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Suppose that in the city of Shenzhen there are three polluters (say, paper mills).
Currently each polluter emits 900 units of NOX per year (E1 = E2 = E3 = 900).
These polluters are currently not subject to emissions control, i.e., zero
abatement (A1 = A2 = A3 = 0). Their total abatement cost functions are given by
Polluter 1: C1(A1) = A
2
1
1
4
+ 100
Polluter 2: C2(A2) = A
2
2
1
2
+ 90
Polluter 3: C3(A3) =
A
2
3
3
4
+ 80

As a result of the NOX emissions, people in the city of Shenzhen suffer from
asthma. Suppose that the city’s target NOX emission level is E = E1 + E2 + E3 =
1,600. Thus the three polluters must jointly reduce their emission level by A =
A1 + A2 + A3 =1,100 units.

Q2. Using the Lagrangian method, derive the abatement levels for the three
polluters A1*, A2*, and A3* that will minimise the total abatement cost for the city.

Step 1: Set up the Lagrangian optimisation problem. (2 points)

Reply 1

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