Calculations Lecture 6
A tiny European country has two firms which have CO2-emissions of firm A=75tons and firm B=75tons.
The country is subject to the plan of the European parliament to reduce emissions by 60%. The
government wants to achieve this by introducing marketable pollution permits.
The abatement cost functions of the firms are
Ca=500+5Za² ; Cb=1000+2,5Zb²
Since firm B has very talented and well-connected lobbyists, they have been guaranteed by
government officials that they will receive 75% of the newly created permits.
Please calculate and provide the following figures in whole numbers.
1. Cost-efficient abatement of Firm A
2. Cost-efficient abatement of Firm B
3. Number of permits traded between the two firms.
A
B
A+B
Uncontrolled emissions
75
75
150
Uncontrolled abatement
0
0
0
Emissions (cap)
60 (=150*0.4)
Initial permit allocation
15 (=60*0.25)
45 (=60*0.75)
60 (see line above)
Efficient abatement
?
?
?
Final permit allocation
?
?
?
𝐶𝐴 = 500 + 5 ∗ 𝑍𝐴2
𝐶𝐵 = 1000 + 2.5 ∗ 𝑍𝐵2
First Derivatives
𝑀𝐶𝐴 = 10 ∗ 𝑍𝐴
𝑀𝐶𝐵 = 5 ∗ 𝑍𝐵
Efficiency Condition
𝑀𝐶𝐴 = 𝑀𝐶𝐵
Abatement Condition
𝑍 = 𝑍𝐴 + 𝑍𝐵
Cost-efficient solution
2𝑍𝐴 = 𝑍𝐵
𝑍𝐴 + 2𝑍𝐴 = 90
𝑍𝐴 = 30
𝑍𝐵 = 60
Trade
𝑍𝐴 has to reduce from 75, by 30, to 45
𝑍𝐵 has to reduce from 75, by 60, to 15
Of the 60 permits, A has been given 15
Of the 60 permits, B has been given 45
➔ 30 permits are traded from B to A
A
B
A+B
Uncontrolled emissions
75
75
150
Uncontrolled abatement
0
0
0
Emissions (cap)
60 (=150*0.4)
Initial permit allocation
15 (=60*0.25)
45 (=60*0.75)
60 (see line above)
Efficient abatement
30
60
90
Final permit allocation
45 (30 permits
bought)
15 (30 permits
sold)
60