Problem Set 1: Posted on December 31
Advanced Microeconomics II (Fall, 2nd, 2013)
1. Question 1 (4 points)
Consider the following vNM utility function, u(x) = α + βx12.
(a) What restrictions must be placed on parameters α and β for this function to express risk aversion?
(b) Given the restrictions derived in (a), show that u(x) displays decreasing abso- lute risk aversion.
2. Question 2 (5 points)
Consider the following three lotteries, L1, L2 and L3:
L1 :
{ 50 dollars with probability 12 150 dollars with probability 12
L2 :{ 100 dollars with probability 23
200 dollars with probability 13
L3 :
50 dollars with probability 13
150 dollars with probability 59 300 dollars with probability 19 Answer the following questions:
(a) Suppose that a decision maker prefers for sure return of 90 dollars rather than L1. Then, can we conclude that she is (i) risk averse or (ii) not risk loving? Explain.
(b) Show that any risk averse decision maker whose preference satisfies indepen- dence axiom must prefer L2 to L3.
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 ǫA and Group B’s is ǫB. At the same level of demand in each group, ǫA> ǫB always holds. Then, which group will face a higher price? Explain.
Remark: The elasticity of demand depends on the amount of the good demanded. 4. Question 4 (6 points)
(a) Suppose every player has a strictly dominant strategy. Then, show that the strategy profile in which everyone takes this strictly dominant strategy be- comes a unique Nash equilibrium.
1
(b) Suppose every player has a weakly dominant strategy. Then, is the strategy profile in which everyone takes this weakly dominant strategy a unique Nash equilibrium? If yes, explain your reason. If not, construct the counter example. (c) Provide an example of static game (with infinitely many strategies) which does
not have any Nash equilibrium, including mixed strategy equilibrium. 5. Question 5 (6 points, Review)
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 payoff if no one calls. If someone (including herself) makes a call, she receives v while making a call costs c. We assume v > c so that each person has an incentive to call if no one else will call.
(a) Derive all pure-strategy Nash equilibria.
(b) Derive a symmetric mixed strategy Nash equilibrium in which every person decides to make a call with the same probability p.
2
