the assumptions of the theorem: markets exist for all possible goods, there are no externalities, markets are perfectly competitive, and market participants have perfect information. In the absence of perfect information or complete markets, outcomes will generally be Pareto-inefficient, per the Greenwald–Stiglitz theorem. The second welfare theorem is essentially the reverse of the first welfare theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth. Variants Weak Pareto efficiency Weak Pareto efficiency is a situation that cannot be strictly improved for every individual. Formally, a strong Pareto improvement is defined as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-efficient if it has no strong Pareto improvements. Any strong Pareto improvement is also a weak Pareto improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at {10, 0}, and George values at {5, 5}. Consider the allocation giving all resources to Alice, where the utility profile is (10, 0): It is a weak PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements). But it is not a strong PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) its utility profile is (10, 5). A market doesn't require local nonsatiation to get to a weak Pareto optimum. Constrained Pareto efficiency Constrained Pareto efficiency is a weakening of Pareto optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents. An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse