### Midterm Exam

Date: June 23, 2010

Subject: Advanced Microeconomics II (ECO601E) Professor: Yosuke YASUDA

1. True or False (9 points)

Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.

(a) When a monopoly …rm chooses the pro…t maximizing price, the elasticity of market demand at this monopoly price is ALWAYS (weakly) greater than 1. (b) A zero-sum game ALWAYS has more than one Nash equilibrium.

(c) If a player randomizes pure strategies X and Y in a (mixed strategy) Nash equilibrium, she MUST be indi¤erent between choosing X and Y .

2. Monopoly (10 points)

Suppose a monopoly …rm operates in two di¤erent markets, A and B. Inverse demand for each market is given as follows.

p_{(q}_{A}_{) = 200} q_{A}
p(q_{B}) = 120 q_{B}
The cost function is given by

C(q_{A}; q_{B}) = ^{1}

2^{(q}^{A}^{+ q}^{B}^{)}

2

(a) Derive the pro…t function of this monopoly …rm, (q_{A}^{; q}_{B}).

(b) What are the optimal (i.e., pro…t maximizing) quantities q_{A} and q_{B}?
3. Nash Equilibrium (16 points)

Monica and Nancy have formed a business partnership. Each partner must make
her e¤ort decision without knowing what e¤ort decision the other player has made.
Let m be the amount of e¤ort chosen by Monica and n be the amount of e¤ort
chosen by Nancy. The joint pro…ts are given by 4m + 4n + mn, and two partners
split these pro…ts equally. However, they must each separately incur the costs of
their own e¤ort, which is a quadratic function of the amount of e¤ort, i.e., m^{2} and
n^{2} respectively.

1

(a) Derive each partner’s payo¤ function.

(b) Derive each partner’s best reply function and graphically draw them in a …gure. (Taking m in the horizontal axis and n in the vertical axis.)

(c) Is this game strategic complementarity, strategic substitution, or neither of them? Explain why.

(d) What is the Nash equilibrium of this game? 4. Mixed Strategy (15 points)

Three …rms (1, 2 and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A …rm advertises exactly once per day. If more than one …rm advertises at the same time, their pro…ts become 0. If exactly one …rm advertises in the morning, its pro…t is 1; if exactly one …rm advertises in the evening, its pro…t is 2. Firms must make their daily advertising decisions simultaneously.

(a) Derive all pure strategy Nash equilibria.

(b) Show the following type of Nash equilibria does NOT exist: One …rm chooses pure strategy M , and other two …rms use mixed strategies.

(c) Derive a symmetric mixed strategy Nash equilibria. You can assume that each

…rm chooses M with probability p and E with probability 1 p, then calculate an equilibrium probability, p.

2