1. Oligopoly Theory1 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

2. Oligopoly Theory2 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

3. Oligopoly Theory3 Cournot Equilibrium Y1Y1 the reaction curve of firm 2 0 Y2Y2 the reaction curve of firm 1 Y1CY1C Y2CY2C

4. Oligopoly Theory4 Shift of the Reaction Curve Y1Y1 the reaction curve of firm 2 0 Ｙ2Ｙ2 the reaction curve of firm 1 (after) Y1CY1C Ｙ2CＹ2C the reaction curve of firm 1 (before) the shift reduces the rival's output →resulting in the increase of its profits

5. Oligopoly Theory5 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

6. Oligopoly Theory6 Cost-Reducing Investments Brander and Spencer (1983) Model Duopoly, 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: Π 1 = P(Y 1 + Y 2 )Y 1 - c 1 (I 1 )Y 1 - I 1

7. Oligopoly Theory7 backward induction Second Stage: Cournot Competition Y 1 C (I 1,I 2 ), Y 2 C (I 2,I 1 ) 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'Y 1 (∂Y 1 C /∂I 1 + ∂Y 2 C /∂I 1 ) + P∂Y 1 C /∂I 1 - c 1 '(I 1 )Y 1 - c 1 ∂Y 1 C /∂I 1 -1 = 0

8. Oligopoly Theory8 First stage At the first stage The first order condition is P'Y 1 (∂Y 1 C /∂I 1 + ∂Y 2 C /∂I 1 )+P∂Y 1 C /∂I 1 - c 1 '(I 1 )Y 1 - C 1 ∂Y 1 C /∂I 1 -1=0 P‘Y 1 + P - c 1 =0 (from the second stage optimization, envelope theorem) ⇒ P'Y 1 ∂Y 2 C /∂I 1 - c 1 '(I 1 )Y 1 - 1 = 0. Cost-Minimizing Level - c 1 '(I 1 )Y 1 - 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.

9. Oligopoly Theory9 Shift of the Reaction Curve Y1Y1 The reaction curve of firm 2 0 Ｙ2Ｙ2 The reaction curve of firm 1 (after) Y1CY1C Ｙ2CＹ2C

10. Oligopoly Theory10 Shifts of the Reaction Curves Y1Y1 the reaction curve of firm 2 (after) 0 Ｙ2Ｙ2 the reaction curve of firm 1 (after) Y1*Y1* Ｙ2*Ｙ2* the shifts reduce the profit of both firms ~ Prisoner's Dilemma

11. Oligopoly Theory11 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)

12. Oligopoly Theory12 Welfare Implication ∂W/∂I 1 = ∂Π 1 /∂I 1 + ∂Π 2 /∂I 1 + ∂CS/∂I 1 ∂Π 1 /∂I 1 must be zero. The investment level is excessive (insufficient) if - ∂Π 2 /∂I 1 > (<) ∂CS/∂I 1 →In the case of linear demand, they happen to be canceled out.

13. Oligopoly Theory13 Risk Linear demand, constant marginal cost, duopoly, homogeneous product market. Firm i chooses Δ i independently and then two firms face Cournot competition.

14. Oligopoly Theory14 Risk The firms' marginal costs are (c-Δ 1, c-Δ 2 ) if both firms succeed in innovation. This takes place with probability q 2. The marginal costs are (c-Δ 1, c) if only firm 1 succeeds in innovation. This takes place with probability q(1-q). They are (c, c-Δ 2 ) 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.

15. Oligopoly Theory15 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.

16. Oligopoly Theory 16 Expected profit of firm 1 q 2 Π 1 (c-Δ 1,c-Δ 2 ) + q(1 - q) Π 1 (c-Δ 1,c) + q(1 - q) Π 1 (c,c-Δ 2 ) + (1 - q) 2 Π 1 (c,c) - I 1 (Δ 1 ). The first order condition is q 2 ∂Π 1 (c-Δ 1,c-Δ 2 ) /∂ Δ 1 + q(1 - q)∂Π 1 (c-Δ 1,c) /∂ Δ 1 = ∂ I 1 (Δ 1 )/ ∂ Δ 1. Expected welfare is q 2 W (c-Δ 1,c-Δ 2 ) + q(1 - q)W(c-Δ 1, c) + q(1 - q)W (c,c-Δ 2 )+(1 - q) 2 W (c,c) - I 1 (Δ 1 ) - I 2 (Δ 2 ). The first order condition is q 2 ∂W (c-Δ 1,c-Δ 2 ) /∂ Δ 1 + q(1 - q)∂W(c-Δ 1,c) /∂ Δ 1 = ∂ I 1 (Δ 1 )/ ∂ Δ 1.

