Additive utility explained

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

Additive utility

A	u(A)
\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\to\R} , where 2 ^{S} is the power set of S , is additive if for any A, B\subseteq S ,u(A) + u(B) = u(A\cup B) + u(A\cap B). It follows that for any A\subseteq S ,u(A) = u(\emptyset) + \sum _{x\in} ( u(\{x\}) - u(\emptyset) ) . 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