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Oligopoly Theory 1
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
Oligopoly Theory 2
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 Behavior7-4 Delegation Game 7-5 Debt Equity Ratio and Strategic
Commitment
Oligopoly Theory 3
Cournot Equilibrium
Y1
the reaction curve of firm 2
0
Y2
the reaction curve of firm 1
Y1C
Y2C
Oligopoly Theory 4
Shift of the Reaction Curve
Y1
the reaction curve of firm 2
0
Y 2
the reaction curve of firm 1 (after)
Y1C
Y 2C
the reaction curve of firm 1 (before)
the shift reduces the rival's output →resulting in the increase of its profits
Oligopoly Theory 5
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
Oligopoly Theory 6
Cost-Reducing InvestmentsBrander and Spencer (1983)
ModelDuopoly, homogeneous goods market First stage: Each firm i independently chooses I i
(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
Oligopoly Theory 7
backward induction
Second Stage: Cournot CompetitionY1
C(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(∂Y1
C/∂I1 + ∂Y2C/∂I1) + P∂Y1
C/∂I1 - c1'(I1)Y1 - c1 ∂Y1C/∂I1 -
1 = 0
Oligopoly Theory 8
First stageAt the first stage The first order condition is P'Y1(∂Y1
C/∂I1 + ∂Y2C/∂I1)+P∂Y1
C/∂I1 - c1'(I1)Y1 - C1 ∂Y1C/∂I1 -
1=0P‘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 = 0Investment level exceeds cost minimizing level under
strategic substitutes~ strategic effecta 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. ⇒
Oligopoly Theory 9
Shift of the Reaction Curve
Y1
The reaction curve of firm 2
0
Y 2
The reaction curve of firm 1 (after)
Y1C
Y 2C
Oligopoly Theory 10
Shifts of the Reaction Curves
Y1
the reaction curve of firm 2 (after)
0
Y 2
the reaction curve of firm 1 (after)
Y1*
Y 2*
the shifts reduce the profit of both firms ~ Prisoner's Dilemma
Oligopoly Theory 11
Welfare Implication
The equilibrium investment level exceeds the profit-maximizing 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)
Oligopoly Theory 12
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.
Oligopoly Theory 13
RiskLinear demand, constant marginal cost,
duopoly, homogeneous product market. Firm i chooses Δi independently and then two
firms face Cournot competition.
Oligopoly Theory 14
RiskThe 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.
Oligopoly Theory 15
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.
Oligopoly Theory16
Expected profit of firm 1q2Π1(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.
Oligopoly Theory 17
Welfare-improving production substitution
Y2
Y 1
the reaction curve of firm 1 (before)
the reaction curve of firm 1 (after)
the reaction curve of firm 2
0
Oligopoly Theory 18
Welfare-reducing production substitution
Y2
Y 1
the reaction curve of firm 1
the reaction curve of firm 2 (before)
0
the reaction curve of firm 2 (after)
Oligopoly Theory 19
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 = q2
E = qE be the equilibrium probability of success at the symmetric equilibrium.
Let q1s = q2
s = qs be the second best probability of success.
Oligopoly Theory 20
UncertaintyThe 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).
Oligopoly Theory 21
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)
Oligopoly Theory 22
Question: Suppose that firm 2's objective is convex combination
of sales and profits.
Y1
0
Y2
the reaction curve of the profit maximizer
Oligopoly Theory 23
managerial incentive
Deviation from the profit-maximizing behavior ~ a more aggressive behavior than the profit-maximizing 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).
Oligopoly Theory 24
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.
Oligopoly Theory 25
If both firms deviate from the profit-maximizing behavior, then
Y10
Y2
competition is accelerated and the resulting profits become smaller.
Oligopoly Theory 26
Delegation and CooperationIs 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.
Oligopoly Theory 27
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.
Oligopoly Theory 28
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).
Oligopoly Theory29
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).
Oligopoly Theory 30
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.
Oligopoly Theory 31
DivisionsFirm 1 competes against Firm 2 in Cournot
fashion.⇒Firm 1→Firm 1-1, Firm 1-2Firm 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'.
Oligopoly Theory 32
Merger ParadoxFirm 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.
Oligopoly Theory 33
DivisionsIf 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.
Oligopoly Theory 34
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.
Oligopoly Theory 35
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* .
Oligopoly Theory 36
Monopoly ProducerP
Y
MRD
0
MC
D
MR
YL YHY*
Oligopoly Theory 37
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* .
Oligopoly Theory 38
Firm's profit and payoff of the stockholders
Limited liability effect→the payoff function becomes convex even when the stockholders are risk neutral ~like the payoff function of the risk lover.
payoff of the stockholders
profits of the firm
10
Oligopoly Theory 39
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.
Oligopoly Theory 40
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 α i
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).
Oligopoly Theory 41
strategic complements case
Y1
The reaction curve of firm 2
0
Y2
The reaction curve of firm 1
Y2C
Y1C
Cournot equilibrium
Oligopoly Theory 42
strategic complements case
Y1
The reaction curve of firm 2
0
Y2
The reaction curve of firm 1
Y2C
Y1C
Oligopoly Theory 43
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).