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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!

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    $\begingroup$ 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) $\endgroup$
    – Bertrand
    Commented Feb 9, 2021 at 21:28
  • $\begingroup$ 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.) $\endgroup$ Commented Feb 9, 2021 at 22:01
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    $\begingroup$ This may help: economics.stackexchange.com/questions/8519/… $\endgroup$
    – Bertrand
    Commented Feb 10, 2021 at 10:40

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