<p>Recovering the utility function given Marshallian demands only is called the 'Integrability problem'.</p>

<p>In general, it is quite difficult. However, there are some conditions under which it is possible. Kim Border has some <a href="http://people.hss.caltech.edu/~kcb/Notes/Demand4-Integrability.pdf" rel="nofollow noreferrer">notes</a> on this topic on his website.</p>

CC BY-SA 3.0
2018-01-22T18:12:30.660Z
2018-01-22T18:12:30.660Z
Theoretical Economist
<p>I have been trying to derive the indirect utility function $(V(p,y))$, where $p$ is price and $w$ is wage, given the Marshallian demand functions $x(p,y)$ with the help of Roy’s identity (the utility function is not known. We only know the Marshallian Demand functions)</p>

<p>I would need to take the integral of the marshallian demand to obtain V. But this seems rather complicated to me.</p>

<p>If Roy’s Identity is:</p>

<p>$$ x(p,y) = - \frac{\partial V(p,y)/ \partial p}{\partial V(p,y)/ \partial y}$$ </p>

<p>So if I know $x$ how can I derive $V$?</p>

<p>I have been thinking about this and trying some equations, but unfortunately I could not come up with a solution.</p>

<p>Thank you for your help!</p>

<microeconomics><mathematical-economics>
How to derive the Indirect Utility Function from the Marshallian Demand Function?
krystal
# How to derive the Indirect Utility Function from the Marshallian Demand Function?

(Asked by: username_0 on 2018-01-20)

I have been trying to derive the indirect utility function $(V(p,y))$, where $p$ is price and $w$ is wage, given the Marshallian demand functions $x(p,y)$ with the help of Roy’s identity (the utility function is not known. We only know the Marshallian Demand functions)

I would need to take the integral of the marshallian demand to obtain V. But this seems rather complicated to me.

If Roy’s Identity is:

$$ x(p,y) = - \frac{\partial V(p,y)/ \partial p}{\partial V(p,y)/ \partial y}$$

So if I know $x$ how can I derive $V$?

I have been thinking about this and trying some equations, but unfortunately I could not come up with a solution.

Thank you for your help!

**Tags:** microeconomics, mathematical-economics

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## 1 answers

### Answer by username_1 on 2018-01-22. Score: 3

Recovering the utility function given Marshallian demands only is called the 'Integrability problem'.

In general, it is quite difficult. However, there are some conditions under which it is possible. Kim Border has some [notes](http://people.hss.caltech.edu/~kcb/Notes/Demand4-Integrability.pdf) on this topic on his website.


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