310-2ps1

# 310-2ps1 - Economics 310-2 Spring 2011 Professor Jeﬀ Ely...

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Unformatted text preview: Economics 310-2 Spring 2011 Professor Jeﬀ Ely Problem Set 1 Due Friday April 8 5PM 1. Come up with an example of a Social Welfare Function that satisﬁes UD, IIA and non-dictatorship but fails Pareto. 2. In class we showed that when there are two alternatives A and B and an odd number of individuals, the majority opinion SWF satisﬁed UD, Pareto, and IIA. When there is an even number of individuals, the majority rule is not fully deﬁned because it does not specify what happens in the case of a tie. Extend the deﬁnition of majority rule by coming up with a tie-breaking rule so that the resulting SWF continues to satisfy UD, Pareto, and IIA. (“Tossing a coin” is not allowed because that produces a random outcome and we want deterministic outcomes. We have no way of describing preferences over random outcomes so we could not check whether “tossing a coin” satisﬁes Pareto, for example.) 3. Can you do the same thing with pairwise majority opinion when there are three alternatives? (That is, can you come up with a tie-breaking rule so that the resulting SWF satisﬁes UD, Pareto, and IIA?) 4. “Serial Dictatorship” is a social welfare function deﬁned as follows. We line the individuals up according to an arbitrary order. The ﬁrst individual in line tells us his top choice. That alternative is placed at the top of society’s ranking. The second individual tells us his top choice among the alternatives that remain and that alternative is place second in society’s ranking. We continue in this way until all alternatives are ranked (possibly running throught the line again if there are more alternatives than agents.) Serial dictatorship seems somewhat better than pure dicatorship. But Arrow’s theorem says that even serial dictatorship is ﬂawed. Which of the requirements (UD, IIA, Pareto) does it violate? Construct an example to show why. 1 ...
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