Question

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

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

= R21/2 ..... equation 2

From eq 1 and eq 2

R21/2 + 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