Question

Firms produce a quantity of a good Q, with marginal social benefits (demand curve) of PMB(Q) = SMB(Q) = 6 − Q. Suppose that the marginal costs of producting the amount Q of the good is PMC(Q) = 2 + Q. Suppose that the industry also creates as a byproduct of the production process a carbon sink, which soaks up excess C02. Suppose the marginal benefit to society of producing Q units is MB(Q) = Q/3.

a. What is the equilibrium quantity produced if there is no government intervention?

b. What is the socially optimal quantity?

c. What is a way to get to the social optimum with a price instrument?

d. What is the deadweight loss if we are not at the optimum?

Answer 1

Solution :-

(a) :-

Firms produce a quantity of a good Q,

with marginal social benefits (demand curve) of PMB(Q) = SMB(Q) = 6 − Q.

The marginal costs of producting the amount Q of the good is = PMC(Q) = 2 + Q.

*The equilibrium quantity produced if there is no government intervention

At equilibrium PMB = PMC

So,

6 - Q = 2 + Q

6 - 2 = Q + Q

4 = 2Q

Q = 4/2

[ Q = 2 ]

Then ,

P = 6 - Q

= 6 - 2

[ P = 4 ]

(b) :-

The socially optimal quantity :-

We have,

The marginal benefit to society of producing Q units is MB(Q) = Q/3.

In social optimal ,

PMB + External benefit ( MEB) = PMC

6 - Q + Q/3 = 2 + Q

6 - 2 = Q + Q - Q/3

4 = 2Q - Q/3

4 = (6Q - Q)/3

4 = 5Q/3

Q = (4 x 3)/5

Q = 12/5

[Q = 2.4 ]

Now,

P = 2 + Q

= 2 + 2.4

[ P = 4.4 ]

(c) :-

A way to get to the social optimum with a price instrument:-

The social optimal can be reached with a Pigouvian subsidy.

When Q = 2.4

MEV = Q/3

= 2.4/3

= 0.8

Per unit subsidy = 0.8

(d) :-

The deadweight loss if we are not at the optimum :-

Now,

When Q = 2

MEB = Q/3

= 2/3

= 0.67

So,

Deadweight loss = 1/2 x MEB x change in Q

= 1/2 x 0.67 x ( 2.4 - 2)

= 1/2 x 0.67 x 0.4

= 0.67 x 0.2

Deadweight loss = 0.134