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+ u_I
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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