Pareto efficiency

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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). The set of SPO solutions is a subset of the set of WPO solutions, because an SPO satisfies the stronger requirement that there is no allocation that is strictly preferred by one individual and weakly preferred by the rest (i.e., no individual loses out, and at least one individual gains).

1 Pareto efficiency in economics
2 Formal representation
- 2.1 Pareto frontier
- 2.2 Relationship to marginal rate of substitution
3 Criticisms
4 See also
5 References

Pareto efficiency in economics

An economic system that is Pareto inefficient implies that a certain change in allocation of income (etc) may result in some individuals be 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 "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 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, 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 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, and there must be no externalities).

Formal representation

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Pareto frontier

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Example of Pareto frontier, given that lower values are preferred to higher values. Point C is not on the Pareto Frontier because it is dominated by both point A and point B. Points A and B are non-inferior.

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 Failed to parse (Missing texvc executable; please see math/README to configure.): f: \mathbb{R}^n \rightarrow \mathbb{R}^m , where X is a compact set of feasible decisions in the metric space Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n , and Y is the feasible set of criterion vectors in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^m , such that Failed to parse (Missing texvc executable; please see math/README to configure.): Y = f(X) = \{ y \in \mathbb{R}^m:\; \exists x \in X: y = f(x)\} .

We assume that the preferred directions of criteria values are known. A point Failed to parse (Missing texvc executable; please see math/README to configure.): y^{\prime\prime} \in \mathbb{R}^m\;

is preferred to (strictly dominating) another point Failed to parse (Missing texvc executable; please see math/README to configure.): y^{\prime} \in \mathbb{R}^m\;

, written as Failed to parse (Missing texvc executable; please see math/README to configure.): y^{\prime\prime} \succ y^{\prime} . The Pareto frontier is thus written as:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): z_i=f^i(x^i)

where Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{i=1}^m x_j^i = b_j^0
for Failed to parse (Missing texvc executable; please see math/README to configure.): j=1,\ldots,n

. To optimize this problem, the Lagrangian is used:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda
and Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial L}{\partial x_j^i} = f_{x^1}^1-\Gamma_j^0=0

for j=1,...,n.

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 will converge to Pareto efficient, but inequitable, distributions.^{[citation needed]} 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 (when none had pie); 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, as welfare can be increased without reducing the welfare of any individual (e.g. just give the quarter pie to someone). The origin of the pie is conceived as immaterial in these examples, i.e. the pie was "free." 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.

Another alleged key drawback of Pareto optimality is its localisation and partial ordering.^{[citation needed]} 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.^{[citation needed]}

Despite its drawbacks (or perhaps precisely because of these drawbacks), Pareto-efficient improvements to real-world economic systems are extremely rare, and considered "the holy grail" of economic improvements. They can be thought of as "free money" in some sense- someone's lot is improved without taking anything from anyone else. One of the other benefits of such improvements is that they do not require the social planner to define a "social welfare function," a (necessarily arbitrary) weighting system which reflects the priorities given to the welfare of each individual in society.

References

Fudenberg, D. and Tirole, J. (1983). Game Theory. MIT Press, Chapter 1, Section 2.4.
Ng, Yew-Kwang (1983). Welfare Economics. Macmillan.
Osborne, M.J. and Rubenstein, A. (1994). A Course in Game Theory. MIT Press, p. 7. ISBN 0-262-65040-1.

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