Simon Fraser University
ECON 804
Spring 2016
S. Lu
Problem Set 2
Due on February 22, in class
1. We saw in class that Arrow’s theorem does not imply that the social preferences exactly
match the dictator’s: when the dictator is indifferent between two alternatives, the social
preference between them may be strict and still satisfy UD, P and IIA. The SWF that we used
broke ties by using a predetermined order of alternatives.
Consider this additional axiom (call it “I”): “If 𝑥 ∼𝑖 𝑦 for all 𝑖 ∈ 𝐼, then 𝑥 ∼𝑠 𝑦.”
Note that the SWF proposed in class does not satisfy I. Find another SWF that satisfies UD,
P, IIA and I, and where ≿𝑠 does not exactly match ≿𝑖 for any 𝑖 ∈ 𝐼.
2. Show that strict single crossing does not imply single peakedness, even only considering
strict preference profiles. That is, provide a strict preference profile that:
has the strict single crossing property for at least one linear order;
does not have the single peakedness property for any linear order.
Hint: Try something with 3 players and 3 alternatives.
3. MWG 21.C.2
4. MWG 21.D.5
5. MWG 21.D.10
(The statement that you are asked to prove in part (d) is erroneous. Please correct it.)
6. MWG 22.C.1
7. MWG 22.E.3
8. MWG 18.AA.9