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Corporate Social Responsibility or Asymmetry of Payoff ?: An

Investigation of Endogenous Timing Game

joint work with Akira Ogawa

2012/3/4 2

Our works related to this paper

(1) Payoff Dominance and Risk Dominance in the Observable Delay Game: A Note (JoE 2009, joint work with Akira Ogawa).(2) On the Robustness of Private Leadership in Mixed Duopoly (AEP 2010, with Akira Ogawa)(3) Randomized Strategy Equilibrium and Simultaneous-Move Outcome in the Action Commitment Game with Small Cost of Leading (ORL 2011, with Takeshi Murooka and Akira Ogawa).(4) Price leadership in a homogeneous product market(JoE 2011, with Daisuke Hirata)

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price leadership

The leader announces the price change first, and then other firms follow this price change.

Some researchers suspect that this is a collusive pricing, implicit cartel.

However, if we regard this as a price version Stackelberg, it is natural that a higher price of the leader induces a higher price of the follower (strategic complements)

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price leadership (Ono, 1978)

Homogeneous product market, no supply obligation, duopoly, increasing marginal cost, price-setting.

One firm chooses the price and then the other firm chooses its price after observing the price of the rival. (Stackelberg)

He compares the equilibrium payoffs when the firm with higher cost is the leader to those when the firm with lower cost is the leader.

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Asymmetric Costs

Y

the MC of the lower cost firm

0

MCthe MC of the higher cost firm

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Follower's pricing(1) Suppose that the leader's price is higher than the

monopoly price of the follower. Then, the follower names its monopoly price and obtains the whole market.

(2) Suppose that the leader's price is lower than the monopoly price of the follower. Then,

(a) names a higher price than the leader and obtains the residual demand, or

(b) the follower names the price slightly lower than the rival's and obtains the whole market. (price under-cutting)

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Firm 1's pricing

Ono (1978) assume that the follower undercuts the leader's price. Predicting this behavior of the follower, the leader chooses its price.

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Residual demand

Y

D

0

MC

P Follower's MC

P1

Y2

residual demand of the leader

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residual demand

Y

D

0

MC

P Follower's MC

residual demand of the leader

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Residual Demand

Y

D

0

MC

PFollower's MC

residual demand of the leader

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price leadershipSuppose that the firm with lower cost becomes the

follower.→It produces a lot as a price taker

→Predicting this aggressive behavior, the firm with higher cost names a low price.

Suppose that the firms with higher cost becomes the follower.→It does not produces a lot as a price taker

→Predicting this less aggressive behavior, the firm with higher cost names a high price. ~ beneficial for both firms.

He concludes that the lower cost firm takes price leadership if the cost difference between two firms is large.

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Contribution of Ono (1978)

(1) pioneering work on Timing Game

(2) pioneering work on Price Leadership

～ the lower cost firm becomes the leader

(3) Mutually beneficial price leadership can appear when the cost difference between two firms are sufficiently large.

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Subsequent Works・ Ono (1982) Oligopoly Version

・ Denekere and Kovenock (1992)

~Capacity Constraint~Capacity Constraint

→→The firm with more capacity becomes the leader.The firm with more capacity becomes the leader.

・ ・ Amir and Stepanova (2006)Amir and Stepanova (2006) ～～ differentiated differentiated product marketproduct market

→→The firm with lower cost firm becomes the leader The firm with lower cost firm becomes the leader and it is mutually beneficial if cost difference is large. and it is mutually beneficial if cost difference is large.

・・ Ishibashi (2007) Ishibashi (2007) ~Capacity Constraint~Capacity Constraint ＋＋ repeated gamerepeated game→→The firm with more capacity becomes the leader.The firm with more capacity becomes the leader.

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Problems in Ono(1978)(1) Is mutually beneficial leadership is always realized

in equilibrium ？・・ He did not formulate the Timing Game. He did not formulate the Timing Game.

(a) Is the outcome where the lower cost firm becomes (a) Is the outcome where the lower cost firm becomes the leader always an equilibriumthe leader always an equilibrium ？？

(b) Is it a unique equilibrium?(b) Is it a unique equilibrium?

(c) If not, the equilibrium with lower cost firm's (c) If not, the equilibrium with lower cost firm's leadership is robustleadership is robust ？？

~No game theoretic foundation ~No game theoretic foundation

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Problems in Ono(1978)(2) Is price undercutting is always best reply?

The answer is NO. undercutting is not always best reply.

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Dastidar (2004)Consider a Stackelberg duopoly with common

increasing marginal cost in a homogeneous product market.

