Question

In a competitive market the demand function is QD=100-P and the supply function is QS=P-20.

In consequence of a new government decision there is a per unit tax on the good.

How large is the tax if the consumer surplus decreased by 75% in consequence of the government decision?

Answer 1

Before tax, QD = QS

100 - P = P - 20

2P = 120

P = 60

Q = 100 - 60 = 40

When QD = 0, P = 100

Consumer surplus (CS) = (1/2) x (100 - 60) x 40 = 20 x 40 = 800

After tax decreases CS by 75%, new CS = 800 x 25% = 200

Let tax per unit be T. Then new supply function is:

QS = P - T - 20

Setting QD = New QS

100 - P = P - T - 20

2P = 120 + T

P = 60 + 0.5T

Q = 100 - 60 - 0.5T = 40 - 0.5T

New CS = (1/2) x [100 - (60 + 0.5T)] x (40 - 0.5T) = 200

(40 - 0.5T) (40 - 0.5T) = 400

(40 - 0.5T)² = 400

40 - 0.5T = 20

0.5T = 20

T = 40