Additive utility

{{Verification|date=March 2021}}

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

class="wikitable" style="float:right" |+Additive utility
$A$	$u(A)$
$\backslash emptyset$	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 $S$ be a finite set of items. A cardinal utility function $u:2^S\backslash to\backslash R$, where $2^S$ is the power set of $S$, is additive if for any $A,\; B\backslash subseteq\; S$,

:$u(A)+u(B)=u(A\backslash cup\; B)+u(A\backslash cap\; B).$

It follows that for any $A\backslash subseteq\; S$,

:$u(A)=u(\backslash emptyset)+\backslash sum\_\{x\backslash in\; A\}\backslash big(u(\backslash \{x\backslash \})-u(\backslash emptyset)\backslash big).$

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.