# How to derive the Indirect Utility Function and Marshallian Demand from Homothetic Preferences

I need to prove the following relationships:

1 - If preferences are homothetic, then the indirect utility function can be written as $$v(p, w) = v(p) · w$$.

2 - If preferences are homothetic, then the Marshallian demand functions take the form $$x_i(p, w) = x_i(p) · w$$, i.e., they are linear functions of income.

I know that I can use the Roy's Identity and the connections below: $$e(p,v(p,w))=w$$ $$v(p,e(p,u))=u,$$ but I can't organize this answer.

Thanks!

• What happens if you apply Roy's identity to the utility function of point 1 ? (it is embarassing btw to use the same notation $v$ for denoting two different functions) – Bertrand Feb 9 at 21:28
• If I use a Roy's Identity in point 1, I derive the Marshallian demand from item 2. I am wondering how to prove that point 1 is true. (Thanks for the tip! It was just to indicate a function of p, but I can use another notation.) – Rômullo Eduardo Feb 9 at 22:01
• This may help: economics.stackexchange.com/questions/8519/… – Bertrand Feb 10 at 10:40
• Thank you, @Bertrand! – Rômullo Eduardo Feb 10 at 15:07