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.