Oligopoly Theory (7)
Multi-Stage Strategic Commitment
Games
Aim of this lecture
(1) To understand the relationship between payoff
function and reaction curve.
(2) To understand the ideas of strategic
commitments
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Outline of the 7th Lecture
7-1 Two-Stage Strategic Commitment Games
7-2 Strategic Cost-Reducing Investments
7-3 Strategic Deviation from Profit-Maximizing
Behavior
7-4 Delegation Game
7-5 Debt Equity Ratio and Strategic Commitment
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Cournot Equilibrium
the reaction curve of firm 1
Y2
the reaction curve of firm 2
Y2C
0
Y1C
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Y1
3
Ｙ2
Shift of the Reaction Curve
the reaction curve of firm 1 (after)
the reaction curve of firm 2
the reaction curve of firm 1 (before)
Ｙ2 C
0
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Y1
Y1
the shift reduces the rival's output
→resulting in the increase of its profits 4
C
Stories of the shifting reaction
curve
(1) Cost-Reducing Investments
(2) Delegation, Reward Contracts
(3) Divisions, Franchising
(4) Financial Contract
(5) Inventory→6th (two production period
model) ,9th, and 11th lectures
(6) Capacity Investment→9th lecture
(7) long-Term Contract→9th lecture
(8) Durability→13th lecture
(9) Product Differentiation→8th lecture
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Cost-Reducing Investments
Brander and Spencer (1983)
Model
Duopoly, homogeneous goods market
First stage: Each firm i independently chooses Ii
(R&D investment level), which affects its
production costs.
Second stage: After observing firms' production
costs, firms face Cournot competition.
Payoff: Π1 = P(Y1 + Y2)Y1 - c1(I1)Y1 - I1
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backward induction
Second Stage: Cournot Competition
Y1C(I1,I2), Y2C(I2,I1)
The output of the firm is increasing in its own
investment level and is decreasing in the rival's.
(Remember the discussions in 2nd and 4th lectures)
First Stage:
The first order condition for firm 1 is
P'Y1(∂Y1C/∂I1 + ∂Y2C/∂I1) + P∂Y1C/∂I1 - c1'(I1)Y1 - c1
∂Y1C/∂I1 -1 = 0
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First stage
At the first stage The first order condition is
P'Y1(∂Y1C/∂I1 + ∂Y2C/∂I1)+P∂Y1C/∂I1 - c1'(I1)Y1 - C1
∂Y1C/∂I1 -1=0
P‘Y1 + P - c1 =0 (from the second stage optimization,
envelope theorem) ⇒ P'Y1∂Y2C/∂I1 - c1'(I1)Y1 - 1 = 0.
Cost-Minimizing Level
- c1'(I1)Y1 - 1 = 0
Investment level exceeds cost minimizing level under
strategic substitutes~ strategic effect
a decrease of its own marginal cost → a reduction of
rival's production → an increase in the price → gain
of the profit ⇒ strategic use of R&D.
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Shift of the Reaction Curve
Ｙ2
The reaction curve of firm 1 (after)
The reaction curve of firm 2
Ｙ2 C
0
Y1C
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Y1
9
Shifts of the Reaction Curves
the reaction curve of firm 1 (after)
Ｙ2
the reaction curve of firm 2 (after)
Ｙ2*
0
Y1*
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Y1
the shifts reduce the profit of both
firms ~ Prisoner's Dilemma
10
Welfare Implication
The equilibrium investment level exceeds the profitmaximizing level.
An increase in the investment improves consumer
surplus.
Is the equilibrium investment level excessive or
insufficient from the viewpoint of social welfare?
→ Suppose that the demand is linear. Consider a
symmetric equilibrium. Then the equilibrium
investment level is equal to the second best
investment level. ~ Brander and Spencer (1981)
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Welfare Implication
∂W/∂I1 = ∂Π1/∂I1 + ∂Π2/∂I1 + ∂CS/∂I1
∂Π1/∂I1 must be zero.
The investment level is excessive (insufficient) if
- ∂Π2/∂I1 > (<) ∂CS/∂I1
→In the case of linear demand, they happen to be
canceled out.
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Risk
Linear demand, constant marginal cost, duopoly,
homogeneous product market.
Firm i chooses Δi independently and then two firms
face Cournot competition.
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Risk
The firms' marginal costs are (c-Δ1, c-Δ2) if both firms
succeed in innovation. This takes place with
probability q2 .
The marginal costs are (c-Δ1, c) if only firm 1 succeeds
in innovation. This takes place with probability q(1-q).
They are (c, c-Δ2) if only firm 2 succeeds in innovation.
This takes place with probability q(1-q).
They are (c, c) if no firm succeeds in innovation. This
takes place with probability (1-q)2.
If q=1, this model is equal to that of Brander and
Spencer mentioned above.
