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The quickest ways to go between these functions are using the duality results and some theorems. I'll list a few for you and tell you what they find. You can Google the theorems if you wish to read more about them

Some duality results -

e(p, v(p, y)) = y

v(p, e(p, u)) = u

xh (p, u) = x(p, e(p, u))

x(p, y) = xh (p, v(p, y))

e(p, u) = v-1 (p, y)

Where e(p, u) is the expenditure function. v(p, y) is the indirect utility function. y is income. xh are the hicksian demands. x is he marshallian demands. p is a vector of prices.

Roy's identity - lets you go from the indirect utility function to the marshallian demand functions

Shepard's lemma - lets you go from the expenditure function to the hicksian demand functions

That is, if you know the marshallian demands you can easily find the hicksian demands. One way to do this is to use the marshallian demands to find the indirect utility function, then use the 5th theorem which states that the expenditure function and Indirect utility function are inverses. Then using shepards lemma.

Similarly, if you have the hicksian demands and wish to find the marshallian demands. You can use the hicksian demands to find the expenditure function then invert it for the indirect utility function then apply roys identity.

Hope that helps

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