Pareto efficiency

Pareto efficiency, or Pareto optimality, is an important notion in neoclassical economics with broad applications in game theory, engineering and the social sciences. Given a set of alternative allocations and a set of individuals, a movement from one allocation to another that can make at least one individual better off, without making any other individual worse off, is called a Pareto improvement or Pareto optimization. An allocation of resources is Pareto efficient or Pareto optimal when no further Pareto improvements can be made.

A strongly Pareto optimal (SPO) allocation (X) is one for which there cannot be any other feasible allocation (say X') such that the allocation (X') is strictly preferred by at least one person, and weakly preferred (not opposed) by everyone else. A (weakly) Pareto optimal (WPO) allocation is one where there is no feasible reallocation that would be strictly preferred by all agents.

The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.

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 dominating) another point $$y^{\prime} \in \mathbb{R}^m\;$$, written as $$y^{\prime\prime} \vdash y^{\prime}$$. The Pareto Frontier is thus written as:

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

Pareto efficiency in economics
If an economic system is Pareto efficient, then it is the case that no individual can be made better off without another being made worse off. It is commonly accepted that outcomes that are not Pareto efficient are to be avoided, and therefore Pareto efficiency is an important criterion for evaluating economic systems and political policies.

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

In the real world ensuring that nobody is disadvantaged by a change aimed at improving economic efficiency may require compensation of one or more parties. For instance, if a change in economic policy dictates that a legally protected monopoly ceases to exist and that market subsequently becomes competitive and more efficient, the monopolist will be made worse off. However, the loss to the monopolist will be more than offset by the gain in efficiency. This means the monopolist can be compensated for its loss whilst still leaving an efficiency gain to be realised by others in the economy. Thus, the requirement of nobody being made worse off for a gain to others is met.

In real-world practice, the compensation of the monopolist (or other loser) is hardly ever made. It is left hypothetical. The change is thus not Pareto efficient. The theory of hypothetical compensation is part of the Kaldor-Hicks concept of efficiency.

Under certain idealised conditions, it can be shown that a system of free markets will lead to a Pareto efficient outcome. This was first demonstrated mathematically by economists Kenneth Arrow and Gerard Debreu. This is called the first welfare theorem. However, the result likely does not reflect the workings of real economies because of the restrictive assumptions necessary for the proof (markets exist for all possible goods, markets are perfectly competitive, and transaction costs are negligible). Among other things, there must be no externalities.

A key drawback of Pareto optimality is its localisation and partial ordering. In an economic system with millions of variables there can be very many local optimum points. The Pareto improvement criterion does not define any global optimum. Given a reasonable criterion which compares all points, many Pareto-optimal solutions may be far inferior to the global best solution.

Criticisms
Pareto efficiency does not require a "just" or equitable distribution of wealth. An economy in which the wealthy hold the vast majority of resources can be Pareto efficient. This should, of course, not be understood as criticism of Pareto efficiency itself, but rather of the idea that Pareto efficiency is desirable or even only Pareto efficiency is desirable.

Amartya Sen has elaborated the mathematical reasons for this criticism, pointing out that under relatively plausible starting conditions, systems of social choice will converge on Pareto efficient, but inequitable, distributions. A simple example is dividing a pie into pieces to distribute among three people. The most equitable distribution is each person getting one third. However the solution of two people getting half a pie and the third person getting none is also Pareto optimal despite not being equitable, because the only way for the person with no piece to get a piece is for one or both of the other two to get less, which is not a Pareto improvement. A Pareto inefficient distribution of the pie might be each person getting one-quarter of the pie with the remainder discarded. Of course, this example completely ignores the origin of the pie, so it breaks down to the criticism that Pareto efficiency does not really help in determining the optimal allocation of windfalls that nobody involved actually produced, such as land, inherited wealth, broadcast spectrum, the environment, or a pie miraculously falling from the sky.

A more generalized form of this criticism lies in the definition of Pareto efficiency itself. By requiring that no participants be worse off, Pareto efficiency protects the status quo and therefore any inequity or other problems currently existing.