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 (in the sense that no further exchange would make one person better off without making another worse off). The requirements for perfect competition are these: There are no externalities and each actor has perfect information. Firms and consumers take prices as given (no economic actor or group of actors has market power).
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 (e.g. one person may own everything and everyone else nothing).
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.See the discussion on pp. 556 f of Mas-Colell et al.
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.'Wealth of nations' (1776), Book IV, Chap II.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) Walras 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'.Paraphrased from Leçon 18 (first ed.) of the Éléments. 'L'échange de deux marchandises entre elles sur un marché régi par la libre concurrence est une opération par laquelle tous les porteurs soit de l'une des deux marchandises, soit de l'autre, soit de toutes les deux, obtiennent la plus grande satisfaction de leurs besoins compatible avec cette condition de donner de la marchandise qu'ils vendent et de recevoir de la marchandise qu'ils achètent dans une proportion commune et identique.' According to Wicksell this passage moved to Leçon 10 in the 4th ed.
F. Y. Edgeworth (1881) Edgeworth 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.Paul Samuelson backed him up, saying that the locus of Paretian optima can be obtained under multilateral monopoly. 'Foundations of Economic Analysis' (1947), p. 214. 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 [the utilities of buyer and seller] do not increase together, but that, while one increases, the other decreases.p. 21. 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...p. 25. See also John Creedy, 'Francis Ysidro Edgeworth and Philip Henry Wicksteed' (2010), https://core.ac.uk/reader/6561724.
Vilfredo Pareto (1906/9) Pareto stated the first fundamental theorem in his Manuale (1906) and with more rigour in its French revision (Manuel, 1909).Manuale di Economia Politica con una Introduzione alla Scienza Sociale (1906) / Manuel d'Économie Politique (1909). 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 pressuresManuale / Manuel Chap III, §22. 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.§116.
His definition of optimality is given in Chap. VI: We will say that the members of a collectivity enjoy a maximum of ophelimity [i.e. of utility] 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. [He has previously defined an increase in individual ophelimity as a move onto a higher indifference curve.] That is to say that any small step is bound to increase the ophelimity of some individuals while diminishing that of others.§33.
The following paragraph gives us a theorem: For phenomena of type I [i.e. perfect competition], 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.§35.
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.K. Wicksell, 'Lectures on Political Economy' I (1906), Eng. tr. (1934), pp. 82 f.
Pareto didn't find it so straightforward. He gives a diagrammatic argument in his text, applying solely to exchange,§35. and a 32-page mathematical argument in the AppendixManuel, §109–end. which Samuelson found 'not easy to follow'.P. A. Samuelson, 'Foundations of Economic Analysis' (1947), p. 212. Pareto was hampered by not having a concept of the production–possibility frontier, whose development was due partly to his collaborator Enrico Barone.Thomas M. Humphrey, 'The Trade Theorist's Sacred Diagram: its Origin and Early Development' (1988). 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.This and the preceding quotation have been condensed from §53 and §55 of Chap. VI.
Enrico Barone (1908) Barone, an associate of Pareto, proved an optimality property of perfect competition,E. Barone, 'Il Ministro della Produzione nello Stato Colletivistica' (1908). 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.In fact he divides this value by the price of an arbitrarily chosen item, but since prices are assumed fixed this merely introduces an irrelevant asymmetry. 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 SamuelsonP. A. Samuelson, 'Foundations of Economic Analysis' (1947), pp. 214–217. 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 (PPF, to which he gave the alternative name of 'productive indifference curve') 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.A. Lerner, 'The Diagrammatical Representation of Cost Conditions in International Trade' (1932), cited in Thomas M. Humphrey, 'The Trade Theorist's Sacred Diagram: its Origin and Early Development' (1988). 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 (i) price should equal marginal cost and (ii) output should be maximised subject to (i). 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.E.g. 'Nothing but prime cost [i.e. marginal cost] enters necessarily and directly into the supply price for short periods': Alfred Marshall, 'Principles of Economics', V.v.6 (eighth ed. consulted). And cf. Knut Wicksell's 'provisional conclusion that free competition is normally a sufficient condition to ensure maximization of production', 'Lectures on Political Economy' I (1906), Eng. tr. (1934), p. 141.
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.P. A. Samuelson, 'Foundations of Economic Analysis' (1947), p. 217.
Harold Hotelling (1938) Hotelling put forward a new argument to show that 'sales at marginal costs are a condition of maximum general welfare' (under Pareto's definition). 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 (for example, in a natural monopoly).H. Hotelling, 'The General Welfare Problem in Relation to Problems of Taxation and of Railway and Utility rates' (1938), Econometrica, pp. 260, 267.
Oscar Lange (1942)