Preference (economics)

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Example of a preference relation
A simple example of a preference order over three goods, in which an orange is preferred to a banana, but an apple is preferred to an orange

In economics and other social sciences, preference is the order that a person (an agent) gives to alternatives based on their relative utility, a process which results in an optimal "choice" (whether real or theoretical). Preferences are evaluations, they concern matters of value, typically in relation to practical reasoning.[1] Instead of the prices of goods, personal income, or availability of goods, the character of the preferences is determined purely by a person's tastes. However, persons are still expected to act in their best (that is, rational) interest.[2] Rationality suggests that people act in ways which serves their best interest when they are faced with a decision. Meaning when given a set of options, you must necessarily have a set of preferences. [3]

The belief of preference plays a key role in many disciplines, including moral philosophy and decision theory. The logical properties that preferences possess also have major effects on rational choice theory which has a carry over effect to all modern economic topics.[4]

Using the scientific method, social scientists try to model how people make practical decisions in order to test predictions about human behavior. Although economists are usually not interested in what causes a person to have certain preferences, they are interested in the theory of choice because it gives a background to empirical demand analysis.[5]

History[edit]

In 1926 Ragnar Frisch developed for the first time a mathematical model of preferences in the context of economic demand and utility functions.[6] Up to then, economists had developed an elaborated theory of demand that omitted primitive characteristics of people. This omission ceased when, at the end of the 19th and the beginning of the 20th century, logical positivism predicated the need of theoretical concepts to be related with observables.[7] Whereas economists in the 18th and 19th centuries felt comfortable theorizing about utility, with the advent of logical positivism in the 20th century, they felt that it needed more of an empirical structure. Because binary choices are directly observable, it instantly appealed to economists. The search for observables in microeconomics is taken even further by revealed preference theory, which holds consumers' preferences can be revealed by what they purchase under different circumstances, particularly under different income and price circumstances.[8]

Despite utilitarianism and decision theory, many economists have differing definitions of what a rational agent is. In the 18th century, utilitarianism gave insight into the utility-maximizing versions of rationality, however, economists still have no single definition or understanding of what preferences and rational actors should be analyzed by.[9]

Since the pioneer efforts of Frisch in the 1920s, one of the major issues which has pervaded the theory of preferences is the representability of a preference structure with a real-valued function. This has been achieved by mapping it to the mathematical index called utility. Von Neumann and Morgenstern 1944 book "Games and Economic Behaviour" treated preferences as a formal relation whose properties can be stated axiomatically. These type of axiomatic handling of preferences soon began to influence other economists: Marschak adopted it by 1950, Houthakker employed it in a 1950 paper, and Kenneth Arrow perfected it in his 1951 book "Social Choice and Individual Values".[10]

Gérard Debreu, influenced by the ideas of the Bourbaki group, championed the axiomatization of consumer theory in the 1950s, and the tools he borrowed from the mathematical field of binary relations have become mainstream since then. Even though the economics of choice can be examined either at the level of utility functions or at the level of preferences, to move from one to the other can be useful. For example, shifting the conceptual basis from an abstract preference relation to an abstract utility scale results in a new mathematical framework, allowing new kinds of conditions on the structure of preference to be formulated and investigated.

Another historical turnpoint can be traced back to 1895, when Georg Cantor proved in a theorem that if a binary relation is linearly ordered, then it is also isomorphically embeddable in the ordered real numbers. This notion would become very influential for the theory of preferences in economics: by the 1940s prominent authors such as Paul Samuelson, would theorize about people having weakly ordered preferences.[11]

Notation[edit]

There are two fundamental comparative value concepts, namely strict preference (better) and indifference (equal in value to).[12] These two concepts are expressed in terms of an agents best wishes, however, they also express objective or intersubjectively valid betterness that does not coincide with the pattern of wishes of any individual person.

Suppose the set of all states of the world is and an agent has a preference relation on . It is common to mark the weak preference relation by , so that means "the agent wants y at least as much as x" or "the agent weakly prefers y to x".

The symbol is used as a shorthand to denote an indifference relation: , which reads "the agent is indifferent between y and x" , meaning they receive the same level of benefit from each.

The symbol is used as a shorthand to the strong preference relation: .

