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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 the marginal rate of substitution'''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'''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'''是商品的矢量。可行性约束为'''<font color="#32CD32">此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数 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 the marginal rate of substitution'''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<math>z_i=f^i(x^i)</math>来推导出。在这个效用方程中，对所有的''i''，<math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math>是商品的矢量。可行性约束为<math>\sum_{i=1}^m x_j^i = b_j</math>。为了找到帕累托最优分配，我们最大化'''拉格朗日函数 Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''<font color="#32CD32">此处需插入公式'''和'''<font color="#32CD32">此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

其中<math>(\lambda_k)_k</math>和<math>(\mu_j)_j</math>是乘子的向量。取关于商品的拉格朗日函数的偏导数<math>x_j^k</math> （<math>j=1,\ldots,n</math> ，<math>k=1,\ldots, m</math>），并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation  计算===

=== Computation  计算===