Question

Suppose the demand and supply for a product are described by the equations

Qd = 1080 -4P and Qs = -120+8P.

a. Find the equilibrium P, Q and elasticities of demand and supply

b. If a $6 per unit tax is levied on the demand for the product, find new P, Q and the percent of the tax incidence that falls on consumers and firms.

c. Find the tax revenue and welfare loss associated with the tax.

d. Now double the tax to $12. Find new P, Q, tax revenue and welfare loss.

e. What happens to the efficiency of taxation as we increase the amount of the tax? What are the implications of this result?

Answer 1

(a)

Setting Qd = Qs:

1080 - 4P = - 120 + 8P

12P = 1200

P = 100

Q = 1080 - 4 x 100 = 1080 - 400 = 680

Elasticity of demand = (dQd/dP) x (P/Qd) = - 4 x (100/680) = - 0.59

Elasticity of supply = (dQs/dP) x (P/Qs) = 8 x (100/680) = 1.18

(b)

After tax, new demand function is

Qd1 = 1080 - 4(P + 6) = 1080 - 4P - 24 = 1056 - 4P

Setting Qd1 = Qs,

1056 - 4P = - 120 + 8P

12P = 1176

P = 98 (price received by sellers, market price)

Price paid by buyers = 98 + 6 = 104

Q = - 120 + 8 x 98 = - 120 + 784 = 664

% of tax incidence on buyers = (104 - 100) / 6 = 4/6 = 66.67%

% of tax incidence on sellers = (100 - 98) / 6 = 2/6 = 33.33% [= (100 - 66.67)%]

(c)

Tax revenue = 6 x 664 = 3,984

DWL = (1/2) x Tax x Change in Q = (1/2) x 6 x (680 - 664) = 3 x 16 = 48

(d)

After tax, new demand function is

Qd1 = 1080 - 4(P + 12) = 1080 - 4P - 48 = 1032 - 4P

Setting Qd1 = Qs,

1032 - 4P = - 120 + 8P

12P = 1152

P = 96 (price received by sellers, market price)

Price paid by buyers = 96 + 12 = 108

Q = - 120 + 8 x 96 = - 120 + 768 = 648

Tax revenue = 12 x 648 = 7,776

DWL = (1/2) x Tax x Change in Q = (1/2) x 12 x (680 - 648) = 6 x 32 = 192

(e)

As tax rate increases, DWL increases, implying higher efficiency loss. So optimal tax should attempt raise maximum tax revenue at lowest efficieny loss.