Editing Utility (section)

=={{Visible anchor|Functions|Utility functions}}==

'''Utility functions''', expressing utility as a function of the amounts of the various goods consumed, are treated as either ''cardinal'' or ''ordinal'', depending on whether they are or are not interpreted as providing more information than simply the rank ordering of preferences among bundles of goods, such as information concerning the strength of preferences.

===Cardinal===
{{Main|Cardinal utility}}

Cardinal utility states that the utilities obtained from consumption can be measured and ranked objectively and are representable by numbers.<ref name=":0">{{Cite book|last=Dominick|first=Salvatore|title=Principles Of Microeconomics|publisher=Oxford Higher Education/Oxford University Press|year=2008|isbn=9780198062301|location=New Delhi|pages=60}}</ref> There are fundamental assumptions of cardinal utility. Economic agents should be able to rank different bundles of goods based on their preferences or utilities and sort different transitions between two bundles of goods.<ref>{{Cite journal|last1=Lin|first1=Chung-Cheng|last2=Peng|first2=Shi-Shu|date=2019|title=The role of diminishing marginal utility in the ordinal and cardinal utility theories|url=https://onlinelibrary.wiley.com/doi/abs/10.1111/1467-8454.12151|journal=Australian Economic Papers|volume=58|issue=3|pages=233–246|doi=10.1111/1467-8454.12151|s2cid=159308055|via=Wiley Online Library}}</ref>

A cardinal utility function can be transformed to another utility function by a positive linear transformation (multiplying by a positive number, and adding some other number); however, both utility functions represent the same preferences.<ref>{{Cite journal|last=Moscati|first=Ivan|date=2013|title=How Cardinal Utility Entered Economic Analysis, 1909-1944|journal=SSRN Electronic Journal|doi=10.2139/ssrn.2296881 |doi-access=free |hdl=10419/149700 |s2cid=55651414 |s2cid-access=free |issn=1556-5068|hdl-access=free }}</ref>

When cardinal utility is assumed, the magnitude of utility differences is treated as an ethically or behaviorally significant quantity. For example, suppose a cup of orange juice has utility of 120 "utils", a cup of tea has a utility of 80 utils, and a cup of water has a utility of 40 utils. With cardinal utility, it can be concluded that the cup of orange juice is better than the cup of tea by the same amount by which the cup of tea is better than the cup of water. This means that if a person has a cup of tea, they would be willing to take any bet with a probability, p, greater than .5 of getting a cup of juice, with a risk of getting a cup of water equal to 1-p. One cannot conclude, however, that the cup of tea is two-thirds of the goodness of the cup of juice because this conclusion would depend not only on magnitudes of utility differences but also on the "zero" of utility. For example, if the "zero" of utility were located at -40, then a cup of orange juice would be 160 utils more than zero, a cup of tea 120 utils more than zero. Cardinal utility can be considered as the assumption that quantifiable characteristics, such as height, weight, temperature, etc can measure utility.

[[Neoclassical economics]] has largely retreated from using cardinal utility functions as the basis of economic behavior. A notable exception is in the context of analyzing choice with conditions of risk (see [[#Expected utility|below]]).

Sometimes cardinal utility is used to aggregate utilities across persons, to create a [[social welfare function]].

===Ordinal===
{{Main|Ordinal utility}}

Instead of giving actual numbers over different bundles, ordinal utilities are only the rankings of utilities received from different bundles of goods or services.<ref name=":0" /> For example, ordinal utility could tell that having two ice creams provide a greater utility to individuals in comparison to one ice cream but could not tell exactly how much extra utility received by the individual. Ordinal utility, it does not require individuals to specify how much extra utility they received from the preferred bundle of goods or services in comparison to other bundles. They are only needed to tell which bundles they prefer. 

When ordinal utilities are used, differences in utils (values assumed by the utility function) are treated as ethically or behaviorally meaningless:  the utility index encodes a full behavioral ordering between members of a choice set, but tells nothing about the related ''strength of preferences''. For the above example, it would only be possible to say that juice is preferred to tea to water. Thus, ordinal utility utilizes comparisons, such as "preferred to", "no more", "less than", etc.

If a function <math>u(x)</math> is ordinal and non-negative, it is equivalent to the function <math>u(x)^2</math>, because taking the square is an increasing [[Monotonic function#Monotonic transformation|monotone (or monotonic) transformation]]. This means that the ordinal preference induced by these functions is the same (although they are two different functions). In contrast, if <math>u(x)</math> is cardinal, it is not equivalent to <math>u(x)^2</math>.

===Examples===

In order to simplify calculations, various alternative assumptions have been made concerning details of human preferences, and these imply various alternative utility functions such as:

* [[constant elasticity of substitution|CES]] (''constant elasticity of substitution'').
* [[Isoelastic utility]]
* [[Exponential utility]]
* [[Quasilinear utility]]
* [[Homothetic preferences]]
* [[Stone–Geary utility function]]
* [[Gorman polar form]]
** [[Greenwood–Hercowitz–Huffman preferences]]
** [[King–Plosser–Rebelo preferences]]
* [[Hyperbolic absolute risk aversion]]

Most utility functions used for modeling or theory are '''well-behaved.''' They are usually monotonic and quasi-concave. However, it is possible for rational preferences not to be representable by a utility function. An example is [[lexicographic preferences]] which are not continuous and cannot be represented by a continuous utility function.<ref>{{cite book |first=Jonathan E. Jr. |last=Ingersoll |title=Theory of Financial Decision Making |url=https://archive.org/details/theoryoffinancia1987inge |url-access=registration |location=Totowa |publisher=Rowman and Littlefield |year=1987 |page=[https://archive.org/details/theoryoffinancia1987inge/page/21 21] |isbn=0-8476-7359-6 }}</ref>