Question

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Q) Two firms produce a good which leads to pollution :

Assume that the marginal cost for cleaning up in one firm is given by ??1 = 40 − 4?

and for the other firm it is given by ??2 = 40 − (4/3)? where q is pollution.

The marginal damage curve for the society is given by p=q. Derive and show in a figure the optimal level of pollution for the society, the optimal tax level and how much the firms clean up given this tax. Assume now that the government instead uses regulation so that the firms have to clean up an equal amount. Show in the figure why regulation is not cost efficient and derive the society’s costs of regulation compared to taxation.

Answer 1

The total marginal cost of cleaning for the both firms combined would be (40-4q)+(40-4q/3). The optimal level of pollution would be where the marginal cost of cleaning is equal to marginal damage. Since marginal damage is p=q, we can say that at optimal level

40-4q+40-4q/3=q

Solving for q, we get

q=240/19

This is shown in the graph below

As shown, the total cleanup is 240/19 and the cost is also 240/19. This is shown at point E. The MD curve is p=q and the MC curve is (40-4q)+(40-4q/3)= 80-16q/3.

Both firms will clean the amount where there marginal costs are equal. Lets say firm 1 will clean q₁ and firm 2 will clean q₂, then

40-4q₁=40-4q₂/3

q₁=q₂/3

q₂=3q₁

Firm 2 will clean up 3 times firm 1. Since total is 240/19,

firm 1 will clean= 240/19*4=60/19.

firm 2 will clean= 240*3/19*4=180/19.

Now, if the government introduces the law that each of the firms have to clean equal amount, this means each firm will clean

240/2*19=120/19.

At this amount,

MC of firm 1=40-4*120/19=40-480/19=280/19

MC of firm 2=40-4/3*120/19=600/19

Total marginal cost= 880/19.

Now, at this marginal cost, the companies would've cleaned, in market equilibrium,

80-16q/3=880/19

q=120/19 pollution. But they are forced to clean 240/19. This results in deadweight loss (cost to society). This is shown in figure below.

In earlier situation, the surplus to the public was the triangle shaded in by slanted lines (triangle EPO). The firm's surplus was triangle AEP. In the new situation, the surplus to public remains same, but the surplus to the firms becomes triangle ABC, which is much lesser than earlier. Cost to society=

Area of triangle AEP-area of triangle ABC.

=(.5*(80-240/19)*240/19)-(.5*(80-880/19)*120/19)

=319.11.