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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!

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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 on this topic on his website.

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