Calculating fat-tax

Question

Question:

Max has a utility function U =√ x1x2 where x1 is litres of ice-cream and x2 is boxes of strawberries. The marginal utility of a litre if ice-cream is MU1 =0.5 √x2/ x1 and the marginal utility of a box of strawberries is MU2 =0.5 √x1 /x2. The prices of x1 and x2 are both $2 and Max has a budget of $80.

(a) How much of each good will Max demand?

(b) A fat-tax of $2 per litre is placed on ice-cream so that it now costs Max $4 per litre. Everything else remains the same. How much of each good does Max now consume? How much tax does he pay?

(c) Now suppose that, instead of imposing a $2 tax on ice-cream, the government imposes a $20 income tax, reducing Max's budget to $60. Would Max prefer the $2 tax on ice-cream or the $20 reduction is his budget?

Solution:

U = (x₁x₂)^0.5, P₁ = 2, P₂ = 2, m = 80

Therefore, the budget equation is:

2x₁ + 2x₂ = 80

MU₁ = 0.5(x₂/x₁)^0.5, MU₂ = 0.5(x₁/x₂)^0.5

Therefore, MRS = MU₁/MU₂ = x₂/x₁

Setting MRS = P₁/P₂ = 1, we get,

x₂/x₁ = 1 => x₂ = x₁

a) Using the budget equation:

2x₁ + 2x₂ = 80

1.   x₁ + x₂ = 40
2.   2x₁ = 40
3.   x₁ = 20 = x₂

Therefore, he will demand 20 units each of both the goods.

b) Now, P₁ = 4

MRS = P₁/P₂

1.   x₂/x₁ = 4/2
2.   x₂/x₁= 2
3.   x₂ =2x₁

Putting it into the budget equation:

x₁ + x₂ = 40

1.   x₁ + 2x₁ = 40
2.   x₁ = 40/3
3.   x₂ = 80/3

Therefore, tax paid = 40/3 x 2 = 80/3

c) The new budget equation:

x₁ + x₂ = 60/2 = 30

Putting, x₁ = x₂ in the budget equation, we get,

1. 2x₁ = 30
2.   x₁ = 15
3.   x₂ = 15

Utility with fat-tax = (40 x 80/9)^0.5 = [40Ö2]/3= 18.86

Utility with income tax = 15

Therefore, Max will prefer the fat-tax on ice cream.