Consider an economy that consists of just three individuals, {a, b, c} and four possible social states of the world. Each state of the world is characterised by a monetary payoff yh thus:
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a
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b
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c
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3
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3
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3
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1
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4
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4
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5
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1
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3
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2
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6
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1
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Suppose that person h has a utility function Uh log(yh).
1. Show that if individuals know the payoffs that will accrue to them under each state of the world, then majority voting will produce a cyclic decision rule.
2. Show that the above conditions can rank unequal states over perfect equality.
3. Show that if people did not know which one of the identities {a, b, c} they were to have before they vote, if they regard any one of these three identities as equally likely and if they are concerned to maximise expected utility, then majority voting will rank the states strictly in the order of the distribution of the payoffs.
4. A group of identical schoolchildren are to be endowed at lunch time with an allocation of pie. When they look through the dining hall window in the morning they can see the slices of pie lying on the plates: the only problem is that no child knows which plate he or she will receive. Taking the space of all possible pie distributions as a complete description of all the possible social states for these schoolchildren, and assuming that ex ante there are equal chances of any one child receiving any one of the plates discuss how a von Neumann-Morgenstern utility function may be used as a simple social-welfare function.
5. What determines the degree of inequality aversion of this social-welfare function?
6. Consider the possible problems in using this approach as a general method of specifying a social-welfare function.