More direct way to derive indirect utility function from expenditure function

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?

1 Answer

Make e (p, V (p, r))=r, hence f (V (p, r)) g (p)=r. Then use the inverse function theorem: V (p, r)=h (r/g (p)) Where h is the inverse of f.

• Thanks a lot! Do you know how could I prove that income elasticity is unitary for each $x_{i}$ ? – Renzo Mauricio Guzmán Anaya Oct 19 '16 at 13:00