If utility is U(x,y) = x^0.3 y^0.7 (i.e x* = 0.3I/Px and y* = 0.7I/Py). What is the indirect utility function and expenditure function?

Answer 1

You are almost already there. Getting x* and y* is the hard part.

The indirect utility function is:
v(Px,Py,I) where I is income
Simply substitute your x* and y* in the original utility equation
maximum U = (0.3I/Px)^.3 * (0.7I/Py)^.7
collapse terms
=(.6968I^.3)/Px^.3 * (.7791I^.7)/Py^.7
= (.5428 * I)/(Px^.3 * Py^7)
= v(Px,Py,I)

I presume for the expenditure function you want the functional form E(Px,Py,U). We know I=PxX + PyY. Here, simply use the above equation and get income-I all by itself.
I = (U * (Px^.3 * Py^.7))/.6968
= E(Px,Py,U)

I hope this helps

Answer 2

Oops

I = (U * (Px^.3 * Py^.7))/.5428