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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!

Rômullo Eduardo
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    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 09 '21 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 09 '21 at 22:01
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    This may help: https://economics.stackexchange.com/questions/8519/how-to-show-that-a-homothetic-utility-function-has-demand-functions-which-are-li?rq=1 – Bertrand Feb 10 '21 at 10:40
  • Thank you, @Bertrand! – Rômullo Eduardo Feb 10 '21 at 15:07

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