17. Oligopoly Theory17 Welfare-improving production substitution Y2Y2 Y１Y１ the reaction curve of firm 1 (before) the reaction curve of firm 1 (after) the reaction curve of firm 2 0

18. Oligopoly Theory18 Welfare-reducing production substitution Y2Y2 Y１Y１ the reaction curve of firm 1 the reaction curve of firm 2 (before) 0 the reaction curve of firm 2 (after)

19. Oligopoly Theory19 Another type of uncertainty Linear demand. constant marginal cost, symmetric duopoly, homogeneous product market. Firm i chooses q i independently and then two firms face Cournot competition, where q i is probability of success of firm i's innovation. Firm i's investment cost is I(q i ). I’ > 0 and I’’ > 0. Let q 1 E = q 2 E = q E be the equilibrium probability of success at the symmetric equilibrium. Let q 1 s = q 2 s = q s be the second best probability of success.

20. Oligopoly Theory20 Uncertainty The firms' marginal costs are (c-Δ,c-Δ) if both firms succeed in innovation. This takes place with probability q 1 q 2. The marginal costs are (c-Δ,c) if only firm 1 succeeds in innovation. This takes place with probability q 1 (1 - q 2 ). They are (c,c-Δ) if only firm 2 succeeds in innovation. This takes place with probability q 2 (1 - q 1 ). They are (c, c) if no firm succeeds in innovation. This takes place with probability (1 - q 1 )(1 - q 2 ).

21. Oligopoly Theory21 Another type of uncertainty q E > q s if and only if q E > 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 q 1 is insufficient. Matsumura (2003) and Kitahara and Matsumura (2006)

22. Oligopoly Theory22 Question: Suppose that firm 2's objective is convex combination of sales and profits. Y1Y1 0 Y2Y2 the reaction curve of the profit maximizer

23. Oligopoly Theory23 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).

24. Oligopoly Theory24 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 α i ) U i (α i PY i + (1 - α i )Π i ) where α i ∈ [0,1] and U i is increasing. In the second stage the management of firm i chooses Y i so as to maximize α i PY i + (1 - α i )Π i. In equilibrium owners chose positive α.

25. Oligopoly Theory25 If both firms deviate from the profit-maximizing behavior, then Y1Y1 0 Y2Y2 competition is accelerated and the resulting profits become smaller.

26. Oligopoly Theory26 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.

27. Oligopoly Theory27 Delegation and Cooperation Let Π M be the monopoly profit. The following simple reward contract can yield collusion. The management obtains bonus if its profit is Π M /2. An increase of its profit from Π M /2 does not increase the reward. If its profit is smaller than Π M /2, the reward is proportional to its profit.

28. Oligopoly Theory28 Delegation and Cooperation If the rival offers the same contract, the management has an incentive for choosing collusive output, resulting profit is Π M /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 Π M /2).

29. Oligopoly Theory 29 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 Π M /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 Π M /2).

30. Oligopoly Theory30 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.

31. Oligopoly Theory31 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 PY 11 - c 1 Y 11 Firm 1-2 maximizes its own profit PY 12 - c 1 Y 12 Firm 2 maximizes its own profit PY 2 - c 2 Y 2 Y 11 + Y 12 > Y 1 ~ 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'.

32. Oligopoly Theory32 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.

33. Oligopoly Theory33 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.

34. Oligopoly Theory34 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.

35. Oligopoly Theory35 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*.

36. Oligopoly Theory36 Monopoly Producer P Y MR D 0 MC D MR YLYL YHYH Y*

37. Oligopoly Theory37 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*.

38. Oligopoly Theory38 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 １ ０

39. Oligopoly Theory39 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.

40. Oligopoly Theory40 relative profit approach Consider a symmetric duopoly in a homogeneous product market. Consider a quantity-setting competition. Suppose that U 1 = π 1 - α 1 π 2, α 1 ∈ [-1,1] and that U 2 = π 2 - α 2 π 1, α 2 ∈ [-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 Y i so as to maximize U i. Consider a strategic substitute case. Suppose that firm 1's owner chooses α 1 given α 2. Then α 1 is (positive, negative, zero).

41. Oligopoly Theory41 strategic complements case Y1Y1 The reaction curve of firm 2 0 Y2Y2 The reaction curve of firm 1 Y2CY2C Y1CY1C Cournot equilibrium

42. Oligopoly Theory42 strategic complements case Y1Y1 The reaction curve of firm 2 0 Y2Y2 The reaction curve of firm 1 Y2CY2C Y1CY1C

43. Oligopoly Theory43 relative profit approach Consider a symmetric duopoly in a homogeneous product market. Consider a quantity-setting competition. Suppose that U 1 = π 1 - α 1 π 2. α 1 ∈ [-1,1]. Consider a strategic complement case. Suppose that firm 1's owner chooses α given α 2. Then α is (positive, negative zero).