A system of preferences or preference structure refers to the set of qualitative relations between different alternatives of consumption. For example, if the alternatives are:

  • Apple
  • Orange
  • Banana

In this example, a preference structure would be:

"The apple is at least as preferred as the orange", and "The orange is as least as preferred as the Banana". One can use to symbolize that some alternative is "at least as preferred as" another one, which is just a binary relation on the set of alternatives. Therefore:

  • Apple Orange
  • Orange Banana

The former qualitative relation can be preserved when mapped into a numerical structure, if we impose certain desirable properties over the binary relation: these are the axioms of preference order. For instance: Let us take the apple and assign it the arbitrary number 5. Then take the orange and let us assign it a value lower than 5, since the orange is less preferred than the apple. If this procedure is extended to the banana, one may prove by induction that if is defined on {apple, orange} and it represents a well-defined binary relation called "at least as preferred as" on this set, then it can be extended to a function defined on {apple, orange, banana} and it will represent "at least as preferred as" on this larger set.

Example:

  • Apple = 5
  • Orange = 3
  • Banana = 2

5 > 3 > 2 = u(apple) > u(orange) > u(banana)

and this is consistent with Apple Orange, and with Orange Banana.

Axiom of order (completeness):

In terms of preference completeness simply means that when a consumer is making a choice between two different options, the consumer can rank them so either, A is preferred to B, B is preferred to A or they are indifferent between the two. [13]

For all and we have or or .

Without completeness of preferences, consumers would not be able to come to a decision when given multiple options, as they wouldn't be able to rank them. Making completeness a necessity for the decision model.

In order for preference theory to be useful mathematically, we need to assume the axiom of continuity. Continuity simply means that there are no ‘jumps’ in people's preferences. In mathematical terms, if we prefer point A along a preference curve to point B, points very close to A will also be preferred to B. This allows preference curves to be differentiated. The continuity assumption is "stronger than needed" in the sense that it indeed guarantees the existence of a continuous utility function representation. Continuity is, therefore, a sufficient condition, but not a necessary one, for a system of preferences.[14]

Although commodity bundles come in discrete packages, economists treat their units as a continuum, because very little is gained from recognizing their discrete nature, the two approaches are reconcilable by this rhetorical device: When a consumer makes repeated purchases of a product, the commodity spaces can get converted from the discrete items to the time rates of consumption. Instead of, say, noting that a consumer purchased one loaf of bread on Monday, another on Friday and another the following Tuesday, we can speak of an average rate of consumption of bread equal to 7/4 loaves per week. There is no reason why the average consumption per week cannot be any real number, thus allowing differentiability of the consumer's utility function. We can speak of continuous services of goods, even if the goods themselves are purchased in discrete units.

Although some authors include reflexivity as one of the axioms required to obtain representability (this axiom states that ), it is redundant inasmuch as the completeness axiom implies it already.[15]


Non-satiation of Preferences

A simple example of non-satiated preference, in which a large amount of oranges are preferred to a single orange.

Non-satiation refers to the belief any commodity bundle with at least as much of one good and more of the other must provide a higher utility, showing that more is always better, always wanting more is known as non-satiation. This assumption is believed to hold as when consumers are able to discard excess goods at no cost, then consumers can be no worse off with extra goods.[16]

Example

Option A

  • Apple = 5
  • Orange = 3
  • Banana = 2

Option B

  • Apple = 6
  • Orange = 4
  • Banana = 2

In this Situation, utility from Option B > A, as it contains more apples and oranges with bananas being constant.


Transitivity[edit]

Transitivity of preferences is a fundamental principle shared by most major contemporary rational, prescriptive, and descriptive models of decision making.[17] Argubaly the most discussed logical property of preferences is the following:

A≽B ∧ B≽C → A≽C (transitivity of weak performance) A∼B ∧ B∼C → A∼C (transitivity of indifference) A≻B ∧ B≻C → A≻C (transitivity of strict preference)

In order to have transitivity preferences, a person, player, or agent that prefers choice option B to A and A to F must prefer B to F. Claims of violations of transitivity by any decision maker (in particular individuals) requires evidence beyond any reasonable doubt, with the onus placed on the individual.[18]

When transitivity does not hold, it results in an endless loop of indecision, as the agent will always have an outcome that is preferred no matter what choice they make. Which is why the assumption of transitivity preference is believed to hold in most situations.

