Demand curve for peanuts

Question

Question:

a) Johnny consumes peanuts (x1) and a composite good (x2). His utility function is U = x1x2. His marginal utilities are MU1 = x2 and MU2 = x1. Johnny's budget is $20 and the price of the composite good is $1. Derive Johnny's demand function for peanuts.

b) Ambrose consumes peanuts (x1) and a composite good (x2).He has a utility functionU = 4 x1 + x2. This means his MU1 = 2/√x1 and his MU2 = 1 . The price of the composite good is p2 = 1. His budget is $20 per month. Derive Ambrose's demand function for peanuts. How does it compare with Johnny's demand curve for peanuts?

Solution:

a) U = x₁x₂

MRS = MU₁/MU₂ = x₂/x₁

Now, MRS = P₁/P₂ = P/1 = x₂/x₁ => x₂ = 2Px₁ {taking P₁ = P}

Putting this value in budget equation:

Px₁ + x₂ = 20

1.   Px₁ + 2Px₁ = 30
2.   Px₁ = 10
3.   x₁ = 10/P
4.   x₂ = 20

b) U = 4 Öx₁ + x2.

MRS = MU₁/ MU₂ = (2/Öx₁)/1 = 2/Öx₁

Now, MRS = P₁/P₂ = P/1 = 2/Öx₁

1.   Öx₁=2/P
2.   x₁ = 4/P²

Therefore, Ambrose's demand for peanuts does not depend upon his income, while Johnny's demand for peanuts does depend upon his income.