Suppose there are two factories whose production of electricity results in carbo

Question

Suppose there are two factories whose production of electricity results in carbon emissions. The net marginal private benefit (consumer's benefit less firm's private costs of production) as a function of the firm's emissions are:     MBE(E1)=150-15E1 and MBE (E2)=100-10E2

Suppose prior to any intervention to require firms to internalize the costs of emissions, the firms emit 10 units each. Furthermore, suppose there are constant marginal damages of 30.

a) What is each firm's marginal abatement cost function, as a function of their abatement? Hint : Remember, the marginal cost of abatement is the mirror image of the net marginal private benefit of emissions.

b) What is the marginal benefit of abatement? Hint: Remember, the marginal benefit of abatement is the mirror image of the marginal damage from emissions.

c)What is the socially optimal level of abatement for each firm? how many units will each firm emit?

d) If the government must impose a uniform quota for all firm a la command and control, it is possible to achieve the socially optimal level of abatement? Why or why not?

*My solution: Please check my work and correct it if it makes sense.

a) E= Q-A, where E=Emission, Q= quantity, and A= abatement

so:

=150-15 (10-a)

150-150+15a

=15a

MAC1=15a.

b) Since marginal damage is a constant and doesn't change and it the mirror image of marginal benefit then i got MB= 30.

c) For c, MAC1= MB(A1)

15a=30

a= 2 * so firm 1 unit of emission is 2.

For Firm 2, we do the same as

10a=30

a=30/10

a=3

d) i believe it wouldn't be possible for both firms to achieve socially optimal level of abatement their difference unit of emission which are 2 and 3 compare to 10 (unit each firm emits). We can see a huge difference of 8 units for firm 1 and 7 units for firm 2.

Explanation / Answer

(part a)

You forgot the 2nd firm's MAC function:

= 100 - 10 (10 - a2)    [a2: Firm 2's abatement function]

= 100 - 100 + 10a2

MAC2 = 10a2

(part b)

I find it correct.

(part c) I find it correct.

(part d) Requires a little more elaboration. Your answer is correct ig government fixes the uniform quota at an emission level of 10. In that case, as you solved correctly, none of the firms can achieve this quota and will fall short by 8 & 7 units respectively. But if the quota is lower than 10 (for example, 2 units), then both firms can achieve it.

So the exact answer to this part depends on the level at which the quota is fixed.

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