Most commonly used axioms[edit]

  • Order-theoretic: acyclicity, the semiorder property, completeness
  • Topological: continuity, openness or closedness of the preference sets
  • Linear-space: convexity, homogeneity

Normative interpretations of the axioms[edit]

Everyday experience suggests that people at least talk about their preferences as if they had personal "standards of judgment" capable of being applied to the particular domain of alternatives that present themselves from time to time.[19] Thus, the axioms are an attempt to model the decision maker's preferences, not over the actual choice, but over the type of desirable procedure (a procedure that any human being would like to follow). Behavioral economics investigates inconsistent behavior (i.e. behavior that violates the axioms) of people. Believing in axioms in a normative way does not imply that everyone is asserted to behave according to them. Instead, they are a basis for suggesting a mode of behavior, one that people would like to see themselves or others following.[7]

Here is an illustrative example of the normative implications of the theory of preferences:[7] Consider a decision maker who needs to make a choice. Assume that this is a choice of where to live or whom to marry and that the decision maker has asked an economist for advice. The economist, who wants to engage in normative science, attempts to tell the decision maker how she should make decisions.

Economist: I suggest that you attach a utility index to each alternative, and choose the alternative with the highest utility.

Decision Maker: You've been brainwashed. You think only in terms of functions. But this is an important decision, there are people involved, emotions, these are not functions!

Economist: Would you feel comfortable with cycling among three possible options? Preferring x to y, and then y to z, but then again z to x?

Decision Maker: No, this is very silly and counterproductive. I told you that there are people involved, and I do not want to play with their feelings.

Economist: Good. So now let me tell you a secret: if you follow these two conditions making decision, and avoid cycling, then you can be described as if you are maximizing a utility function.

Consumers whose preference structures violate transitivity would get exposed to being exploited by some unscrupulous person. For instance, Maria prefers apples to oranges, oranges to bananas, and bananas to apples. Let her be endowed with an apple, which she can trade in a market. Because she prefers bananas to apples, she is willing to pay, say, one cent to trade her apple for a banana. Afterwards, Maria is willing to pay another cent to trade her banana for an orange, and again the orange for an apple, and so on. There are other examples of this kind of irrational behaviour.

Completeness implies that some choice will be made, an assertion that is more philosophically questionable. In most applications, the set of consumption alternatives is infinite and the consumer is not conscious of all preferences. For example, one does not have to choose over going on holiday by plane or by train: if one does not have enough money to go on holiday anyway then it is not necessary to attach a preference order to those alternatives (although it can be nice to dream about what one would do if one would win the lottery). However, preference can be interpreted as a hypothetical choice that could be made rather than a conscious state of mind. In this case, completeness amounts to an assumption that the consumers can always make up their mind whether they are indifferent or prefer one option when presented with any pair of options.

Under some extreme circumstances, there is no "rational" choice available. For instance, if asked to choose which one of one's children will be killed, as in Sophie's Choice, there is no rational way out of it. In that case, preferences would be incomplete, since "not being able to choose" is not the same as "being indifferent".

The indifference relation ~ is an equivalence relation. Thus we have a quotient set S/~ of equivalence classes of S, which forms a partition of S. Each equivalence class is a set of packages that is equally preferred. If there are only two commodities, the equivalence classes can be graphically represented as indifference curves. Based on the preference relation on S we have a preference relation on S/~. As opposed to the former, the latter is antisymmetric and a total order.

Factors which effect Consumer Preferences[edit]

1. Indifference Curve


An indifference curve is so named because the consumer would be indifferent between choosing any commodity bundles.[20] Indifference Curves explain an agent behaviour in terms of their preferences for different combinations of two goods, for example X & Y. An indifference curve can be detected in a market when the economics of scope is not overly diverse, or the goods and services are a part of a perfect market. One example of this is deodorant. Deodorant is similarly priced throughout a number of different brands. Deodorant also does not have any major differences in use; therefore, consumers really have no preference in which they should use.


2. Monopolised Markets

A monopolised market almost always has a direct effect on consumer preferences. A monopolised market refers to when a company and its product offerings dominate a sector or industry.[21] A monopolised market in turn, means that the business has complete control of supply and demand of a good or service. Businesses which have a monopoly can also use a number of strategies which can be used to ensure they remain control of the industry by refusing entry to the market, these are, blockaded entry, accommodated entry and deterred entry.[22] When a business has a monopoly, the business has a massive advantage in having a large majority of consumer preferences.


3. Changes in new technology


New changes in technology is a big factor in changes of consumer preferences. When an industry has a new competitor in the market who has found ways to make the goods or services work more effectively, has the ability to completely change the market. Some examples of changes in technology is Android phones. Five years ago, android was struggling to compete with Apple for market share. With the advances in technology throughout the last five years they have now passed the stagnant apple brand. Changes in technology examples are, but not limited to, increased efficiency, longer lasting batteries and new easier interface for consumers.

