Pareto efficiency

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Pareto efficiency

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From Wikipedia, the free encyclopedia

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In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves everyone in a society better-off (or at least as well-off as they were before). A situation is called Pareto efficient or Pareto optimal if all possible Pareto improvements have already been made; in other words, there are no longer any ways left to make one person better-off, without making some other person worse-off.[1]

In social choice theory, the same concept is sometimes called the unanimity principle, which says that if everyone in a society (non-strictly) prefers A to B, society as a whole also non-strictly prefers A to B.

The Pareto front consists of all Pareto-efficient situations.[2]

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto-efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.[3]

Besides economics, the notion of Pareto efficiency has also been applied to selecting alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

History[edit]

The concept is named after Vilfredo Pareto (1848–1923), an Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution.

Pareto originally used the word "optimal" for the concept, but this is somewhat of a misnomer: Pareto's concept more closely aligns with an idea of "efficiency", because it does not identify a single "best" (optimal) outcome. Instead, it only identifies a set of outcomes that might be considered optimal, by at least one person.[4]

Overview[edit]

Formally, a state is Pareto-optimal if there is no alternative state where at least one participant's well-being is higher, and nobody else's well-being is lower. If there is a state change that satisfies this condition, the new state is called a "Pareto improvement". When no Pareto improvements are possible, the state is a "Pareto optimum".

In other words, Pareto efficiency is when it is impossible to make one party better off without making another party worse off.[5] This state indicates that resources can no longer be allocated in a way that makes one party better off without harming other parties. In a state of Pareto Efficiency, resources are allocated in the most efficient way possible.[5]

Pareto efficiency is mathematically represented when there is no other strategy profile s' such that ui (s') ≥ ui (s) for every player i and uj (s') > uj (s) for some player j. In this equation s represents the strategy profile, u represents the utility or benefit, and j represents the player.[6]

Efficiency is an important criterion for judging behavior in a game. In a notable and often analyzed game known as Prisoner's Dilemma, depicted below as a normal-form game, this concept of efficiency can be observed, in that the strategy profile (Cooperate, Cooperate) is more efficient than (Defect, Defect).[6]

The Prisoner's DilemmaPlayer 2
Player 1CooperateDefectCooperateDefect

-1, -1	-5, 0
0, -5	-2, -2

Using the definition above, let s = (-2, -2) (Both Defect) and s' = (-1, -1) (Both Cooperate). Then ui(s') > ui(s) for all i. Thus Both Cooperate is a Pareto improvement over Both Defect, which means that Both Defect is not Pareto-efficient. Furthermore, neither of the remaining strategy profiles, (0, -5) or (-5, 0), is a Pareto improvement over Both Cooperate, since -5 < -1. Thus Both Cooperate is Pareto-efficient.

In zero-sum games, every outcome is Pareto-efficient.

A special case of a state is an allocation of resources. The formal presentation of the concept in an economy is the following: Consider an economy with n agents and k goods. Then an allocation {x1,…,xn}, where xi∈Rk for all i, is Pareto-optimal if there is no other feasible allocation {x1′,…,xn′} where, for utility function ui for each agent i, ui(xi′)≥ui(xi) for all i∈{1,…,n} with ui(xi′)>ui(xi) for some i.[7] Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

Under the assumptions of the first welfare theorem, a competitive market leads to a Pareto-efficient outcome. This result was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.[8] However, the result only holds under the assumptions of the theorem: markets exist for all possible goods, there are no externalities, markets are perfectly competitive, and market participants have perfect information.

In the absence of perfect information or complete markets, outcomes will generally be Pareto-inefficient, per the Greenwald–Stiglitz theorem.[9]

The second welfare theorem is essentially the reverse of the first welfare theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.[7]

Pareto efficiency and market failure[edit]

An ineffective distribution of resources in a free market is known as market failure. Given that there is room for improvement, market failure implies Pareto inefficiency.

For instance, excessive use of negative commodities (such as drugs and cigarettes) results in expenses to non-smokers as well as early mortality for smokers. Cigarette taxes may help individuals stop smoking while also raising money to address ailments brought on by smoking.

Pareto efficiency and equity[edit]

A Pareto improvement may be seen, but this does not always imply that the result is desirable or equitable. After a Pareto improvement, inequality could still exist. However, it does imply that any change will violate the "do no harm" principle, because at least one person will be worse off.

A society may be Pareto efficient but have significant levels of inequality. The most equitable course of action would be to split the pie into three equal portions if there were three persons and a pie. The third person does not lose out (even if he does not partake in the pie), hence splitting it in half and giving it to two individuals would be considered Pareto efficient.

On a frontier of production possibilities, Pareto efficiency will happen. It is impossible to raise the output of products without decreasing the output of services when an economy is functioning on a basic production potential frontier, such as at point A, B, or C.

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https://en.wikipedia.org/wiki/Pareto_efficiency