Editing Utility (section)

== Utility function ==
Consider a set of alternatives among which a person has a preference ordering. A '''utility function''' represents that ordering if it is possible to assign a [[real number]] to each alternative in such a manner that ''alternative a'' is assigned a number greater than ''alternative b'' if and only if the individual prefers ''alternative a'' to ''alternative b''. In this situation, someone who selects the most preferred alternative must also choose one that maximizes the associated utility function. 

Suppose James has utility function <math>U = \sqrt{xy}</math> such that <math>x</math> is the number of apples and <math>y</math> is the number of chocolates. Alternative A has <math>x = 9</math> apples and <math>y = 16</math> chocolates; alternative B has <math>x = 13</math> apples and <math>y = 13</math> chocolates. Putting the values <math>x, y</math> into the utility function yields <math>\sqrt{9 \times 16} = 12</math> for alternative A and <math>\sqrt{13 \times 13} = 13</math> for B, so James prefers alternative B. In general economic terms, a utility function ranks preferences concerning a set of goods and services.

[[Gérard Debreu]] derived the conditions required for a preference ordering to be representable by a utility function.<ref>{{citation|last=Debreu|first=Gérard|title=Decision processes|pages=159–167|year=1954|postscript=.|editor-last1=Thrall|editor-first1=Robert M.|contribution=Representation of a preference ordering by a numerical function|location=New York|publisher=Wiley|oclc=639321|editor-last2=Coombs|editor-first2=Clyde H.|editor-last3=Raiffa|editor-first3=Howard|editor-link2=Clyde H. Coombs|editor-link3=Howard Raiffa}}</ref> For a finite set of alternatives, these require only that the preference ordering is complete (so the individual can determine which of any two alternatives is preferred or that they are indifferent), and that the preference order is [[Transitive relation|transitive]].

Suppose the set of alternatives is not finite (for example, even if the number of goods is finite, the quantity chosen can be any real number on an interval). In that case, a continuous utility function exists representing a consumer's preferences if and only if the consumer's preferences are complete, transitive, and continuous.<ref>{{citation|last1=Jehle|first1=Geoffrey|last2=Reny|first2=Philipp|title=Advanced Microeconomic Theory|pages=13–16|year=2011|postscript=.|publisher=Prentice Hall, Financial Times|isbn=978-0-273-73191-7}}</ref>