More direct way to derive indirect utility function from expenditure function

Question

I have this general form for a expenditure function $e(p,u)=f(u)\cdot g(p)$ where $f(u)$ is increasing monotonic. How can I derive a functional form for an indirect utility function from this expenditure function?

Answer 1

Let $e (p, v (p, u))=r$, hence $f (v (p, u)) g (p)=r$.

Then use the inverse function theorem, i.e. $(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}$

$v (p, r)=h (r/g (p))$, where $h=f^{-1}$.

Answer 2

To answer your question in the comments,

Simply use the Walras' law:

$ p'x(p,w) = w $ and differentiate with respect to w.

$p$ is the $L\times1$ price vector.