Pareto efficiency

Pareto efficiency

Overview
Pareto efficiency, or Pareto optimality, is a concept in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

 with applications in engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 and social sciences
Social sciences
Social science is the field of study concerned with society. "Social science" is commonly used as an umbrella term to refer to a plurality of fields outside of the natural sciences usually exclusive of the administrative or managerial sciences...

. The term is named after Vilfredo Pareto
Vilfredo Pareto
Vilfredo Federico Damaso Pareto , born Wilfried Fritz Pareto, was an Italian engineer, sociologist, economist, political scientist and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices....

, an Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

 economist who used the concept in his studies of economic efficiency and income distribution
Income distribution
In economics, income distribution is how a nation’s total economy is distributed amongst its population.Income distribution has always been a central concern of economic theory and economic policy...

.

Given an initial allocation of goods among a set of individuals
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....

, a change to a different allocation that makes at least one individual better off
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

 without making any other individual worse off is called a Pareto improvement.
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Encyclopedia
Pareto efficiency, or Pareto optimality, is a concept in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

 with applications in engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 and social sciences
Social sciences
Social science is the field of study concerned with society. "Social science" is commonly used as an umbrella term to refer to a plurality of fields outside of the natural sciences usually exclusive of the administrative or managerial sciences...

. The term is named after Vilfredo Pareto
Vilfredo Pareto
Vilfredo Federico Damaso Pareto , born Wilfried Fritz Pareto, was an Italian engineer, sociologist, economist, political scientist and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices....

, an Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

 economist who used the concept in his studies of economic efficiency and income distribution
Income distribution
In economics, income distribution is how a nation’s total economy is distributed amongst its population.Income distribution has always been a central concern of economic theory and economic policy...

.

Given an initial allocation of goods among a set of individuals
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....

, a change to a different allocation that makes at least one individual better off
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

 without making any other individual worse off is called a Pareto improvement. An allocation is defined as "Pareto efficient" or "Pareto optimal" when no further Pareto improvements can be made.

Pareto efficiency is a minimal notion of efficiency and does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society.

Pareto efficiency in short


An economic system that is not Pareto efficient implies that a certain change in allocation of goods (for example) may result in some individuals being made "better off" with no individual being made worse off, and therefore can be made more Pareto efficient through a Pareto improvement. Here 'better off' is often interpreted as "put in a preferred position." 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
Economic system
An economic system is the combination of the various agencies, entities that provide the economic structure that defines the social community. These agencies are joined by lines of trade and exchange along which goods, money etc. are continuously flowing. An example of such a system for a closed...

s and public policies.

If economic allocation in any system is not Pareto efficient, there is potential for a Pareto improvement—an increase in Pareto efficiency: through reallocation, improvements to at least one participant's well-being can be made better 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 realized 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 compensations have substantial frictional costs. They can also lead to incentive distortions over time since most real-world policy changes occur with players who are not atomistic, rather who have considerable market power (or political power) over time and may use it in a game theoretic
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

 manner. Compensation attempts may therefore lead to substantial practical problems of misrepresentation and moral hazard
Moral hazard
In economic theory, moral hazard refers to a situation in which a party makes a decision about how much risk to take, while another party bears the costs if things go badly, and the party insulated from risk behaves differently from how it would if it were fully exposed to the risk.Moral hazard...

 and considerable inefficiency as players behave opportunistically and with guile.

In real-world practice, the compensation principle
Compensation principle
In welfare economics, the compensation principle refers to a decision rule used to select between pairs of alternative feasible social states. One of these states is the hypothetical point of departure...

 often appealed to is hypothetical. That is, for the alleged Pareto improvement (say from public regulation of the monopolist or removal of tariffs) some losers are not (fully) compensated. The change thus results in distribution effects in addition to any Pareto improvement that might have taken place. The theory of hypothetical compensation is part of Kaldor-Hicks efficiency
Kaldor-Hicks efficiency
Kaldor–Hicks efficiency, named for Nicholas Kaldor and John Hicks, also known as Kaldor–Hicks criterion, is a measure of economic efficiency that captures some of the intuitive appeal of Pareto efficiency, but has less stringent criteria and is hence applicable to more circumstances...

, also called Potential Pareto Criterion.

Under certain idealized conditions, it can be shown that a system of free market
Free market
A free market is a competitive market where prices are determined by supply and demand. However, the term is also commonly used for markets in which economic intervention and regulation by the state is limited to tax collection, and enforcement of private ownership and contracts...

s will lead to a Pareto efficient outcome. This is called the first welfare theorem. It was first demonstrated mathematically by economists Kenneth Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....

 and Gerard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...

