Econ 407.3 Spring 2016 Homework 1

1. (5 points) Complete the proof of Case 3 in Lemma 1 of Arrow's Impossibility Theorem.

2. (5 points) Complete the proof of Case 4 in Lemma 1 of Arrow's Impossibility Theorem.

3. (4 points) Suppose that there are two alternatives and three individuals (A = fa; bg

and I = f1; 2; 3g). If S I is decisive for a over b, is S necessarily decisive for b over a?

If your answer is YES, give a proof. If your answer is NO, give a counterexample, that is,

an example in which S is decisive for a over b but not decisive for b over a.

4. Let A = fa; b; cg and I = f1; 2g. Consider the following social welfare function:

f(P1; P2) =

P1 if aP1c;

P2 if cP1a.

i) (2 points) Does f respect unanimity?

ii) (2 points) Does f satisfy IIA?

iii) (2 points) is f dictatorial?