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Encyclopedia > Pareto efficiency

Pareto efficiency, or Pareto optimality, is an important notion in neoclassical economics with broad applications in game theory, engineering and the social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution. Neoclassical economics refers to a general approach (a metatheory) to economics based on supply and demand which depends on individuals (or any economic agent) operating rationally, each seeking to maximize their individual utility or profit by making choices based on available information. ... Game theory is often described as a branch of applied mathematics and economics that studies situations where multiple players make decisions in an attempt to maximize their returns. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... The social sciences are a group of academic disciplines that study human aspects of the world. ... Vilfredo Federico Damaso Pareto [vilfre:do pare:to] (July 15, 1848, Paris â€“ August 19, 1923, Geneva) was a French-Italian sociologist, economist and philosopher. ... Economic efficiency is a general term for the value assigned to a situation by some measure designed to capture the amount of waste or friction or other undesirable economic features present. ... This graphic shows the distribution of gross annual household income. ...

Given a set of alternative allocations and a set of individuals, a movement from one allocation to another that can make at least one individual better off, without making any other individual worse off, is called a Pareto improvement or Pareto optimization. An allocation of resources is Pareto efficient or Pareto optimal when no further Pareto improvements can be made. In economics, factors of production are resources used in the production of goods and services. ...

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 (not opposed) by everyone else. A (weakly) Pareto optimal (WPO) allocation is one where there is no feasible reallocation that would be strictly preferred by all agents.

Example of Pareto frontier, given that lower values are preferred to higher values. Point C is not on the Pareto Frontier because it is dominated by both point A and point B. Points A and B are non-inferior.

For a given system, the Pareto frontier or Pareto set 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 can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters. Image File history File links Front_pareto. ... Image File history File links Front_pareto. ...

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function $f: mathbb{R}^n rightarrow mathbb{R}^m$, 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 . In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

We assume that the preferred directions of criteria values are known. A point is preferred to (strictly dominating) another point $y^{prime} in mathbb{R}^m;$, written as $y^{primeprime} succ y^{prime}$. The Pareto frontier is thus written as:

$P(Y) = { y^{prime} in Y: ; {y^{primeprime} in Y:; y^{primeprime} succ y^{prime}, y^{primeprime} neq y^{prime} ; } = empty }$.

Pareto efficiency in economics

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 systems and public policies. This article does not cite any references or sources. ...

If economic allocation in any system (in the real world or in a model) is not Pareto efficient, there is theoretical potential for a Pareto improvement - an increase in Pareto efficiency: through reallocation, improvements to at least one participant's well-being can be made without reducing any other participant's well-being.

In the real world ensuring that nobody is disadvantaged by a change aimed at improving economic efficiency may require compensation of one or more parties. For instance, if a change in economic policy dictates that a legally protected monopoly ceases to exist and that market subsequently becomes competitive and more efficient, the monopolist will be made worse off. However, the loss to the monopolist will be more than offset by the gain in efficiency. This means the monopolist can be compensated for its loss while still leaving an efficiency gain to be realised by others in the economy. Thus, the requirement of nobody being made worse off for a gain to others is met.

In real-world practice, the compensation of the monopolist (or other loser) is hardly ever made. It is left hypothetical. The change is thus not Pareto efficient. The theory of hypothetical compensation is part of the Kaldor-Hicks concept of efficiency. Kaldor-Hicks efficiency is a type of economic efficiency that occurs only if the economic value of social resources is maximized. ...

Under certain idealised conditions, it can be shown that a system of free markets will lead to a Pareto efficient outcome. This was first demonstrated mathematically by economists Kenneth Arrow and Gerard Debreu. This is called the first welfare theorem. However, the result likely does not 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, transaction costs are negligible, and there must be no externalities). A free market is an idealized market, where all economic decisions and actions by individuals regarding transfer of money, goods, and services are voluntary, and are therefore devoid of coercion and theft (some definitions of coercion are inclusive of theft). Colloquially and loosely, a free market economy is an economy... Kenneth Joseph Arrow (born August 23, 1921) is an American economist, joint winner of the Nobel Prize in Economics with John Hicks in 1972, and the youngest person ever to receive this award, at 51. ... Gerard Debreu was a naturalized US citizen from France Gerard Debreu (July 4, 1921 â€“ December 31, 2004) was a French economist and mathematician (In July 1975, he became a naturalized citizen of the United States). ... In welfare economics, the First Welfare Theorem is that a system of free markets will lead to a Pareto efficient equilibrium. ... In economics, an externality is a cost or benefit resulting from an economic transaction that is borne or received by parties not directly involved in the transaction. ...

A key drawback of Pareto optimality is its localisation and partial ordering. In an economic system with millions of variables there can be very many local optimum points. The Pareto improvement criterion does not define any global optimum. Given a reasonable criterion which compares all points, many Pareto-optimal solutions may be far inferior to the global best solution. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... Polynomial of degree 4, on the right one finds a local optimum, on the left is the global optimum. ...

Relationship to marginal rate of substitution

An important fact about the Pareto frontier in economics is that at a Pareto efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as zi = fi(xi) where $x^i=(x_1^i, x_2^i, ldots, x_n^i)$ is the vector of goods, both for all i. The supply constraint is written $sum_{i=1}^m x_j^i = b_j^0$ for $j=1,ldots,n$. To optimize this problem, the Lagrangian is used: A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

$L(x, lambda, Gamma)=f^1(x^1)+sum_{i=2}^m lambda_i(z_i^0 - f^i(x^i))+sum_{j=1}^n Gamma_j(b_j^0-sum_{i=1}^m x_j^i)$ where λ and Γ are multipliers.

