Problem - Suppose you have a total income of I to spend on three goods x₁, x₂ and x₃, with unit prices p₁, p₂ and p₃ respectively. Your taste can be represented by the utility function u(x₁, x₂, x₃) = x₁^ax₂^bx₃^1-a-b where a and b are between 0 and 1, and a + b < 1.

(a) What is your optimal choice for x₁, x₂ and x₃? Use the Lagrange Method.

(b) What are the shares of income spent on the three goods respectively?

(c) Derive your indirect utility function.

(d) Derive your expenditure function.

(e) If there are n goods and the utility function is: u(x₁, x₂, ... ,x_n) = x₁^a1x₂^a2 · · · x_n^{1- a1 -a2 - · · · - an-1} where a₁, a₂,..., a_n-1 are all between 0 and 1, and a₁+a₂+...+a_n-1 < 1.

The unit prices are p₁ , p₂,..., p_n respectively.

Without calculations, write down the demand function for x₁ and the share of income spent on it.