Question

Suppose the market for grass seed can be expressed as Q_D = 100 - 5p and Q_S = 40 + 5p. Answer the following:

a.If the government collects a $5 specific tax from the consumers, what price will consumers pay after the tax? What price will the sellers receive after the tax? How much the tax revenue?

b.If the government wants to collect from an ad-valorem tax levied on the consumers that will generate the same amount of tax revenue as a $5 specific tax levied on the consumers, how big will the ad-valorem tax need to be?     

Answer 1

(a) The $5 tax will increase the effective price paid by buyers, shifting demand curve leftward. New demand function is

QD = 100 - 5 x (P + 5) = 100 - 5P - 25 = 75 - 5P

Equating with QS,

75 - 5P = 40 + 5P

10P = 35

P = $3.5 (Price received by producers)

Price paid by buyers = $3.5 + $5 = $8.5

Q = 75 - (5 x 3.5) = 75 - 17.5 = 57.5

Tax revenue = $5 x 57.5 = $287.5

(b) Let ad-valorem tax be t% of price, so new price paid by consumers is [P x (1 + t)].

100 - 5 x [P x (1 + t)]. = 40 + 5P

100 - 5P x (1 + t) = 40 + 5P

100 - 5P - 5t = 40 + 5P

10P = 60 - 5t

P = 6 - 0.5t

Q = 40 + 5 x (6 - 0.5t) = 40 + 30 - 2.5t = 70 - 2.5t

Tax revenue = t x (70 - 2.5t) = 70t - 2.5t²

For equivalence,

70t - 2.5t² = 287.5

2.5t² - 70t + 287.5 = 0

t² - 28t + 115 = 0

t² - 23t - 5t + 115 = 0

t x (t - 23) - 5 x (t - 23) = 0

(t - 5) x (t - 23) = 0

Therefore, Ad-valorem tax rate (t as % of price) = 5% or 23%