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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.
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.
除了分配效率的背景之外，帕累托最优的概念也出现在'''<font color="#ff8000">生产效率efficiency in production</font>'''对比于'''<font color="#ff8000">x-低效率x-inefficiency</font>'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>
除了分配效率的背景之外，帕累托最优的概念也出现在'''<font color="#ff8000">生产效率 efficiency in production</font>'''对比于'''<font color="#ff8000">x-低效率 x-inefficiency</font>'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>
Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly 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).
Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly 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).
除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''<font color="#ff8000">多目标优化multi-objective optimization</font>'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。
除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''<font color="#ff8000">多目标优化 multi-objective optimization</font>'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。
Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.
Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.
在'''<font color="#ff8000">福利经济学第一定理the first welfare theorem</font>'''的理想条件下，一个'''<font color="#ff8000">自由市场free market</font>'''系统，也称为“'''<font color="#ff8000">竞争均衡competitive equilibrium</font>'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。
在'''<font color="#ff8000">福利经济学第一定理 the first welfare theorem</font>'''的理想条件下，一个'''<font color="#ff8000">自由市场 free market</font>'''系统，也称为“'''<font color="#ff8000">竞争均衡 competitive equilibrium</font>'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。
However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.
However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.
然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''<font color="#ff8000">完全信息perfect information</font>'''。
然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''<font color="#ff8000">完全信息 perfect information</font>'''。
In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.
In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.
根据'''<font color="#ff8000">格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem</font>'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<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=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>
根据'''<font color="#ff8000">格林沃德-斯蒂格利茨定理 the Greenwald-Stiglitz theorem</font>'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<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=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</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.
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.
'''<font color="#ff8000">福利经济学第二定理The second welfare theorem</font>'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。
'''<font color="#ff8000">福利经济学第二定理 The second welfare theorem</font>'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。
Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.
Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.
'''<font color="#ff8000">部分帕累托最优Fractional Pareto optimality </font> '''是在物品公平分配的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''<font color="#ff8000">部分帕累托最优 fractionally Pareto-optimal(fPO)</font> '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>
'''<font color="#ff8000">部分帕累托最优 Fractional Pareto optimality </font> '''是在物品公平分配的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''<font color="#ff8000">部分帕累托最优 fractionally Pareto-optimal(fPO)</font> '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>
图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]
图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]
一个'''<font color="#ff8000">生产可能性边界production-possibility frontier</font>'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。
一个'''<font color="#ff8000">生产可能性边界 production-possibility frontier</font>'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。
The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.
The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.
帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''<font color="#32CD32">此处需插入公式</font>'''的系统，其中''X''是'''<font color="#ff8000">度量空间metric space</font>''' '''<font color="#32CD32">此处需插入公式</font>'''中可行决策的'''<font color="#ff8000">紧集compact set</font>'''，''Y''是'''<font color="#32CD32">此处需插入公式</font>'''中标准向量的可行集，使得'''<font color="#32CD32">此处需插入公式</font>'''。
帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''<font color="#32CD32">此处需插入公式</font>'''的系统，其中''X''是'''<font color="#ff8000">度量空间 metric space</font>''' '''<font color="#32CD32">此处需插入公式</font>'''中可行决策的'''<font color="#ff8000">紧集 compact set</font>'''，''Y''是'''<font color="#32CD32">此处需插入公式</font>'''中标准向量的可行集，使得'''<font color="#32CD32">此处需插入公式</font>'''。
A significant aspect of 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 <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:
A significant aspect of 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 <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:
经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''<font color="#ff8000">边际替代率the marginal rate of substitution</font>'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''<font color="#32CD32">此处需插入公式</font>'''来推导出。在这个效用方程中，对所有的''i''，'''<font color="#32CD32">此处需插入公式</font>'''是商品的矢量。可行性约束为'''<font color="#32CD32">此处需插入公式</font>'''。为了找到帕累托最优分配，我们最大化'''<font color="#ff8000">拉格朗日函数Lagrangian</font>''':
经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''<font color="#ff8000">边际替代率 the marginal rate of substitution</font>'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''<font color="#32CD32">此处需插入公式</font>'''来推导出。在这个效用方程中，对所有的''i''，'''<font color="#32CD32">此处需插入公式</font>'''是商品的矢量。可行性约束为'''<font color="#32CD32">此处需插入公式</font>'''。为了找到帕累托最优分配，我们最大化'''<font color="#ff8000">拉格朗日函数 Lagrangian</font>''':
The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.
The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.
阿马蒂亚·森 Amartya Sen阐述的'''<font color="#ff8000">自由主义悖论The liberal paradox</font>'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>
阿马蒂亚·森 Amartya Sen阐述的'''<font color="#ff8000">自由主义悖论 The liberal paradox</font>'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>