# Pareto efficiency

Pareto efficiency, or Pareto optimality, is an important concept in economics with broad applications in game theory, engineering and the social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.

Given a set of alternative allocations of, say, goods or income for a set of individuals, a change 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. An allocation is Pareto efficient or Pareto optimal when no further Pareto improvements can be made. This is often called a strong Pareto optimum (SPO).

A weak Pareto optimum (WPO) satisfies a less stringent requirement, in which a new allocation is only considered to be a Pareto improvement if it is strictly preferred by all individuals (i.e., all must gain with the new allocation). In other words, when an allocation is WPO there are no possible alternative allocations where every individual would gain. An SPO is a WPO, because at an SPO, we can rule out alternative allocations where at least one individual gains and no individual loses out, and these cases where "at least one individual gains" include cases like "all individuals gain", the latter being the cases considered for a weak optimum. Clearly this first condition for the SPO is more restrictive than for a WPO, since at the latter, other allocations where one or more (but not all) individuals would gain (and none lose) would still be possible.

Formally, a (strong/weak) Pareto optimum is a maximal element for the partial order relation of Pareto improvement/strict Pareto improvement: it is an allocation such that no other allocation is "better" in the sense of the order relation.

A common criticism of a state of Pareto efficiency is that it does not necessarily result in a socially desirable distribution of resources, as it makes no statement about equality or the overall well-being of a society; notably, allocating all resources to one person and none to anyone else is Pareto efficient.[1][2]

## Pareto efficiency in economics

An economic system that is Pareto inefficient implies that a certain change in allocation of goods (for example) may result in some individuals being made "better off" with no individual being made worse off, and therefore can be made more Pareto efficient through a Pareto improvement. Here 'better off' is often interpreted as "put in a preferred position." 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 public 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 while still leaving an efficiency gain to be realized 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 principle often appealed to is hypothetical. That is, for the alleged Pareto improvement (say from public regulation of the monopolist or removal of tariffs) some losers are not (fully) compensated. The change thus results in distribution effects in addition to any Pareto improvement that might have taken place. The theory of hypothetical compensation is part of Kaldor-Hicks efficiency, also called Potential Pareto Criterion. (Ng, 1983).

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 Gerard Debreu. However, the result does not rigorously establish welfare results for real economies because of the restrictive assumptions necessary for the proof (markets exist for all possible goods, all markets are in full equilibrium, markets are perfectly competitive, transaction costs are negligible, there must be no externalities, and market participants must have perfect information). Moreover, it has since been demonstrated mathematically that, in the absence of perfect competition or complete markets, outcomes will generically be Pareto inefficient (the Greenwald-Stiglitz Theorem).[3]

## Formal representation

### Pareto frontier

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.

Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set is the set of choices that are Pareto efficient. The Pareto frontier is particularly useful in engineering: 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.

The Pareto frontier is defined formally as follows.

Consider a design space with n real parameters, and for each design-space point there are m different criteria by which to judge that point. Let $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$ be the function which assigns, to each design-space point x, a criteria-space point f(x). This represents the way of valuing the designs. Now, it may be that some designs are infeasible; so let X be a set of feasible designs in ${\mathbb{R}}^n$, which must be a compact set. Then the set which represents the feasible criterion points is f(X), the image of the set X under the action of f. Call this image Y.

Now construct the Pareto frontier as a subset of Y, the feasible criterion points. It can be assumed that the preferable values of each criterion parameter are the lesser ones, thus minimizing each dimension of the criterion vector. Then compare criterion vectors as follows: One criterion vector x strictly dominates (or "is preferred to") a vector y if each parameter of x is no greater than the corresponding parameter of y and at least one parameter is strictly less: that is, $\mathbf{x}_i \le \mathbf{y}_i$ for each i and $\mathbf{x}_i < \mathbf{y}_i$ for some i. This is written as $\mathbf{x} \succ \mathbf{y}$ to mean that x strictly dominates y. Then the Pareto frontier is the set of points from Y that are not strictly dominated by another point in Y.

Formally, this defines a partial order on Y, namely the (opposite of the) product order on $\mathbb{R}^m$ (more precisely, the induced order on Y as a subset of $\mathbb{R}^m$), and the Pareto frontier is the set of maximal elements with respect to this order.

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science. There, this task is known as the maximum vector problem or as skyline query.

### 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 zi = fi(xi) where $x^i=(x_1^i, x_2^i, \ldots, x_n^i)$ is the vector of goods, both for all i. The supply constraint is written $\sum_{i=1}^m x_j^i = b_j^0$ for $j=1,\ldots,n$. To optimize this problem, the Lagrangian is used:

$L(x, \lambda, \Gamma)=f^1(x^1)+\sum_{i=2}^m \lambda_i(z_i^0 - f^i(x^i))+\sum_{j=1}^n \Gamma_j(b_j^0-\sum_{i=1}^m x_j^i)$ where λ and Γ are multipliers.

Taking the partial derivative of the Lagrangian with respect to one good, i, and then taking the partial derivative of the Lagrangian with respect to another good, j, gives the following system of equations:

$\frac{\partial L}{\partial x_j^i} = f_{x^1}^1-\Gamma_j^0=0$ for j=1,...,n. $\frac{\partial L}{\partial x_j^i} = -\lambda_i^0 f_{x^1}^1-\Gamma_j^0=0$ for i = 2,...,m and j=1,...,m, where fx is the marginal utility on f' of x (the partial derivative of f with respect to x).

$\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}$ for i,k=1,...,m and j,s=1,...,n...,.

## Criticism

Pareto efficiency does not consider the equity of resource allocations. It may be that one economic agent owns all of the world's resources; it would be impossible to make anyone else better off without making said agent worse off, so this situation is described as "Pareto optimal", even though it is inequitable.

More generally, it can be misleading, in that "not Pareto optimal" implies "can be improved" (making someone better off without hurting anyone), but "Pareto optimal" does not imply "cannot be improved" by some measure—it only implies that someone must receive less. Thus if an allocation is not Pareto optimal, it means that one can improve it, but does not mean that one should categorically reject it for a Pareto optimal solution. More importantly, not all Pareto optimal states are equally desirable from the standpoint of society in general. For instance, a one-time transfer of wealth from the very wealthy to the very poor may not be a Pareto improvement but may nevertheless result in a new Pareto optimal state that is more socially desirable than the previous one.

## Notes

1. ^ Barr, N. (2004). Economics of the welfare state. New York, Oxford University Press (USA).
2. ^ Sen, A. (1993). Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms. Oxford Economic Papers, 45(4), 519-541.
3. ^ Greenwald, Bruce; Stiglitz, Joseph E. (1986), "Externalities in economies with imperfect information and incomplete markets", Quarterly Journal of Economics 101: 229–264