Question

Suppose firm A and firm B are the only two firms in an industry. Each firm’s Marginal Abatement cost functions is given by:

MACa = 200-Ea

MACb = 200-2Eb

Also, there are four people, each with marginal damage function:

MDi = 1/3Et , Where Et = Ea+Eb

a) What is the uncontrolled emission levels of each firm?

b) Find the aggregate MAC function

c) Find the aggregate MD function

d) Determine the socially optimal level of emissions ?t ∗ and the MD.

e) Suppose the government decided to use a fair standard that required each firm to produce half of the socially optimal emissions. What would be each firm’s MAC and total abatement cost?

f) Compute the total social cost.

Answer 1

a) The firm A has an abatement cost function as MAC(a)=200-E(a) and firm B has an abatement cost function as MAC(b)=200-2E(b). Now, the uncontrolled emission levels of each firm implies an emission level by the individual firms when their respective marginal abatement costs are equal to 0.

Therefore, based on the economic condition for the uncontrolled emission level, we can state:-

MAC(a)=0

200-E(a)=0

-E(a)=-200

E(a)=200

Therefore, the uncontrolled emission level by firm A would be 200 units of emission.

Again, based on the economic condition for the uncontrolled emission, it can be stated:-

MAC(b)=0

200-2E(b)=0

-2E(b)=-200

E(b)=-200/-2

E(b)=100

Hence, the uncontrolled emission level of firm B would be 100 units of emission, in this case.

b) The aggregate MAC function, in this case would be MAC(a)+MAC(b)=(200-E(a))+(200-2E(b))=200-E(a)+200-2E(b)=400-3E(t) where E(t) is the total or aggregate emission level combining the emission level of both firm A and B or E(t)=E(a)+E(b).

c) Now, there are 4 people having individual marginal damage function as MD(i)=1/3E(t). Therefore, the aggregate MD function=4*MD(i)=4*1(3)E(t)=4/3E(t)

d) Based on the condition to establish the socially optimal level of emission E(t)*, the MD function has to be equal to the MAC function.

Therefore, based on the condition to determine the socially optimal level of emission, we can state:-

MAC=MD

400-3E(t)=4/3E(t)

400-3E(t)=1.33E(t)

-3E(t)-1.33E(t)=-400

-4.33E(t)=-400

E(t)=-400/-4.33

E(t)*=92.38 approximately

Thus, the socially optimal level emission would be approximately 92.38 units.

Now, plugging the value of E(t)* into the MD function, we obtain:-

4/3E(t)

=1.33E(T)

=1.33*(92.38 units)

=122.86 approximately

Hence, the socially optimal level of marginal damage or MD would be 122.86 approximately.

e) The socially optimal level of emission or E(t)* has been calculated or derived as 92.38 units approximately. Now, if the government mandates each firm to produce exactly half of E(t)* then the individual emission level of firm A and B would be=92.38 units/2=46.19 units implying that E(a) and E(b) both will be 46.19 units.

Therefore, plugging the new value of E(a) into MAC(a), we get:-

200-E(a)

=200-46.19 units

=153.81

Thus, the firm A's MAC, in this case would be 153.81.

Now, plugging the new value of E(b) into MAC(b), we obtain:-

200-2E(b)

=200-2*(46.19 units)

=200-92.38 units

=107.62

Hence, the firm B's MAC, in this case is 107.62.

The aggregate MAC function has been derived as MAC=(153.81+107.62)=261.43

f) The aggregate MD function has been derived as MD=4/3E(t) and the socially optimal level of emission is E(t)*=92.38 units.

Hence, plugging the value of E(t)* into the MD function, we can get:-

4/3E(t)

=1.33E(t)

=1.33*(92.38 units)

=122.86

Thus, the socially optimal level of MD is 122.86 and therefore, the total social cost=(122.86+261.43)=384.29