Monotonic transformation

Summary

• Given a sequence of $n$ numbers $\{ x_1,…,x_n \}$ a monotonic transformation is any transformation of the numbers $\{ x_1,…,x_n \}$ into the numbers $\{ y_1,…,y_n \}$ such that order is preserved.
• Example. Transforming $\{ 2,8,9 \}$ into $\{ 1,2,11 \}$ is a monotonic transformation
• Given a sequence of $n$ numbers $\{ x_1,…,x_n \}$ and a strictly increasing function $f$ , a transformation defined by $y_i=f\left( x_i \right)$ is a monotonic transformation
• Given a utility function $u=u\left( x_1,x_2 \right)$ , a transformation by a strictly increasing function $f$ , $v=f\left( u \right)$ is a monotonic transformation.
• Example:
• $u\left( x_1,x_2 \right)=x_1x_2$ and $v=u^2$ is a monotonic transformation.
• Combining, $v\left( x_1,x_2 \right)=x_1^2x_2^2$
• Result: Any monotonic transformation of a utility function will represent the same preferences.