Pareto efficiency

Pareto efficiency, also known as Pareto optimality or Pareto optimality, is a concept in economics that has applications in engineering and different social sciences. The term gets its name from the Italian economist Vilfredo Pareto, who used this concept in his studies on economic efficiency and income distribution.^{[citation needed]}

Given an initial allocation of goods among a set of individuals, a change to a new allocation that at least improves the situation of one individual without making it worse off for others is called Pareto improvement. An allocation is defined as "pareto-efficient" or "pareto-optimal" when no further Pareto improvements can be achieved.

Pareto efficiency is a minimal notion of efficiency and does not necessarily result in a socially desirable distribution of resources. It does not pronounce itself on equality, or on the well-being of the whole of society.

Use and technical considerations

The technical definition could be as follows: P a multi-objective optimization problem. It is then said that a solution S1{displaystyle S_{1}} is pareto-optim when there is no other solution S2{displaystyle S_{2}} such that it improves in a goal without worsening at least one of the others.

It is important to keep in mind that the concept does not refer, in economics, to the efficiency of production or even to the distribution (exchange and consumption) of goods in general or wealth in a society but to a description of a &# 34;desideratum" It has been argued that in more general economic terms, "efficiency" includes or should include aspects of both productive and distributive efficiency. (see allocative efficiency)

It has been argued that the concept of Pareto efficiency is minimalist. It does not necessarily imply or result in a socially desirable distribution of resources, nor does it refer to equality or a general state of social welfare. It only implies a situation that cannot be modified without harming at least one individual. Additionally, it does not imply that if something generates or produces profit, comfort, fruit or interest without harming another, it will cause a natural optimization process until reaching the sweet spot.

Consequently, it has been said that the criterion poses a dilemma between efficiency and equity, since although it solves the individual optimum, it does not solve the problem of the social optimum where not only the allocation of resources is relevant, but also the distribution of the rent. Additionally, it presents a practical difficulty since any political-economic change would be unfeasible if any member of society felt harmed.

Additionally, Amartya Sen points out that there can be many situations that are Pareto efficient without all of them being equally desirable or acceptable from the point of view of society (or its members).

Furthermore, there may be situations that are not Pareto optimal but are nevertheless generally preferable. For example, that hypothetical situation in which 1% of the population owns 99% of the overall wealth and the remaining 99% of the population owns 1% of the wealth, redistributive measures could generally be viewed as not just equitable, but could have a positive effect on the general economy, to the extent that an increase in demand can increase production. An argument to that effect is advanced by Davis (see also Keynesianism).

As a consequence of the above, the concept of "Social Optimum" or "Best Pareto Optimum" has been proposed, which is supposed to synthesize the preferences of society through of a Social Welfare Function, incorporating ethical considerations. However, it has been argued that it is not clear what would be the method to determine such a "social preference". Consequently Kenneth Arrow raises doubts about the viability of the project. On the other hand Sen argues that individual preferences are similar in a certain sense: there is a preference for economic growth, effective use of resources, equitable distribution of products and other benefits, etc.

Sen proposes a formulation to consistently solve the impossibility raised by Arrow; that is, he suggests a coherent and satisfactory way to deduce the preferences of society through individual preferences; This allows you to find the social state resulting from collective elections, specifically, it allows you to order and evaluate social states based on the construction of welfare indicators, which necessarily require interpersonal comparisons to study the distributive consequences (poverty, inequality, etc..) of certain types of society.

Another alternative is the proposal of Abba Lerner, who suggested using distributive efficiency — which is measured in relation to the efficiency with which those who need the goods and services receive them Lerner argues that to the greater efficiency of distribution, the greater general well-being. But this better distribution of goods and services implies in turn a better distribution of the means of access to such goods and services in society, or, more formally: "assuming that a fixed amount of income, a concave social welfare function, Individual welfare functions are also of the concave type, and that these are distributed in an equiprobabilistic manner among the members of society, the maximization of the mathematical expectation of the welfare of society is reached only when income is distributed equally. (A proof of this theorem is found in Sen, A.K. On economic inequality. Editorial Crítica. (1979).” (see also Discussion on fundamental theorems of welfare economics)

Under certain idealized conditions a free market system can be shown to lead to outcomes that are Pareto efficient. (see the first of the fundamental theorems of welfare economics). However, this result does not really reflect a real economic situation, since the conditions it assumes are too restrictive. The theorem assumes that there are markets -perfectly competitive and in equilibrium- for all possible goods, that transaction costs are negligible, that there are no externalities and that the participants have perfect information. It has been shown (Greenwald-Stiglitz Theorem) that in the absence of such conditions, the results are Pareto inefficient.

Formal aspects

The formalization of the Pareto proposal has allowed it to be applied in the areas of operational research and game theory. Its applications are multiple in decision making, in optimization environments with multiple objectives and, in general, cost-benefit analysis.

Example of Frontier de Pareto. The squares represent possible solutions or decisions (lower values are preferred) Option or solution C is not on the Frontier because it is preferred (dominated) by A and B, Those in turn are not dominated by any other, are consequently on the border.

From this point of view, the concept is used to analyze the possible optimal options of an individual given a variety of objectives or desires and one or several evaluation criteria. Given a "universe" of alternatives, the aim is to determine the set that are efficient according to Pareto (that is, those alternatives that satisfy the condition of not being able to better satisfy one of those desires or objectives without worsening another). That set of optimal alternatives establishes a “Pareto set” or the “Pareto Frontier”. The study of solutions at the frontier allows designers to analyze the possible alternatives within the established parameters, without having to analyze the totality of possible solutions.

