Question

Consider two polluting firms that want to abate 100 units of pollution together, in the cheapest way possible. Their total abatement costs are given below:

TAC₁ (x₁) = x₁²    & TAC₂ (x₂) = 3x₂²

1. (1 point) Find an equation for the Aggregate Total Abatement Cost that combines the two firms (ATAC). In other words, sum TAC₁ and TAC₂.
2. (1 point) Using the equality x₁ + x₂ = 100, find the ATAC equation in terms of x₁ only.
3. (1 point) What values of x₁ and x₂ minimize ATAC? Hint: find x₁ first.
4. (1 point) Calculate the Marginal Abatement Costs for each firm, at the level of x₁ and x₂ you found in the previous question. How does MAC₁ relate to MAC₂?

Answer 1

a)

Aggregate total abatement cost will be the sum of invidual firms' total abatement costs. So,

ATAC=TAC1+TAC2

$ATAC=x_{1}^{2}+3x_{2}^{2}$

b)

We are given that

11 + 12 = 100

On rearranging, we get

$x_{2}=100-x_{1}$

So,

AT AC = x + 3(100 – 21)

AT AC = zi + 3(10000 – 20021 + x)

$ATAC=4x_{1}^{2}-600x_{1}+30000$

c)

To minimize ATAC, differentiate ATAC with respect to x1 and set is equal to 0

DAT AC/dri = 8.01 – 600

Set dTATC/dx1=0

8.71 600=0

x1=75 (ATAC minimizing value of x1)

x2=100-x1=100-75=25 (ATAC minimizing value of x1)

d)

MAC DAT AC1/d.c1 231

Set x1=75

MAC1=2*75=$150

Now

MAC2 = dAT AC2/d.x2 6.02

Set x2=25

MAC2=6*25=$150

We find that MAC1=MAC2 at level of x1 and x2 found in earlier part.