# Pareto efficiency

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Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

## Contents

• Given an initial situation, a Pareto improvement is a new situation where some agents will gain, and no agents will lose.
• A situation is called Pareto dominated if there exists a possible Pareto improvement.
• A situation is called Pareto optimal or Pareto efficient if no change could lead to improved satisfaction for some agent without some other agent losing or if there is no scope for further Pareto improvement.

The Pareto front (also called Pareto frontier or Pareto set) is the set of all Pareto efficient situations. [1]

Pareto originally used the word "optimal" for the concept, but as it describes a situation where a limited number of people will be made better off under finite resources, and it does not take equality or social well-being into account, it is in effect a definition of and better captured by "efficiency". [2]

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. [3] :459

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly 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).

## Overview

Formally, an allocation is Pareto optimal if there is no alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the new reallocation is called a "Pareto improvement". When no Pareto improvements are possible, the allocation is a "Pareto optimum".

The formal presentation of the concept in an economy is the following: Consider an economy with ${\displaystyle n}$ agents and ${\displaystyle k}$ goods. Then an allocation ${\displaystyle \{x_{1},...,x_{n}\}}$, where ${\displaystyle x_{i}\in \mathbb {R} ^{k}}$ for all i, is Pareto optimal if there is no other feasible allocation ${\displaystyle \{x_{1}',...,x_{n}'\}}$ where, for utility function ${\displaystyle u_{i}}$ for each agent ${\displaystyle i}$, ${\displaystyle u_{i}(x_{i}')\geq u_{i}(x_{i})}$ for all ${\displaystyle i\in \{1,...,n\}}$ with ${\displaystyle u_{i}(x_{i}')>u_{i}(x_{i})}$ for some ${\displaystyle i}$. [4] 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 vectors 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. [5] [ citation needed ] However, the result only holds under the assumptions of the theorem: markets exist for all possible goods, there are no externalities; 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. [6]

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. [4]

## Variants

### Weak Pareto efficiency

Weak Pareto efficiency is a situation that cannot be strictly improved for every individual. [7]

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):

• It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).
• But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5).

A market doesn't require local nonsatiation to get to a weak Pareto-optimum. [8]

### Constrained Pareto efficiency

Constrained Pareto efficiency is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents. [9] :104

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".

### Fractional Pareto efficiency

Fractional Pareto efficiency is a strengthening of Pareto-efficiency in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-efficient (fPE or fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-efficiency, which only considers domination by feasible (discrete) allocations. [10] [11]

• It is Pareto-efficient, since any other discrete allocation (without splitting items) makes someone worse-off.
• However, it is not fractionally-Pareto-efficient, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).

The following example [10] shows the "price" of fPO. The integral allocation maximizing the product of utilities (also called the Nash welfare) is PO but not fPO. Moreover, the product of utilities in any fPO allocation is at most 1/3 of the maximum product. There are 5 goods {h1,h2,g1,g2,g3} and 3 agents with the following values (where C is a large constant and d is a small positive constant):

Agents ↓ Goods ⇒h1h2g1g2g3
A1CC11-d1-d
A2CC1-d11-d
A3CC1-d1-d1

A max-product integral allocation is {h1},{h2},{g1,g2,g3}, with product ${\displaystyle C^{2}\cdot (3-2d)}$. It is not fPO, since it is dominated by a fractional allocation: agent 3 can give g1 to agent 1 (losing 1-d utility) in return to a fraction of h1 that both agents value at 1-d/2. This trade strictly improves the welfare of both agents. Moreover, in any integral fPO allocation, there exists an agent Ai who receives only (at most) the good gi - otherwise a similar trade can be done. Therefore, a max-product fPO allocation is {g1,h1},{g2,h2},{g3}, with product ${\displaystyle (C+1)^{2}}$. When C is sufficiently large and d is sufficiently small, the product ratio approaches 1/3.

### Ex-ante Pareto efficiency

When the decision process is random, such as in fair random assignment or random social choice or fractional approval voting, there is a difference between ex-post and ex-ante Pareto-efficiency:

• Ex-post Pareto-efficiency means that any outcome of the random process is Pareto efficient.
• Ex-ante Pareto-efficiency means that the lottery determined by the process is Pareto-efficient with respect to the expected utilities. That is: no other lottery gives a higher expected utility to one agent and at least as high expected utility to all agents.

