1. ECON 330 Lecture 13
Wednesday, November 6

2. The midterm exam is next week!
Time: Wednesday November 13, 9:00 to 10:45. Place: Engineering Auditorium ENG Z50 This will be an easy, straightforward exam.

3. The midterm exam is next week!
You may use a simple calculator. Bring your own! No sharing! No smart-phones please.

4. This one is not allowed

5. This one is OK

6. Office hours for the exam
THIS WEEK Today 1 30 PM to 3 30 PM Thursday 1 30 PM to 3 30 PM NEXT WEEK Monday 1 30 PM to 3 30 PM Tuesday to 12 15, and 1 30 PM to 3 30 PM

7. also …
NEXT WEEK Pınar Önen (our TA) will hold a problem solving session for the exam. Time: Tuesday Nov 12, 3 30 to 4 45 PM Place: Room CASE B 38 (or 39?)

8. A few words on Monday lecture
Price competition with 2 firms
Demand for Firm 1: q1 = 10–p1+0.5p2
Demand for Firm 2: q2 = 10–p2+0.5p1
Cost functions:
Firm 1: c(q) = 3q
Firm 2: c(q) = q

9. We computed the Nash equilibrium prices pNE1 and pNE2

10. Definition of the Nash equilibrium prices
π1(pNE1, pNE2) ≥ π1(p1, pNE2) for any p1
π2(pNE1, pNE2) ≥ π2(pNE1, p2) for any p2

11. A nice interpretation of Nash equilibrium: we can imagine that…
a Nash equilibrium is a “law” no one wants to break (unilaterally, by themselves) even in the absence of an effective police force.
So, if the “law” says that firm 1 must set p1 = pNE1 and firm 2 must set p2= pNE2 then no firm will want to break this law as long as the other firm follows the law.

12. But this is not how the analysis of oligopoly competition got started in the first part of the 19th century

13. Now, back to the beginnings..
Augustine Cournot, and the “Cournot equilibrium”
“L'équilibre de Cournot”

14. In his HEA (his pages long, unfinished book) Joseph Schumpeter wrote this: “[Cournot’s result is] the backbone of all further work in oligopoly.” In its many variations Cournot’s model has been used to address an astonishing range of questions in industrial economics.

15. Antoine Augustin Cournot (1801-1877)
The first scholar to apply mathematics to economics in a serious way. His book ''Researches Into the Mathematical Principles of the Theory of Wealth'' was published in 1838.

16. Andrew Carnegie ( ), “the richest man in the world”, once said: ‘It does not pay to pioneer’ Cournot’s life and work is possibly the best illustration of this.
Cournot was a true pioneer, and as such, was neglected in his time.
The impact of his work on modern economics cannot be overstated.

17. So, here is the Cournot model of oligopoly competition

18. The “original” Cournot model
Cournot considers the case of two profit-maximizing spring water suppliers who sell bottles of identical quality at zero costs.

19. Cournot’s spring water duopoly

20. The Cournot model, continued
The market demand for spring water is negatively sloped. Firms choose output levels simultaneously and independently.

21. Now, the confusing part: Once the firms choose quantities, the price adjusts so as to clear the market: that means the quantity demanded equals the total quantity produced by the two firms.

22. Who sets the market price?
The firms choose their quantities, the price adjusts (by some magic mechanism) to clear the market.
Example:
The market demand is Q(p) = 12 – p.
If q1 = 2, and q2 = 5,
then the price that clears the market is…
p = 5.
If q1 = 5, and q2 = 2,

23. Assumption 1
Firms choose output levels simultaneously. What if firms choose quantities sequentially? For example first firm 1 then firm 2?

24. Assumption 2
Firms compete by choosing quantities (not prices). Some refer to the Cournot model as “quantity competition”. What happens when the two firms announce prices and then meet whatever demand they receive?

25. The truth is that this is not exactly what Cournot wrote in Chapter VII in that famous book

26. In chapter V titled “of monopoly”, he studied a … monopoly supplier of mineral water who picks the price to maximize profit. In chapter VII titled “of competition of producers” he wrote …

27. Let us now imagine two proprietors and two springs of which the qualities are identical and which on account of their similar positions supply the same market in competition. In this case the price is necessarily the same for each proprietor. If p is this price, D = F(p) the total sales, D1 the sales from the spring (1), and D2 the sales from the spring (2), then D1 + D2 = D.

28. If, to begin with, we neglect the cost of production, the respective incomes of the proprietors will be pD1 and pD2 and…
each of them independently will seek to make this income as large as possible.
Instead of adopting D = F(p) as before, in this case it will be convenient to adopt the inverse notation p = f(D).

