Practice Questions for Midterm
Subject: Advanced Microeconomics II (ECO600E) Professor: Yosuke YASUDA
1. True or False
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.
(a) The pro…t maximizing monopoly …rm sets price equal to marginal cost. (b) Non-cooperative game theory rules out cooperative behaviors, and hence fails
to analyze cooperation.
(c) Some games have multiple Nash equilibria.
(d) When best reply functions in a game are upward sloping, we call the game as strategic substitution.
2. Monopoly
Consider a monopoly problem with the following demand function, Q= a P.
Assume the …rm’s cost function is quadratic, C(Q) = Q2.
(a) Formulate price setting monopoly problem and derive the monopoly price. (b) Formulate quantity setting monopoly problem and derive the monopoly quan-
tity.
(c) Verify that your answers in (a) and (b) are identical. 3. Price discrimination
Suppose a monopoly …rm with 0 marginal cost operates in two di¤erent markets, A and B. Inverse demand for each market is given as follows.
p(qA) = 200 qA
p(qB) = 120 qB
(a) Solve the optimal quantities and prices when the (3rd degree) price discrimi- nation is allowed.
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(b) How is your answer in (a) di¤erent from the ones when price discrimination is prohibited, i.e., the monopolist needs to charge the same price for both markets?
4. Nash equilibrium
Find all pure strategy Nash equilibria in each of the following games.
(a)
1 2 L R
U 1; 1 0; 1
D 1 ; 0 0; 0
(b)
1 2 L M R
U 200; 0 50; 1 2; 2 D 80; 100 100; 50 1; 1
(c)
1 2 L M R
U 0; 0 2; 1 6; 2 M 1; 2 4; 4 5; 3 D 2; 6 3; 5 3; 4 5. Cournot model
Consider a cournot model with non-linear …rms’ costs. Suppose the inverse demand function is given by
p= a (q1+ q2), and each …rm’s cost function is quadratic,
Ci(qi) = qi2
for i = 1; 2. Then, derive a Cournot Nash equilibrium. 6. Mixed strategies: Theory
Show the following two de…nitions of mixed strategy Nash equilibrium, (1) and (2), are equivalent. A mixed-strategy pro…le is a Nash equilibrium if, for all players i, if
ui( i; i) ui( i; i) for all i 2 i (1) or
ui( i; i) ui(si; i) for all si 2Si. (2) 7. Mixed strategies: Application
A crime is observed by a group of n people. Each person would like the police to be informed but prefers that someone else make the phone call. They choose either
“call” or “not” independently and simultaneously. A person receives 0 payo¤ if no 2
one calls. If someone (including herself) makes a call, she receives v while making a call costs c. We assume v c >0 so that each person has an incentive to call if no one else will call.
(a) Derive all pure strategy Nash equilibria.
(b) Is there any mixed strategy Nash equilibrium in which every person decides to make a call with the same probability p? If yes, derive such p.
(c) In your answer in (b), is the probability that nobody makes a call increasing or decreasing in n?
8. Zero-sum game (Di¢cult!)
A two-person game is called zero-sum if the players’ payo¤s always sum to zero, that is,
u1(s1; s2) = u2(s1; s2)
for all s1 2S1 and s2 2S2. Assume, for simplicity, S1 and S2 are both …nite sets. (a) Prove the minimax theorem. That is, show that thre exists a pair of mixed
strategies 1, 2 such that max
12 1
min
22 2
u( 1; 2) = u( 1; 2) = min
22 2
max
12 1
u( 1; 2). Using this minimax theorem, answer the following questions.
(b) Show that Nash equilibria are interchangeable; if and are two Nash equilibria, then and are also Nash equilibria.
(c) Show that each player’s payo¤ is the same in every Nash equilibrium.
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