Examples

To clearly illustrate its rationale, we propose the following example:

In the automobile market we have multiple vehicles to purchase. Each vehicle has certain technical characteristics and a price, the latter normally related to its quality, although this is not always the case. Before a person who is going to buy a car, there are in principle two possibilities:

1. That the person has money left, that is, that he wishes to acquire the vehicle of greater quality - defined according to any criterion - without taking into account the price. In this case we would be faced with a single-objective problem, i.e. the only objective is to find the vehicle with the most benefits, such as a sports car or a luxury car.
2. Let the person have a tight budget. In this case, apart from the benefits will also consider the price. We are facing a multi-objective problem (in this case with 2 objectives). In the face of this situation there is a question. What is the best vehicle to buy?. The answer is that there is not a single vehicle that is considered the best. A sportsman will be the one that will give better benefits, but will also be the most expensive (the best in the performance target and the worst in the price target). A non-powerful vehicle can be the least available, but the best price you have (the worst in the performance target and the best in the price target). So we cannot say that one is better than the other. (the border becomes a Curve of indifference).

In this situation it is worth considering additional criteria: in addition to a possible main desire for adequate cost and personal transportation (for example to provide convenient transportation to work), does the person want to transport, at least occasionally, others in the car? (for example, her family). Is the function of the car, in addition to satisfying the desire for speed or comfort, demonstrate your professional success? Or is it just going to work - in which case a cheap car, easy to park and with few additional costs, might be more suitable, etc.

The examination of these possible options, within the border or set established by those solutions that are cars, unlike motorcycles or helicopters, etc., and have acceptable prices for the person, allows us to establish the advantages and disadvantages that these cars individuals possess from the point of view of those additional criteria. That is to say, they allow to establish which is the car that maximizes the obtaining of benefits for that person.

So it is said that a car, Corche1{displaystyle car_{1}} is a similar-optim solution when there is no other car, Corche2{displaystyle car_{2}}So you have a better price than Corche1{displaystyle car_{1}} and also provide greater benefits.

That is why it is interesting to have, not one solution, but several, so that when making decisions they consider all possible Pareto-optimal solutions.

Formalization

Then the concepts of domain and Pareto optimization are defined, applied to a minimization problem; The extension to the case of a maximization problem is trivial.

Dominancia de Pareto: Given a vector u=(u1, ,uk){displaystyle mathbf {u} =(u_{1},cdotsu_{k}}}}}}}, it is said to dominate another vector v=(v1, ,vk){displaystyle mathbf {v} =(v_{1},cdotsv_{k}}}}} Yes and only if:

Optimality of couple: A solution x↓ ↓ {displaystyle mathbf {x^{}} } it is said to be a couple-optim if and only if there is no other vector x{displaystyle mathbf {x} } such as v=f(x)=(v1,....,vk){displaystyle mathbf {v} =f(mathbf {x})=(v_{1},...,v_{k})} domine a u=f(x↓ ↓ )=(u1,....,uk){displaystyle mathbf {u} =f(mathbf {x^{}})=(u_{1},....,u_{k})}.

In other words, the previous definition says that the point x↓ ↓ {displaystyle mathbf {x^{}} } is an optimal pair if there is no vector x{displaystyle mathbf {x} } to improve any of the objectives — in respect of the values obtained x↓ ↓ {displaystyle mathbf {x^{}} }— without any of the others getting worse simultaneously. In general, the solution in Pareto's sense to the multi-objective optimization problem will not be unique: the solution will be formed by the set of all undominated vectors, to which it is known by the name of group of non-dominated or front of couple.

Figure 1 represents, with a thick stroke, the pair's front of a function with 2 objectives. The coloured area T represents the image of that objective function. It can be observed that there is no point belonging to T that improves in the sense of Couple, at some point of the Front: choosing a point of T arbitrarily, for example p3{displaystyle p_{3}}, you can trace the vertical to get the cut point with the Couple Front, in this case p1{displaystyle p_{1}}; such cutting point will always have the same value f1{displaystyle f_{1} and a better value f2{displaystyle f_{2}}. It can also be observed that for 2 points any of the Front de Pareto, there will never be one that simultaneously improves the two objectives regarding the other point. Taking for example points p1{displaystyle p_{1}} and p2{displaystyle p_{2}}, it is observed that p1{displaystyle p_{1}} improvement f2{displaystyle f_{2}}But at the cost of getting worse f1{displaystyle f_{1} (a case of minimization is being considered).

In economic analysis, the Pareto optimum is the point of equilibrium in which none of the affected agents can improve their situation without reducing the welfare of any of the other agents. Therefore, while one of the individuals included in the distribution, production or consumption system can improve their situation without harming another, we will find ourselves in non-optimal situations in the Pareto sense. The Pareto optimum is not sensitive to imbalances and injustices in the allocation of resources, factors, goods and services, or in their ownership, since a situation in which 10 units of a good are distributed for consumption between two individuals allows 10 non-Pareto optimal to be obtained regardless of the fairness of such allocation. Both a 10 to 0 type distribution and a 5 to 5 type distribution would be optimal, since once assigned in both cases, to improve the situation of one individual the situation of the other would inevitably worsen by having to give up one of the units of the good or service (even if the first starts from 0 and the last from 10).