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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''是度量空间'''<font color="#32CD32">此处需插入公式</font>'''中可行决策的紧集，''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>''':
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)阐述的自由主义悖论表明，当人们对他人的行为有偏好时，帕累托最优的目标可能与个人自由的目标发生冲突。
阿马蒂亚·森(Amartya Sen)阐述的'''<font color="#ff8000">自由主义悖论(The liberal paradox)'''表明，当人们对他人的行为有偏好时，帕累托最优的目标可能与个人自由的目标发生冲突。