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' which ex-ante Pareto-dominates L.

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:

1. With probability 1/2, give car to Alice and house to George; otherwise, give car to George and house to Alice. The expected utility is (2/2+3/2)=2.5 for Alice and (2/2+9/2)=5.5 for George. Both allocations are ex-post PE, since the one who got the car cannot be made better-off without harming the one who got the house.
2. With probability 1, give car to Alice. Then, with probability 1/3 give the house to Alice, otherwise give it to George. The expected utility is (2+3/3)=3 for Alice and (9*2/3)=6 for George. Again, both allocations are ex-post PE.

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. [12] There are 5 possible outcomes (a,b,c,d,e) and 6 voters. The voters' approval sets are (ac, ad, ae, bc, bd, be). 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.

### Approximate Pareto-efficiency

Given some ε>0, an outcome is called ε-Pareto-efficient if no other outcome gives all agents at least the same utility, and one agent a utility at least (1+ε) higher. This captures the notion that improvements smaller than (1+ε) are negligible and should not be considered a breach of efficiency.

## Pareto-efficiency and welfare-maximization

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

${\displaystyle W_{a}(x):=\sum _{i=1}^{n}a_{i}u_{i}(x)}$.

Let xa be an allocation that maximizes the welfare over all allocations, i.e.:

${\displaystyle x_{a}\in \arg \max _{x}W_{a}(x)}$.

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 [13] 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. [14]

## Use in engineering

The notion of Pareto efficiency has been used in engineering. [15] :111–148 Given a set of choices and a way of valuing them, the Pareto front (or Pareto set or Pareto frontier) is the set of choices that are Pareto efficient. 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. [16] :63–65

## Use in public policy

Modern microeconomic theory has drawn heavily upon the concept of Pareto efficiency for inspiration. Pareto and his successors have tended to describe this technical definition of optimum resource allocation in the context of it being an equilibrium that can theoretically be achieved within an abstract model of market competition. It has therefore very often been treated as a corroboration of Adam Smith's "invisible hand" notion. More specifically, it motivated the debate over "market socialism" in the 1930s. [2]

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.' [17]

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. [18]

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 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. [19]

## Use in biology

Pareto optimisation has also been studied in biological processes. [20] :87–102 In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. [21] :166–169 Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage). [22]

## Common misconceptions

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, [23] :358–364 as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. [24] :10–15 An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution. [25] :95–132

## Criticism

Some commentators contest that Pareto efficiency could potentially serve as an ideological tool. With it implying that capitalism is self-regulated thereof, it is likely that the embedded structural problems such as unemployment would be treated as deviating from the equilibrium or norm, and thus neglected or discounted. [2]

Pareto efficiency does not require a totally equitable distribution of wealth, which is another aspect that draws in criticism. [26] :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. [27] :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. [28] :92–94

Lastly, it is proposed that Pareto efficiency to some extent inhibited discussion of other possible criteria of efficiency. As the scholar Lockhood argues, one possible reason is that any other efficiency criteria established in the neoclassical domain will reduce to Pareto efficiency at the end. [2]

## Related Research Articles

In welfare economics, a social welfare function is a function that ranks social states as less desirable, more desirable, or indifferent for every possible pair of social states. Inputs of the function include any variables considered to affect the economic welfare of a society. In using welfare measures of persons in the society as inputs, the social welfare function is individualistic in form. One use of a social welfare function is to represent prospective patterns of collective choice as to alternative social states. The social welfare function provides the government with a simple guideline for achieving the optimal distribution of income.

Welfare economics is a branch of economics that uses microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level.

Allocative efficiency is a state of the economy in which production is aligned with consumer preferences; in particular, every good or service is produced up to the point where the last unit provides a marginal benefit to consumers equal to the marginal cost of producing.

