# Monotone transformation of utility

We have learned that any "strictly positive monotonous transformation" of utility functions is okay, as long as they preserve the ranking of choices implied by the underlying preferences.

Consider $U(x, y) = (x/y)^\alpha$

The cross derivative is $\frac{\partial^2}{\partial y \partial x} U(x,y) = -\alpha^2 (\frac{x}{y})^{\alpha - 1}$

Now, consider the monotonous transformation $V(x,y) = \log U(x,y) = \alpha \log x - \alpha \log y$. The cross-derivative of $V$ is $0$.

The cross-derivative incorporates important information on how the ranking of choices of $x$ changes, as we change $y$. Clearly, these two different cross-derivatives cannot be generated from the same underlying preference ranking - or am I mistaken?

• I think you see too much meaning into the cross derivative. This might become more clear if you were to formalize this "ranking of choice of $x$". – Giskard Aug 25 '15 at 6:24

The preservation of ranking is over consumption bundles, not individual goods. So the one and only question is whether the ranking of bundles $(x_i,y_i),\; (x_k,y_k); \forall i,k$ is preserved or not.
$$(x_i,y_i)>_{pr}\; (x_k,y_k) \implies \exists \;U, U_i > U_k$$
The logarithm is a strictly monotonic transformation of any amenable to it function as a mathematical fact, irrespective of what we use the function for. So it follows that $$U_i > U_k \implies \ln U_i > \ln U_k ,\; \forall i,k$$