Question

5. Suppose the current equilibrium point for the market of good X is (200, $100). Also we know that ɛ = -2 and n = 1.5 . Suppose the government imposes an unit tax of $3/unit of the good. Calculate the size of government revenue. [You are NOT allowed to use the short cut formula.]

Answer 1

Current equilibrium point is (200 , 100)  

Equilibrium price P = 100

Equilibrium quantity Q = 200

= - 2

= (dQ/dP)(P/Q)

- 2 = (dQ/dp)(100/200)

- 2 = (dQ/dP)(1/2)

dQ/dP = - 4

dP/dQ = - 1/4

slope of demand curve = dP/dQ = - 1/4

Linear equation of demand curve is

(P - 100) = -1/4(Q - 200)  

P - 100 = - Q/4 + 50

P = 50 + 100 - Q/4

P = 150 - Q/4  

Elasticity of supply n = 1.5  

n = (dQ/dP)(P/Q)

1.5 = (dQ/dP)(100/200)

1.5 = (dQ/dP)(1/2)

dQ/dP = 3  

dP/dQ = 1/3

slope of supply curve = dP/dQ = 1/3

Linear equation of supply curve is

(P - 100) = 1/3(Q - 200)

P - 100 = Q/3 - 200/3

P = Q/3 - 200/3 + 100

P = Q/3 + 100/3

Suppose the government imposes an unit tax of $3/unit of the good so demand curve after tax is

P = 150 - Q/4 - t

P = 150 - Q/4 - 3

P = 147 - Q/4

P = Q/3 + 100/3

147 - Q/4 = Q/3 + 100/3 (dd = ss)

(588 -Q)/4 = (Q + 100)/3

1764 - 3Q = 4Q + 400

1764 - 400 = 4Q + 3Q

1364 = 7Q  

Q = 1364/7

Q = 194.85

Therefore equilibrium quantity after tax is Q*_t = 194.85

Tax revenue collected by government = tQ*_t

= 3194.85

= 584.55