Types of Preferences[edit]

A simple graph showing convex preferences, as the indifference line, curves in

Convex Preferences:[edit]

Convex preferences relate to averages between two points on an indifference curve. It comes in two forms, weak and strong. In its weak form, convex preferences states that if . Then the average of A and B is at least as good as A. Where as in its strong form the average of A and B would be preferred. Which is why in its strong form, the indifference line curves in, meaning that the average of any two points would result in a point further away from the origin, thus giving a higher utility. [23]

Concave Preferences:[edit]

Concave preferences is the opposite of convex, where when , then the average of A and B is worst than A. This because concave curves slope outwards, meaning an average between two points on the same indifference curve would result in a point that is closer to the origin thus giving a lower utility. [24]

Straight Line indifference:[edit]

Straight line indifferences, occur when there are perfect substitutes. Perfect substitutes are goods and/or services that can be used in the same way as the good or service it replaces. Meaning when , then the average of A and B will fall on the same indifference line, and will give the exact same utility. [25]

An example of straight line indifference curves, where Good X and Good Y are perfect substitutes.

Types of Goods effecting Preferences:[edit]

When a consumer is faced with a choice between different goods, the type of goods they are choosing between will affect how they make their decision process. To begin with, when there are normal goods, these goods have a direct correlation with the income the consumer makes, meaning as they make more money they will choose to consume more of this good and as their income decreases they will consume less of the good. The opposite to this however is inferior goods, these have a negative correlation with income, so as a consumer makes less money, they’ll decide to consume more inferior goods as they are seen as less desirable meaning they come with a reduced cost and as they make more money, they'll consume less inferior goods as they'll have the money available to buy more desirable goods. [26]An example of a normal good would be branded clothes, as they are more expensive compared to their inferior good counterparts which are non-branded clothes. Goods that are not affected by income as referred to as a necessity good, which are product(s) and services that consumers will buy regardless of the changes in their income levels, these usually include medical care, clothing and basic food. Finally, there are also luxury goods, which are the most expensive goods and deemed the most desirable. Just like normal goods, as income is increased, so is the demand for luxury goods, however in the case for luxury goods, the greater the increase in income, the greater the increase in demand for luxury goods. [27]

Applications to theories of utility[edit]

In economics, a utility function is often used to represent a preference structure such that if and only if . The idea is to associate each class of indifference with a real number such that, if one class is preferred to the other, then the number of the first one is greater than that of the second one. When a preference order is both transitive and complete, then it is standard practice to call it a rational preference relation, and the people who comply with it are rational agents. A transitive and complete relation is called a weak order (or total preorder). The literature on preferences is far from being standardized regarding terms such as complete, partial, strong, and weak. Together with the terms "total", "linear", "strong complete", "quasi-orders", "pre-orders" and "sub-orders", which also have a different meaning depending on the author's taste, there has been an abuse of semantics in the literature.[19]

According to Simon Board, a continuous utility function always exists if is a continuous rational preference relation on .[28] For any such preference relation, there are many continuous utility functions that represent it. Conversely, every utility function can be used to construct a unique preference relation.

All the above is independent of the prices of the goods and services and of the budget constraints faced by consumers. These determine the feasible bundles (which they can afford). According to the standard theory, consumers chooses a bundle within their budget such that no other feasible bundle is preferred over it; therefore their utility is maximized.

Primitive equivalents of some known properties of utility functions[edit]

Lexicographic preferences[edit]

Lexicographic preferences are a special case of preferences that assign an infinite value to a good, when compared with the other goods of a bundle.[29]

Georgescu-Roegen pointed out that the measurability of the utility theory is limited as it excludes lexicographic preferences. Causing an amplified level of awareness placed upon lexicographic preferences as a substitute hypothesis on consumer behaviour. [30]

Strict versus weak[edit]

The possibility of defining a strict preference relation as distinguished from the weaker one , and vice versa, suggests in principle an alternative approach of starting with the strict relation as the primitive concept and deriving the weaker one and the indifference relation. However, an indifference relation derived this way will generally not be transitive.[6] The conditions to avoid such inconsistencies were studied in detail by Andranik Tangian.[29] According to Kreps "beginning with strict preference makes it easier to discuss noncomparability possibilities".[31]

Elicitation of preferences[edit]

The mathematical foundations of most common types of preferences — that are representable by quadratic or additive utility functions — laid down by Gérard Debreu[32][33] enabled Andranik Tangian to develop methods for their elicitation. In particular, additive and quadratic preference functions in variables can be constructed from interviews, where questions are aimed at tracing totally 2D-indifference curves in coordinate planes.[34][35]

Aggregation[edit]

Under certain assumptions, individual preferences can be aggregated onto the preferences of a group of people. However, Arrow's impossibility theorem states that voting systems sometimes cannot convert individual preferences into desirable community-wide acts of choice.

