Additive utility

In economics, '''additive utility''' is a cardinal utility function with the sigma additivity property.<ref>{{Cite ComSoc Handbook 2016}}</ref>{{rp|287-288}}

{| class="wikitable" style="float:right" |+Additive utility |- ! <math>A</math> !! <math>u(A)</math> |- | <math>\emptyset</math> || 0 |- | apple || 5 |- | hat || 7 |- | apple and hat || 12 |} Additivity (also called ''linearity'' or ''modularity'') means that "the whole is equal to the sum of its parts." That is, the utility of a set of items is the sum of the utilities of each item separately. Let <math>S</math> be a finite set of items. A cardinal utility function <math>u:2^S\to\R</math>, where <math>2^S</math> is the power set of <math>S</math>, is additive if for any <math>A, B\subseteq S</math>, :<math>u(A)+u(B)=u(A\cup B)+u(A\cap B).</math> It follows that for any <math>A\subseteq S</math>, :<math>u(A)=u(\emptyset)+\sum_{x\in A}\big(u(\{x\})-u(\emptyset)\big).</math> An additive utility function is characteristic of independent goods. For example, an apple and a hat are considered independent: the utility a person receives from having an apple is the same whether or not he has a hat, and vice versa. A typical utility function for this case is given at the right.<ref>{{Cite journal |last=Fuhrken |first=Gebhard |last2=Richter |first2=Marcel K. |date=1991-03-01 |title=Additive utility |url=https://doi.org/10.1007/BF01210575 |journal=Economic Theory |language=en |volume=1 |issue=1 |pages=83–105 |doi=10.1007/BF01210575 |issn=1432-0479|url-access=subscription }}</ref>

== Notes == * As mentioned above, additivity is a property of cardinal utility functions. An analogous property of ordinal utility functions is weakly additive. * A utility function is additive if and only if it is both submodular and supermodular.

== References == {{reflist}}

Category:Utility function types