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The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., 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', ..., x_n'\} </math> such that, 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, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. 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 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.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., 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', ..., x_n'\} </math> such that, 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, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. 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 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.

这个概念在一个经济体中的正式表现如下: 考虑一个有数学代理和数学 k / 数学商品的经济体。然后一个分配数学，... ，xn / math，其中 mathbb ^ k / math 对所有 i，是 Pareto 最优的，如果没有其他可行的分配数学 x1’ ，... ，xn’ / math 这样的数学，对于效用函数数学 u i / math 对于每个 agent 数学 i / math, math u i (xi’) geq u i (xi) / math for all math i  in 1，... ，n  n / math with math u i (xi’) u i (xi) / math for some math i / math。在这个简单的经济体系中，「可行性」是指每种商品的分配总额不超过该经济体系中该商品的分配总额。在一个有生产的更为复杂的经济体中，分配将包括消费载体和生产载体，而且可行性要求每种消费品的总量不大于最初的禀赋加上生产量。

这个概念在一个经济体系中的正式表现如下: 考虑一个有数学主体和数学 k / 数学商品的经济体系。如果没有其他可行的分配数学 x1’ ，... ，xn’ / math 使得数学，对于效用函数数学 u i / math 对于每个 agent 数学 i / math, math u i (xi’) geq u i (xi) / math for all math i  in 1，... ，n  n / math with math u i (xi’) u i (xi) / math for some math i / math，一个分配数学，... ，xn / math，其中 mathbb ^ k / math 对所有 i，是 Pareto 最优的。在这个简单的经济体系中，「可行性」是指每种商品的分配总额不超过该经济体系中所有商品的总额。在一个有生产能力的更为复杂的经济体中，一个分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。