Problem Set 1: Solutions
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.
(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.
Answers
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?
Answer: No restriction on α while β must be strictly positive.
(b) Given the restrictions derived in (a), show that u(x) displays decreasing abso- lute risk aversion.
Answer: The absolute risk aversion is calculated as follows:
−u
′′(x)
u′(x) = −
−14x−
3 2
1 2x
−12
= 1 2x
−1,
which is clearly decreasing in x. 2. Question 2 (5 points)
(a) (i) No. Although her choice is not inconsistent with risk aversion, this infor- mation alone is not enough for us to judge if she is risk averse or not. (ii) Yes. Any risk loving agent must choose L1 rather than for sure return of $X when X≤ 100.
(b) Let L4 be the following lottery:
L4 : 150 dollars with probability 23
300 dollars with probability 13
Then, L3 could be written as L3 =
2 3L1⊕
1 3L4.
When % is risk averse, [100] % L1 and [200] % L4 must hold. By independence axiom, we obtain
L2 =
2
3[100] ⊕ 1
3[200] % 2 3L1⊕
1
3L4 = L3. 3. Question 3 (4 points)
pi− c 1
4. Question 4 (6 points)
(a) Let s∗ = (s∗1, ..., s∗n) be a vector of strictly dominant strategies. By definition of strictly dominant strategy, the following condition holds for each player i:
ui(s∗i, s−i) > ui(si, s−i) for any si 6= s∗i and s−i ∈ S−i. This implies
ui(s∗i, s∗−i) ≥ ui(si, s∗−i)
for all si ∈ Si, which is identical to Nash equilibrium condition. To establish uniqueness, assume on the contrary that there is another Nash equilibrium s∗∗6= s∗. Pick player j with s∗∗j 6= s∗j. Since s∗∗j is a Nash equilibrium strategy, it must satisfy
uj(s∗∗j , s∗∗−j) ≥ ui(sj, s∗∗−j) for all sj ∈ Sj, which implies
uj(s∗∗j , s∗∗−j) ≥ ui(s∗j, s∗∗−j).
This is contradiction to the presumption that s∗j is a dominant strategy. (b) No, it is not. The following is a counter example:
12 L R
U 2, 2 1, 1 D 1, 1 1, 1
Note that U and L are weakly dominant strategies while we have multiple NE, (U, L) and (D, R).
(c) Integer game: Every player chooses an integer number simultaneously, and the person whose number is the highest wins.
5. Question 5 (6 points)
(a) This game has n asymmetric pure-strategy Nash equilibria, in each of which exactly one person calls.
(b) By the indifference property, each person must be indifferent between choosing
“call” and “not.”
ucall= v − c = Pr{nobody else calls} · 0 + [1 − Pr{nobody else calls}]v = unot,
which becomes
v− c = [1 − (1 − p)n−1]v.
⇒ (1 − p)n−1 = c v.
⇒ p = 1 −c v
1
n−1.
Since c
v < 1 by assumption, p is decreasing in n. Note that, the probability such that nobody makes a call can be expressed by
Pr{nobody calls} = (1 − p)n=c v
n−1n ,
which is increasing in n. That is, the larger the group, the less likely the police are informed of the crime!