The Second Fundamental Welfare Theorem goes further, to state that any Paretooptimal allocation can be achieved as a competitive equilibrium, provided there is an appropriate redistribution of initial endowments^{} (and through optimising behaviour on the part of society).
These two theorems imply that efficiency can be dealt with separately from equity; if a particular distribution of welfare is pursued, this should be done by altering the distribution of initial endowments, or by redistributing purchasing power, for example using lumpsum taxes and allowances, rather than by interfering with markets.
The Edgeworth Box^{} is used in general equilibrium analysis and shows the allocations of two goods between two ‘agents’ using indifference curves and a contract curve.^{} The area enclosed in the box represents all possible combinations of the goods, and therefore any point represents some distribution of the goods between the agents.
Consumer A’s utility increases the further from OA it is, and the utility for consumer B increases the further away from OB it is.
The contract curve (OAOB) joins all the consumptionefficient allocations, which are all the points of tangency between consumer A and consumer B’s indifference curves. At each of these points, the slopes of the indifference curves must be the same; hence marginal rates of substitution are equal.
One method of reallocating resources is for the government to “transfer the purchasing power of endowments”^{}  that is, the government could tax consumer A according to the value of his endowment and transfer that money to consumer B. This would result in a shift of the budget line from M0 to M1, so the endowment would change from R to W, as seen below:
This movement from R to W is possible due to the Second Welfare Theorem. It enables consumers to move from one budget set to another. The mechanisms of a competitive market would then ensure that the Paretooptimal allocation (at E) is achieved. The consumers can therefore benefit from an allocation that still maximises utility, yet is also more equitable.
Algebraically, we can show this reallocation of resources as:
M1A = M0A  T  shift of budget line for A
(where T = lump sum transfer)
And therefore, conversely:
M1B = M0B + T  shift of budget line for B
Due to the difficulty in measuring consumers’ endowments, we must use a nondistortionary form of lump sum taxation – this measures the potential (rather than actual) value. The tax is therefore based on the endowment itself, which cannot be changed, and not on consumers’ choices, as inefficiencies result when taxes depend on choice^{}.
We have seen that the Second Welfare Theorem proves that any Pareto efficient allocation can be achieved by redistribution of endowment. However, there is an exception to this; if one agent’s preferences are nonconvex, then it is possible to have a Paretooptimal allocation which is not in equilibrium, as seen below. Consumer A wants allocation M, but consumer B wants allocation N – the optimal demands are in disequilibrium.
CONCLUSION
Welfare economics is a framework for deciding on the optimal use of scarce resources. Paretooptimal allocation occurs when it is not possible to redistribute goods to increase the welfare of one consumer without reducing the welfare of another. Conversely, therefore, a situation is not Paretooptimal if you can make someone better off without making anyone else worse off. Such inefficient outcomes are to be avoided, and therefore Pareto efficiency is important for evaluating economic systems and political policies.
In terms of efficiency alone, Paretooptimality is best, but it is important to remember that efficiency does not guarantee equity, so Paretooptimality does not necessarily ensure the maximisation of social welfare. The First Theorem of Welfare Economics only guarantees that an efficient outcome will occur in a perfectly competitive market. As a result, the government might have a role to play. The Second Theorem of Welfare Economics tells us that one way of achieving a particular Pareto efficient allocation is to "adjust" the budget constraints, i.e. redistribute endowments, and then let competitive markets ‘work’. That is, the only intervention that is needed to achieve any Pareto efficient allocation is the redistribution of initial endowments. The Second Welfare Theorem effectively separates efficiency and distribution issues. Distribution objectives are achieved by the redistribution of endowment, while efficiency is achieved by market mechanism.
The social welfare theorems indicate that under conditions of perfect competition, market exchanges will result in a distribution of goods and services that maximize the overall welfare or utility of individuals in society, so long as there is some form of intervention to redistribute the endowment.
It is true, then, that any pareto efficient allocation can be achieved by lump sum redistribution of endowment, provided the preferences (diagrammatically, the indifference curves) of all consumers involved are convex.
BIBLIOGRAPHY
WEB RESOURCES:
 www.tutor2u.com
 www.bized.ac.uk

The History of Economic Thought website  http://cepa.newschool.edu/het/
 Walrasian General Equilibrium Theory http://cepa.newschool.edu/het/essays/get/getcont.htm
 The Paratian System (parts I, II and IV) http://cepa.newschool.edu/het/essays/paretian/paretocont.htm

Ted Bergstrom’s Economics Homepage 

‘The Fundamental Theorems of Welfare Economics’ 

‘Wageningen Economics Papers: Chapter 2  Origins’ 
TEXTBOOKS:
M. L. Katz and H. S. Rosen: Microeconomics (third edition, McGrawHill)
H. R. Varian: Intermediate Microeconomics (sixth edition, Norton)
D. Rutherford: Routledge Dictionary of Economics (Routledge)
 D. Laidler and S. Estrin: Introduction to Microeconomics (third edition)
Routledge Dictionary of Economics pg 418 (‘Pareto Optimum’)
Microeconomics (third edition) – Katz and Rosen, pg 379 (‘Welfare Economics’)
Routledge Dictionary of Economics pg 208 (‘First Welfare Theorem’)
 The History of Economic Thought website (The Paretian System: IV Social Welfare)
Misnamed after the 19thcentury economist F. Y. Edgeworth – it was in fact invented by V. Pareto
‘Implications of the Second Welfare Theorem’ – Intermediate Microeconomics, H. R. Varian pg 562
‘Implications of the Second Welfare Theorem’ – Intermediate Microeconomics, H. R. Varian pg 563