[2] The second theorem states that any Pareto optimum can be supported as a competitive equilibrium for some initial set of endowments.
[3] The theorems can be visualized graphically for a simple pure exchange economy by means of the Edgeworth box diagram.
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...
[5] 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'.
[6] Edgeworth took a step towards the first fundamental theorem in his 'Mathematical Psychics', looking at a pure exchange economy with no production.
[8]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...[9]Pareto stated the first fundamental theorem in his Manuale (1906) and with more rigour in its French revision (Manuel, 1909).
[citation needed] He defines equilibrium more abstractly than Edgeworth as a state which would maintain itself indefinitely in the absence of external pressures[11] 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.
[13]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.
[18] Pareto was hampered by not having a concept of the production–possibility frontier, whose development was due partly to his collaborator Enrico Barone.
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.
[20]Barone, an associate of Pareto, proved an optimality property of perfect competition,[21] 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 1934 Lerner restated Edgeworth's condition for exchange that indifference curves should meet as tangents, presenting it as an optimality property.
[24] 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.
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.
[25] Lerner ascribes to his LSE colleague Victor Edelberg the credit for suggesting the use of indifference curves.
[26] Hotelling put forward a new argument to show that 'sales at marginal costs are a condition of maximum general welfare' (under Pareto's definition).
[27] Lange's paper 'The Foundations of Welfare Economics' is the source of the now-traditional pairing of two theorems, one governing markets, the other distribution.
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.
Samuelson (crediting Abram Bergson for the substance of his ideas) brought Lange's second welfare theorem to approximately its modern form.
[30] He follows Lange in deriving a set of equations which are necessary for Pareto optimality, and then considers what additional constraints arise if the economy is required to satisfy a genuine social welfare function, finding a further set of equations from which it follows 'that all of the action necessary to achieve a given ethical desideratum may take the form of lump sum taxes or bounties'.
[31] Arrow's and Debreu's two papers[32] (written independently and published almost simultaneously) sought to improve on the rigour of Lange's first theorem.
Arrow motivated his paper by reference to the need to extend proofs to cover equilibria at the edge of the space, and Debreu by the possibility of indifference curves being non-differentiable.
[33] The paper establishes that a competitive equilibrium of an economy with asymmetric information is generically not even constrained Pareto efficient.
A government facing the same information constraints as the private individuals in the economy can nevertheless find Pareto-improving policy interventions.
Another instance in which the welfare theorems fail to hold is in the canonical Overlapping generations model (OLG).
A further assumption that is implicit in the statement of the theorem is that the value of total endowments in the economy (some of which might be transformed into other goods via production) is finite.
[37] In the OLG model, the finiteness of endowments fails, giving rise to similar problems as described by Hilbert's paradox of the Grand Hotel.
This follows from the fact that the minimal marginal price and finite wealth limits the maximum feasible production (0 limits the minimum) and Tychonoff's theorem ensures the product of these compacts spaces is compact ensuring us a maximum of whatever continuous function we desire exists.
These two statements, along with the feasibility of the allocation at the Pareto optimum, satisfy the three conditions for a price quasi-equilibrium with transfers supported by wealth levels