Question

2.

The inverse demand curve of a monopolist is given by:   

P = 200 − Q

and the marginal cost is constant at ​$10, how does charging the monopoly a specific tax of τ=​$12 per unit affect the monopoly optimum and the welfare of​ consumers, the​ monopoly, and society​ (where society's welfare includes the tax​ revenue)?

Calculate the following:

1. equilibrium price and quantity before the tax
2. equilibrium price and quantity after the tax
3. the change in consumer surplus
4. the change in producer surplus
5. the tax revenue
6. the change in society’s welfare
7. the incidence of the tax on​ consumers

Please show numeric work

Answer 1

P= 200-Q

MC= $10

a.

P=200-Q

Total revenue(TR)= P x Q= 200Q-Q²

Marginal revenue(MR)= Differentiation of TR with respect to Q= 200-2Q

Equilibrium in monopoly:

MC=MR

10= 200-2Q

2Q= 190

Q*= 95 Equilibrium Quantity before tax

P*= 200-Q= 200-95= 105 Equilibrium price before tax

b.

A specific tax on monopoly of T= $12 per unit cause MC to increase by the equal amount:

New MC= MC'= 22

Equilibrium after tax:

MC'= MR

22= 200-2Q

2Q= 178

Q**= 89 Equilibrium quantity after tax

P**= 200-89= 111 Equilibrium price after tax

c.

P=200-Q

If Q= 0, P= 200(Pm)

Consumer surplus(Before tax)= 1/2 (Pm-P*)(Q*)= 1/2 (200-105)(95)= 4512.5

Consumer surplus (after tax)= 1/2 (Pm-P**)(Q**)= 1/2 (200-111)(89)= 3960.5

Change consumer surplus= 3960.5-4512.5= -552

d.

Producer surplus (before tax)= (P*-MC)(Q*)= (105-10)(95)= 9025

Producer surplus (after tax)= (P**-MC')(Q**)= (111-22)(89)= 7921

Change in producer surplus= 7921-9025= -1104

e.

Tax revenue= T x Q**= 12 (89)= 1068