Cheapter33

1. Suppose that we say an allocation x is socially preferred to an allocation y only if everyone prefers x to y. (This is sometimes called the Pareto ordering，since it is closely related to the idea of Pareto efficient.) What shortcoming does this have as a rule for making social decisions？
   A：The major shortcoming is that there are many allocations that cannot be compared –there is no way to decide between any two Pareto efficient allocations.
   
2. A Rawlsian welfare function counts only the welfare of the worst off agent.The opposite of the Rawlsian welfare function might be called the "Nietzschean" welfare function－a welfare function that says the value of an allocation depends only on the welfare of the best off agent.What mathematical form would the Nietzchean welfare function take?
   Ans: It would have the form:W(u1,...,un)=max(u1,...,un).
   
3. Suppose that the utility possibilities set is a convex set and that consumers care only about their own consumption. What kind of allocations represent welfare maxima of the Nietzschean welfare function？
   A：Since the Nietzschean welfare function cares only about the best off individual，welfare maxima for this allocation would typically involve one person getting everything.
   
4. Suppose that an allocation is Pareto efficient, and that each individual only cares about his own consumption. Prove that there must be some individual that envies no one, in the sense described in the text.(This puzzel requires some thought,but it is worth it.)
   Ans: Suppose that this is not the case.Then each individual envies someone else.Let's construct a list of who envies whom. Person A envies someone－call him person B. Person B in turn envies someone－say person C. And so on. But eventually we will find someone who envies someone who came earlier in the list. Suppose the cycle is "C envies D envies E envies C." Then consider the following swap: C gets what D has, D gets what E has, and E gets what C has. Each person in the cycle gets a bundle that he prefers, and thus each person is made better off. But then the original allocation couldn't have been Pareto efficient!
   
5. The ability to set the voting agenda can often be a powerful asset. Assuming that social preferences are decides by pair-wise majority voting and that the preferences given in Table 30.1 hold，demonstrate this fact by producing a voting agenda results in allocation y winning. Find an agenda that has z as the winner. What property of the social preferences is responsible for this agenda-setting power？
   A：First vote between x and z，and then vote between the winner (z) and y. First pair x and y，and then vote between the winner (x) and z. The fact that the social preferences are intransitive is responsible for this agenda-setting power.