Suppose you have a total income ofIto spend on three goodsx₁,x₂andx₃, with unit pricesp₁,p₂andp₃respectively. Your taste can be represented by the utility function

u(x₁,x₂,x₃) =x₁^ax₂^bx₃^1-a-b

whereaandbare between 0 and 1, anda+b< 1.

(a) What is your optimal choice forx₁,x₂andx₃? 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^a₂x_n^{1- a}₁-a₂-...- a_n-1

wherea₁,a₂,..., a_n-1are all between 0 and 1, anda₁+a₂+...+a_n-1<1

The unit prices arep₁,p₂,...,p_nrespectively.

Without calculations, write down the demand function forx₁and the share of income spent on it.

Continue from question 1, return to the 3 good case.

(a) Derive the compensated demand functions for the three goods using the Shepard's Lemma.

(b) Use your results in (a) to verify the expenditure function you obtained in question 1(d).

(c)Supposea=b=1/3,I= 60,p2= 2,p3= 2. Calculate the substitution and income effects onx1whenp1increases from 2 to 3.