Bài tập kinh tế môi trường

Extraction Path
1. Recently a large cave with about 200.000 tons of guano was rediscovered in
north-eastern Zimbabwe (this is truel). Suppose that the government wishes to dig up
and sell the guano over a period of three years, and that it faces the following demand
functions:
In year 1
Qd = 161.000 – 100 P
In year 2
Qd = 180.000 – 100 P
In year 3
Qd = 190.000 – 100 P
The marginal extraction cost is $200/ton, the interest rate is 10%, and there is a fixed,
up-front cost (access roads, drying plinths, etc.) of $40 million.
a. How much guano should be mined in each year (the “optimal extraction
path”)?
b. Is this proposed project profitable? Explain, by setting up the cost-benefit
analysis.
c. How would the answer to a. be changed if the deposit has only 150.000 tons?

2. a. Assume that marginal extraction costs are zero. Sine user cost = P – MEC we
have UC = P here. We expect UC to rise with the rate of interest, so here we expect P
to rise with the rate of interest, giving
P2 = P1 (1 + 0,1)

and

P3 = P1 (1 + 0,1)(1 + 0,1)

We also know that since the resource is to be used up in the three years,
Q1 + Q2 + Q3 = 200.000
Substitution gives

161.000 – 100.P1 + 180.000 – 100P2 + 190.000 – 100.P3 = 200.000
Replacing P2 and P3 gives
531.000 – 100P1 – 100P1.(1,1) – 100P1. (1,21) = 200.000
Whish may be solved to give
P1 = $1.000
and so Q1 = 61.000
P2 = $1.100
and so Q1 = 61.000
P3 = $1.210
and so Q1 = 61.000
Note: If the MEC = 200, as the problem originally indicated, then we have that
(P2 – 200) = 1,1 (P1 – 200) and (P3 – 200) = 1,2 (P1 – 200).


Appropriate substitutions yield prices of $1018,73; $1100,60 and $1190,66 in the
three years, and quantities sold of 59, 127, 69, 940 and 70,934 respectively. [try the
exercise!].
b. A cost-benefit analysis sets out the cash flow, which indicates the inflow,
and outflow, of cash in each year. These should then be discounted to give the net
present value. Here is how we would set it up for the mine, in the case where MEC =
$200/ton. The figures are in millions of dollars.
Year 1
Year 2
Year 3
Inflows: Sales
60,23
76,98
84,46
Outflows: Investment
40,00

Extraction
11,83
13,99
14,19
Net cash lows
8,40
62,99
70,27
This is clearly a very profitable project, for the revenues cover the costs in year 1!
Repeat the computations of part a. to give
P1 = $1.151,06
and so Q1 = 45.894
P2 = $1.266,16
and so Q1 = 53.384
P3 = $1.392,78
and so Q1 = 50.772
Assuming that the marginal extraction cost is zero here too.

3. Coal and External Costs
a. Suppose that the demand curve for coal is given by Qd = 480 – 2P, the supply
curve by Qs = 10p, and that the industry is competitive. Graph the demand and
supply curves, and find the market price and quantity.
b. Expert estimate that for every ton of coal used, the rest of society has to bear
costs of $15/ton (= “marginal external costs”). So the government puts a
$15/ton tax coal. Draw the new supply curve, and show and find the price
which consumers of coal will now have to pay, and the quantity produced.
c. Briefly comment on the pros and cons of using each of the following to deal
with pollution caused by using coal:
i) Taxing the output of coal
ii) Taxing the pollution caused by using coal

iii) Regulating/limiting the output of coal-generated pollution directly.

4. a. See diagram 1. here Qd = 480 – 2P and Qs = 10P. In equilibrium demand
equals supply, so Qd = Qs so 480 – 2P = 10P, giving P = $40, Q = 400.
b. When the consumer pays Pd, the producer gets Pd – tax = Pd – 15, which is
the “supply price” (Ps). So Qs = 10 (Pd – 15). Set this equal to Qd, and solve to
find P = $52,50 and Q = 375. Note that although the tax is $15, consumers pay
only $12,50 more than before; suppliers absorb the other $2,50.


c. Tax coal output. Easy to do. But puts a burden on clean coal as well as dirty
(i.e. high sulfur) coal. And provides no incentive to install pollution control
devices.
Tax pollution. Economists prefer this; it provides the right incentive to use
cleaner fuel, and/or clean up. Problem: how set the tax rate.
Regulation. The U.S relies mainly on this (e.g. no-lead gasoline, etc.). Can be
effective, but often at high cost. No incentive to continually improve.

5. Forests
a. You plant a tree which grows 50% in the first year, 49% in the second year,
and so on – i.e. it grown 1% less quickly in each succeeding year. The real
interest rate you face is 6%. When should you cut the tree? Explain.
You are interested in harvesting trees from a 15 square kilometer area of
tropical forest. Two types of trees are worth cutting; species A trees can be sold
for $100/cubic meter and species B for $70/cubic meter. There are 2000 stems
of species A and 5000 of species B which are potentially worth harvesting; the
rest of the forest will be left more or less intact, so this level or harvesting is
considered sustainable. Your overhead fixed costs are $ 80.000 (to build a
logging road, buy some equipment, etc.); it then costs $50 per cubic meter
harvested.

i) What is the maximum amount you would pay for permission to work this
concession (i.e. what is the rent generated by this forest)?
ii) How much tax revenue would the government get if it put a tax of 30% on
the selling price of the wood? (Assume you cannot sell the wood for more
than $100 and $70 respectively). Explain.
iii) How much tax revenue would the government get from a 45% tax on the
profits of the company? Is this a better or a worse tax than the tax in part
ii)? why?

6. a. The tree grows 50% in the first year, 49% in the second, and so on. It will
grow at 6% in year 45, at which time it should be cut; after that your money
would grow more quickly in the bank (at 6%) than in the form of the tree (at
5%, 4%, etc.). Algebraically,
Tree growth = (51 – year); cut when tree growth = 6%; so (51 – year) = 6.
Therefore year = 45.
b. (i) The number s are as follows:
Specie
s
A
B

Price

Output

Revenue

Extraction cost

Gross profit

$100
$70

2.000
5.000

$200.000
$350.000

$100.000
$250.000
Less overhead
= net profit

$100.000
$100.000
$80.000
$120.000


You would pay a maximum of $120.000 for a permit to cut; this is the resource
rent.
(ii) A tax of 30% on species 2 would leave an after-tax price of $49, which
is less than the extraction cost of $50 per cubit meter. Thus you would not cut
this species. The tax would leave an after tax gross profit of $20 (= $100 - $30 $50) on species 2, for a gross profit of $60.000. But this is not enough to cover
the overhead costs. Thus the firm would shut down, and the government would
collect no revenue.
(iii) A tax of 45% on $120.000 would yield $54.000, and would not
discourage cutting the trees. This is clearly preferable to the situation in (ii).