# Pareto efficiency

(Redirected from Pareto optimal)

Pareto efficiency, or Pareto optimality, is a state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off. The term is named after Vilfredo Pareto (1848–1923), an 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.

Given an initial allocation of goods among a set of individuals, a change to a different allocation that makes at least one individual better off without making any other individual worse off is called a Pareto improvement. 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.[1][2] 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 is identified with the property that no other option can categorically outperform any of its members.

## Overview

A production-possibility frontier is an example of a Pareto Efficient Frontier. The connected line of red points represents Pareto optimal choices of production.

If economic allocation in any system is not Pareto efficient, there is potential for a Pareto improvement—an increase in Pareto efficiency: through reallocation, improvements can be made to at least one participant's well-being without reducing any other participant's well-being.

It is important to note, however, that a change from an inefficient allocation to an efficient one is not necessarily a Pareto improvement. Thus, in practice, ensuring that nobody is disadvantaged by a change aimed at achieving Pareto efficiency may require compensation of one or more parties. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive and efficient, the monopolist will be made worse off. However, the loss to the monopolist will be more than offset by the gain in efficiency, in the sense that the monopolist could hypothetically be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement.

In real-world practice, such compensations have unintended consequences. They can lead to incentive distortions over time as agents anticipate such compensations and change their actions accordingly. Under certain idealized conditions, it can be shown that a system of free markets will lead to a Pareto efficient outcome. This is called the first welfare theorem. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu. However, the result only holds under the restrictive assumptions necessary for the proof (markets exist for all possible goods so there are no externalities, all markets are in full equilibrium, markets are perfectly competitive, transaction costs are negligible, and market participants have perfect information). In the absence of perfect information or complete markets, outcomes will generically be Pareto inefficient, per the Greenwald-Stiglitz theorem.[3]

## Weak Pareto efficiency

A "weak Pareto optimum" (WPO) is an allocation for which there are no possible alternative allocations whose realization would cause every individual to gain. Thus an alternative allocation is considered to be a Pareto improvement only if the alternative allocation is strictly preferred by all individuals. When contrasted with weak Pareto efficiency, a standard Pareto optimum as described above may be referred to as a "strong Pareto optimum" (SPO).

Weak Pareto-optimality is "weaker" than strong Pareto-optimality in the sense that any SPO also qualifies as a WPO, but a WPO allocation is not necessarily an SPO.

## Use in engineering

Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto Frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence do lie on the frontier.

The notion of Pareto efficiency is also useful in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

### Formal representation

#### Pareto frontier

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, where X is a compact set of feasible decisions in the metric space $\mathbb{R}^n$, and Y is the feasible set of criterion vectors in $\mathbb{R}^m$, such that $Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}$.

We assume that the preferred directions of criteria values are known. A point $y^{\prime\prime} \in \mathbb{R}^m\;$ is preferred to (strictly dominates) another point $y^{\prime} \in \mathbb{R}^m\;$, written as $y^{\prime\prime} \succ y^{\prime}$. The Pareto frontier is thus written as:

$P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^\prime, y^{\prime\prime} \neq y^\prime \; \} = \empty \}.$

#### Relationship to marginal rate of substitution

An important fact about 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 $z_i=f^i(x^i)$ where $x^i=(x_1^i, x_2^i, \ldots, x_n^i)$ is the vector of goods, both for all i. The feasibility constraint is $\sum_{i=1}^m x_j^i = b_j$ for $j=1,\ldots,n$. To find the Pareto optimal allocation, we maximize the Lagrangian:

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

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

$\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,$
$\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,$

where $f_{x^i_j}$ denotes the partial derivative of $f$ with respect to $x_j^i$. Now, fix any $k\neq i$ and $j,s\in \{1,\ldots,n\}$. The above first-order condition imply that

$\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.$

Thus, in a Pareto optimal allocation, the marginal rate of substitution must be the same for all consumers.

#### Computation

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science, power engineering,[4] sometimes referred to as the maximum vector problem or the skyline query.[5][6]

## Criticisms

It would be invalid to treat Pareto efficiency as equivalent to social optimality, since the latter is a normative concept that is a matter of opinion but typically would take into account the degree of inequality of a distribution.

Pareto efficiency does not require an equitable distribution of wealth. An economy in which the wealthy hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distributions. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin of the pie is conceived as immaterial in these examples. In such cases, in which a "windfall" that none of the potential distributees actually produced is to be allocated (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation.

Amartya Sen has elaborated a mathematical basis for this criticism, pointing out that under relatively plausible starting conditions, systems of social choice will converge to Pareto efficient, but inequitable, distributions.[7]