Expected utility theory[edit]

Preference relations were initially applied only to alternatives that involve no risk and uncertainties because this is an assumption of the homo economicus model of behaviour. Nonetheless, a very similar theory of preferences has also been applied to the space of simple lotteries, as in expected utility theory. In this case a preference structure over lotteries can also be represented by a utility function.

Criticism[edit]

Some critics say that rational theories of choice and preference theories rely too heavily on the assumption of invariance, which states that the relation of preference should not depend on the description of the options or on the method of elicitation. But without this assumption, one's preferences cannot be represented as maximization of utility.[36]

Milton Friedman said that segregating taste factors from objective factors (i.e. prices, income, availability of goods) is conflicting because both are "inextricably interwoven".

The concept of Transitivity is a highly debated topic, with many examples being presented to suggest that it does not hold in general. One of the most well-known being the Sorites Paradox. Where if people are indifferent between small changes in value, then this can be continued on infinitely to the point where they’d be indifferent between two largely different values. [37]

See also[edit]

References[edit]

  1. ^ Broome John[disambiguation needed] (1993) "Can a Humean be moderate?" In value, Welfare and Morality, R.G. Frey and Christopher Morris (eds.) (Cambridge University Press)>
  2. ^ Blume, Lawrence (15 December 2016). The New Palgrave Dictionary of Economics. London: Palgrave Macmillan. doi:10.1007/978-1-349-58802-2. ISBN 978-1-349-95121-5.
  3. ^ Falk, Armin; Becker, Anke (30 May 2018). "Global Evidence on Economic Preferences". The Quarterly Journal of Economics.
  4. ^ Hansson, Sven Ove; Grüne-Yanoff, Till (May 4, 2018). "Preferences". Stanford Encyclopedia of Economics.
  5. ^ Arrow, Kenneth (1958). "Utilities, attitudes, choices: a review note". Econometrica. 26 (1): 1–23. doi:10.2307/1907381. JSTOR 1907381.
  6. ^ a b Barten, Anton and Volker Böhm. (1982). "Consumer theory", in: Kenneth Arrow and Michael Intrilligator (eds.) Handbook of mathematical economics. Vol. II, p. 384
  7. ^ a b c Gilboa, Itzhak. (2009). Theory of Decision under uncertainty Archived 2018-02-19 at the Wayback Machine. Cambridge: Cambridge university press
  8. ^ Roper, James and David Zin. (2008). "A Note on the Pure Theory of Consumer's Behaviour"
  9. ^ Blume, Lawrence E.; Easley, David (2008). "Rationality". The New Palgrave Dictionary of Economics. pp. 1–13. doi:10.1057/978-1-349-95121-5_2138-1. ISBN 978-1-349-95121-5.
  10. ^ Moscati, Ivan (2004). "Early Experiments in Consumer Demand Theory" (PDF). 128.118.178.162. Wayback Machine. Archived from the original (PDF) on 2014-03-02.
  11. ^ Fishburn, Peter (1994). "Utility and subjective probability", in: Robert Aumann and Sergiu Hart (eds). Handbook of game theory. Vol. 2. Amsterdam: Elsevier Science. pp. 1397–1435.
  12. ^ = Halldén Sören, (1957) "On the Logic of Betterm Lund: Library of Theoria" (10).
  13. ^ Galaabaatar, Tsogbadral (September 2018). "Completeness and Transitivity of Preferences on Mixture Sets∗". Journal of Economic Literature.
  14. ^ Gallego, Lope (2012). "Policonomics. Economics made simple". Preferences. Open Dictionary. Retrieved 16 March 2013.
  15. ^ Mas-Colell, Andreu, Michael Whinston and Jerry Green (1995). Microeconomic theory. Oxford: Oxford University Press ISBN 0-19-507340-1