Firm 1 names the price and after observing the price firm names the price.

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price-undercuttinprice-undercutting

Y

D

0

P

MC Firm 2's MC

P1

Y2

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no price-undercutting

Y

D

0

P

MCFirm 2's MC

P1

Y1

Residual Demand

P2

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price-undercutting vs non-undercutting

A decrease in the price of the leader makes the undercutting strategy more profitable and non-undercutting strategy less profitable.

→There exists p* such that the follower takes non-undercutting strategy if and only if p p*. ≦

In equilibrium, firm 1 names P1= p* , firm 2 takes non-undercutting strategy, and two firms obtain the same profits.

Two prices appear in the homogeneous product market. The leader engages in marginal cost-pricing, while the follower is not.

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Problems in Ono(1978)(3) The assumption of price-undercutting is innocuous?

The answer is NO. This assumption changes the results

In equilibrium, the follower does not undercut the price.

(i) The price leadership by the higher price firm is mutually beneficial even when the cost difference is small.

(ii) It is a unique equilibrium, or it is the risk-dominant equilibrium.

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Hirata and Matsumura (2011)(i) The price leadership by the higher price firm is

mutually beneficial even when the cost difference is small.

(ii) It is a unique equilibrium, or it is the risk-dominant equilibrium.

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price-undercuttinprice-undercutting

Y

D

0

P

MC Firm 2's MC

P1

Y2

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no price-undercutting

Y

D

0

P

MCFirm 2's MC

P1

Y1

Residual Demand

P2

Firm 1's MC

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Intuition behind the results

Suppose that the leader has a higher cost. ～ It is easy to induce the follower to take non-undercutting strategy (taking a residual demand).

It can name the relatively higher price, and it is mutually beneficial.

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Stackelberg or CournotStackelberg or CournotCournot (Bertrand) model and Stackelberg model yield

different results.

Simultaneous move model and sequential move model yield different results.

Which model should we use? Which model is more realistic?

An incumbent and a new entrant competes →sequential-move model

There is no such asymmetry between firms →simultaneous-move model

However, in reality, firms can choose both how much they produce and when they produce.

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Timing Games

Firms can choose when to produce.

Formulating a model where Cournot outcome and Stackelberg outcome can appear, and investigating whether Cournot or Stackelberg appear in equilibrium.

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Stackelberg DuopolyFirm 1 and firm 2 compete in a homogeneous

product market.

Firm 1 chooses its output Y1 [0, ∞)∈ . After observing Y1, firm 2 chooses its output Y2 [0, ∞)∈ .

Each firm maximizes its own profit Πi.

Πi ＝ P(Y)Yi ー Ci(Yi), P: Inverse demand function,

Y: Total output, Yi: Firm i's output, Ci: Firm i's cost function

I assume that P'+P''Y1<0 (strategic substitutes)

⇒First-Mover Advantage

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Stackelberg's discussion on the market instability

In the real world, it is not predetermined which firm becomes the leader.

Because of the first-mover advantage, both firms want to be the leaders.

Straggle for becoming the leader make the market instable.

～ This is just the idea for endogenous timing game.

But he himself did not present a model formally.

Some papers discussing this problem appeared at the end of 70s.

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Four representative timing games

(1) Observable delay game (2) Action commitment game(3) Infinitely earlier period model (4) Seal or disclose (5) Two production period model

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Observable Delay Game DuopolyFirst stage: Two firm choose period 1 or

period 2. Second Stage: After observing the timing, the firm choosing period 1 chooses its

action. Third Stage: After observing the actions

taking at the second stage, the firm choosing period 2 chooses its action.

Payoff depends only on its action (not period).

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Possible OutcomesPossible Outcomes Both firms choose period 1 ⇒CournotBoth firms choose period 2 ⇒CournotOnly firm 1 chooses period 1 ⇒Stackelberg Only firm 2 chooses period 1 ⇒Stackelberg

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Equilibrium in Observable Delay Game

Strategic Substitutes ⇒Both firms choose period 1 (Cournot) since Leader ≫ Cournot ≫ Follower

Strategic Complements ⇒Only firm1 chooses period 1 (Stackelberg) or Only firm2 chooses period 1 (Stackelberg)since Leader ≫ Cournot and Follower ≫ Cournot.

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Asymmetric Cases

It is possible that two firms have different payoff ranking.

e.g., Price Leadership Firm 1 Leader ≫ Follower >> BertrandFirm 2 Follower >> Leader >> BertrandIt is quite natural to think that firm 1 becomes a

leader in equilibrium. cf Ono (1978,1982) Is it true?

