Editing Pareto efficiency (section)

== Overview ==

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.<ref name=":1">{{Cite web |title=Pareto Efficiency |url=https://corporatefinanceinstitute.com/resources/economics/pareto-efficiency/ |access-date=2022-12-10 |website=Corporate Finance Institute |language=en-US}}</ref> 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.<ref name=":1" />

Pareto efficiency is mathematically represented when there is no other strategy profile ''s''' such that ''u<sub>i</sub> (s') ≥ u<sub>i</sub> (s)'' for every player ''i'' and ''u<sub>j</sub> (s') > u<sub>j</sub> (s)'' for some player ''j''. In this equation ''s'' represents the strategy profile, ''u'' represents the utility or benefit, and ''j'' represents the player.<ref name=":2">{{Cite book |last=Watson |first=Joel |title=Strategy: An Introduction to Game Theory |publisher=W. W. Norton and Company |year=2013 |edition=3rd}}</ref>

Efficiency is an important criterion for judging behavior in a game. 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 <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, \dots, x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto-optimal'' if there is no other feasible allocation <math> \{x_1', \dots, x_n'\} </math> where, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, \dots, n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}.</ref> 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 [[Vector space|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]].<ref>{{cite journal |last1=Gerard |first1=Debreu |title=Valuation Equilibrium and Pareto Optimum |year=1959 |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=40 |issue=7 |pages=588–592 |doi=10.1073/pnas.40.7.588 |jstor=89325 |pmid=16589528 |pmc=528000 |doi-access=free}}</ref> However, the result only holds under the assumptions of the theorem: markets exist for all possible goods, there are no [[externality|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 [[Joseph Stiglitz#Information asymmetry|Greenwald–Stiglitz theorem]].<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–264 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114|doi-access=free }}</ref>

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.<ref name="AndreuMas95"/>
===Pareto efficiency and market failure===

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 tax]]es may help individuals stop smoking while also raising money to address ailments brought on by smoking.

===Pareto efficiency and equity===
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.