Second welfare theorem

The second welfare theorem states that, under assumptions such as convex preferences and production sets, any Pareto-efficient allocation can be supported as a competitive equilibrium after suitable redistribution of initial endowments.

Pareto efficiency: An allocation is Pareto efficient if no feasible change can improve someone’s welfare without reducing another person’s welfare. The theorem’s target allocations are exactly those that satisfy this efficiency criterion.
Edgeworth box: A diagram of two consumers’ allocations of two goods, with each consumer’s origin at an opposite corner. Its contract curve displays the efficient allocations the theorem aims to support.
Lump-sum tax: A tax whose amount does not vary with a person’s choices or economic activity. Non-distorting transfers approximate the redistribution needed to implement an efficient allocation.
Market failure: A market failure occurs when decentralized markets do not produce an efficient allocation. Externalities, public goods, and other failures can prevent the competitive equilibrium required by the theorem.
Competitive equilibrium: A market outcome in which agents optimize at given prices and markets clear. The theorem shows how redistribution can make an efficient allocation emerge as this kind of market outcome.
Contract curve: The set of allocations in an exchange diagram where consumers’ indifference curves are tangent or efficiency otherwise holds. It visualizes the theorem’s collection of Pareto-efficient targets.
Redistributive taxation: Tax policy that shifts resources from some people or activities toward others. It is the practical policy counterpart to the theorem’s redistribution of initial resources.
Nonconvexity: Nonconvexity occurs when a feasible set or preference structure lacks the mathematical properties of convex sets. Nonconvexities can block price support for efficient allocations and undermine the theorem’s conclusion.
First welfare theorem: The first welfare theorem says that, under suitable assumptions, competitive equilibria are Pareto efficient. It runs the implication in the opposite direction: equilibrium implies efficiency, rather than efficiency being supportable by equilibrium.
Walrasian auctioneer: A hypothetical market coordinator who adjusts prices until supply equals demand, without allowing trade at disequilibrium prices. The theorem’s supporting prices are equilibrium prices, often introduced through this device.

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