Chapter 28 Oligopoly Key Concept: We have the first taste of how to solve a game !

1. Chapter 28 Oligopoly • Key Concept: We have the first taste of how to solve a game! • Stackelberg, Cournot, Bertrand.

2. Chapter 28 Oligopoly • It is the case that lies between two extremes, i.e., pure competition and pure monopoly. • Often there are a number of competitors but not so many as to regard each of them as having a negligible effect on price.

3. This is the situation known as oligopoly. • We will look at how strategic interactions arise.

4. There are several relevant models since firms may behave in different ways in an oligopolistic environment. • It is unreasonable to expect one grand model since many different behavior patterns are observed in real life. • We will introduce some models to understand some possible patterns.

5. We will restrict to the case of two firms or duopoly. • The duopoly case captures important essence of strategic interactions. • We will also focus on cases in which each firm is producing an identical product so that strategic interactions instead of product differentiation is the key.

6. The first model is the Stackelberg model, for instance, Apple and its follower. • It is used to describe industries with a dominant firm or a natural leader. • A typical pattern is smaller firms wait for the leader’s announcement of new products and then adjust their decisions accordingly.

7. In this case we could model the dominant firm as the Stackelberg leader and the other as followers.

8. It is a sequential quantity setting game. • 1 is the leader who chooses to produce y1 while after seeing that, the follower, 2 decides to produce y2. • The total output of the market is therefore y1+y2 and the price is p(y1+y2).

9. What should the leader do? • Depends on how the leader thinks that follower will react to its choice. • The leader should expect that the follower will attempt to maximize profits given the choice of the leader. • Hence for the leader to have a sensible decision, it has to consider the follower’s maximization.

10. This suggests we could solve backwards. • Follower’s problem is • maxy2 p(y1+y2)y2-c2(y2) • FOC: p(y1+y2)+p’(y1+y2)y2 = c2’(y2)

11. FOC: p(y1+y2)+p’(y1+y2)y2=c2’(y2) • The marginal unit gives p, but it pushes all units sold before • So basically, from FOC, we can derive the reaction function of 2. • That is, given y1, there is an optimal level of y2 or y2 =f(y1).

12. FOC: p(y1+y2)+p’(y1+y2)y2=c2’(y2). • In the following we will work with the case where p(Y)=a-bY and c2(y2)=0 for all y2. • 2=(a-b(y1+y2))y2=ay2-by1y2-by22 • FOC: a-by1-2by2=0 or y2=(a-by1)/2b, 2’s reaction function.

13. 2=(a-b(y1+y2))y2=ay2-by1y2-by22 • FOC: a-by1-2by2=0 or y2=(a-by1)/2b, 2’s reaction function. • The isoprofit line 2=ay2-by1y2-by22=k • This looks similar in shape to -y1-y22=0 or y2=-x (verify by yourself).

14. 2=(a-b(y1+y2))y2=ay2-by1y2-by22 • Given y2, if y1 is smaller, the profit of firm 2 is bigger. • Hence firm 2’s profits increase as we move further to the left.

15. Two points worth mentioning. • When y1=a/b (1 has flooded the market), then y2=0. • On the other hand, when y1=0, it is as if 2 is the monopolist, so y2=a/2b.

16. For each y1, 2 will choose its output to make its profit as large as possible or pick y2 so that its isoprofit line is furthest to the left. • The optimum will satisfy the tangency. The slope of isoprofit line will be vertical at the optimal choice.

17. Fig. 27.1

18. Let us suppose c1(y1)=0 for all y1 and work out the leader’s problem. • maxy1 p(y1+y2)y1-c1(y1) • s.t. y2 =f(y1) • Now, we can plug in 2’s reaction curve. So 1=p[y1+ f(y1)]y1

19. For the linear demand case, we can plug in 2’s reaction curve. • So 1=(a-b[y1+(a-by1)/2b])y1=(a-by1)y1/2. • FOC: a/2-by1=0 So y1=a/2b. • Plugging this into 2’s reaction curve, we get y2=[a-b(a/2b)]/2b=a/4b.

