3  Utility

Economists often use numbers to represent strength of preference. This is done through utility functions.

A utility function associates a number with each member of the universe. For example:

• Banana: 3
• Orange: 2
• Apple: 1

This does not mean that I rate bananas three times higher than apples. It simply means that I prefer bananas to apples. This utility scale is ordinal, not cardinal. The following is equivalent:

• Banana: 300
• Orange: 2
• Apple: 1

Formally, the utility function u(\cdot):

• maps the set of alternatives into the set of real numbers
• assigns larger numbers to preferred alternatives.

For example, we might write:

\begin{align*} u(\text{banana})&=3 \\[6pt] u(\text{orange})&=2 \\[6pt] u(\text{apple})&=1 \end{align*}

The rank of those numbers gives us the preference relation:

x\succcurlyeq y \Longleftrightarrow u(x)\geq u(y)

x\succ y \Longleftrightarrow u(x)>u(y)

x\sim y \Longleftrightarrow u(x)=u(y)

Again, following from the above:

u(\text{banana})=3>2=u(\text{orange})\Longleftrightarrow \text{banana}\succ \text{orange}

This calculation of utility is not how the mind actually works. But under the axioms of completeness and transitivity, the consumer behaves as if they have a utility function u(x_i) over outcomes x_i.