Economics ECON0059: Advanced Microeconomics 2019-20
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Economics ECON0059: Advanced Microeconomics
2019-20
Question no. 1
Two profit-maximizing firms independently choose whether to offer a product of high or low quality. When a firm offers a high-quality product, it incurs a fixed cost (independently of whether it sells the product) of c 
(0,½). Offering a low-quality product entails no cost. Simultaneously with its quality choice, each firm also decides whether to advertise its product at a cost m.
A consumer is initially assigned to one of the two firms (with probability ½ each). If the other firm offers a higher-quality product and advertises it, the consumer switches to it. Otherwise, the consumer sticks to her initially assigned firm. The firm that is eventually selected by the consumer earns a gross revenue of 1, while the other firm earns zero gross revenues.
1. Describe the firms’ interaction as a two-player strategic game.
2. Let m=0. Characterize the set of symmetric pure-strategy Nash equilibria in the game.
3. Now suppose m 
(0, ½-c).
a. Show there exists no pure-strategy Nash equilibrium.
b. Is there a symmetric mixed-strategy Nash equilibrium in which the equilibrium strategy has full support (i.e., every pure strategy is played with positive probability)? Explain.
c. Find a symmetric mixed-strategy Nash equilibrium. Explain your calculations.
Question no. 2
An office party is about to be held, and two workers disagree over the important question of whether music will be played at the event. Worker 1 wants to have music, whereas worker 2 does not want any music. Specifically, if music is played, worker 1 will earn a gross utility v1, whereas worker 2 will experience a gross disutility –v2 . The values v1 and v2 are independently and uniformly drawn from [0,1].
The two workers try to influence the plans for the party. Each worker chooses whether to remain silent or express her opinion. If the worker chooses the latter course of action, she incurs a cost of ¼. Music will be played if and only if agent 1 requests it and agent 2 remains silent.
1. Describe the interaction between the two workers as a Bayesian game.
2. Show that a pure-strategy Nash equilibrium must be in cutoff strategies: Each worker i=1,2 expresses her opinion if and only if vi is above some threshold (which may be different for each player).
3. Find the game’s pure-strategy Nash equilibrium. What is the probability that music will be played at the party? Explain your calculations.
Question no. 3
Consider the following infinitely repeated prisoner’s dilemma, with discount factor δ ∈ (0,1) and payoff matrix:
C D
C 2,2 - 1,3
D 3,- 1 0,0
1. Describe all terminal histories of this game.
2. For each of the following strategies, characterize the set of δ (if any) for which there is a symmetric subgame-perfect equilibrium as described below:
a. Play C in the first period. Then play C if in the last period the players both played C or both played D, and play D if last period one played C and the other played D.
b. In odd-numbered periods, always play D. In even-numbered periods, play C if and only if both players played C is all previous even-numbered periods.
c. Play C in the first period. After any subsequent history, play the action the opponent played in the previous period.
3. Are all of the strategies in parts (a)-(c) above sustained in a subgame-perfect equilibrium when δ is close to one? Provide an intuitive explanation why or why not.
Question no. 4
Consider the following signaling game.
Nature moves first and chooses player 1's type θ which is either A (with probability p)
or B (with probability 1-p).
Player 1 observes Nature's move and chooses an action 
1 ∈ {
, 
}.
Player 2 sees 1's move but not 1's type and chooses 
2 ∈ {
, 
}.
The payoffs for each θ are given by the following matrices:
When θ =A When θ =B
L R L R
_______
U 3,3 0,0 U 1,- 1 - 1,1
D 0,0 2,2 D - 1,1 1,- 1
1. Draw the tree of this game. Make sure to indicate all information sets.
2. For what values of p does the game have a separating Perfect Bayesian Equilibrium (PBE)? Fully characterize at least one PBE for each of those values (if any).
3. For what values of p does the game have a pooling PBE? Fully characterize at least one PBE for each of those values (if any).
4. For what values of p does the game have a partially separating PBE in which type B plays U with probability 1 and type A assigns strictly positive probabilities to both actions?
2022-05-04