Price vs. Quantity under Uncertainty

Classwork 5

Suppose that the (external) damage done by polluting goods \(Q\) is known to be \(MD = 300 + 5Q\), and the (private) cost and benefit are given by \(MC = 100 + 2Q\) and \(MB = D_{0} + 2Q\), where \(D_{0}\) is not precisely known.

Q1a.

If \(D_{0} = 1,000\), what would be the optimal quantity? What tax would be necessary in order for that to be the equilibrium quantity?

Q1b.

Suppose that, based on the result from Q1a, a cap-and-trade system is imposed to allow the optimal quantity of pollution to be produced. If \(D_{0} = 900\), what would be the deadweight loss associated with having the wrong quantity?

Q1c.

Suppose that, based on the result from part Q1a, a tax is imposed to allow the optimal quantity of pollution to be produced. If \(D_{0} = 900\), what would be the deadweight loss associated with having the wrong tax level?

Q1d.

If \(D_{0}\) is not exactly known, which is likely to give better results, a cap-and-trade system or a tax? What would be the answer to this question if the marginal damage were \(300 + 3Q\) instead of \(300 + 5Q\)?