Question

Suppose that the (external) damage done by pollution is known to be MD = 300 + 5Q, and the (private) cost and benefit are given by MC = 100 + 2Q and MB = D0 − 2Q, where D0 is not precisely known.

a. If D0 = 1,000, what would be the optimal quantity? What tax would be necessary in order for that to be the equilibrium quantity?

Question (b) => Suppose that, based on the result from part a, a cap-and-trade system is imposed to allow the optimal quantity of pollution to be produced. If D0 = 900, what would be the deadweight loss associated with having the wrong quantity?

Answer 1

When D = 900, MB = 900 - 2Q

In market equilibrium, MB = MC.

900 - 2Q = 100 + 2Q

4Q = 800

Q = 200 (market quantity)

P = 900 - 2 x 200 = 900 - 400 = 500 (market price)

In efficient outcome, MB = MC + MD

900 - 2Q = (100 + 2Q) + (300 + 5Q)

900 - 2Q = 400 + 7Q

9Q = 500

Q = 55.56 (efficient quantity)

P = 900 - 2 x 55.56 = 900 - 111.12 = 788.88 (efficient price)

At market quantity of Q = 200,

MD = 300 + 5 x 200 = 300 + 1,000 = 1,300

So,

Deadweight loss = (1/2) x MD x (Market quantity - Efficient quantity)

= (1/2) x 1,300 x (200 - 55.56)

= 650 x 144.44

= 93,886