# Why might a monotone increasing but nonlinear transformation of a utility function not represent the same preferences?

According to a textbook, a monotone increasing but nonlinear transformation of a utility function might not represent the same preferences. Why is it so?

An example of such preference would be appreciated.

• Maybe I am underthinking this, but the transformation has to be strictly monotone increasing. I think it is simple to see why, as only the strictly monotone function would keep the ordering of any preference relation intact. But you probably meant something else? – IMA Oct 21 '19 at 9:09
• Thank you. I somehow thought that monotone increasing the same as strictly monotone increasing. – Aqqqq Oct 21 '19 at 9:27

One reason I could think of is regarding convexity of the function (and hence risk aversion). For example, $$u(x) = x$$ is risk neutral, but $$v(x) = \ln(u(x)) = \ln(x)$$ is risk averse.