Suppose 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=D-2q, where D is not known

A.

At D=1000, equilibrium quantity and Marginal Private Benefit (MPB) would be:

MB=MC100+2q=D−2q100+2q=1000−2q4q=900q=225MB=100+2(225)=550MB=MC100+2q=D−2q100+2q=1000−2q4q=900q=225MB=100+2(225)=550

Optimum quantity is:

MD=MB300+5q=D−2q300+5q=1000−2q7q=700q=100MD=300+5(100)=800MD=MB300+5q=D−2q300+5q=1000−2q7q=700q=100MD=300+5(100)=800

To bring q to the equilibrium quantity to 225 from 100, tax equivalent to external damage (MD) or external cost is needed. That is MD-MC

=MD−MC=800−550=250=MD−MC=800−550=250

B.

If D=900, optimal quantity is:

MD=MB300+5q1=D−2q1300+5q1=900−2q17q=600q1=85.71MD1=300+5(85.71)=728.55MD=MB300+5q1=D−2q1300+5q1=900−2q17q=600q1=85.71MD1=300+5(85.71)=728.55

Deadweight loss associated would be:

=0.5(MD−MD1)(q−q1)=0.5(800−728.55)(1000−900)=3572.5=0.5(MD−MD1)(q−q1)=0.5(800−728.55)(1000−900)=3572.5

C.

With D=900 the correct amount of tax imposed should be:

=MD−MC=728.55−550=178.55=MD−MC=728.55−550=178.55

and tax imposed=250, the extra amount of tax would be:

=250−178.55=71.45DWL=0.5(71.45)(1000−900)=3572.5=250−178.55=71.45DWL=0.5(71.45)(1000−900)=3572.5

Where DWL is deadweight loss.

D.

Considering D is not known, cap and trade system is better one. The deadweight loss would be minimum because the sellers can trade whatever marginal damage they created. Tax is ideal only for scenario when exact D is known, otherwise there would be deadweight loss.

If MD=300+3q, optimal quantity is:

MD=MB300+3q=D−2q300+3q=1000−2q5q=700q=140MD=300+3(140)=720MD=MB300+3q=D−2q300+3q=1000−2q5q=700q=140MD=300+3(140)=720

Tax would be:

=720−550=170=720−550=170

As here, marginal damage is less, tax can be imposed. Deadweight loss would be less.