# How to show that Roy's identity holds in the case of a monotonic increasing transformation?

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?

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## migrated from economics.stackexchange.comMay 1 '12 at 19:56

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I've edited your question so that it is readable, but I'd recommend just looking this up. When posting (as here and in your other question), please be sure to define things (e.g. notice how I specified for you what V(), $V_i$ and so on mean. Otherwise your reader is left to guess, and most people won't have the patience for that. –  Jason B Nov 14 '11 at 4:34

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. $\square$