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Matsumura and Ogawa (2009)

Assumption UiL U≧ i

C

Result If U1L ＞ U1

F and U2F ＞ U2

L, (i) firm 1's leadership is the unique equilibrium

outcome,(ii) equilibrium outcomes other than firm 1's

leadership is supported by weakly dominated strategies,

or (iii) firm 1's leadership is risk dominant⇒Pareto dominance implies risk dominance in the

observable delay game. ～ foundation for Ono's discussion.

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Pareto efficient outcome can fail to be an equilibrium in

general contextsＣ Ｄ

Ｃ （ 3,3 ） （ 0,4)

Ｄ （ 4,0 ） （ 1,1 ）１

２

Pareto Dominance →(C,C)Risk Dominance →(D,D)

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Pareto dominant equilibrium can fail to be the risk

dominant equilibrium in general contexts

C D

C (3,3) (-100,-1)

D (-1,-100) (1,1)

１

２

Pareto Dominance →(C,C)Risk Dominance →(D,D)

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risk dominance

C D

C (3,3) (-100,-1)

D (-1,-100) (1,1)１

２

Consider a mixed strategy equilibrium. Suppose that in the mixed strategy equilibrium each firm independently chooses C with probability q. Then (C,C) is risk dominant if and only if q <1/2.

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Observable Delay

1 2

1 (A,a) (C,b)

2 (B,c) (A,a)１

２

C A, c a. ≧ ≧

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Asymmetric Cases

It is possible that two firms have different payoff ranking.

e.g., Mixed Duopoly

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Pal (1998):mixed duopoly, domestic private firm, quantity-competition

1 2

1 (A,a) (C,b)

2 (B,c) (A,a)１

２ (private firm)

B>C>A, c>a, b>a Question: Derive the equilibrium outcome.

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Matsumura(2003): mixed duopoly, foreign private firm, quantity-

competition

1 2

1 (A,a) (C,b)

2 (B,c) (A,a)１

２

C>A>B, c>a, b>a Question: Derive the equilibrium outcome.

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Observable Delay, Matsumura (2003)

1 2

1 (A,a) (C,b)

2 (B,c) (A,a)１

２

C>A>B, c>a, b>a

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Barcena-Ruiz (2007) :mixed duopoly, domestic private firm,

price-competition

1 2

1 (A,a) (C,b)

2 (B,c) (A,a)１

２

B>A>C, c>a>b Question: Derive the equilibrium outcome.

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Barcena-Ruiz (2007)

1 2

1 (A,a) (C,b)

2 (B,c) (A,a)１

２

B>A>C, c>a>b

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Mixed DuopolyMixed duopoly Quantity-competition→ StackelbergPrice-Competition→ Bertrand

Private duopolyQuantity-competition→ CournotPrice-Competition→ Stackelberg

Does asymmetry in objectives or welfare-maximizing objective yield contrasting results in mixed duopoly?

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Social responsibility approach

Ghosh and Mitra (2010)Payoff θΠ ＋ (1-θ)WThe same payoff of partially privatized firm Difference ~ Both firm has the same payoff (no

asymmetry)

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Social responsibility approach

Quantity-competition→ CournotPrice-competition→ Stackelberg

Asymmetry in objectives, not welfare-maximizing objective, yields contrasting results in mixed duopoly.

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Endogenous Choice of Price-Quantity Contract

Firms choose whether to adopt price contract or quantity contract, and then choose the prices or quantities. Singh and Vives (1984) showed that choosing the quantity (price) contract is a dominant strategy for each firm if the goods are substitutes (complements).

Intuition (substitutable goods case) ： Choosing a price contract increases the demand elasticity of the rival, resulting in a more aggressive action of the rival.

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Endogenous Choice of Price-Quantity Contract in Mixed Duopoly

For the private firm, choosing a price contract increases the demand elasticity of the rival, resulting in a less aggressive action of the rival (substitutable goods case). Thus, the private firm has an incentive to choose the price contract, as opposed to the private duopoly. For the public firm, choosing a price contract increases the demand elasticity of the rival, resulting in a more aggressive action of the rival . Thus, the public firm has an incentive to choose the price contract.→Bertrand competition appears in Mixed Duopoly (Matsumura and Ogawa, 2012)

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Social responsibility approach

Quantity contract is chosen.

Asymmetry in objectives, not welfare-maximizing objective, yields contrasting results in mixed duopoly.