1 Eco 600E Advanced Microeconomics II
Term: Spring (2nd), 2009 Lecturer: Yosuke Yasuda
Problem Set 1 Due in class on July 16
1. Question 1 (10 points)
A monopolist faces two kinds of consumers: students and non‐students. The demand curve of each student is q = 200 ‐ 2p. The demand of each non‐student is given as q = 200 ‐ p. There are x students and y non‐students. There is a zero marginal cost of production.
a) First suppose that the monopolist must set a single price to sell to all consumers. What price would the monopolist charge? How much would each student and each non‐student consume?
b) Now suppose that the monopolist can charge different prices to students and non‐students. What price would the monopolist charge in each market? How much would each student and each non‐student consume?
2. Question 2 (15 points) Consider the following 2‐2 game.
P1 / P2 L R
U 3, 0 0, 2
D 0, 1 1, 0
a) Is there any strictly dominated strategy or a strictly dominant strategy in this game? b) Find all pure‐strategy Nash equilibria in this game. If there is no pure strategy
equilibrium, explain why.
c) Suppose player 1 takes U with probability q and D with probability (1‐q). Likewise, player 2 takes L with probability p and R with probability (1‐p). Find a combination of p and q which constitutes a mixed strategy Nash equilibrium.
3. Question 3 (15 points) See the following game tree.
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a) Translate this game into normal‐form and draw the corresponding payoff bi‐matrix. Hint: Remember that a strategy in dynamic games is a complete plan of action. b) Find all pure‐strategy Nash equilibria. How many are there?
c) Solve this game by backward induction.
4. Question 4 (15 points) See the following game tree.
a) How many information sets (containing two or more decision nodes) does this game have?
b) How many subgames (including the entire game) does this game have? c) Find all (pure‐strategy) subgame perfect Nash equilibria.
5. Question 5 (15 points)
To produce output of a homogenous good, each firm must pay a fixed cost of $f and a marginal cost of $c per unit. The demand curve for this good is p = a ‐ bQ, where Q is
1
2
2 A
B
C
D E
F
(4, 1) (1, 4) (2, 3) (0, 2)
1
2
2
1 1
1
A F
C
K
L
M
N
D E
G B
H
I H
I
(2, 0) (3, 4)
(1, 3)
(1, 1) (0, 4) (4, 0) (3, 3) (1, 4) (0, 2)
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the total output in the industry. Assume a – c > 0 and b > 0.
a) First suppose that there are n firms in the industry who have paid the fixed cost. Suppose that they compete as Cournot quantity setting oligopolists. How must will each firm produce? What will be the market price and the total quantity produced? b) Now suppose that firms will exit the industry if their profit (net of fixed cost) is
negative and that identical firms may enter if there are profits to be made. How many firms will enter? Remember that your answer must be an integer.
c) What happen to the number of firms in the industry and prices as f becomes small? Give some economic intuition for your answer.
6. Question 6 (15 points)
Two firms produce an identical good. The inverse demand curve for the good is P = 101
‐ X, where X is the total quantity produced by the two firms. Firm 1 has a constant marginal cost 1 of producing the good. Firm 2 has a constant marginal cost 1 + c of producing the good, with 0 < c < 100.
a) Suppose each firm i produces and sells xi units of the good. Write down an expression for firm i’s profits (as a function of the output of each firm).
b) Suppose that each firm compete as quantity setting duopolists. What quantities will they produce, what is the market price and how much profit does each firm earn? c) Suppose that firm 1 decides how much to produce first; firm 2 chooses only after
observing firm 1’s choice. What quantities will they produce, what is the market price and how much profit does each firm earn?
7. Question 7 (15 points)
Consider the following game depicting the process of standard setting in high‐definition television (HDTV). The United States and Japan must simultaneously decide whether to invest a high or a low value into HDTV research. Each country’s payoffs are summarized in the following figure.
US / JAPAN Low High
Low 4, 3 2, 4
High 3, 2 1, 1
a) Are there any dominant strategies in this game? What is the Nash equilibrium of the game?
b) Suppose now that the United States has the option of committing to a strategy
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before Japan’s decision is reached. Model this new situation by a game tree and solve it by backward induction.
c) Comparing the answers to (a) and (b), what can you say about the value of commitment for the United States?
