Editing Cardinal utility (section)

== Applications ==

===Welfare economics===
Among welfare economists of the utilitarian school it has been the general tendency to take satisfaction (in some cases, pleasure) as the unit of welfare. If the function of welfare economics is to contribute data which will serve the social philosopher or the statesman in the making of welfare judgments, this tendency leads perhaps, to a hedonistic ethics.<ref>{{cite journal |last1=Viner |first1=Jacob |date=December 1925 |title=The Utility Concept in Value Theory and Its Critics II |journal=Journal of Political Economy |volume=33 |issue=6 |pages=638–659 |doi=10.1086/253725 |jstor=1822261|s2cid=222430888 }}</ref>

Under this framework, actions (including production of goods and provision of services) are judged by their contributions to the subjective wealth of people. In other words, it provides a way of judging the "greatest good to the greatest number of persons". An act that reduces one person's utility by 75 utils while increasing two others' by 50 utils each has increased overall utility by 25 utils and is thus a positive contribution; one that costs the first person 125 utils while giving the same 50 each to two other people has resulted in a net loss of 25 utils.

If a class of utility functions is cardinal, intrapersonal comparisons of utility differences are allowed. If, in addition, some comparisons of utility are meaningful interpersonally, the linear transformations used to produce the class of utility functions must be restricted across people. An example is cardinal unit comparability. In that information environment, admissible transformations are increasing affine functions and, in addition, the scaling factor must be the same for everyone. This information assumption allows for interpersonal comparisons of utility differences, but utility levels cannot be compared interpersonally because the intercept of the affine transformations may differ across people.<ref>{{cite book |last1=Blackorby |first1=Charles |last2=Bossert |first2=Walter |last3=Donaldson |first3=David |title=Utilitarianism and the Theory of Justice |editor1-last=Arrow |editor1-first=Kenneth |editor2-last=Sen |editor2-first=Amartya |editor3-last=Suzumura |editor3-first=Kotaru |work=Handbook of Social Choice and Welfare |date=2002 |publisher=Elsevier |isbn=978-0-444-82914-6 |page=552 |url=https://books.google.com/books?id=rh10cOpltLsC&pg=PA552}}</ref>

===Marginalism ===
{{Details|Marginal utility}} 
*Under cardinal utility theory, the ''sign'' of the marginal utility of a good is the same for all the numerical representations of a particular preference structure.
*The ''magnitude'' of the marginal utility is not the same for all cardinal utility indices representing the same specific preference structure.
*The ''sign'' of the second [[derivative]] of a differentiable utility function that is cardinal, is the same for all the numerical representations of a particular preference structure. Given that this is usually a negative sign, there is room for a ''law of diminishing marginal utility'' in cardinal utility theory.
*The ''magnitude'' of the second derivative of a differentiable utility function is not the same for all cardinal utility indices representing the same specific preference structure.

=== Expected utility theory ===
{{Details|Expected utility theory}}
This type of indices involves choices under risk. In this case, {{math|''A'', ''B''}}, and {{math|''C''}}, are [[Lottery (probability)|lotteries]] associated with outcomes. Unlike cardinal utility theory under certainty, in which the possibility of moving from preferences to quantified utility was almost trivial, here it is paramount to be able to map preferences into the set of real numbers, so that the operation of mathematical expectation can be executed. Once the mapping is done, the introduction of additional assumptions would result in a consistent behavior of people regarding fair bets. But fair bets are, by definition, the result of comparing a gamble with an expected value of zero to some other gamble. Although it is impossible to model attitudes toward risk if one doesn't quantify utility, the theory should not be interpreted as measuring strength of preference under certainty.<ref>{{cite journal |last=Shoemaker |first=Paul |date=June 1982 |title=The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations |journal=Journal of Economic Literature |volume=20 |issue=2 |pages=529–563 |jstor=2724488}}</ref>

===Construction of the utility function===

Suppose that certain outcomes are associated with three states of nature, so that ''x''<sub>3</sub> is preferred over ''x''<sub>2</sub> which in turn is preferred over ''x''<sub>1</sub>; this set of outcomes, {{math|''X''}}, can be assumed to be a calculable money-prize in a controlled game of chance, unique up to one positive proportionality factor depending on the currency unit.

Let {{math|''L''<sub>1</sub>}} and {{math|''L''<sub>2</sub>}} be two lotteries with probabilities ''p''<sub>1</sub>, ''p''<sub>2</sub>, and ''p''<sub>3</sub> of ''x''<sub>1</sub>, ''x''<sub>2</sub>, and ''x''<sub>3</sub> respectively being

:<math>L_1 =(0.6, 0, 0.4),</math>
:<math>L_2 =(0,1,0)\ .</math>

Assume that someone has the following preference structure under risk:

:<math>L_{1} \succ L_{2},</math>

meaning that {{math|''L''<sub>1</sub>}} is preferred over {{math|''L''<sub>2</sub>}}. By modifying the values of {{math|''p''<sub>1</sub>}} and {{math|''p''<sub>3</sub>}} in {{math|''L''<sub>1</sub>}}, eventually there will be some appropriate values ({{math|''L''<sub>1'</sub>}}) for which she is found to be indifferent between it and {{math|''L''<sub>2</sub>}}&mdash;for example

:<math>L_{1}' =(0.5, 0, 0.5).</math>

Expected utility theory tells us that

:<math>EU(L_{1}') = EU(L_2)</math>

and so

:<math>(0.5) \times u(x_1)+(0.5) \times u(x_{3}) = 1 \times u(x_{2}).</math>

In this example from Majumdar<ref>{{cite journal |last1=Majumdar |first1=Tapas |date=February 1958 |title=Behaviourist Cardinalism in Utility Theory |journal=Economica |volume=25 |issue=97 |pages=26–33 |doi=10.2307/2550691 |jstor=2550691}}</ref> fixing the zero value of the utility index such that the utility of {{math|''x''<sub>1</sub>}} is 0, and by choosing the scale so that the utility of {{math|''x''<sub>2</sub>}} equals 1, gives

:<math>(0.5) \times u(x_{3})=1.</math>
:<math>u(x_{3}) = 2.</math>

===Intertemporal utility===
{{Details|Intertemporal choice}}
Models of utility with several periods, in which people discount future values of utility, need to employ cardinalities in order to have well-behaved utility functions. According to Paul Samuelson the maximization of the discounted sum of future utilities implies that a person can rank utility differences.{{sfnp|Moscati|2012|p=20}}