The liberal paradox, also Sen paradox or Sen's paradox, is a logical paradox proposed by Amartya Sen which shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly mild conditions:

1. The Unrestrictedness Condition, or U: every possible ranking of each individual's preferences and all outcomes of every possible voting rule will be considered equally,
2. The Pareto Condition, or P: if everybody individually likes some choice better at the same time, the society in its voting rule as a whole likes it better as well, and
3. Liberalism, or L : all individuals in a society must have at least one possibility of choosing differently, so that the social choice under a given voting rule changes as well. That is, as an individual liberal, you can exert your freedom of choice at least in some decision with tangible results.

There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal. The requirements for perfect competition are these:

1. There are no externalities and each actor has perfect information.
2. Firms and consumers take prices as given.

Competitive equilibrium is a concept of economic equilibrium introduced by Kenneth Arrow and Gérard Debreu in 1951 appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal standard by which other market structures are evaluated.

Bayesian efficiency is an analog of Pareto efficiency for situations in which there is incomplete information. Under Pareto efficiency, an allocation of a resource is Pareto efficient if there is no other allocation of that resource that makes no one worse off while making some agents strictly better off. A limitation with the concept of Pareto efficiency is that it assumes that knowledge about other market participants is available to all participants, in that every player knows the payoffs and strategies available to other players so as to have complete information. Often, the players have types that are hidden from the other player.

Efficient cake-cutting is a problem in economics and computer science. It involves a heterogeneous resource, such as a cake with different toppings or a land with different coverings, that is assumed to be divisible - it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible, etc. The allocation should be economically efficient. Several notions of efficiency have been studied:

Efficiency and fairness are two major goals of welfare economics. Given a set of resources and a set of agents, the goal is to divide the resources among the agents in a way that is both Pareto efficient (PE) and envy-free (EF). The goal was first defined by David Schmeidler and Menahem Yaari. Later, the existence of such allocations has been proved under various conditions.

Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among n partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide a cake fairly without compromising on economic efficiency.

Utilitarian cake-cutting is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different cardinal utility functions, such that the sum of the utilities of the partners is as large as possible. It is a special case of the utilitarian social choice rule. Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is often in conflict with fair cake-cutting.

Envy-free (EF) item allocation is a fair item allocation problem, in which the fairness criterion is envy-freeness - each agent should receive a bundle that they believe to be at least as good as the bundle of any other agent.

Approximate Competitive Equilibrium from Equal Incomes (A-CEEI) is a procedure for fair item assignment. It was developed by Eric Budish.

Fair random assignment is a kind of a fair division problem.

Random priority (RP), also called Random serial dictatorship (RSD), is a procedure for dividing indivisible items fairly among people.

The Probabilistic Serial rule (PS), also called serial eating algorithm, is a rule for fair random assignment. It yields a randomized allocation of indivisible items among several agents that is ex-ante envy-free and Pareto efficient. It was developed by Hervé Moulin and Anna Bogomolnaia.

The Price of Anarchy (PoA) is a concept in game theory and mechanism design that measures how the social welfare of a system degrades due to selfish behavior of its agents. It has been studied extensively in various contexts, particularly in auctions.

Egalitarian equivalence (EE) is a criterion of fair division. In an egalitarian-equivalent division, there exists a certain "reference bundle" such that each agent feels that his/her share is equivalent to .

When allocating objects among people with different preferences, two major goals are Pareto efficiency and fairness. Since the objects are indivisible, there may not exist any fair allocation. For example, when there is a single house and two people, every allocation of the house will be unfair to one person. Therefore, several common approximations have been studied, such as maximin-share fairness (MMS), envy-freeness up to one item (EF1), proportionality up to one item (PROP1), and equitability up to one item (EQ1). The problem of efficient approximately-fair item allocation is to find an allocation that is both Pareto-efficient (PE) and satisfies one of these fairness notions. The problem was first presented at 2016 and has attracted considerable attention since then.

In social choice and operations research, the egalitarian rule is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the minimum utility of all individuals in society. It is a formal mathematical representation of the egalitarian philosophy. It also corresponds to John Rawls' principle of maximizing the welfare of the worst-off individual.