. However, the result does not rigorously establish welfare results for real economies because of the restrictive assumptions necessary for the proof (markets exist for all possible goods, all markets are in full equilibrium, markets are perfectly competitive, transaction costs are negligible, there must be no externalities
Externality
In economics, an externality is a cost or benefit, not transmitted through prices, incurred by a party who did not agree to the action causing the cost or benefit...

, and market participants must have perfect information
Information asymmetry
In economics and contract theory, information asymmetry deals with the study of decisions in transactions where one party has more or better information than the other. This creates an imbalance of power in transactions which can sometimes cause the transactions to go awry, a kind of market failure...

). Moreover, it has since been demonstrated mathematically that, in the absence of perfect information or complete markets, outcomes will generically be Pareto inefficient (the Greenwald-Stiglitz theorem).

Pareto improvements and microeconomic theory


Note that microeconomic analysis does not assume additive utility nor does it assume any interpersonal utility tradeoffs. To engage in interpersonal utility tradeoffs leads to greater good problems faced by earlier utilitarians
Utilitarianism
Utilitarianism is an ethical theory holding that the proper course of action is the one that maximizes the overall "happiness", by whatever means necessary. It is thus a form of consequentialism, meaning that the moral worth of an action is determined only by its resulting outcome, and that one can...

. It also creates a question as to how weights are assigned and who assigns them, as well as questions regarding how to compare pleasure or pain across individuals.

Efficiency – in all of standard microeconomics – therefore refers to the absence of Pareto improvements.
It does not in any way opine on the fairness of the allocation (in the sense of distributive justice
Distributive justice
Distributive justice concerns what some consider to be socially just allocation of goods in a society. A society in which incidental inequalities in outcome do not arise would be considered a society guided by the principles of distributive justice...

 or equity
Equity (economics)
Equity is the concept or idea of fairness in economics, particularly as to taxation or welfare economics. More specifically it may refer to equal life chances regardless of identity, to provide all citizens with a basic minimum of income/goods/services or to increase funds and commitment for...

). An 'efficient' equilibrium could be one where one player has all the goods and other players have none (in an extreme example).

Weak and strong Pareto optimum


A "weak Pareto optimum" (WPO) nominally satisfies the same standard of not being Pareto-inferior to any other allocation, but for the purposes of weak Pareto optimization, an alternative allocation is considered to be a Pareto improvement only if the alternative allocation is strictly preferred by all individuals. In other words, when an allocation is WPO there are no possible alternative allocations whose realization would cause every individual to gain.

Weak Pareto-optimality is "weaker" than strong Pareto-optimality in the sense that the conditions for WPO status are "weaker" than those for SPO status: any allocation that can be considered an SPO will also qualify as a WPO, but a WPO allocation won't necessarily qualify as an SPO.

Under any form of Pareto-optimality, for an alternative allocation to be Pareto-superior to an allocation being tested—and, therefore, for the feasibility of an alternative allocation to serve as proof that the tested allocation is not an optimal one—the feasibility of the alternative allocation must show that the tested allocation fails to satisfy at least one of the requirements for SPO status. One may apply the same metaphor to describe the set of requirements for WPO status as being "weaker" than the set of requirements for SPO status. (Indeed, because the SPO set entirely encompasses the WPO set, with respect to any property the requirements for SPO status are of strength equal to or greater than the strength of the requirements for WPO status. Therefore, the requirements for WPO status are not merely weaker on balance or weaker according to the odds; rather, one may describe them more specifically and quite fittingly as "Pareto-weaker.")
  • Note that when one considers the requirements for an alternative allocation's superiority according to one definition against the requirements for its superiority according to the other, the comparison between the requirements of the respective definitions is the opposite of the comparison between the requirements for optimality: To demonstrate the WPO-inferiority of an allocation being tested, an alternative allocation must falsify at least one of the particular conditions in the WPO subset, rather than merely falsify at least one of either these conditions or the other SPO conditions. Therefore, the requirements for weak Pareto-superiority of an alternative allocation are harder to satisfy (in other words, "stronger") than are the requirements for strong Pareto-superiority of an alternative allocation.

  • It further follows that every SPO is a WPO (but not every WPO is an SPO): Whereas the WPO description applies to any allocation from which every feasible departure results in the NON-IMPROVEMENT of at least one individual, the SPO description applies to only those allocations that meet both the WPO requirement and the more specific ("stronger") requirement that at least one non-improving individual exhibit a specific type of non-improvement, namely doing worse.

  • The "strong" and "weak" descriptions of optimality continue to hold true when one construes the terms in the context set by the field of semantics: If one describes an allocation as being a WPO, one makes a "weaker" statement than one would make by describing it as an SPO: If the statements "Allocation X is a WPO" and "Allocation X is a SPO" are both true, then the former statement is less controversial than the latter in that to defend the latter, one must prove everything to defend the former "and then some." By the same token, however, the former statement is less informative or contentful in that it "says less" about the allocation; that is, the former statement contains, implies, and (when stated) asserts fewer constituent propositions about the allocation.

