Environmental economics: exercises for practice

Environmental economics – mathematical exercises for practice
(solutions can be found at the end of the document)
1. A factory producing chemical fertilisers (MNPB=4-0,5q) is polluting the nearby
river, causing environmental damage: MEC=1,5q.
a) How much will the factory produce to maximise its profit, and how much
profit will it then have?
b) If the goverment decides to regulate the company using a tax, how high
should the tax rate be, how much tax will the company be paying,and how
much profit will it then have?
2. A paper company (MNPB = 4 - q) is operating next to a river, dumping harmful
chemicals into it. As a result, the soft drink factory operating further down the river
is suffering losses (MEC = 1/3 q) because of the incerased water purification
costs.
a)If the government now passes a law that everybody has the right for a clean
environment, how much would the paper company have to pay (minimum and
maximum) to the soft drink company in order to be allowed to continue operation?
What will be the output of the paper company at the end of the bargaining
process? Please make a drawing showing the situation!
b) If the government would like to ensure the pollution reduction using a norm or a
tax, which instrument should it choose and why? (Taking into account that the
goverment does not know exactly the MNPB curve of the paper company) (You
do not need to draw or make any calculations here!)
3. The profit curve of a steel plant is MNPB=9-q, the associated environmental
pollution causes social costs in the order of MEC=1/2q. The authority tries to
estimate the company’s profit curve and arrives at the following result: MNPB’=6-q.
a) If the government decides to regulate using a norm, where will it set the
norm, and what will be loss caused by the mistake?
b) If the government decides to regulate using a tax, where will it set the tax,
and what will be loss caused by the mistake?
4. Two large factories are operating in the same area, causing serious SO 2-emissions
(70 and 40 units). The authorities would like to reduce the emissions of SO 2 to 60

units using a tax. The total abatement cost curves of the factories are:
TAC1=2q12
TAC2=3q22
(where q means the amount of removed pollution).
a) How many units of pollution will each company remove and what is the tax
rate necessary to achieve this? What will be the cost of this policy solution for
the companies?


b) What will happen if another factory starts operation in the same area?
(Please describe the nature of the effect on the two existing factories, the
authorities and the environment!)
And what is the effect of a new entry if there is an emissions trading system
instead of a tax?
5. The marginal abatement cost curves of two companies are the folowing: MAC1 =
5q1; MAC2 = 4q2 (where q means the amount of removed emissions). The
companies are regulated by a tax, and the revenue of the authorities from the two
companies is 880 and 1200. The total cost of the regulation for the two companies
is 1040 and 1400. What were their original emissions?

6. Three coal power plants are operating in the same area, causing serious SO 2emissions (40, 60 and 20 units). In order to protect the health of the local
population, the authority would like to bring down the total emissions of SO2 to
60 units. The marginal abatement cost curves of the power plants are:
MAC1=2q1
MAC2=2q2 and MAC3=5q3
(where q means the amount of pollution removed by the company).
The authority is considering two instruments to achieve the desired goal:
a) Setting a norm requiring equal remaining emissions.
b) Introducing an emissions trading system, where the SO2 permits are auctioned
to the companies (they do not get any for free).

Questions:
How many units of pollution will each company remove in the two situations?
In situation a), how high does the penalty at least have to be in order to ensure
that all three power plants comply with the norm?
In situation b), what will be the permit price?
Please compare the financial effect of the two solutions for
-the authorities
-the companies
-the society as a whole!
7. Two companies’ pollution abatement costs can be described by the following
curves: MAC1=30q1 MAC2=20q2. Their original emissions are 54 and 40 units. The
authorities would like to reduce the total pollution by 30 units, and they do this by
issuing tradable permits, which are equally distributed between the two companies.
How many permits will change hands between the companies and at what price?


Solutions
1. a) It will produce 8 units, and the profit will be 16.
b) The social optimum is 2. The necessary tax rate is 3, the tax payed by the
company is 6, the remaining profit is 1.
2. a) At the end of the bargaining process, the output of the paper company will be
3, and it will have to pay minimum 1,5; maximum 7,5.
b) The government should use a tax, because MEC is flatter than MNPB, so using
a tax will lead to a smaller loss if the goverment makes a mistake when estimating
the paper company’s MNPB curve.
3. a) The norm will be 4, the loss is 3.
b) The tax will be 2, the loss is 0,75.
4. a) q1=30 q2=20, the tax rate is 120. Costs: company 1: reduction cost=1800 + tax
paid=4800; total cost 6600. Company 2: reduction cost 1200 + tax paid 2400;
total cost 3600.

b) If another company starts operation, the first two companies will not be
affected, the authorities will have more income from the taxes – but there will be
more environmental damage (unless the government raises the tax rate). If there
is a permit trading system instead of a tax, then the permit price will automatically
go up (because the number of permits stays the same). This means higher costs
for the companies and no effect on the environment.
5. Company1: removed emission:8, remaining emission 22, original emission 30
Company2: removed emission:10, remaining emission 30,original emission 40
6. Norm situation: equal rest emissions mean 20 for all companies, so the removed
pollution is: 20; 40; 0. The minimum penalty necessary is 80 (the 2nd company
will only respect the norm if the penalty is at least this high). The costs for the
companies are: 400, 1600 and 0 (in this case the reduction cost is the only cost).
Emissions trading: removed pollution is: 25, 25, 10. The permit price is 50.
Financial effects: The costs for the companies are: company 1: reduction cost:
625 + buying permits: 750, total 1375. Company 2: reduction cost 625 + buying
permits 1750, total 2350. Company 3: reduction costs 250 + buying permits 500,
total 750. So, for each company, the total costs are lower if there is a norm. For
the authorities, the permit system results in an income of 3000, if there is a norm,
they have no income. For society, the permit system is better, because the total
cost of the reduction is lower in this case (1500 vs. 2000).
7. Company 1 will buy 10 permits from company 2 a the price of 360/permit.