Understand different theories of social welfare.
We suppose that each individual's well-being/welfare from any consumption bundle is measured by a (non-ordinal) utility function, where this function may of course differ across individuals. As usual, utility is increasing in any good. We assume that a social planner knows people's utility functions and has the authority and power to distribute (allocate) all commodities to promote the social good. This assumption about a social planner is there purely for concreteness; in principle, we could talk about people's welfare levels without anyone knowing them, indeed without them being externally measurable.
Exercise 1
1) If the social planner allocates according to the utilitarian principle, the allocation is necessarily Pareto-efficient.
2) If the social planner allocates according to the maximin principle (by maximizing the utility of the worst off, ignoring others' utilities), the allocation is necessarily Pareto-efficient.
3) An allocation brought about by free trade (in Walrasian equilibrium) maximizes total utility, i.e., is utilitarian optimal, since otherwise some additional trade could further raise someone's utility, in contradiction with Walrasian equilibrium.
4) If everyone is allocated the same amount from each good, the allocation is necessarily Pareto-efficient since any re-allocation would reduce at least one person's allocation.
5) If everyone is allocated the same amount from each given good, we necessarily achieve perfect equality in welfare (utility).
6) If everyone holds the same utility function, then equalizing people's endowments implies equalizing their utilities.
7) If everyone holds the same utility function, then to equalize people's utilities the social planner necessarily has to equalize their endowments.
Exercise 2
1) Every allocation is Pareto-efficient, but the recommendations of the various other principles (utilitarianism, welfare egalitarianism, etc.) may differ.
2) If everyone has a linear utility function, then the social planner can maximize sum-total utility (i.e., achieve utilitarian optimality) by giving everything to one, suitably chosen, person.
3) Utilitarianism necessarily requires assigning the full amount - call it T - to one person, namely to the person whose utility of T is largest.
4) If everyone has the same strictly concave (and of course increasing) utility function, then utilitarianism and (objective and subjective) egalitarianism make the same distributional recommendation of giving the same amount to everyone.
5) If all have the same linear utility function, then all allocations are equivalent from a utilitarian perspective.