micro doubt

[tanudas]

I have a small doubt from utility theory. Why do utility functions that represent monotonic transformation, have the same preferences ? Their indifferent curves may be above or below each other, so does the statement not contradict itself ?

[Dreyfus]

I guess the monotonic transformation only changes the position of level curves (i.e. Indifference curves) but monotonically transformed utility functions retains all the properties of the original utility function. The consumer's preferences among different goods remains the same even after the transformation of utility function the only change is in the level curves position!
Take for an example U(x,y)=x⅔.y⅓ and its monotonically transformed version V(x,y)=⅔lnx + ⅓lny
The slope of both the function is -2y/x the only change is in the level curve, for function 1 level curve is C1=x⅔.y⅓ and K1=⅔lnx + ⅓lny which can be represented as e^K1=C2 = x⅔.y⅓.