Additive utility
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In economics, additive utility is a cardinal utility function with the sigma additivity property.
| A {\displaystyle A} | u ( A ) {\displaystyle u(A)} |
|---|---|
| ∅ {\displaystyle \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 {\displaystyle S} be a finite set of items. A cardinal utility function u : 2 S → R {\displaystyle u:2^{S}\to \mathbb {R} }, where 2 S {\displaystyle 2^{S}} is the power set of S {\displaystyle S}, is additive if for any A , B ⊆ S {\displaystyle A,B\subseteq S},
u ( A ) + u ( B ) = u ( A ∪ B ) + u ( A ∩ B ) . {\displaystyle u(A)+u(B)=u(A\cup B)+u(A\cap B).}
It follows that for any A ⊆ S {\displaystyle A\subseteq S},
u ( A ) = u ( ∅ ) + ∑ x ∈ A ( u ( { x } ) − u ( ∅ ) ) . {\displaystyle u(A)=u(\emptyset )+\sum _{x\in A}{\big (}u(\{x\})-u(\emptyset ){\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.
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