<p>Let <span class="math-container">$e (p, v (p, u))=r$</span>, hence <span class="math-container">$f (v (p, u)) g (p)=r$</span>.</p>
<p>Then use the inverse function theorem, i.e. <span class="math-container">$(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}$</span></p>
<p><span class="math-container">$v (p, r)=h (r/g (p))$</span>, where <span class="math-container">$h=f^{-1}$</span>.</p>

CC BY-SA 3.0
2016-10-19T13:00:48.390Z
Thanks a lot! Do you know how could I prove that income elasticity is unitary for each $x_{i}$ ?
Renzo Mauricio Guzmán Anaya
CC BY-SA 4.0
2016-10-03T05:10:23.440Z
2022-10-14T00:56:36.983Z
2022-10-14T00:56:36.983Z
Rodrigo Penaloza
<p>To answer your question in the comments,</p>
<p>Simply use the Walras' law:</p>
<p><span class="math-container">$ p'x(p,w) = w $</span> and differentiate with respect to w.</p>
<p><span class="math-container">$p$</span> is the <span class="math-container">$L\times1$</span> price vector.</p>

CC BY-SA 4.0
2022-10-14T19:25:03.353Z
2022-10-14T19:25:03.353Z
Matteo Bulgarelli
<p>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?</p>

<utility><consumer-theory>
More direct way to derive indirect utility function from expenditure function
Renzo Mauricio Guzmán Anaya
# More direct way to derive indirect utility function from expenditure function

(Asked by: username_0 on 2016-10-02)

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?

**Tags:** utility, consumer-theory

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## 2 answers

### Answer by username_1 on 2016-10-03. Score: 2 | Accepted answer

Let $e (p, v (p, u))=r$, hence $f (v (p, u)) g (p)=r$.

Then use the inverse function theorem, i.e. $(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}$

$v (p, r)=h (r/g (p))$, where $h=f^{-1}$.

> **Comments:**
> **username_0**: Thanks a lot! Do you know how could I prove that income elasticity is unitary for each $x_{i}$ ?  

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### Answer by username_2 on 2022-10-14. Score: 0

To answer your question in the comments,

Simply use the Walras' law:

$ p'x(p,w) = w $ and differentiate with respect to w.

$p$ is the $L\times1$ price vector.


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