Finance:Additive utility

In economics, additive utility is a cardinal utility function with the sigma additivity property.

Additive utility

[math]\displaystyle{ A }[/math]	[math]\displaystyle{ u(A) }[/math]
[math]\displaystyle{ \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]\displaystyle{ S }[/math] be a finite set of items. A cardinal utility function [math]\displaystyle{ u:2^S\to\R }[/math], where [math]\displaystyle{ 2^S }[/math] is the power set of [math]\displaystyle{ S }[/math], is additive if for any [math]\displaystyle{ A, B\subseteq S }[/math],

[math]\displaystyle{ u(A)+u(B)=u(A\cup B)+u(A\cap B). }[/math]

It follows that for any [math]\displaystyle{ A\subseteq S }[/math],

[math]\displaystyle{ 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.

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.

See also

• Utility functions on indivisible goods
• Independent goods
• Submodular set function
• Supermodular set function