organization
Joe Chen
6.3
67
Supergames
Consider the standard setup in the Bertrand game: Two firms produce perfect substitutes
with the same marginal cost, c. The difference is that we replicate the basic Bertrand
game T + 1 times. The game is then called a repeated game, or a supergame, with
Bertrand game as the stage (constituent) game. Let Πi (pit , pjt ) be firm i’s profit at
time t, t = 0, 1, ..., T , when it charges pit and its rival charges pjt . Each firm maximizes
XT
δ t Πi (pit , pjt ); where: δ = e−rτ , with an
the present discounted value of its profits,
t=0
instantaneous interest rate r and the time between periods τ . Denote the history of the
the game at time t as: Ht ≡ (pi0 , pj0 ; pi1 , pj1 ; ...; pi,t−1 , pj,t−1 ). The price strategy pit (Ht )
depends on the history of the game. We look for strategies to form a perfect equilibrium:
for any history Ht , firm i’s strategy from date t on maximizes the present discounted value
of profits given firm j 0 s strategy from t on.
If the horizon is finite, i.e., T < ∞, the game is a finite replication of the simple Bertrand
game. For finitely repeated games, to find a perfect equilibrium, it is easiest to proceed by
backward induction.
• Given HT , what are the firms’ choices of prices at the last period? For all HT ,
pi,T = pj,T = c.
— HT does not affect the profits in period T ;
— Each firm maximizes: Πi (piT , pjT ).
• What will be the equilibrium prices in period T − 1? Again, for all HT −1 ,
pi,T −1 = pj,T −1 = c.
— Price choices at T does not depend on what happens at T − 1 ;
— Everything is as if T − 1 were the last period.
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Joe Chen
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Following the same argument, the outcome of the (T + 1)-period price game is the
Bertrand solution repeated T + 1 times. The dynamic element contributes nothing to the
model.
Things change dramatically when the horizon is infinite; i.e., T = ∞. On the one hand,
the above result: pit = pjt = c, for all t, is still an equilibrium. In particular, consider
a strategy to price at the marginal cost regardless of the history of the game. On the
other hand, an interesting feature arises: the repeated Bertrand result is no longer the only
equilibrium.
Let pm be the monopoly price, and Πm be the monopoly profit. Consider a trigger
strategy as follows:
pit (Hit ) =
⎧
⎪
⎪
pm if t = 0
⎪
⎨
pm if t > 0, and Ht = (pm , pm ; ...; pm , pm )
⎪
⎪
⎪
⎩ c otherwise.
The strategy is called a trigger strategy because a single deviation triggers a halt to the
“cooperation”. Note that the temptation to deviate from pm is a current profit of Πm /2,
the retaliation is a loss of future profits of: (δ + δ 2 + ...)Πm /2; hence, as long as:
Πm /2 ≤ (δ + δ 2 + ...)Πm /2;
or, equivalently,
δ ≥ 1/2,
the trigger strategy is an equilibrium strategy. This equilibrium is “collusive” in that the
equilibrium price pit = pjt = pm > c; it is a “tacit” collusion because the collusion is
enforced through a purely noncooperative manner.
Naturally, this raises the question that the results of the supergame framework are not
robust to finite-length price interaction. Note that the infinite horizon assumption need
not be taken too seriously. Suppose that at each period there is a (constant) probability x
(1 − x) that the market continues (ends). Everything is as if the horizon were infinite and
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Joe Chen
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the firms’s discount factor were equal to e
δ ≡ xδ. Hence, if both δ and x are sufficiently
high, supergame collusion can be enforced.
As a matter of fact, one can replace pm with any p ∈ [c, pm ], and Πi ∈ [0, Πm ], the
above result still holds. This is a special case of a very general result, known as the Folk
theorem. The Folk theorem asserts that any pair of profits (Πi , Πj ) such that: Πi , Πj > 0
and Πi + Πj ≤ Πm , is a per-period equilibrium payoff for δ sufficiently close to one. This is
to say: there exist strategies {pit (Ht ), pjt (Ht )} that form a perfect equilibrium where firm
i (j) makes a per-period payoff: Πi (Πj ).
6.4
Folk theorem
Actually, several results are called by game theory people — the “Folk theorem”. Here
we are interested in the version (infinitely repeated games of complete information) given
by Friedman (1971). Consider an infinitely repeated n-player game with the static game
(the constituent game) defined by action spaces Ai for all i = 1, 2, ..., n, and payoff functions for player i, Πi (a1 , ..., ai , ..., an ), where aj ∈ Aj for all j = 1, ..., n. Denote: a−i ≡
(a1 , ..., ai−1 , ai+1 , ..., an ). Since the game is repeated infinitely many times and we allow the
X∞
δ t Πi (a1 (t), ..., an (t)),
strategies to be functions of past history, the total payoff is: V i =
t=0
with average payoffs, v i , defined as: for all i,
vi ≡ (1 − δ)V i .
