Maximizing Social Welfare:
• The problem afterwards is to allocate consumption goods across individuals to maximize social welfare w. That is,
• The following theorem institute that this problem amounts to finding the best allocation within the Pareto set.
• Theorem: If x *maximizes a social welfare function, then x *is Pareto optimal.
• Proof: The FOCs are:
Take the ratio of these for each agent I to find that marginal rates of substitution are equated across agents, i.e., x * is a Pareto optimum.
• Solution depends on the type of the social welfare function which in principle is supplied by social philosophers.
• A utilitarian simply total the utilities.
w = u1 + u2 + L+ uI Therefore the utilitarian solution equalizes marginal utilities of income across individuals.
• More in general the utilitarian could sum weighted utilities.
Where the weights i α are positive and sum to one.• A Rawlsian weights only the least fortunate therefore the social welfare function would be:
w = min(u1 ,L,uI )
Therefore the Rawlsian solution equalizes utilities across individuals.
• Theorem: If x *maximizes a social welfare function, then there exist weights α i such that x *maximizes the utilitarian social welfare function.
• We utilized the utilitarian social welfare function to derive the Pareto set simply by varying the weights.
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