1. ECON 330 Lecture 13
Tuesday, November 6

2. Today last 20 minutes of class time KOLT will come to class and conduct a mid-semester course evaluation.

3. Midterm Exam: (very easy) Next week, Thursday November 15, 9:00 to 11:00. Engineering Auditorium ENG Z50 You can use a simple pocket calculator only if you bring one to the exam. No borrowing from others. No smartphones, no mobiles.

4. This one is not allowed

5. Neither this one

6. This is OK

7. Nash equilibrium, Oligopoly models
Today’s lecture is a continuation of last week’s lecture which was the most important lecture of the semester.
Nash equilibrium, Oligopoly models
definition of,
computing the,

8. The Cournot Model 2 firms Firms choose output levels simultaneously.
After output levels are chosen the price adjusts so that the quantity demanded equals to the sum of the output levels of the two firms.
Firms know the market demand and their rival’s costs.

9. Example
Both firms have the same cost function C(q) = 30q; (this means AC = MC = 30)
The inverse demand is p = 120–Q, where Q is the total output.
If you were the CEO of firm 1 what would you do?

10. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 10 20 30 40 50 60 70
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11. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 10 20 30 40 50 60 70
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12. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 10 20 30 40 50 60
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13. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 10 20 30 40 50 60
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1000
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14. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 10 20 30 40 50
700
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15. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 10 20 30 40 50
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-500

16. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 20 30 40 50
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-500

17. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 20 30 40 50
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18. Quantity chosen by firm 2 Quantity chosen by firm 1
Profits for Firm 1
Quantity chosen by firm 2
Quantity chosen by firm 1 20 30 40
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19. Computing the Nash Equilibrium by using best response functions
Find the Nash equilibrium output levels q1* and q2* for these two firms for the Cournot's model.

20. The cost function is C(q) = 30q. The inverse demand is p = 120–Q.
Step 1 Write the profit function for firm 1 as a function of q1 and q2. (remember: Q = q1 + q2) π1(q1, q2) = P(Q)q1 –30q1 = (120 – Q)q1 –30q1. Since Q = q1+q2, the profit function becomes π1(q1, q2) = (120 –q1–q2)q1 –30q1 Step 2 Find the best response function for firm 1. To do this we differentiate the profit function with respect to q1, set equal to 0: 120 –2q1–q2 –30 = 0. Solve for q1= (90 – q2)/2 Step 3 Firms have identical cost functions, so the Nash equilibrium output levels q1* and q2* will be equal. Use q1* = q2* in the best response function and compute q1*. You are done. q* = (90 – q*)/2  q* = 30. In the Nash equilibrium each firm produces 30 units of output.

21. The 7 steps of finding the Nash Equilibrium by using best response functions
Identify the strategic variable the two firms choose.
In the Cournot model the strategic variables are the quantity levels.
In the Bertrand model the strategic variables are the prices.
In other set ups they can be advertising levels, R&D levels, etc.
Let X denote the strategic variable chosen by firm 1, and Y the strategic variable chosen by firm 2.

22. Write the profit function of firm 1 as the function of X and Y.
Find firm 1’s best response function:
Differentiate firm 1’s profits function with respect X (the strategic variable chosen by firm 1), set equal to 0.
Solve this equation for X. Note that X will be a function of Y. This is the best response function of firm 1.

23. Repeat steps 3 and 4 for firm 2.
Differentiate firm 2’s profits function with respect Y (the strategic variable chosen by firm 2), set equal to 0.
Solve this equation for Y. Note that Y will be a function of X. This is the best response function of firm 2.
You have two equations (the two best response functions) in two unknowns (X and Y). Solve these equations simultaneously for the two unknowns.
When you do this you find the pair (X*, Y*) with the property that firm 1's action X* is a best response to firm 2's action Y*, and firm 2's action Y* is a best response to firm 1's action X.

24. Another example
Two firms are competing by choosing advertising levels.
If firm 1 chooses the amount a1 of advertising and firm 2 chooses the amount a2 of advertising,
Firm 1’s profit will be π1(a1, a2) =
Firm 2’s profit will be π2(a1, a2) =
S and c are positive constants.

25. Solution
π1(a1, a2) =
Differentiate with respect to a1. Remember the rule of differentiation: f(x)/g(x) differentiated with respect to x is (f’g – g’f)/g2 where f’ is the derivative of f and g’ is the derivative of g.
This is what we get
Use the shortcut that the Nash equilibrium values of advertising levels will be equal: a1* = a2* = a*.

26. This short cut gives us the following equation:
Solving this for a* gives us the solution:

27. Now the Bertrand Model

28. In 1883 Bertrand wrote a review in a journal on Cournot’s book published in (Cournot died in 1877.) Betrand argued that Cournot’s argument was faulty. He claimed that, as a result, Cournot had reached the wrong conclusions about the determination of duopoly prices.

29. Bertrand maintained all other assumptions of the Cournot model and “reworked” the model with prices as the strategic variables (rather than quantities).
Firms have zero marginal and fixed cost.
Firms produce an identical product.
Firms choose their strategic variables (prices) simultaneously.
He showed that prices will be driven immediately down to the perfectly competitive solution.

30. Demand is Q = 20 – P
Costs: c(q) = 0.5q for both firms. (MC = AC = 0.5)
Firms must choose prices from positive integers. (0 is not allowed.)

31. If one firm has a lower price that firm gets the entire quantity demanded at that lower price. The higher price firm gets 0 demand.
If both firms have the same price they equally share the quantity demanded at that price.

32. Your firm’s profit table
Opponent’s Price 1 2 3 4 5 6 7 8 9
Your Price
4,75
9,5
13,5 27
21,3
42,5 43 28 56
33,8
67,5 68
38,5 77 42 85 45 90 47