We can define player i’s reservation payoff, Πi∗ , as the worst outcome (on average) that
she can be forced to take; i.e.,
Πi∗ = min max Πi (ai , a−i ).
a−i
ai
A payoff vector Π = (Π1 , ..., Πi , ..., Πn ) is individually rational if: for all j, Πj ≥ Πj∗ .
It is feasible if: there exists strategy a = (a1 , ..., ai , ..., an ), such that: for all j, aj ∈ Aj ;
and, Πj (a) = Πj . The individually rational profits in the Bertrand price game or the
Cournot quantity game are zeros. Moreover, any set of profits whose sum does not exceed
the monopoly profit is feasible.
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Joe Chen
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The Folk theorem states that: Any average payoff vector that is better, for all players,
than a Nash equilibrium payoff vector of the stage game can be sustained as the outcome of
a perfect equilibrium of the infinitely repeated game if the players are sufficiently patient.
N
i N N
i
N
Mathematically, denote ΠiN = Πi (aN
1 , ..., an ), where: Π (ai , a−i ) ≥ Π (ai , a−i ), for all
ai ∈ Ai , and for all i = 1, ..., n. Let v = (v 1 , ..., v n ) be feasible, and for all i, vi ≥ ΠiN .
Then there exists δ 0 < 1, such that: for all δ ≥ δ 0 , v is an equilibrium payoff vector.
The proof of the Folk theorem is not difficult. Construct a trigger strategy with Nash
threats (threats to revert to Nash behavior forever). By deviating today, a players gains at
most a bounded amount; on the other hand, she loses:
(v i − ΠiN )(δ + δ 2 + ...).
Clearly, the loss goes to infinity as δ goes to 1.
When the Nash equilibrium payoffs are the same as the reservation payoffs for the stage
game (e.g., the Bertrand price game), this version of the Folk theorem gives a full description
of the set of equilibria for δ close to 1. Fudenberg and Maskin (1986) show that: Every
individually rational and feasible payoff vector can be enforced in perfect equilibrium for δ
sufficiently close to 1.
The multiplicity of equilibria is too successful in explaining tacit collusion. The multiplicity of equilibria is an “embarrassment of riches”. In many applications, we need to pick
one out of these equilibria. It is natural to focus on symmetric equilibrium if the game is a
symmetric one. In addition, it is natural to assume that firms coordinate on the equilibrium
that is Pareto-optimal. In our content, it is the monopoly price, pm .
Lecture Notes: Industrial Organization
Joe Chen
6.5
6.5.1
71
A few direct applications
Market concentration
On the one hand, more firms in the market sharing the profits implies a higher profit from
undercutting. On the other hand, a large number of firms reduces the profit per firm
and thus the “cost” of being punished for undercutting. Hence, concentration facilitates
collusion. Mathematically, consider a homogenous good industry with n-firms facing the
same constant marginal cost. The constituent game is the Bertrand price competition.
Suppose the (perfect) equilibrium is the one where all firms charge the monopoly price and
share the market. It is enforced by Nash threats. Then, the benefit of undercutting is:
Πm − Πm /n;
and, the cost of undercutting is:
(δ + δ 2 + ...)Πm /n.
Hence, it requires:
δ ≥ (n − 1)/n = 1 − 1/n.
The higher the concentration is (a smaller n), the lower the value of δ that is required to
sustain collusion.
6.5.2
Information lags and infrequent interaction
The latter is straightforward because it decreases δ and thus the cost of undercutting. The
former (an information lag) increases the temptation to undercut since it takes time for the
rival to detect undercutting (if happened). Both factors hinder collusion.
For the former, consider the setup of 2 firms producing perfect substitutes with the same
marginal cost. Suppose prices are observed two periods after they are chosen. Note that
collusion is sustainable if and only if:
Πm /2 + δΠm /2 ≤ (δ 2 + δ 3 + ...)Πm /2,
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Joe Chen
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or,
√
δ ≥ 1/ 2 > 1/2.
6.5.3
Multimarket contact
Consider two identical and independent markets, and both firms participate in both markets.
Assume further that market A meets more frequently than market B. To be concise, let
market A meets every period and market B meets every even periods. The implicit discount
factor of market B is: δ 2 . Suppose δ 2 < 1/2 < δ.
In the absence of multimarket contact, collusion is sustainable in market A but not in
market B. With multimarket contact, full collusion on both markets is sustainable if:
2
Πm
Πm
Πm
≤ (δ + δ 2 + ...)
+ (δ 2 + δ 4 + ...)
;
2
2
2
or, when: δ ≥ 0.593. Hence, for δ = 0.6, full collusion in both markets can be sustained
under multimarket contact, whereas no amount of collusion is sustainable in market B under
single-market contact.
