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?

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

Oops

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

To find the indirect utility function and the expenditure function, we need to follow these steps:

1. Start with the utility function U(x, y) = x^0.3 * y^0.7.

2. Since the indirect utility function represents the maximum utility that can be achieved for a given level of income and prices, we need to solve the consumer's optimization problem by maximizing utility subject to the budget constraint.

3. The budget constraint is given by: Pxx + Pyy = I, where Px and Py are the prices of goods x and y, and I is the consumer's income.

4. To maximize utility subject to the budget constraint, we can use the method of Lagrange multipliers.

5. Define the Lagrangian function L(x, y, λ) as follows: L(x, y, λ) = U(x, y) + λ(I - Px*x - Py*y).

6. Take partial derivatives of L with respect to x, y, and λ, and set them equal to zero to find the optimal values of x, y, and λ.

7. First, take the partial derivative of L with respect to x: ∂L/∂x = 0.3x^(-0.7) * y^0.7 - λPx = 0.

8. Second, take the partial derivative of L with respect to y: ∂L/∂y = 0.7x^0.3 * y^(-0.3) - λPy = 0.

9. Third, take the partial derivative of L with respect to λ: ∂L/∂λ = I - Px*x - Py*y = 0.

10. Solve these three equations simultaneously to find the optimal values of x, y, and λ.

11. Once you obtain the optimal values, substitute them back into the utility function U(x, y) to find the indirect utility function V(Px, Py, I).

12. The indirect utility function represents the maximum utility that can be attained given the prices and income: V(Px, Py, I) = U(x*, y*), where x* and y* are the optimal values obtained in the previous steps.

13. Finally, to find the expenditure function, substitute the optimal values of x* and y* back into the budget constraint to solve for I: Px*x* + Py*y* = I.

Following these steps should allow you to find the indirect utility function V(Px, Py, I) and the expenditure function I(Px, Py, V).