# Pareto efficiency

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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 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. 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, at a WPO, alternative allocations where every individual would gain over the WPO are ruled out. 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 later 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 later, other allocations where one or more (but not all) individuals would gain (and none lose) would still be possible.

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 may lead to unjust and inefficient inequities.Barr, N. (2004). "Economics of the welfare state". New York, Oxford University Press (USA).] Sen, A. (1993). Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms. "Oxford Economic Papers, 45"(4), 519-541.]

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 (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 always be Pareto inefficient (the Greenwald-Stiglitz Theorem).

Formal representation

Pareto 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.

Suppose we have a design space with "n" real parameters, and for each design-space point we have "m" different criteria by which to judge that point. Let $f : mathbb\left\{R\right\}^n ightarrow mathbb\left\{R\right\}^m$ be the function which assigns, to each design-space point x, a criteria-space point "f"(x). This represents our way of valuing the designs. Now, it may be that some designs are infeasible; so let "X" be a set of feasible designs in $mathbb\left\{R\right\}^n$, 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 we will construct the Pareto frontier as a subset of "Y", the feasible criterion points. We can assume that the preferable values of each criterion parameter are the lesser ones, thus we wish to minimize each dimension of the criterion vector. We can then compare criterion vectors as follows: we say that 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, $mathbf\left\{x\right\}_i le mathbf\left\{y\right\}_i$ for each "i" and $mathbf\left\{x\right\}_i < mathbf\left\{y\right\}_i$ for some "i". We write $mathbf\left\{x\right\} succ mathbf\left\{y\right\}$ 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".

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

$L\left(x, lambda, Gamma\right)=f^1\left(x^1\right)+sum_\left\{i=2\right\}^m lambda_i\left(z_i^0 - f^i\left(x^i\right)\right)+sum_\left\{j=1\right\}^n Gamma_j\left(b_j^0-sum_\left\{i=1\right\}^m x_j^i\right)$ where $lambda$ and $Gamma$ 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:

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

Rearranging these to eliminate the multipliers gives the wanted result:

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

Criticism

A drawback of Pareto optimality is its localisation and partial ordering.Fact|date=September 2007 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.Fact|date=September 2007

ee also

* Abram Bergson
* Bayesian efficiency
* Compensation principle
* Constrained Pareto efficiency
* Efficiency (economics)
* Kaldor-Hicks efficiency
* Multidisciplinary design optimization
* Multiobjective optimization
* Social Choice and Individual Values for the '(weak) Pareto principle'
* Welfare economics

References

*cite book|title=Game Theory|author=Fudenberg, D. and Tirole, J.|year=1983|publisher=MIT Press|pages=Chapter 1, Section 2.4
*
*cite book|title=A Course in Game Theory|author=Osborne, M. J. and Rubenstein, A.|year=1994|publisher=MIT Press|pages=p. 7|id=ISBN 0-262-65040-1

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