Question

Suppose there are two plants that emit a pollutant. Plant 1 can abate emissions by the level 

A1 at costs C1 (A1) = 3. A1 3 . Plant 2 can abate emissions by the level A2 at costs C2 (A2) 

= 1 +A2. You have the objective to reduce total emissions by A = A1+ A2, with A = 1, at the 

lowest total costs C = C1 + C2. 

1. Formulate this as a constrained optimasation problem and write down the lagrangian. 

2. Compute the marginal abatement costs from both plants. 

3. Compute the cost minimal of abatement plant one and plant two 

Answer 1

$$C(A_1)=3\times A_1^3$$

$$C(A_2)=1+A_2$$

$$A=A_1+A_2=1$$

1.

$$C=3\times A_1^3+1+A_2- \lambda(A_1+A_2)$$

2.

$$MC_1=9A_1^2$$

$$MC_2=1$$

3.

$$C=3A_1^3+2-A_1$$

$$C^/=9A_1^2-1$$

$$A_1=\frac {1}{3}$$

$$A_2=\frac {2}{3}$$