  16. ^ Bertoletti, Paolo; Etro, Federico (2016). "Preferences, entry, and market structure". RAND Journal of Economics.
  17. ^ Davis-Stober P. Clintin (2011, 42-56) "Transitivity of Preferences"http://www.chapman.edu/research/institutes-and-centers/economic-science-institute/_files/ifree-papers-and-photos/michel-regenwetter1.pdf
  18. ^ J, Dana, M, Regenwetter (2011) "Transitivity of preferences"
  19. ^ a b Shapley, Lloyd and Martin Shubik. (1974). "Game theory in economics". RAND Report R-904/4
  20. ^ University of Southern Indiana. (2021). Retrieved 26 April 2021, from http://www.usi.edu/business/cashel/331/CONSUMER.pdf
  21. ^ Kenton, W. (2021). Monopoly. Retrieved 26 April 2021, from http://www.investopedia.com/terms/m/monopoly.asp
  22. ^ Board, S. (2021). Competitive Strategy: Week 8 Entry. Retrieved 26 April 2021, from http://www.econ.ucla.edu/sboard/teaching/strategy_05b/strategy_slides8_05b.pdf
  23. ^ Richter, Michael; Rubinst, Ariel (December 2019). "Convex preferences: A new definition". Theoretical Economics.
  24. ^ Lahiri, Somdeb (September 2015). "Concave Preferences Over Bundles and the First Fundamental Theorem of Welfare Economics". Social Science Research Network.
  25. ^ Lipatov, Vilen (September 2020). "Preempting the Entry of Near Perfect Substitute". Journal of Competition Law & Economics. Volume 17: 194–210. |volume= has extra text (help)
  26. ^ Cherchye, Laurens (August 2020). "Revealed Preference Analysis with Normal Goods". American Economic Journal.
  27. ^ Mortelmans, D. (2005). "Sign values in processes of distinction: The concept of luxury". Semantic Scholar. Vol. 157: 497–520. |volume= has extra text (help)
  28. ^ Board, Simon. "Preferences and Utility" (PDF). UCLA. Retrieved 15 February 2013.
  29. ^ a b Tanguiane (Tangian), Andranick (1991). "2. Preferences and goal functions". Aggregation and representation of preferences: introduction to mathematical theory of democracy. Berlin-Heidelberg: Springer. pp. 23–50. doi:10.1007/978-3-642-76516-2. ISBN 978-3-642-76516-2.
  30. ^ Hayakawa, Hiroaki. "LEXICOGRAPHIC PREFERENCES AND CONSUMER THEORY". Cite journal requires |journal= (help)
  31. ^ Kreps, David. (1990). A Course in Microeconomic Theory. New Jersey: Princeton University Press
  32. ^ Debreu, Gérard (1952). "Definite and semidefinite quadratic forms". Econometrica. 20 (2): 295–300. doi:10.2307/1907852.
  33. ^ Debreu, Gérard (1960). "Topological methods in cardinal utility theory". In Arrow, Kenneth (ed.). Mathematical Methods in the Social Sciences,1959. Stanford: Stanford University Press. pp. 16–26. doi:10.1016/S0377-2217(03)00413-2.
  34. ^ Tangian, Andranik (2002). "Constructing a quasi-concave quadratic objective function from interviewing a decision maker". European Journal of Operational Research. 141 (3): 608–640. doi:10.1016/S0377-2217(01)00185-0.
  35. ^ Tangian, Andranik (2004). "A model for ordinally constructing additive objective functions". European Journal of Operational Research. 159 (2): 476–512. doi:10.1016/S0377-2217(03)00413-2.
  36. ^ Slovic, P. (1995). "The Construction of Preference". American Psychologist, Vol. 50, No. 5, pp. 364–371.
  37. ^ Luce, Duncan. "SEMIORDERS AND A THEORY OF UTILITY DISCRIMINATION" (PDF). Cite journal requires |journal= (help)

[1][2][3]



  1. ^ Board, S. (2021). Competitive Strategy: Week 8 Entry. Retrieved 26 April 2021, from http://www.econ.ucla.edu/sboard/teaching/strategy_05b/strategy_slides8_05b.pdf
  2. ^ Kenton, W. (2021). Monopoly. Retrieved 26 April 2021, from http://www.investopedia.com/terms/m/monopoly.asp
  3. ^ University of Southern Indiana. (2021). Retrieved 26 April 2021, from http://www.usi.edu/business/cashel/331/CONSUMER.pdf