Pareto Optimum

The term is named after Vilfredo Pareto , an Italian economist who used the concept in his studies of Economic Efficiency and Income Distribution .

If an economic system is Pareto efficient, then it is the case that no individual can be made better off without another being made worse off. It is commonly accepted that outcomes that are not Pareto efficient are to be avoided, and therefore Pareto efficiency is an important criterion for evaluating Economic System s and political policies.

In particular, it can be shown that, under certain idealised conditions, a system of Free Market s will lead to a Pareto efficient outcome. This was first demonstrated mathematically by economists Kenneth Arrow and Gerard Debreu , although the result may not necessarily reflect the workings of real economies because of the restrictive assumptions necessary for the proof (markets exist for all possible goods, markets are perfectly competitive, and transaction costs are negligible). This is called the First Welfare Theorem .

A strongly Pareto optimal (SPO) allocation (X) is one for which there cannot be any other feasible allocation (say X') such that the allocation (X') is strictly preferred by at least one person, and weakly preferred by everyone else. A '''weakly Pareto optimal''' (WPO) allocation is one where a feasible reallocation would be strictly preferred by all agents.

A key drawback of Pareto optimality is its localization. In an economic system with millions of variables there can be very many Local optimum points. The Pareto improvement criterion does not even define any Global Optimum . Under a reasonable criterion, many Pareto-optimal solutions may be far inferior to the global solution.

PARETO FRONTIER

For a given system, the Pareto Frontier is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto Frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer to can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

The Pareto Frontier, ''P(Y)'', may be more formally described as follows. Consider a system with function

f: \mathbb{R}^n 
ightarrow \mathbb{R}^m

, where ''X'' is a compact set of feasible decisions in the metric space

\mathbb{R}^n

, and ''Y'' is the feasible set of criterion vectors in

\mathbb{R}^m

, such that

Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}

.

We assume that the preffered directions of criteria values are known. A point

y{\prime\prime} \in \mathbb{R}^m\;

is preffered to (strictly dominating) another point

y{\prime} \in \mathbb{R}^m\;

, written as

y{\prime\prime} dash y\prime

(note: wikipedia does not support the preceeding operator character, which is typically used). The Pareto Frontier is thus written as,

P(Y) = \{ y\prime \in Y: \; \{y{\prime\prime} \in Y:\; y{\prime\prime} dash y\prime, y{\prime\prime} 
eq y\prime \; \} = \empty \}

.

CRITICISMS

Many undesirable systems are Pareto efficient. For example, a dictatorship where the dictator gets every resource is Pareto efficient because any redistribution would decrease the wealth of the dictator.

Another problem with Pareto efficiency is that it only considers private property and private income but does not consider the Commons , Natural Environment , and effects of some Externalities .

SEE ALSO

Deadweight Loss

First Welfare Theorem

Kaldor-Hicks Efficiency

Multidisciplinary Design Optimization

Welfare Economics

Liberal Paradox

REFERENCES

Fudenberg, D. and Tirole, J. (1993) ''Game Theory'', MIT Press. (see Chapter 1, sect 2.4)

Osborne, M.J. and Rubenstein, A. (1994) ''A Course in Game Theory'', MIT Press (see p7)

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Pareto Optimum