The intuition is that the loss of collusion on market A can be so large so that it actually
facilitates collusion in market B. The concern of a “general” price warfare in all markets
discourages firms from undercutting in any of the markets that they participate.
Technically (mathematically), the incentive constraints for these two markets are pooled
into a single constraint. If (originally) they are both satisfied, then the pooled constraint is
satisfied. However when one of them is “highly” satisfied while the other is just a little bit
“short”, it can be the case that the pooled constraint is satisfied.
6.5.4
Demand fluctuation
Rotemberg and Saloner (1986) propose a theory of price wars during booms. Consider the
typical setup of two (symmetric) firms with constant marginal cost competing in prices in
a market where demand is stochastic with i.i.d. shocks over time. Let DL (p) < DH (p) for
all p. At time t, demand can be low, DL (p), with probability 1/2, or high, DH (p), with
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Joe Chen
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probability 1/2. In each period, firms learn the state of the demand (low or high) before
they make their pricing decisions. We look for a pair of prices {pL , pH } such that:
• Both firms charge ps where the state s ∈ {L, H};
• {pL , pH } is sustainable in equilibrium;
• The expected present discounted profit is not Pareto dominated.
m
m
m
Let pm
s ≡ arg maxp (p − c)Ds (p), and Πs = (ps − c)Ds (ps ). The first question to ask
m
would be: Can the fully collusive outcome, {pm
L , pH }, be sustainable in equilibrium? Note
that the payoff V when both firms charge the price configuration {pL , pH } is:
V =
∞
X
t=0
δt
¸
∙
¸¾
½ ∙
1
1
DL (pL )
DH (pH )
(pL − c)
+
(pH − c)
2
2
2
2
(pL − c)DL (pL ) + (pH − c)DH (pH )
4(1 − δ)
ΠL (pL ) + ΠH (pH )
.
=
4(1 − δ)
=
m
So, when the fully collusive outcome is sustainable, V m = (Πm
L + ΠH )/[4(1 − δ)]. The cost
of undercutting at anytime t is: δV m ; and, the benefit of undercutting is (note that firms
already know the state s): Πm
s /2. Hence, if the fully collusive outcome is sustainable, then
m
m
m
we must have: Πm
s /2 ≤ δV , for all s = L, H. Since ΠL /2 < ΠH /2, we need to consider
m
m
only: Πm
H /2 ≤ δV . Substitute for V , we derive:
δ≥
2Πm
H
.
Πm
+
3Πm
L
H
m
m
Denote 2Πm
H /(ΠL + 3ΠH ) as δ 0 , we have: δ 0 ∈ (1/2, 2/3). Therefore, when: δ ∈ [1/2, δ 0 ),
the fully collusive outcome is not sustainable. What would be the price configuration when:
δ ∈ [1/2, δ 0 )? The firms’ problem is:
max
pL ,pH
s.t.
ΠL (pL ) + ΠH (pH )
4(1 − δ)
ΠL (pL )
)+ΠH (pH )
≤ δ ΠL (pL4(1−δ)
2
ΠH (pH )
)+ΠH (pH )
≤ δ ΠL (pL4(1−δ)
.
2
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Joe Chen
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Since δ ∈ [1/2, δ 0 ), it has to be the case that one of the two incentive constraints is binding.
Intuitively, the temptation to undercut is higher when demand is high. So, the binding
constraint should be the second one. This gives us:
¶
µ
δ
ΠL (pL ) ≡ KΠL (pL );
ΠH (pH ) =
2 − 3δ
and, the whole problem reduces to:
max
pL
(ΠL (pL ) + KΠL (pL ))
1+K
≡ max
ΠL (pL ).
pL
4(1 − δ)
4(1 − δ)
m
m
Hence, pL = pm
L and pH can be solved by: ΠH (pH ) = KΠL (pL ) = KΠL . The important
m
m
question to ask is: pH R pm
H ? Note that K is increasing in δ, and when δ = δ 0 , K = ΠH /ΠL .
m
m
So, when δ < δ 0 , ΠH (pH ) = KΠm
L < ΠH . This implies pH < pH . Observe also that the
2 m
first incentive constraint, ΠL (pL ) ≤ KΠH (pH ), is satisfied: Πm
L ≤ K ΠL (when δ ≥ 1/2,
K ≥ 1).
m
We can conclude: When δ ∈ [1/2, δ 0 ), although the fully collusive outcome {pm
L , pH } is
not sustainable, some kind of collusion is still possible. In the low state of demand, pL = pm
L;
in the high state of demand, pH < pm
H . This is interpreted as price wars during booms —
i.e., in booms, firms are forced to lower the price to prevent undercutting. Countercyclical
price movement is consistent with this theory (though not necessarily because: pH R pm
L ).