In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse off than they were before. A situation is called Pareto efficient or Pareto optimal if all possible Pareto improvements have already been made; in other words, there are no longer any ways left to make one person better off without making some other person worse-off.
In social choice theory, the same concept is sometimes called the unanimity principle, which says that if everyone in a society (non-strictly) prefers A to B, society as a whole also non-strictly prefers A to B. The Pareto frontier consists of all Pareto-efficient situations.
In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto-efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.Black, J. D., Hashimzade, N., Gareth Myles (eds.), A Dictionary of Economics, 5th ed. (Oxford: Oxford University Press, 2017), p. 459.
Besides economics, the notion of Pareto efficiency has also been applied to selecting alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).
Pareto originally used the word "optimal" for the concept, but this is somewhat of a misnomer: Pareto's concept more closely aligns with an idea of "efficiency", because it does not identify a single "best" (optimal) outcome. Instead, it only identifies a set of outcomes that might be considered optimal, by at least one person.
In other words, Pareto efficiency is when it is impossible to make one party better off without making another party worse off. This state indicates that resources can no longer be allocated in a way that makes one party better off without harming other parties. In a state of Pareto Efficiency, resources are allocated in the most efficient way possible.
Pareto efficiency is mathematically represented when there is no other strategy profile s' such that ui (s') ≥ ui (s) for every player i and uj (s') > uj (s) for some player j. In this equation s represents the strategy profile, u represents the utility or benefit, and j represents the player.
Efficiency is an important criterion for judging behavior in a game. In zero-sum games, every outcome is Pareto-efficient.
A special case of a state is an allocation of resources. The formal presentation of the concept in an economy is the following: Consider an economy with agents and goods. Then an allocation , where for all i, is Pareto-optimal if there is no other feasible allocation where, for utility function for each agent , for all with for some .. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption Vector space and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.
Under the assumptions of the first welfare theorem, a competitive market leads to a Pareto-efficient outcome. This result was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu. However, the result only holds under the assumptions of the theorem: markets exist for all possible goods, there are no externality, markets are perfectly competitive, and market participants have perfect information.
In the absence of perfect information or complete markets, outcomes will generally be Pareto-inefficient, per the Greenwald–Stiglitz theorem.
The second welfare theorem is essentially the reverse of the first welfare theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.
For instance, excessive use of negative commodities (such as drugs and cigarettes) results in expenses to non-smokers as well as early mortality for smokers. may help individuals stop smoking while also raising money to address ailments brought on by smoking.
A society may be Pareto efficient but have significant levels of inequality. For example, if there are three persons and a pie, the most equitable course of action would be to split the pie into three equal portions. By contrast, splitting the pie in half and giving each piece only to two individuals would be considered Pareto efficient, too, because the third person who receives no pie at all is nonetheless not any worse off than before the pie became available.
On a frontier of production possibilities, Pareto efficiency will happen. It is impossible to raise the output of products without decreasing the output of services when an economy is functioning on a basic production potential frontier, such as at point A, B, or C.
Consider a vector-valued minimization problem: Pareto dominates if and only if:
If , then this defines a preorder in the search space and we say Pareto dominates the alternative and we write .
Formally, a strong Pareto improvement is defined as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-efficient if it has no strong Pareto improvements.
Any strong Pareto improvement is also a weak Pareto improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at {10, 0}, and George values at {5, 5}. Consider the allocation giving all resources to Alice, where the utility profile is (10, 0):
A market does not require local nonsatiation to get to a weak Pareto optimum.Markey‐Towler, Brendan and John Foster. " Why economic theory has little to say about the causes and effects of inequality ", School of Economics, University of Queensland, Australia, 21 February 2013, RePEc:qld:uq2004:476.
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".
As an example, consider an item allocation problem with two items, which Alice values at {3, 2} and George values at {4, 1}. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3, 1):
The opposite is not true: ex-ante PE is stronger that ex-post PE. For example, suppose there are two objects a car and a house. Alice values the car at 2 and the house at 3; George values the car at 2 and the house at 9. Consider the following two lotteries:
While both lotteries are ex-post PE, the lottery 1 is not ex-ante PE, since it is Pareto-dominated by lottery 2.