Formal representation


Formally, a (strong/weak) Pareto optimum is a maximal element for the partial order relation of Pareto improvement/strict Pareto improvement: it is an allocation such that no other allocation is "better" in the sense of the order relation.

Pareto frontier


Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. The Pareto frontier is particularly useful in engineering: by restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

The Pareto frontier is defined formally as follows.

Consider a design space with n real parameters, and for each design space point there are m different criteria by which to judge that point. Let be the function which assigns, to each design space point x, a criteria space point f(x). This represents the way of valuing the designs. Now, it may be that some designs are infeasible; so let X be a set of feasible designs in , which must be a compact set
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

. Then the set which represents the feasible criterion points is f(X), the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

 of the set X under the action of f. Call this image Y.

Now construct the Pareto frontier as a subset of Y, the feasible criterion points. It can be assumed that the preferable values of each criterion parameter are the lesser ones, thus minimizing each dimension of the criterion vector. Then compare criterion vectors as follows: One criterion vector y strictly dominates (or "is preferred to") a vector y* if each parameter of y is no greater than the corresponding parameter of y* and at least one parameter is strictly less: that is, for each i and for some i. This is written as to mean that y strictly dominates y*. Then the Pareto frontier is the set of points from Y that are not strictly dominated by another point in Y.

Formally, this defines a partial order on Y, namely the product order
Product order
In mathematics, given two ordered sets A and B, one can induce a partial ordering on the Cartesian product A × B. Giventwo pairs and in A × B, one sets ≤...

 on (more precisely, the induced order on Y as a subset of ), and the Pareto frontier is the set of maximal elements with respect to this order.

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science, being sometimes referred to as the maximum vector problem or the skyline query.

Relationship to marginal rate of substitution


At a Pareto efficient allocation (on the Pareto frontier), the marginal rate of substitution
Marginal rate of substitution
In economics, the marginal rate of substitution is the rate at which a consumer is ready to give up one good in exchange for another good while maintaining the same level of utility.-Marginal rate of substitution as the slope of indifference curve:...

 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 where is the vector of goods, both for all i. The supply constraint is written for . To optimize this problem, the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

 is used:


where and are multipliers.

Taking the partial derivative 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:



where ƒx is the marginal utility on ƒ' of x (the partial derivative of ƒ with respect to x).

See also


  • Admissible decision rule
    Admissible decision rule
    In statistical decision theory, an admissible decision rule is a rule for making a decision such that there isn't any other rule that is always "better" than it, in a specific sense defined below....

    , analog in decision theory
    Decision theory
    Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...

  • Bayesian efficiency
    Bayesian efficiency
    Bayesian efficiency addresses an appropriate economic definition of Pareto efficiency where there is incomplete information. Under Pareto efficiency, an allocation of a resource is Pareto efficient if there is no other allocation of that resource that makes no one worse off while making some agents...

  • Fundamental theorems of welfare economics
    Fundamental theorems of welfare economics
    There are two fundamental theorems of welfare economics. The first states that any competitive equilibrium or Walrasian equilibrium leads to a Pareto efficient allocation of resources. The second states the converse, that any efficient allocation can be sustainable by a competitive equilibrium...

  • Constrained Pareto efficiency
    Constrained Pareto efficiency
    The condition of Constrained Pareto optimality is a weaker version of the standard condition of Pareto Optimality employed in Economics which accounts for the fact that a potential planner may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient...

  • Deadweight loss
    Deadweight loss
    In economics, a deadweight loss is a loss of economic efficiency that can occur when equilibrium for a good or service is not Pareto optimal...

  • Efficiency (economics)
    Efficiency (economics)
    In economics, the term economic efficiency refers to the use of resources so as to maximize the production of goods and services. An economic system is said to be more efficient than another if it can provide more goods and services for society without using more resources...

  • Kaldor–Hicks efficiency
  • Maximal element
    Maximal element
    In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually...

    , concept in order theory
    Order theory
    Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

  • Multiobjective optimization
    Multiobjective optimization
    Multi-objective optimization , also known as multi-criteria or multi-attribute optimization, is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints....

  • Nash equilibrium
    Nash equilibrium
    In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

  • Social Choice and Individual Values
    Social Choice and Individual Values
    Kenneth Arrow's monograph Social Choice and Individual Values and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor...

     for the '(weak) Pareto principle'
  • Welfare economics
    Welfare economics
    Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...

  • Game Theory
    Game theory
    Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...