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).