29. So far, so good. But here comes the troubling part…

30. Proprietor (1) can have no direct influence on the determination of D2
Proprietor (1) can have no direct influence on the determination of D2. All that he can do when D2 has been determined by proprietor (2) is to choose for D1 the value which is best for him. This he will be able to accomplish by properly adjusting his price,… except as proprietor (2), who seeing himself forced to accept this price and this value of D1, may adopt a new value for D2 more favorable to his interests than the preceding one.

31. So, it was not clear what Cournot had in mind
So, it was not clear what Cournot had in mind! Is it a … dynamic model in which firms adjust their choices in response to their rival’s earlier choices so that this process terminates at some point where no further adjustments are needed Or is it a … static model in which quantities are chosen simultaneously and once and for all.

32. And also.. Why say something like “proprietor (1) adjust his price…”

33. The modern approach. Use Nash’s method
The modern approach. Use Nash’s method. Firms make their quantity decisions simultaneously. Focus on the Nash equilibrium quantities.

34. Definition of the Nash equilibrium quantities
π1(qNE1, qNE2) ≥ π1(q1, qNE2) for any q1
π2(qNE1, qNE2) ≥ π2(qNE1, q2) for any q2

35. This is all too abstract
Let’s do an example.

36. An example
Demand Q(P) = 12 – P; Inverse demand P(Q) = 12 – Q q1 : output level of firm 1; q2 : output level of firm 2 For both firms all costs are zero. (In memory of Cournot) P is determined by Q = q1 + q2, so  We can write P = 12 – (q1 + q2) Firm 1 chooses q1 to maximize {12 – (q1 + q2)}xq1 Firm 2 chooses q2 to maximize {12 – (q1 + q2)}xq2

37. Firm 1 Profit table q2 , your rival’s quantity q1 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 11 10 20 18 16 14 12 27 24 21 15 32 28 -4 35 30 25 -5
-10 36 -6
-12
-18 -7
-14
-21
-28 -8
-16
-24
-32
-40 -9
-27
-36
-45
-54 q1

38. Firm 2 Profit table q1 , your rival’s quantity q2 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 11 10 20 18 16 14 12 27 24 21 15 32 28 -4 35 30 25 -5
-10 36 -6
-12
-18 -7
-14
-21
-28 -8
-16
-24
-32
-40 -9
-27
-36
-45
-54 q2

39. Computing the Cournot-Nash equilibrium
Start with firm 1. Write the profit function (remember: all costs are zero) π1(q1 , q2) = P x q1 P = 12 – Q, and Q = q1 + q2. P = 12 – (q1 + q2) π1(q1 , q2) = {12 – (q1 + q2)} x q1 Differentiate π1 with respect to q1, and , set equal to 0: dπ1/dq1 = 12 – 2q1 – q2 = 0 Best response function for firm 1: q1 = 6 – q2/2

40. By definition of the Nash equilibrium, we must have q1 = 6 – qNE2/2
π1(qNE1, qNE2) ≥ π1(q1, qNE2) for any q1

41. By definition of the Nash equilibrium, we must have qNE1 = 6 – qNE2/2 Both firms have the same cost function (zero cost), so, it is most likely that in the Nash Equilibrium we have qNE1 = qNE2 Use qNE1 = qNE2 And … qNE1 = qNE2 = 4. In the Nash equilibrium each firm will produce 4 units of output.

42. qNE1 = 4, and qNE2 = 4 (and P = 4)
These are the Nash equilibrium quantities of the Cournot model. The Nash Equilibrium is a law that no one wants to break alone even without enforcement. If q2 = 4, the best (profit maximizing) quantity for firm 1 is q1 = 4.

43. Consider this: Both firms can make more profit by producing a smaller quantity each, say 3 units. Suppose the law says that each firm must produce 3 units. This law must be enforced by punishing violations. Why? If q2 = 3, the best quantity for firm 1 is q1 = 4,5 (not 3) So unless there is police force that punishes firms that produce more than 3 units, this is not going to be self-sustaining situation.

44. Your turn now

45. Both firms have the same cost function: TC(q) = 30q
Both firms have the same cost function: TC(q) = 30q. The inverse demand function is p(Q) = 120–Q, where Q is the total output. Compute the Nash equilibrium output levels and the price of the Cournot model. Don’t panic HELP is coming (see next slide)

46. Cost function: C(q) = 30q. The inverse demand p = 120–Q.
A. Write the profit function for firm 1 as a function of q1 and q2. (remember: Q = q1 + q2) B. Find the best response function for firm 1. C. Firms have identical cost functions, so the Nash equilibrium output levels,q1* and q2*, will be the same for both firms. Use q1* = q2* in the best response function and compute q1*. You are done.

47. End of the lecture.