Editing Utility (section)

==Preference==
While [[Preference (economics)|preference]]s are the conventional foundation of choice theory in [[microeconomics]], it is often convenient to represent preferences with a utility [[Function (mathematics)|function]].  Let ''X'' be the '''consumption set''', the set of all mutually exclusive baskets the consumer could consume.  The consumer's '''utility function''' <math> u\colon X\to \R</math> ranks each possible outcome in the consumption set.  If the consumer strictly prefers ''x'' to ''y'' or is indifferent between them, then <math>u(x)\geq u(y)</math>.

For example, suppose a consumer's consumption set is ''X'' =&nbsp;{nothing, 1&nbsp;apple,1&nbsp;orange, 1&nbsp;apple and 1&nbsp;orange, 2&nbsp;apples, 2&nbsp;oranges}, and his utility function is ''u''(nothing)&nbsp;=&nbsp;0, ''u''(1&nbsp;apple)&nbsp;=&nbsp;1, ''u''(1&nbsp;orange)&nbsp;=&nbsp;2, ''u''(1&nbsp;apple and 1&nbsp;orange)&nbsp;=&nbsp;5, ''u''(2&nbsp;apples)&nbsp;=&nbsp;2 and ''u''(2&nbsp;oranges)&nbsp;=&nbsp;4.  Then this consumer prefers 1 orange to 1 apple but prefers one of each to 2 oranges.

In micro-economic models, there is usually a finite set of L commodities, and a consumer may consume an arbitrary amount of each commodity.  This gives a consumption set of <math>\R^L_+</math>, and each package <math>x \in \R^L_+</math> is a vector containing the amounts of each commodity.  For the example, there are two commodities: apples and oranges.  If we say apples are the first commodity, and oranges the second, then the consumption set is <math>X =\R^2_+</math> and ''u''(0,&nbsp;0)&nbsp;=&nbsp;0, ''u''(1,&nbsp;0)&nbsp;=&nbsp;1, ''u''(0,&nbsp;1)&nbsp;=&nbsp;2, ''u''(1,&nbsp;1)&nbsp;=&nbsp;5, ''u''(2,&nbsp;0)&nbsp;=&nbsp;2, ''u''(0,&nbsp;2)&nbsp;=&nbsp;4 as before.  For ''u'' to be a utility function on&nbsp;''X'', however, it must be defined for every package in&nbsp;''X'', so now the function must be defined for fractional apples and oranges too. One function that would fit these numbers is <math> u(x_\text{apples}, x_\text{oranges}) = x_\text{apples} + 2 x_\text{oranges} + 2x_\text{apples} x_\text{oranges}. </math>

Preferences have three main '''properties''':

* '''Completeness''' 

Assume an individual has two choices, A and B. By ranking the two choices, one and only one of the following relationships is true: an individual strictly prefers A (A&nbsp;>&nbsp;B); an individual strictly prefers B (B>A); an individual is indifferent between A and B (A&nbsp;=&nbsp;B).
Either ''a'' ≥ ''b'' OR ''b'' ≥ ''a'' (OR both) for all (''a'',''b'')

*  '''Transitivity''' 

Individuals' preferences are consistent over bundles. If an individual prefers bundle A to bundle B and bundle B to bundle C, then it can be assumed that the individual prefers bundle A to bundle C.
(If ''a'' ≥ ''b'' and ''b'' ≥ ''c'', then ''a'' ≥ ''c'' for all (''a'',''b'',''c'')).

* '''Non-satiation''' or '''monotonicity'''
If bundle A contains all the goods that a bundle B contains, but A also includes more of at least one good than B. The individual prefers A over B.<ref>{{Cite web |title=nonsatiation |url=https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100237926 |access-date=2024-07-18 |website=Oxford Reference |language=en }}</ref> If, for example, bundle A&nbsp;=&nbsp;{1&nbsp;apple,2&nbsp;oranges}, and bundle B&nbsp;=&nbsp;{1&nbsp;apple,1&nbsp;orange}, then A is preferred over B.

===Revealed preference===
It was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer relative utilities from observed choice. These 'revealed preferences', as termed by [[Paul Samuelson]], were revealed e.g. in people's willingness to pay: 
<blockquote>
Utility is assumed to be correlative to Desire or Want. It has been argued already that desires cannot be measured directly, but only indirectly, by the outward phenomena which they cause: and that in those cases with which economics is mainly concerned the measure is found by the price which a person is willing to pay for the fulfillment or satisfaction of his desire.<ref>{{cite book |first=Alfred |last=Marshall |year=1920 |title=Principles of Economics. An introductory volume |edition=8th |location=London |publisher=Macmillan }}</ref>{{RP|78}}</blockquote>