A total income of I is given to spend on two goods x₁ and x₂ with prices p₁ and p₂ respectively. Your utility function for x₁ and x₂ is:

U (x₁, x₂) = x₁³ x₂²

Using this information, solve the following questions:

(a) Using the Lagrange Method, solve for your optimal choice for x₁ and x₂ as functions of p₁ and p₂ and I

(b) What is the maximum utility you can attain given prices p₁ and p₂ with an income I? What is this function called?

(c) Using your result from (b), what is the minimum expenditure you need to attain a utility level u given prices p₁ and p₂? What is this function called?

(d) Derive the compensated demand functions h₁ and h₂ using the Shephard's Lemma. Use your result in (c) to solve this.

(e) Find the compensated demand functions by setting up an expenditure minimization problem. Verify that your answers are the same as in (d).