Suppose the supply curve of portable radio rentals in Golden Gate Park is given by P=5+0

a).

Given that;

P=5+0.1QsP=20−0.2QdP=5+0.1QsP=20−0.2Qd

Let's find the free market output and price, before internalization of externality:

Pd=Pd5+0.1Q=20−0.2QQ=50P=$10Pd=Pd5+0.1Q=20−0.2QQ=50P=$10

Without internalizing externality, 50 units will be rented per day at a price of $10 each.

Internalizing the externality shift the supply curve (marginal private cost) upward and to the left (marginal social cost curve), with an amount equal to damage caused (external cost). The socially optimal output and price will be obtained at the intersection of the demand curve (marginal benefit) and the new supply curve (marginal social cost):

Damage=External costMB=MPC+EMB=MSCMSC=P=5−0.1Qs+3MSC=P=8+0.1Qs8+0.1Q=20−0.2QQ=40P=$12Difference=50−40=10Damage=External costMB=MPC+EMB=MSCMSC=P=5−0.1Qs+3MSC=P=8+0.1Qs8+0.1Q=20−0.2QQ=40P=$12Difference=50−40=10

The equilibrium number of portable radios rented will exceed the socially optimal number by 10 units.

b).

Imposition of a per unit tax will shift the supply curve upward and to the left:

P=5+0.1QQ=−50+10PQ=−50+10(P+3)Q=−20+10PP=20−0.2(−20+10P)P=$8Q=30P=5+0.1QQ=−50+10PQ=−50+10(P+3)Q=−20+10PP=20−0.2(−20+10P)P=$8Q=30

Imposition of a per unit tax of $30 will not achieve social optimal output.