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Relationship between optimal and
equilibrium investment levels under
uncertainty
If q＝1, the equilibrium investment level is optimal.
(Brander and Spencer, 1981)
Question：Suppose that 0 < q < 1.
The equilibrium investment level is (too large, optimal,
too small) for social welfare.
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Expected profit of firm 1
q2Π1(c-Δ1,c-Δ2) + q(1 - q)Π1(c-Δ1,c)
+ q(1 - q)Π1 (c,c-Δ2) + (1 - q)2Π1(c,c) - I1(Δ1).
The first order condition is
q2∂Π1(c-Δ1,c-Δ2) /∂Δ1 + q(1 - q)∂Π1(c-Δ1,c)/∂Δ1
= ∂I1(Δ1)/∂Δ1.
Expected welfare is
q2W(c-Δ1,c-Δ2) + q(1 - q)W(c-Δ1, c)
+ q(1 - q)W (c,c-Δ2)+(1 - q)2W (c,c) - I1(Δ1) - I2(Δ2).
The first order condition is
q2∂W(c-Δ1,c-Δ2)/∂Δ1 + q(1 - q)∂W(c-Δ1,c)/∂Δ1
= ∂I1(Δ1)/∂Δ1.
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Oligopoly Theory
Welfare-improving production
substitution
the reaction curve of firm 1 (after)
Y2
the reaction curve of firm 2
the reaction
curve of firm 1
(before)
0
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Y１
17
Welfare-reducing production
substitution
the reaction curve of firm 1
Y2
the reaction curve of
firm 2 (before)
the reaction curve
of firm 2 (after)
0
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Y１
18
Another type of uncertainty
Linear demand. constant marginal cost, symmetric
duopoly, homogeneous product market.
Firm i chooses qi independently and then two firms
face Cournot competition, where qi is probability of
success of firm i's innovation.
Firm i's investment cost is I(qi). I’ > 0 and I’’ > 0.
Let q1E = q2E = qE be the equilibrium probability of
success at the symmetric equilibrium.
Let q1s = q2s = qs be the second best probability of
success.
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Uncertainty
The firms' marginal costs are (c-Δ,c-Δ) if both firms
succeed in innovation. This takes place with
probability q1q2.
The marginal costs are (c-Δ,c) if only firm 1 succeeds
in innovation. This takes place with probability q1(1 q2).
They are (c,c-Δ) if only firm 2 succeeds in innovation.
This takes place with probability q2(1 - q1).
They are (c, c) if no firm succeeds in innovation. This
takes place with probability (1 - q1)(1 - q2).
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Another type of uncertainty
qE > qs if and only if qE > 1/2.
→Risky projects should be subsidized.
Intuition
When firm 2 fails, the change from failure to success
of the firm 1's project induces welfare-improving
production substitution. →the incentive for
increasing q1 is insufficient.
Matsumura (2003) and Kitahara and Matsumura
(2006)
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Question: Suppose that firm 2's
objective is convex combination
of sales and profits.
Y2
0
Oligopoly Theory
the reaction curve of the profit
maximizer
Y1
22
managerial incentive
Deviation from the profit-maximizing behavior ~ a
more aggressive behavior than the profitmaximizing behavior
→resulting in a decrease of the rival's output, yielding
an increase of its profit, through strategic interaction
Delegation Game (Fershtman and Judd (1987))
The owners have incentives for offering a strategic
reward contract (hiring an agent (management)
who does not maximize the payoff of the principal).
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managerial incentive
The game runs as follows.
In the first stage, the owner of firm i (i = 1,2)
independently offers the reward contract (chooses
αi ) Ui(αiPYi + (1 - αi)Πi) where αi ∈[0,1] and Ui is
increasing.
In the second stage the management of firm i
chooses Yi so as to maximize αiPYi + (1 - αi)Πi.
In equilibrium owners chose positive α.
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If both firms deviate from the
profit-maximizing behavior, then
Y2
0
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competition is accelerated
and the resulting profits
become smaller.
Y1
25
Delegation and Cooperation
Is it possible to use managerial incentive for increasing
the profits of firms？~Fershtman et al (1991)
Using managerial incentive contract for maintaining the
collusive behavior.
The required conditions for the contract
(1) When the rival has an incentive to cooperate, the
contract must offer an incentive for collusion, too.
(2) When the rival firm does not have an incentive to
cooperate, the contract must not offer an incentive for
collusion.
～The same structure as Repeated Game discussed in
11th lecture.
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Delegation and Cooperation
Let ΠM be the monopoly profit.
The following simple reward contract can yield
collusion.
The management obtains bonus if its profit is ΠM/2.
An increase of its profit from ΠM/2 does not increase
the reward.
If its profit is smaller than ΠM/2, the reward is
proportional to its profit.
