<p>I need to prove the following relationships:</p>
<p>1 - If preferences are homothetic, then the indirect utility function can be written as
<span class="math-container">$v(p, w) = v(p) · w$</span>.</p>
<p>2 - If preferences are homothetic, then the Marshallian demand functions take the form
<span class="math-container">$x_i(p, w) = x_i(p) · w$</span>, i.e., they are linear functions of income.</p>
<p>I know that I can use the Roy's Identity and the connections below:
<span class="math-container">$$ e(p,v(p,w))=w$$</span> <span class="math-container">$$v(p,e(p,u))=u,$$</span>
but I can't organize this answer.</p>
<p>Thanks!</p>

CC BY-SA 4.0
2021-02-09T21:28:30.127Z
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
CC BY-SA 4.0
2021-02-09T22:01:01.777Z
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
CC BY-SA 4.0
2021-02-10T10:40:34.227Z
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
<utility><demand>
How to derive the Indirect Utility Function and Marshallian Demand from Homothetic Preferences
Rômullo Eduardo
# How to derive the Indirect Utility Function and Marshallian Demand from Homothetic Preferences

(Asked by: username_0 on 2021-02-09)

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!

**Tags:** utility, demand

> **Comments:**
> **username_1**: 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)  
> **username_0**: 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.)  
> **username_1**: 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  

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## 0 answers

