A bargaining situation is described by a tuple hX, D, % 1 , % 2 i: X is a set of possible agreements: a set of possible consequences that the two players can jointly achieve.
D ∈ X is the disagreement outcome: the event that occurs if the players fail to agree.
are differentiated. After the prices are set, consumers demand 24 − p i +
2 units (i 6= j, i = 1, 2) of the good that firm i produces. Assume that each firm’s marginal cost is 6, and the payoff for each firm is equal to the firm’s profit.
(a) Write the payoff functions π 1 and π 2 (as a function of p 1 and p 2 ).
(b) Derive the best response functions and solve the pure-strategy Nash equilib- rium of this game.
(c) Derive the prices (p 1 , p 2 ) that maximize joint-profit, i.e., π 1 + π 2 .
Hint: Your answers in (a) – (c) may change depending on the value of θ.
4. Duopoly (20 points)
Consider a duopoly game in which two firms, denoted by firm 1 and firm 2, simul- taneously and independently select their own price, p 1 and p 2 . The firms’ products are differentiated. After the prices are set, consumers demand 24 − p i +
Three firms (1, 2 and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is 1; if exactly one firm advertises in the evening, its profit is 2. Firms must make their daily advertising decisions simultaneously.
where u i (x, θ i ) is the money-equivalent value of alternative x ∈ X.
This assumes the case of private values in which player i’s payoff does not depend directly on other players’ types. If it does, then it is called common values case. The outcome (of the mechanism) is described by
(a) Find a Bayesian Nash equilibrium of the game in pure strategies in which each player i accepts an exchange if and only if the value v i does not exceed some
threshold θ i
(b) How would your answer to (a) change if the value of player i’s house to the other player j becomes 5
(d) The perfect Bayesian equilibrium puts NO restriction on beliefs at information sets that are not reached in equilibrium.
(e) In the simple moral hazard problem we studied in class, the optimal wage (= s( )) is NOT necessarily increasing in outcome (= x).
Each player’s strategy specifies optimal actions given her beliefs and the strategies of the other players, and
The beliefs are consistent with Bayes’ rule wherever possible. If (4) is not required, the equilibrium concept is called weak perfect Bayesian equilibrium (weak PBE).
4. Simultaneous Game (20 points, moderate)
Suppose three cafe chain companies, i = 1; 2; 3, are considering to open new shops near the Roppongi cross (Each company opens at most one shop). They make the decision independently and simultaneously. A company receives 0 pro…t if it does not open a shop. If opens, then each …rm’s pro…ts depend on the number of shops which are given as follows:
Proof Sketch (2): Existence of Pivotal Voter Lemma 3 (Existence of Pivotal Voter)
There is a voter n ∗ = n(b) who is extremely pivotal in the sense that by changing his vote at some profile he can move b from the very bottom of the social ranking to the very top.
(b) Show that any risk averse decision maker whose preference satisfies indepen- dence axiom must prefer L 2 to L 3 .
3. Question 3 (4 points) Suppose a monopolist with constant marginal costs prac- tices third-degree price discrimination. Group A’s elasticity of demand is ǫ A and
object for each buyer is independently and uniformly distributed between 0 and 1. (a) Suppose that buyer 2 takes a linear strategy, b 2 = v 2 . Then, derive the
probability such that buyer 1 wins as a function of b 1 .
(b) Solve a Bayesian Nash equilibrium.
(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.
Similarly, player 2 must be indi¤erent amongst choosing X and Y , which implies 4q + 6(1 q) = 7(1 q)
, 5q = 1 , q = 1=5.
Thus, the mixed-strategy equilibirum is that player 1 takes A with probability 1=5 (and B with probability 4=5) and player 2 takes X with probability 3=4 (and Y with probability 1=4).
Problem Set 2: Posted on November 4
Advanced Microeconomics I (Fall, 1st, 2014)
1. Question 1 (7 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.