Question

Suppose there are two firms and they pollute the environment. Their total abatement cost is given below.  

TC(R1)=1/2(R1)^2,

TC(R2)=2/3(R2)^3/2

Suppose the government wants to reduce pollution by 12 units.

1. Find the cost-efficient way to abate 12 units of pollution. How many of these 12 units would Firm 1 abate?

2. Find the cost-efficient way to abate 12 units of pollution. How many of these 12 units would Firm 2 abate?

3. Choose all the policies below that do not achieve cost efficiency.

A. The government mandates that Firm 1 abates 6 units and Firm 2 abates 6 units.

B. The government mandates that Firm 1 abates 4 units and Firm 2 abates 8 units.

C. The government mandates that Firm 1 abates 5 units and Firm 2 abates 7 units.

D. The government mandates that Firm 1 abates 3 units and Firm 2 abates 9 units.

E. The government mandates that Firm 1 abates 7 units and Firm 2 abates 5 units.

4. What is the lowest abatement cost to society to reduce pollution by 12 units? Use 2 decimals.

5. Suppose the government wants to reduce pollution by mandating that both firms cut pollution by 6 units. What is the abatement cost of society for such a policy? Use 2 decimals.

Answer 1

1) If 12 units have to be abated, then

R1 + R2 = 12 ...... eq 1

and at cost efficient method, the following takes place -

MAC(R1) = MAC(R2)

dTC(R1)/dR1 = dTC(R2)/R2

R1 = R2^1/2 ..... equation 2

From eq 1 and eq 2

R2^1/2 + R2 = 12

R2 = 9

and thus, R1 = 12-9 = 3

Hence, firm 1 will abate 3 units of total

b) As already solved in part (a), that R2 = 9

thus firm 2 will abate 9 units of total 12 units.

c) All the policies except D do not achieve cost efficiency. Because at cost efficient, R2 = 9 and R1 = 3, which is present in option D.

d) Now put R1 = 3 and R2 = 9 into given cost functions as follows -

Total cost = TC1 + TC2

TC = 1/2(R1)² + 2/3(R2)^3/2

TC = 1/2(3)² + 2/3(9)^3/2

TC = 22.5

e) For such policy total cost will be -

TC = 1/2(6)² + 2/3(6)^3/2

TC = 27.79