Second Welfare Theorem:
• Any allocation within the Pareto set is able to be supported by a competitive equilibrium. Second Welfare Theorem of Economics- Suppose x is an element of the Pareto set such that each agent i receives a positive amount of each good. If preferences are convex (as well as continuous and monotonic), then x is a competitive (i.e. Walrasian) equilibrium allocation for any endowment w satisfying pwi = pxi for i =1, L, I.
• The force of this theorem is that the issue of distribution can be separated from the issue of efficiency.
– Equity unease are handled by moving along the Pareto set.– Any egalitarian allocation is able to be achieved via competition by juggling the endowment.– First society enquires what alternative it likes best, second, it juggles endowments to get that as a competitive equilibrium.– For example, the state might impose “land reform,” which would switch the endowment point to E; from there, everyone would trade freely to alternative D, which is Pareto optimal.
(Second Welfare Theorem)
Social Welfare Function:
1) Pareto and Complete Rankings:
• Pareto standard ranks allocations but not completely.
• Several allocations are just not comparable on Pareto grounds– Pareto needs unanimous agreement for comparison.
• How must a society decide among all the alternatives within the Pareto set?– Pareto principle tenders no guide for social choice.
• Perhaps we are able to construct a social welfare function that represents society’s rankings of allocations.
2) Ranking Social Alternatives:
• The problem is how to aggregate the liking of individuals to determine the best alternative for the group.
• If there exists a solution to this problem in the form of complete and transitive social preferences, then social alternatives (example the allocation of resources) can be ranked.
• Could social liking be represented by a social welfare function, which assigns numbers to these alternatives such that better alternatives receive higher numbers? If complete, transitive, and continuous social preferences exist, then there exists a continuous social welfare function that represents those preferences.
Social welfare function is ordinal.
• Generally, we have w = W(x) =W(x1,L, xI) . If we restrict attention to social welfare as a function of individual utilities,w = W(u1, L, uI)
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