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Pareto efficiency, or
Pareto optimality, is an important notion 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
efficiency (economics) and income distribution.
Given a set of alternative allocations of say goods (economics) 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.
The
Strong Pareto Optimal (SPO) defines a more stringent optimum, where for a new allocation to be better it must be strictly preferred by
all individuals (
all must gain with the new allocation). In the
Weak Pareto Optimal (WPO) a new allocation is considered better if it is strictly preferred by at least one person, and weakly preferred (there is no gain and no loss) by everyone else (as in the example stated above). The SPO is said to be strong, because the set of SPO solutions is a sub-set of the weaker optimal set (
it is easier for an allocation to WPO).
Pareto efficiency in economics
An economic system that is Pareto efficient implies that no individual can be made better off without another being made worse off. Here 'better off' is often interpreted "put in a more 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 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 principle often appealed to is hypothetical. That is, for the alleged Pareto improvement, say from public regulation of the monopolist or removal of tariffs, or other losers from a policy change 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 the
Kaldor-Hicks efficiency (Ng, 1983),
Under certain idealised 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, markets are perfectly competitive, transaction costs are negligible, and there must be no
externality).
An alleged 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.
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 space 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} \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 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 \lambda and \Gamma are multipliers.
Taking the partial derivatve 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 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{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.
==Criticisms==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; by requiring that an allocation leave no participant worse off, Pareto efficiency tends to favor outcomes that do not depart radically from the
status quo.
Amartya Sen has elaborated the mathematical basis for this criticism, pointing out that under relatively plausible starting conditions, systems of
social choice theory will converge to Pareto efficient, but inequitable, distributions. 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 is left worse off than before, 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.
See also
References
Pareto efficiency, or
Pareto optimality, is an important notion 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
efficiency (economics) and
income distribution.
Given a set of alternative allocations of say
goods (economics) 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.
The
Strong Pareto Optimal (SPO) defines a more stringent optimum, where for a new allocation to be better it must be strictly preferred by
all individuals (
all must gain with the new allocation). In the
Weak Pareto Optimal (WPO) a new allocation is considered better if it is strictly preferred by at least one person, and weakly preferred (there is no gain and no loss) by everyone else (as in the example stated above). The SPO is said to be strong, because the set of SPO solutions is a sub-set of the weaker optimal set (
it is easier for an allocation to WPO).
Pareto efficiency in economics
An economic system that is Pareto efficient implies that no individual can be made better off without another being made worse off. Here 'better off' is often interpreted "put in a more 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 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 principle often appealed to is hypothetical. That is, for the alleged Pareto improvement, say from public regulation of the monopolist or removal of tariffs, or other losers from a policy change 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 the
Kaldor-Hicks efficiency (Ng, 1983),
Under certain idealised 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, markets are perfectly competitive, transaction costs are negligible, and there must be no externality).
An alleged 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.
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 space 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} \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 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 \lambda and \Gamma are multipliers.
Taking the partial derivatve 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 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{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.
==Criticisms==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; by requiring that an allocation leave no participant worse off, Pareto efficiency tends to favor outcomes that do not depart radically from the status quo.
Amartya Sen has elaborated the mathematical basis for this criticism, pointing out that under relatively plausible starting conditions, systems of social choice theory will converge to Pareto efficient, but inequitable, distributions. 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 is left worse off than before, 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.
See also
References
Pareto efficiency - Wikipedia, the free encyclopedia
Pareto efficiency, or Pareto optimality, is an important concept in economics with broad applications in game theory, engineering and the social sciences.
Pareto - Wikipedia, the free encyclopedia
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