Second Welfare Theorem and Social Welfare Function

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.

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(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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