Cardinal and ordinal utility

Summary

Cardinal and ordinal utility function

• We say that a utility function is cardinal if the actual value assigned by the u tility f unction can be interpreted.
• Example:
• Say that $$u=u\left( x_1,x_2 \right)=x_1x_2$$ is a cardinal utility function
• Then $$u\left( 2,4 \right)=8$$ and $$u\left( 4,4 \right)=16$$ and the bundle $$\left( 4,4 \right)$$ provides twice as much utility as $$\left( 2,4 \right)$$ .
• We say that a utility function is ordinal if the actual value assigned by the utility function has no interpretation.
• An ordinal utility function can only be used to rank bundles.
• Example:
• Say that $$u=u\left( x_1,x_2 \right)=x_1x_2$$ is an ordinal utility function
• Then $$u\left( 2,4 \right)=8$$ and $$u\left( 4,4 \right)=16$$ but all you can say is that the bundle $$\left( 4,4 \right)$$ is strictly preferred to $$\left( 2,4 \right)$$ (you cannot conclude that is is “twice as good”).

Ordinal utility function from preferences

• Given: A total order $$≽$$ representing weak preferences.
• We can create an ordinal utility function from $$≽$$
• Ordinal utility functions are not unique .
• Example: $$u\left( x_1,x_2 \right)=x_1x_2$$ and $$v\left( x_1,x_2 \right)=x_1^2x_2^2$$ will order bundles identically .
• Even though such an ordinal utility function is not unique, each one will represent exactly the same preferences.