Pareto efficient

Pareto efficiency or Pareto optimality is a state of allocation of resources from which it is impossible to reallocate so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The concept has applications in academic fields such as economics, engineering, and the life sciences.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically.

A Pareto improvement is a change to a different allocation that makes at least one individual or preference criterion better off without making any other individual or preference criterion worse off, given a certain initial allocation of goods among a set of individuals. An allocation is defined as "Pareto efficient" or "Pareto optimal" when no further Pareto improvements can be made.

Pareto efficiency is a minimal notion of efficiency and does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. The notion of Pareto efficiency can also be applied to the selection of alternatives in engineering and similar fields. Each option is first assessed under multiple criteria and then a subset of options are identified with the property that no other option can categorically outperform any of its members.

Pareto optimality is a formally defined concept used to determine when an allocation is optimal. Simply put, an allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a Pareto improvement. When no further Pareto improvements are possible, the allocation is a Pareto optimum.

A formal definition for an economy is as follows: Consider an economy with ${\displaystyle i}$ agents and ${\displaystyle j}$ goods. Then an allocation ${\displaystyle \{x_{1},...,x_{i}\}}$, where ${\displaystyle x_{n}\in \mathbb {R} ^{j}}$, is Pareto optimal if there is no other feasible allocation ${\displaystyle \{x_{1}',...,x_{i}'\}}$ such that, for utility function ${\displaystyle u_{i}}$ for each agent ${\displaystyle i}$, ${\displaystyle u_{n}(x_{n}')\geq u_{n}(x_{n})}$ for all ${\displaystyle n\in \{1,...,i\}}$ with ${\displaystyle u_{n}(x_{n}')>u_{n}(x_{n})}$ for some ${\displaystyle n}$. 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 good consumed is no greater than the initial endowment plus the amount produced.

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