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.