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Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, 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 market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

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; 市场参与者拥有完全的信息。

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; 市场参与者拥有完全的信息。

 

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

帕累托边界的例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

帕累托边界的例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|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.]]

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|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.]]

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，所有消费者的边际替代率是相同的。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，我们最大化拉格朗日函数:

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，所有消费者的边际替代率是相同的。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，我们最大化拉格朗日函数:

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托最优并不需要完全公平的财富分配。一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托最优的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。--[[用户:粲兰|袁一博]]（[[用户讨论:粲兰|讨论]]）   

帕累托最优并不需要完全公平的财富分配。一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托最优的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

 

  --[[用户:粲兰|袁一博]]（[[用户讨论:粲兰|讨论]]）   

 

 

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的一个最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的一个最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

* [[Welfare economics]]  福利经济

* [[Welfare economics]]  福利经济

 

==References  参考文献==

==References  参考文献==