I know that by Roy's identity, the Marshallian demand for a good (i) is $x^*_i = -\frac{V_i}{V_y}$, where $V(Y,P)$ is the indirect utility function, $V_i=\frac{\partial V}{\partial P_i}$, and $V_y=\frac{\partial V}{\partial Y} $.
I also know that a monotonic increasing transformation means strictly increasing. How do I go about proving that Roy's identity holds if the utility function is subjected to a monotonic increasing transformation?
I have shown that Roy's identity holds, but without the "montonic increasing transformation" condition. How do I put it in?
Let $W$ be a monotonic increasing transformation of $V$, i.e., $W=G.V$ for some strictly increasing function $G$. You have to show that, given Roy's identity holds for $V$, it also holds for $W$. By the chain rule $W_i=G'(V(Y,p)).V_i$ and $W_y=G'(V(Y,P)).V_y$, so that $x*=(-)V_i/V_y=(-)W_i/W_y$, which is what you needed to show. $\square$