Monotone transformation of utility

Question

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?

Answer 1

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.

If preferences are rational (complete and transitive), and continuous, then they can be represented by a continuous utility function. So if these hold and

$$(x_i,y_i)>_{pr}\; (x_k,y_k) \implies \exists \;U, U_i > U_k$$

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 $$U_i > U_k \implies \ln U_i > \ln U_k ,\; \forall i,k$$

and the ranking of bundles is preserved. Whether due to the transformation we lose other information or not (that we may have wished we could have kept) does not concern the representation/transformation theorem.