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Delegation and Cooperation
If the rival offers the same contract, the management
has an incentive for choosing collusive output,
resulting profit is ΠM/2. (a deviation can increase
the profit but it does not increase the reward)
If the rival does not offer the same contract, the
management lose an incentive for choosing
collusive output, resulting profit is non-cooperative
one (because the management knows that it
cannot obtain the profit ΠM/2).
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Properties of Cooperation through
Strategic Delegation
Multiple Equilibria ~ Common Property of Repeated
Game.
If the rival offers the same contract, the management
has an incentive for choosing collusive output,
resulting profit is ΠM/2. (a deviation can increase
the profit but it does not increase the reward)
If the rival does not offer the same contract, the
management lose an incentive for choosing
collusive output, resulting profit is non-cooperative
one (because the management knows that it
cannot obtain the profit ΠM/2).
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Oligopoly Theory
A Problem of Cooperation through
Strategic Contract
It is difficult to commit the reward contract.
・After offering the reward contract and making it
public, the owners have incentives for recontracting
and offering alternative reward contract making the
management be profit-maximizer secretly.
→It is difficult to commit that they never make secret
recontracting.
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Divisions
Firm 1 competes against Firm 2 in Cournot fashion.
⇒
Firm 1→Firm 1-1, Firm 1-2
Firm 1-1 maximizes its own profit PY11 - c1Y11
Firm 1-2 maximizes its own profit PY12 - c1Y12
Firm 2 maximizes its own profit PY2 - c2Y2
Y11 + Y12 > Y1 ~ Division of the firm makes the firm more
aggressive → reduction of the rival's output →
increase of its own profit.
We can easily guess this result from `merger paradox'.
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Merger Paradox
Firm 1, firm 2, and firm 3 face Cournot competition.
⇒Firm 1 and firm 2 merge → Firm 1' and Firm 3 face
Cournot competition.
This merger usually increases the profit of firm 3 but
not firms 1 and 2 as long as the cost condition
remains unchanged since the merger increases the
rival's output.
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Divisions
If a firm can be divided into n firms without division
costs and the firm can choose n, then each firm
chooses n as large as it can, resulting in a perfect
competition (Baye et al (1996)).
~Fourth foundation of perfect competition in this
lecture.
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Commitment through Financial
Structure
Brander and Lewis (1983)
An increase of Debt/Equity ration makes the firm
more aggressive (induces upward shift of the
reaction curve), resulting in an increase the profit.
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Risk and Optimal Production
Level
Consider the following situation. Monopoly, Linear
demand, P = a - Y, constant marginal cost, a =
3 with probability 1/2 and a = 1 with probability
1/2.
The monopolist chooses its output before
observing the demand parameter a.
Question: Suppose that the risk neutral
monopolist chooses Y = Y*. The risk averse
monopolist chooses Y' (>,=,<) Y* .
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Monopoly Producer
P
MR
D
D
MR
0
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YL Y* YH
MC
Y
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Risk and Optimal Production
Level
Consider the following situation. Monopoly, Linear
demand, P=a-Y, constant marginal cost, a=3
with probability 1/2 and a=1 with probability 1/2.
The monopolist chooses its output before
observing the demand parameter a.
Answer: Suppose that the risk neutral monopolist
chooses Y=Y*. The risk averse monopolist
chooses Y'<Y* .
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Firm's profit and payoff of the
stockholders
payoff of the stockholders
０
１
profits of the firm
Limited liability effect→the payoff function becomes
convex even when the stockholders are risk neutral
~like the payoff function of the risk lover.
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Commitment through Financial
Structure
Brander and Lewis (1983)
An increase of Debt/Equity ration strengthens the
limited liability effect → It makes the firm more
aggressive (induces upward shift of the reaction
curve), resulting in an increase the profit.
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relative profit approach
Consider a symmetric duopoly in a homogeneous
product market. Consider a quantity-setting
competition. Suppose that U1 = π1 - α1π2, α1∈[-1,1]
and that U2 = π2 - α2π1, α2∈[-1,1].
Consider the following two stage game.
In the first stage, owner of firm i chooses αi
independently. In the second stage, management of
firm i chooses Yi so as to maximize Ui.
Consider a strategic substitute case.
Suppose that firm 1's owner chooses α1 given α2.
Then α1 is (positive, negative, zero).
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strategic complements case
Y2
Cournot
equilibrium
The reaction curve
of firm 1
The reaction
curve of firm 2
Y2C
0
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Y1C
Y1
41
strategic complements case
The reaction curve
of firm 1
Y2
The reaction
curve of firm 2
Y2C
0
Oligopoly Theory
Y1C
Y1
42
relative profit approach
Consider a symmetric duopoly in a homogeneous
product market. Consider a quantity-setting
competition. Suppose that U1 = π1 - α1π2. α1∈[-1,1].
Consider a strategic complement case.
Suppose that firm 1's owner chooses α given α2.
Then α is (positive, negative zero).
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