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Let ''x_a'' be an allocation that maximizes the welfare over all allocations, i.e.:

Let ''x_a'' be an allocation that maximizes the welfare over all allocations, i.e.:

Let x_a be an allocation that maximizes the welfare over all allocations, i.e.:

假设是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation x_a is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of x_a.

It is easy to show that the allocation x_a is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of x_a.

很容易证明分配 x 子 a / sub 是 pareto 有效的: 因为所有的权重都是正的，任何 pareto 改进都会增加和，这与 x 子 a / sub 的定义相矛盾。

很容易证明分配是帕累托有效的: 因为所有的权重都是正的，任何帕累托改进都会增加总和，这与 x 子 a / sub 的定义相矛盾。