20. Show this graphically.

21. Fig. 27.2

22. We draw firm 1’s reaction curve in order to help draw firm 1’s isoprofit. • They have the same shape as those of firm 2’s, except they are rotated 90 degrees.

23. Since firm 2 will choose an output so that 1 and 2’s choices will be on 2’s reaction curve, 1 wants to choose an output combination on the reaction curve that gives it the highest profit. • This means picking an isoprofit that is tangent to the reaction curve.

24. Two firms simultaneously decide output levels. • In this case each firm has to forecast the other firm’s output choice. • We seek an equilibrium in forecasts, a situation where each firm finds its belief about the other firm to be confirmed.

25. This is the Cournot model. • We look for the case where 1’s output is a best response to 2’s and vice versa 2’s is a best response to 1’s. • Graphically, it is the intersection of the two reaction curves.

26. Since y2=(a-by1)/2b and symmetrically y1=(a-by2)/2b, solving these two together, we get y1=y2=a/3b.

27. Fig. 27.2

28. The Cournot equilibrium can be justified by the following dynamics.

29. Fig. 27.4

30. Suppose the two quantity setting firms get together and attempt to set outputs so that their joint profit is maximized, i.e., they collude or form a cartel, what will the output levels be?

31. Their problem becomes • maxy1,y2 (a-b(y1+y2))(y1+y2) • FOC: y1+y2=a/2b • Together they should produce the monopoly output.

32. But there is the temptation to cheat.

33. Look at FOC again. • Profit is PY=(a-bY)Y where Y=y1+y2 • FOC: P+(dP/dY)Y=0 or P+(dP/dY)(y1+y2)=0

34. P+(dP/dY)(y1+y2)=0 • From 1’s perspective, increasing output (if 2’s output is fixed), its profit increases by P+(dP/dY)y1=-(dP/dY)y2>0. • Hence it is a big issue for the cartel to deter members from cheating.

35. P+(dP/dY)(y1+y2)=0 • Intuitively, when 1 consider increasing output, the extra profit from selling the marginal unit is P, but the joint monopoly will have to lower the price for all units sold before. • However, those include sold by firm 2.

36. Fig. 27.5

37. Often some repeated interactions would help. • Consider the punishment strategies. Each firm produces half of the monopoly output and gets profit m.

38. If there is any cheating in the past, switch to Cournot competition forever and each gets c. • So if a firm deviates, it can at best get d for one shot and then forever it gets c.

39. Note that d> m > c. • [1/(1+r)]+[1/(1+r)]2+[1/(1+r)]3+… • =1/r • So a firm is comparing • m+m /r> d+c/r. • As long as r<(m-c)/(d-m) or future important enough, it will not deviate.

40. The intuition is that a deviation in one shot is not worthwhile as long as future is important enough, making punishment severe enough.

41. Sometimes, price matching is one possibility to detect deviation and hence is used to maintain collusion.

42. Lastly, we talk a bit about Bertrand competition.

43. Two firms simultaneously announce prices pi and pj. • There is a market demand x(p).

44. If pi<pj, then xi(pi,pj)=x(pi). • If pi = pj, then xi(pi,pj)=x(pi)/2. • If pi>pj, then xi(pi,pj)=0. • Suppose both firms have marginal cost c. • There is a unique pure Nash equilibrium.

45. Suppose pi≤pj and pi<c, then i can deviate to pic. • Suppose pi=c and pj>c, i can increase the price a bit to earn positive profit. • Suppose c<pi≤pj, j earns at most (pi-c)x(pi)/2, undercutting i a bit, it could earn close to (pi-c)x(pi). • We are left with the case where pi=pj=c.

46. Chapter 28 Oligopoly • Key Concept: We have the first taste of how to solve a game! • Stackelberg, Cournot, Bertrand.