## References

1. proximedia. "Pareto Front". www.cenaero.be. Retrieved October 8, 2018.
2. Lockwood, B. (2008). The New Palgrave Dictionary of Economics (2nd ed.). London: Palgrave Macmillan. ISBN   978-1-349-95121-5.
3. Black, J. D., Hashimzade, N., & Myles, G., eds., A Dictionary of Economics, 5th ed. (Oxford: Oxford University Press, 2017), p. 459.
4. Mas-Colell, A.; Whinston, Michael D.; Green, Jerry R. (1995), "Chapter 16: Equilibrium and its Basic Welfare Properties", , Oxford University Press, ISBN   978-0-19-510268-0
5. Gerard, Debreu (1959). "Valuation Equilibrium and Pareto Optimum". Proceedings of the National Academy of Sciences of the United States of America. 40 (7): 588–592. doi:. JSTOR   89325. PMC  . PMID   16589528.
6. Greenwald, B.; Stiglitz, J. E. (1986). "Externalities in economies with imperfect information and incomplete markets". Quarterly Journal of Economics. 101 (2): 229–64. doi:. JSTOR   1891114.
7. Mock, William B T. (2011). "Pareto Optimality". Encyclopedia of Global Justice. pp. 808–809. doi:10.1007/978-1-4020-9160-5_341. ISBN   978-1-4020-9159-9.
8. 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
9. Magill, M., & Quinzii, M., Theory of Incomplete Markets, MIT Press, 2002, p. 104.
10. Barman, S., Krishnamurthy, S. K., & Vaish, R., "Finding Fair and Efficient Allocations", EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation, June 2018.
11. Sandomirskiy, Fedor; Segal-Halevi, Erel (September 13, 2020). "Efficient Fair Division with Minimal Sharing". arXiv: [cs.GT].
12. Bogomolnaia, Anna; Moulin, Hervé; Stong, Richard (June 1, 2005). "Collective choice under dichotomous preferences". Journal of Economic Theory. 122 (2): 165–184. doi:10.1016/j.jet.2004.05.005. ISSN   0022-0531.
13. Negishi, Takashi (1960). "Welfare Economics and Existence of an Equilibrium for a Competitive Economy". Metroeconomica. 12 (2–3): 92–97. doi:10.1111/j.1467-999X.1960.tb00275.x.
14. Varian, Hal R. (1976). "Two problems in the theory of fairness". Journal of Public Economics. 5 (3–4): 249–260. doi:10.1016/0047-2727(76)90018-9. hdl:.
15. Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., Introduction to Optimization Analysis in Hydrosystem Engineering (Berlin/Heidelberg: Springer, 2014), pp. 111–148.
16. Jahan, A., Edwards, K. L., & Bahraminasab, M., Multi-criteria Decision Analysis, 2nd ed. (Amsterdam: Elsevier, 2013), pp. 63–65.
17. Lockwood B. (2008) Pareto Efficiency. In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London.
18. Lockwood B. (2008) Pareto Efficiency. In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London.
19. Lockwood B. (2008) Pareto Efficiency. In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London.
20. Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., Genetic Programming Theory and Practice X (Berlin/Heidelberg: Springer, 2013), pp. 87–102.
21. Eiben, A. E., & Smith, J. E., Introduction to Evolutionary Computing (Berlin/Heidelberg: Springer, 2003), pp. 166–169.
22. Seward, E. A., & Kelly, S., "Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria", Genome Biology , Vol. 19, 2018.
23. Drèze, J., Essays on Economic Decisions Under Uncertainty (Cambridge: Cambridge University Press, 1987), pp. 358–364
24. Backhaus, J. G., The Elgar Companion to Law and Economics (Cheltenham, UK / Northampton, MA: Edward Elgar, 2005), pp. 10–15.
25. Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. The Finance of Higher Education: Theory, Research, Policy, and Practice (New York: Agathon Press, 2001), pp. 95–132.
26. Bhushi, K., ed., Farm to Fingers: The Culture and Politics of Food in Contemporary India (Cambridge: Cambridge University Press, 2018), p. 222.
27. Wittman, D., Economic Foundations of Law and Organization (Cambridge: Cambridge University Press, 2006), p. 18.
28. Sen, A., Rationality and Freedom (Cambridge, MA / London: Belknep Press, 2004), pp. 92–94.