Fundamental theorems of welfare economics

There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal. The requirements for perfect competition are these:

There are no externalities and each actor has perfect information.
Firms and consumers take prices as given.

The theorem is sometimes seen as an analytical confirmation of Adam Smith's "invisible hand" principle, namely that competitive markets ensure an efficient allocation of resources. However, there is no guarantee that the Pareto optimal market outcome is equitative, as there are many possible Pareto efficient allocations of resources differing in their desirability.
The second theorem states that any Pareto optimum can be supported as a competitive equilibrium for some initial set of endowments. The implication is that any desired Pareto optimal outcome can be supported; Pareto efficiency can be achieved with any redistribution of initial wealth. However, attempts to correct the distribution may introduce distortions, and so full optimality may not be attainable with redistribution.
The theorems can be visualized graphically for a simple pure exchange economy by means of the Edgeworth box diagram.

History of the fundamental theorems

Adam Smith (1776)

In a discussion of import tariffs Adam Smith wrote that:

Every individual necessarily labours to render the annual revenue of the society as great as he can... He is in this, as in many other ways, led by an invisible hand to promote an end which was no part of his intention... By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it.

Note that Smith's ideas were not directed towards welfare economics specifically, as this field of economics had not been created at the time. However, his arguments have been credited towards the creation of the branch as well as the fundamental theories of welfare economics.

Léon Walras (1870)

wrote that 'exchange under free competition is an operation by which all parties obtain the maximum satisfaction subject to buying and selling at a uniform price'.

F. Y. Edgeworth (1881)

took a step towards the first fundamental theorem in his 'Mathematical Psychics', looking at a pure exchange economy with no production. He included imperfect competition in his analysis. His definition of equilibrium is almost the same as Pareto's later definition of optimality: it is a point such that...

in whatever direction we take an infinitely small step, P and Π do not increase together, but that, while one increases, the other decreases.

Instead of concluding that equilibrium was Pareto optimal, Edgeworth concluded that the equilibrium maximizes the sum of utilities of the parties, which is a special case of Pareto efficiency:

It seems to follow on general dynamical principles applied to this special case that equilibrium is attained when the total pleasure-energy of the contractors is a maximum relative, or subject, to conditions...

Vilfredo Pareto (1906/9)

stated the first fundamental theorem in his Manuale and with more rigour in its French revision. He was the first to claim optimality under his own criterion or to support the claim by convincing arguments.
He defines equilibrium more abstractly than Edgeworth as a state which would maintain itself indefinitely in the absence of external pressures and shows that in an exchange economy it is the point at which a common tangent to the parties' indifference curves passes through the endowment.
His definition of optimality is given in Chap. VI:

We will say that the members of a collectivity enjoy a maximum of ophelimity at a certain position when it is impossible to move a small step away such that the ophelimity enjoyed by each individual in the collectivity increases, or such that it diminishes. That is to say that any small step is bound to increase the ophelimity of some individuals while diminishing that of others.

The following paragraph gives us a theorem:

For phenomena of type I , when equilibrium takes place at a point of tangency of indifference curves, the members of the collectivity enjoy a maximum of ophelimity.

He adds that 'a rigorous proof cannot be given without the help of mathematics' and refers to his Appendix.
Wicksell, referring to his definition of optimality, commented:

With such a definition it is almost self-evident that this so-called maximum obtains under free competition, because if, after an exchange is effected, it were possible by means of a further series of direct or indirect exchanges to produce an additional satisfaction of needs for the participators, then to that extent such a continued exchange would doubtless have taken place, and the original position could not be one of final equilibrium.

Pareto didn't find it so straightforward. He gives a diagrammatic argument in his text, applying solely to exchange, and a 32-page mathematical argument in the Appendix which Samuelson found 'not easy to follow'. Pareto was hampered by not having a concept of the production–possibility frontier, whose development was due partly to his collaborator Enrico Barone. His own 'indifference curves for obstacles' seem to have been a false path.
Shortly after stating the first fundamental theorem, Pareto asks a question about distribution:

Consider a collectivist society which seeks to maximise the ophelimity of its members. The problem divides into two parts. Firstly we have a problem of distribution: how should the goods within a society be shared between its members? And secondly, how should production be organised so that, when goods are so distributed, the members of society obtain the maximum ophelimity?

His answer is an informal precursor of the second theorem:

Having distributed goods according to the answer to the first problem, the state should allow the members of the collectivity to operate a second distribution, or operate it itself, in either case making sure that it is performed in conformity with the workings of free competition.

Enrico Barone (1908)

, an associate of Pareto, proved an optimality property of perfect competition, namely that – assuming exogenous prices – it maximises the monetary value of the return from productive activity, this being the sum of the values of leisure, savings, and goods for consumption, all taken in the desired proportions. He makes no argument that the prices chosen by the market are themselves optimal.
His paper wasn't translated into English until 1935. It received an approving summary from Samuelson but seems not to have influenced the development of the welfare theorems as they now stand.

Abba Lerner (1934)

In 1934 Lerner restated Edgeworth's condition for exchange that indifference curves should meet as tangents, presenting it as an optimality property. He stated a similar condition for production, namely that the production–possibility frontier should be tangential with an indifference curve for the community. He was one of the originators of the PPF, having used it in a paper on international trade in 1932. He shows that the two arguments can be presented in the same terms, since the PPF plays the same role as the mirror-image indifference curve in an Edgeworth box. He also mentions that there's no need for the curves to be differentiable, since the same result obtains if they touch at pointed corners.
His definition of optimality was equivalent to Pareto's:

If... it is possible to move one individual into a preferred position without moving another individual into a worse position... we may say that the relative optimum is not reached...

The optimality condition for production is equivalent to the pair of requirements that price should equal marginal cost and output should be maximised subject to. Lerner thus reduces optimality to tangency for both production and exchange, but does not say why the implied point on the PPF should be the equilibrium condition for a free market. Perhaps he considered it already sufficiently well established.
Lerner ascribes to his LSE colleague Victor Edelberg the credit for suggesting the use of indifference curves. Samuelson surmised that Lerner obtained his results independently of Pareto's work.

Harold Hotelling (1938)

put forward a new argument to show that 'sales at marginal costs are a condition of maximum general welfare'. He accepted that this condition was satisfied by perfect competition, but argued in consequence that perfect competition could not be optimal since some beneficial projects would be unable to recoup their fixed costs by charging at this rate.

Oscar Lange (1942)

's paper 'The Foundations of Welfare Economics' is the source of the now-traditional pairing of two theorems, one governing markets, the other distribution. He justified the Pareto definition of optimality for the first theorem by reference to Lionel Robbins's rejection of interpersonal utility comparisons, and suggested various ways to reintroduce interpersonal comparisons for the second theorem such as the adjudications of a democratically elected Congress. Lange believed that such a congress could act in a similar way to a capitalist: through setting price vectors, it could achieve any optimal production plan to have achieve efficiency and social equality.
His reasoning is a mathematical translation of Lerner's graphical argument. The second theorem does not take its familiar form in his hands; rather he simply shows that the optimisation conditions for a genuine social utility function are similar to those for Pareto optimality.