$y= (x_1x_2)^2-x_1x_2$

Let $x_1x_2=z$

then $y=z^2-z$ and $y'=2z-1$

If I can prove that y' is monotonically increasing, does that prove y is a homothetic function? if so then how can I prove that y' is monotonically increasing?

  • $\begingroup$ I added the specific point of my confusion. $\endgroup$ – xyz Jun 9 '17 at 12:01

A homothetic function is a monotonic transformation of a homogenous function. However, that function is not homogeneous.

For $x_1x_2 = y$, take then $f(y) = y^2 - y$.

If $f(y)$ is homogenous of degree $k$, it means that $f(t y) = t^k f(y)$, $\forall t>0$. In this case,

$f(t y) = (t y)^2 - t y = t^{2} y^2 - t y \neq t^k y^2 - t^k y = t^k (y^2 - y) = t^k (f(y))$

This concludes the proof.

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