Taking the partial derivatve of the Lagrangian with respect to one good, i, and then taking the partial derivative of the Lagrangian with respect to another good, j, gives the following system of equations:

$frac{partial L}{partial x_j^i} = f_{x^1}^1-Gamma_j^0=0$ for j=1,...,n. $frac{partial L}{partial x_j^i} = -lambda_i^0 f_{x^1}^1-Gamma_j^0=0$ for i = 2,...,m and j=1,...,m, where fx is the marginal utility on f' of x (the partial derivative of f with respect to x).

Rearranging these to eliminate the multipliers gives the wanted result:

$frac{f_{x_j^i}^i}{f_{x_s^i}^i}=frac{f_{x_j^k}^k}{f_{x_s^k}^k}$ for i,k=1,...,m and j,s=1,...,n.

Criticisms

Pareto efficiency does not require an equitable distribution of wealth. An economy in which the wealthy hold the vast majority of resources can be Pareto efficient. This should, of course, not be understood as criticism of Pareto efficiency itself, but rather of the idea that Pareto efficiency is desirable or even only Pareto efficiency is desirable. This point has been strongly made by Michele Piccione and Ariel Rubinstein in their paper, Equilibrium in the Jungle, which shows that outcomes in a world where strong agents may steal goods from weaker agents are efficient. Ariel Rubinstein (born April 13, 1951) is an economist who works in game theory. ...

Amartya Sen has elaborated the mathematical reasons for this criticism, pointing out that under relatively plausible starting conditions, systems of social choice will converge on Pareto efficient, but inequitable, distributions. A simple example is dividing a pie into pieces to distribute among three people. The most equitable distribution is each person getting one third. However the solution of two people getting half a pie and the third person getting none is also Pareto optimal despite not being equitable, because the only way for the person with no piece to get a piece is for one or both of the other two to get less, which is not a Pareto improvement. A Pareto inefficient distribution of the pie might be each person getting one-quarter of the pie with the remainder discarded. Of course, this example completely ignores the origin of the pie, so it breaks down to the criticism that Pareto efficiency does not really help in determining the optimal allocation of windfalls that nobody involved actually produced, such as land, inherited wealth, broadcast spectrum, the environment, or a pie miraculously falling from the sky. This article does not cite any references or sources. ... Social choice theory studies how individual preferences are aggregated to form a collective preference. ...

A more generalized form of this criticism lies in the definition of Pareto efficiency itself. By requiring that no participants be worse off, Pareto efficiency protects the status quo and therefore any inequity or other problems currently existing. Status Quo are an English rock band whose music is characterised by a strong boogie line. ...

The compensation principle in welfare economics refers to a decision rule used to select between pairs of alternative feasible social states. ... In economics, a deadweight loss (also known as excess burden) is a permanent loss of well being to society that can occur when equilibrium for a good or service is not Pareto optimal, (that at least one individual could be made better off without others being made worse off). ... Economic efficiency is a general term for the value assigned to a situation by some measure designed to capture the amount of waste or friction or other undesirable economic features present. ... The liberal paradox is a logical paradox advanced by Amartya Sen, building on the work of Kenneth Arrow and his general possibility theorem, which showed that within a system of menu-independent social choice, it is impossible to have both a commitment to Minimal Liberty, which was defined as the... There exists two fundamental theorems of welfare economics. ... Kaldor-Hicks efficiency is a type of economic efficiency that occurs only if the economic value of social resources is maximized. ... Multidisciplinary design optimization (MDO) is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. ... Multi-objective optimization (or programming),[1][2] also known as multi-criteria or multi-attribute optimization, is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints. ... There are very few or no other articles that link to this one. ... Kenneth Arrows monograph Social Choice and Individual Values (1951, 2nd ed. ... Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution associated with it. ... Abram Bergson, born Abram Burk (April 21, 1914, New York City - April 23, 2003), was an American economist. ...

References

• Fudenberg, D. and Tirole, J. (1983). Game Theory. MIT Press, Chapter 1, Section 2.4.
• Osborne, M.J. and Rubenstein, A. (1994). A Course in Game Theory. MIT Press, p. 7. ISBN 0-262-65040-1.

Jean Tirole (born 9 August 1953) is a notable contemporary french economist, author of many works in economics, scientific director of the Industrial Economics Institute in Toulouse. ... Ariel Rubenstein (born April 13, 1951) is an economist who works in game theory. ...

Results from FactBites:

 The Pareto Principle - The 80/20 rule (512 words) In 1906, Italian economist and sociologist, Vilfredo Pareto (sometimes misspelled Wilfredo, Alfredo, or Vilfred) created a mathematical formula to describe the uneven income distribution in Switzerland at that time, observing that eighty percent of the wealth was held by a mere twenty percent of the families. Because Pareto's initial discovery involved a distribution of 80% of wealth to 20% of families and it's inverse, the Pareto Principle is often called "The 80/20 rule". Pareto's Principle, or the 80/20 Rule, should serve as a continual reminder to focus eighty percent of your effort on the twenty percent of your tasks that matter the most.
 Reference.com/Encyclopedia/Pareto efficiency (1159 words) Pareto efficiency, or Pareto optimality, is an important notion in neoclassical economics with broad applications in game theory, engineering and the social sciences. 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 systems and public policies. An important fact about the Pareto frontier in economics is that at a Pareto efficient allocation, the marginal rate of substitution is the same for all consumers.
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