Question

4. Say that you have an income of $100, all of which you spend on 2 goods: x1 and x2. The price of x1 is $1, and the price of x2 is $2.

(a) The government imposes a quantity tax of on the consumption of x1. Write down an equation for your budget constraint.

(b) The government initially sets the quantity tax at = 0:5. What is the slope of the budget constraint in this case?

(c) Now say that the government increases the tax to = 0:75. How does this increase in the quantity tax change the slope of the budget line?

Answer 1

Ans)- Given, Income (M) = $100

Price of good X₁ , P₁ = $1

Price of good X₁ , P₂ = $2

So, for this situation, the budget constraint for the consumer will be-

P₁X₁ + P₂X₂ = M, with slope = P₁/P₂ [slope can be calculated by differentiating X₂ w.r.t. X₁]

X₁ + 2X₂ = 100 with slope = P₁/P₂ = ½

a) If government imposes a quantity tax of on the consumption of X₁, then the effective price of good X₁ faced by the consumer will increase the by the quantity tax rate for each consuming each unit of good X₁.

Let ‘t’ be the quantity tax rate then the effective price of good X₁ faced by consumer will be = P₁ + t

Thus, new budget constraint will look like-

(P₁+t) X₁ + P₂X₂ = M with slope = (P₁+t)/P₂

Put prices and income

Equation of budget constraint, [(1+t) X₁ + 2X₂ = 100]   with slope= (1+t)/2

b) If govt. sets the quantity tax at, t=0.5, then

budget constraint will be-

(P₁+t) X₁ + P₂X₂ = M with slope = (P₁+t)/P₂

Put, P₁ =1 , P₂ =2 M=100 and t=0.5

(1+0.5) X₁ + 2X₂ = 100   with slope= (1+0.5)/2

i.e. budget line will be, 1.5X₁ + 2X₂ =100

and Slope of the budget constraint will be,

Slope = (1+t)/2 = (1+0.5)/2 = 1.5/2

[Slope = 0.75]

c) Now, if govt. sets quantity tax at, t=0.75 then

budget constraint will be-

(P₁+t) X₁ + P₂X₂ = M with slope = (P₁+t)/P₂

Put, P₁ =1 , P₂ =2 M=100 and t=0.75

(1+0.75) X₁ + 2X₂ = 100   with slope= (1+0.75)/2

i.e. new budget line will be, 1.75X₁ + 2X₂ =100

and new Slope of the budget constraint will be,

Slope = (1+t)/2 = (1+0.75)/2 = 1.75/2

[Slope = 0.875]

Hence, with the increase in quantity tax from 0.5 to 0.75, slope of the budget constraint increased from 0.75 to 0.875.