1 Date: July 30, 2009
Subject: Advanced Microeconomics II Lecturer: Yosuke Yasuda
Final Exam
1. True or False (10 points)
Answer whether each of the following statement is true (T) or false (F). You do NOT need to explain the reason.
a) In the Stackelberg model (with symmetric firms’ cost), the leader can ALWAYS earn a higher profit than the follower.
b) If strategy spaces are infinite, Nash equilibrium (possibly in mixed‐strategies) may NOT exist.
2. Bertrand Model (15 points)
Consider a duopoly game in which two firms simultaneously and independently select prices, p1 and p2. The firms’ products are differentiated. After the prices are set, consumers demand 10 – pi + pj units of the good that firm i produces. Assume that each firm produces at zero cost, and the payoff for each firm is equal to the firm’s profit.
a) Write the payoff functions of the firms (as a function of their strategies p1 and p2). b) Is this a game of “strategic complements” or “strategic substitutes”?
Hint: A game is called “strategic complements (/substitutes)” if each player’s best reply curve is upward (/downward) sloping.
c) Solve the (pure‐strategy) Nash equilibrium.
3. Dynamic Game (15 points) See the following game tree.
1 2 1
A
B D F
C E
(6, 2)
(5, 5)
(0, 3) (1, 0)
2
a) Translate this game into normal‐form by drawing the payoff bi‐matrix. b) Find all pure‐strategy Nash equilibria. How many are there?
c) Solve this game by backward induction.
4. Repeated Game (20 points) Consider the following static game.
1 / 2 C D
C 4, 3 0, 5
D 5, 0 1, 2
a) Find all the (pure‐strategy) Nash equilibrium. Are there any dominant strategies in this game?
b) Now consider a dynamic game in which the above static game will be played twice. Then, how many subgames (including the entire game) does this game have? c) Solve the subgame perfect Nash equilibrium of the dynamic game in (b).
d) Now suppose that the above static game will be played for infinitely many times, and each player maximizes the average payoff with a discount factor δ. Find the range of δ that can achieve (C, C) by the trigger strategy.
Hint: Under the trigger strategies, players choose (C, C) as long as no one deviates from (C, C); if someone deviates, they will play the NE (solved in (a)) forever after.
5. Focal Point (5 points, bonus!)
This is a bonus question. Choose one country and write down its name. If your answer coincides with the most popular one, you would get 5 points bonus!
Hint: This question is similar to what we had in class. The important thing is that you have to guess which country other students would likely choose.
