Pareto efficiency

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 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. 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, in which case we are assumed to have reached Pareto optimality. 'Pareto efficiency' is considered as a minimal notion of efficiency that 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. It is simply a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization). Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. 'Pareto optimality' is a formally defined concept used to describe when an allocation is optimal. 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.' The formal presentation of the concept in an economy is as follows: Consider an economy with n {displaystyle n} agents and k {displaystyle k} goods. Then an allocation { x 1 , . . . , x n } {displaystyle {x_{1},...,x_{n}}} , where x i ∈ R k {displaystyle x_{i}in mathbb {R} ^{k}} for all i, is Pareto optimal if there is no other feasible allocation { x 1 ′ , . . . , x n ′ } {displaystyle {x_{1}',...,x_{n}'}} such that, for utility function u i {displaystyle u_{i}} for each agent i {displaystyle i} , u i ( x i ′ ) ≥ u i ( x i ) {displaystyle u_{i}(x_{i}')geq u_{i}(x_{i})} for all i ∈ { 1 , . . . , n } {displaystyle iin {1,...,n}} with u i ( x i ′ ) > u i ( x i ) {displaystyle u_{i}(x_{i}')>u_{i}(x_{i})} for some i {displaystyle i} . 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.