Another example involves dichotomous preferences. There are 5 possible outcomes and 6 voters. The voters' approval sets are . All five outcomes are PE, so every lottery is ex-post PE. But the lottery selecting c, d, e with probability 1/3 each is not ex-ante PE, since it gives an expected utility of 1/3 to each voter, while the lottery selecting a, b with probability 1/2 each gives an expected utility of 1/2 to each voter.
Let xa be an allocation that maximizes the welfare over all allocations:
It is easy to show that the allocation xa is Pareto-efficient: since all weights are positive, any Pareto improvement would increase the sum, contradicting the definition of xa.
Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.
However, because the Pareto-efficient outcome is difficult to assess in the real world when issues including asymmetric information, signalling, adverse selection, and moral hazard are introduced, most people do not take the theorems of welfare economics as accurate descriptions of the real world. Therefore, the significance of the two welfare theorems of economics is in their ability to generate a framework that has dominated neoclassical thinking about public policy. That framework is that the welfare economics theorems allow the political economy to be studied in the following two situations: "market failure" and "the problem of redistribution".Lockwood B. (2008) Pareto Efficiency. In: Palgrave Macmillan (eds.) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London.
Analysis of "market failure" can be understood by the literature surrounding externalities. When comparing the "real" economy to the complete contingent markets economy (which is considered efficient), the inefficiencies become clear. These inefficiencies, or externalities, are then able to be addressed by mechanisms, including property rights and corrective taxes.
Analysis of "the problem with redistribution" deals with the observed political question of how income or commodity taxes should be utilized. The theorem tells us that no taxation is Pareto-efficient and that taxation with redistribution is Pareto-inefficient. Because of this, most of the literature is focused on finding solutions where given there is a tax structure, how can the tax structure prescribe a situation where no person could be made better off by a change in available taxes.
Pareto efficiency does not require a totally equitable distribution of wealth, which is another aspect that draws in criticism.Bhushi, K. (ed.), Farm to Fingers: The Culture and Politics of Food in Contemporary India (Cambridge: Cambridge University Press, 2018), p. 222. An economy in which a wealthy few hold the vast majority of resources can be Pareto-efficient. 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 could be made better off without decreasing someone else's share; and there are many other such distribution examples. 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.Wittman, D., Economic Foundations of Law and Organization (Cambridge: Cambridge University Press, 2006), p. 18.
The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.Sen, A., Rationality and Freedom (Cambridge, MA / London: Belknep Press, 2004), pp. 92–94.
Lastly, it is proposed that Pareto efficiency to some extent inhibited discussion of other possible criteria of efficiency. As Wharton School professor Ben Lockwood argues, one possible reason is that any other efficiency criteria established in the neoclassical domain will reduce to Pareto efficiency at the end.
Pareto, V (1906). Manual of Political Economy. Oxford University Press. https://global.oup.com/academic/product/manual-of-political-economy-9780199607952?cc=ca&lang=en&.
/ref> : and We then write , where is the Pareto order. This means that is not worse than in any goal but is better (since smaller) in at least one goal . The Pareto order is a strict partial order, though it is not a product order (neither non-strict nor strict).
Variants
Weak Pareto efficiency
Constrained Pareto efficiency
Fractional Pareto efficiency
Ex-ante Pareto efficiency
If some lottery L is ex-ante PE, then it is also ex-post PE. Proof: suppose that one of the ex-post outcomes x of L is Pareto-dominated by some other outcome y. Then, by moving some probability mass from x to y, one attains another lottery L that ex-ante Pareto-dominates L.
Bayesian Pareto efficiency
Ordinal Pareto efficiency
Pareto efficiency and equity
Pareto efficiency and market failure
Approximate Pareto efficiency
Pareto-efficiency and welfare-maximization
Use in engineering
Use in public policy
Use in biology
Common misconceptions
Criticism
See also
Further reading
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