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Environmental and Natural Resource Economics · ISCTE-IUL, Fall 2017 Workout No.4
Workout No.4 December 7, 2017
Exercise 1 - International environmental problems
(PMMC 10 Ex. 1)
The world consists of two countries, X which is poor and Y which is rich. The total benefits
(B) and total costs (C) of emissions abatement (A) are given by the functions BX = 8(AX + AY),
BY = 5(AX + AY), CX = 10+ 2AX + 0.5A2X and CY = 10+ 2AY + 0.5A
2
Y
(a) Obtain the non-cooperative equilibrium levels of abatement for X and Y.
In the non-cooperative solution (NC), each country individually maximizes its own net benefits wrt to
its own abatement, taking the other country’s abatement as given
MaxAi
{
Bi
(
Ai + Aj
)− Ci (Ai)}
which for X and Y respectively is
[X] : MaxAX
{
8 (AX + AY)−
(
10+ 2AX + 0.5A2X
)}
[Y] : MaxAY
{
5 (AX + AY)−
(
10+ 2AY + 0.5A2Y
)}
The individual maximization yields
[X] : 6− AX = 0→ ANCX = 6
[Y] : 3− AY = 0→ ANCY = 3
The total abatement is ANCT = A
NC
X + A
NC
Y = 9
(b) Obtain the cooperative equilibrium levels of abatement for X and Y. Is the total abatement
higher or lower? Explain.
In the cooperative solution (C), X and Y jointly agree on the abatement levels that maximize total (sum
of) net benefits. In this case the problem is
MaxAX ,AY {BX (AX + AY) + BY (AX + AY)− CX (AX)− CY (AY)}
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Environmental and Natural Resource Economics · ISCTE-IUL, Fall 2017 Workout No.4
which is
MaxAX ,AY
{
13 (AX + AY)−
(
20+ 2 (AX + AY) + 0.5
(
A2X + A
2
Y
))}
The FOCs wrt AX and AY respectively are
[X] : 11− AX = 0→ ACX = 11
[Y] : 11− AY = 0→ ACY = 11
The total abatement in the cooperative (C) solution is ACT = A
C
X + A
C
Y = 11 > A
NC
T . The reason
for it to be higher that under the NC solution is that, in the C solution there is a full internalization of
the total benefits from abatement. Abatement generates public benefits (as in both country’s benefit
from one country’s abatement), but the cost are private (each country pays for its own abatement). In
the NC solution countries do not incorporate the positive effect of their own abatement on the other
country’s benefits. As a result, there is too little total abatement in the NC solution.
(c) Calculate the utility (net benefit) levels enjoyed by X and by Y in the non-cooperative and
cooperative solutions. Does the cooperative solution deliver Pareto improvements for each country,
or would one have to give a side payment to the other to obtain Pareto improvements for each with
cooperation?
For X
NBNCX = 8 (9)−
(
10+ 2 (6) + 0.5 (6)2
)
= 32
NBCX = 8 (22)−
(
10+ 2 (11) + 0.5 (11)2
)
= 83.5
For Y
NBNCX = 5(9)−
(
10+ 2 (3) + 0.5 (3)2
)
= 24.5
NBCX = 5(22)−
(
10+ 2 (11) + 0.5 (11)2
)
= 17.5
Y would need to receive side payments in order to get a Pareto improvement from cooperation
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Environmental and Natural Resource Economics · ISCTE-IUL, Fall 2017 Workout No.4
(d) Suppose that Y acts as a ‘swing abater’, doing whatever (non-negative) amount of abatement is
required to make the combined world abatement equal to the combined total under a full cooperative
solution. How much abatement is undertaken in the two countries
In this case AY = 22− AX. Knowing this, X has no incentives to abate at all. To see this, note that
now under the NC problem, X’s payoff is given by
8 (AX + (22− AX))−
(
10+ 2AX + 0.5A2X
)
= 8(22)−
(
10+ 2AX + 0.5A2X
)
Exercise 2 - Nonrenewable resources
(PMMC 15 Ex. 1)
Consider two consecutive years, labeled 0 and 1. You are currently at the start of year 0. The
following information is available. There is a single fixed stock of a non-renewable resource; the
magnitude of this stock at the start of year 0 is 224 (million tonnes). The inverse resource demand
functions for this resource in each of the years are P0 = a− bR0 and P1 = a–bR1 in which a = 107
and b = 1. The constant marginal cost of resource extraction is 5. All (non-physical) units are in units
of utility. The social welfare function is discounted utilitarian in form, with a social utility discount rate
of 0.1. Given that the objective is to maximize social welfare over periods 0 and 1
(a) Calculate the amounts of resource that should be extracted in each period, subject to the
restriction that at least 104 units of the resource should be left (unextracted) for the future at the end
of period 1.
The social welfare is given by the discounted sum of Net Social Benefits (NSB)
W = NSB0 +
NSB1
1+ ρ
where the NSB in period t are given by the gross social benefits from extraction SB net of the
extraction cost C:
NSBt = SBt − Ct
The gross social benefits from extraction is how much does society value the consumption of R
extracted units. Using the (inverse) demand function this is
SB =
ˆ R
0
(a− bm) dm = aR− bR
2
2
Back into the social welfare function
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Environmental and Natural Resource Economics · ISCTE-IUL, Fall 2017 Workout No.4
W = (a− c) R0 − bR
2
0
2
+
(a− c) R1 − bR
2
1
2
1+ ρ
The problem is then
MaxR0,R1
(a− c) R0 − bR202 + (a− c) R1 −
bR21
2
1+ ρ

subject to
R0 + R1 ≤ S0 − 104 = 120
Note that in the optimum the cumulative extraction constraint will hold with equality. The La-
grangian representing this problem is given by
L = (a− c) R0 − bR
2
0
2
+
(a− c) R1 − bR
2
1
2
1+ ρ
+ λ (120− R0 − R1)
The FOCS are given by
[R0] : (a− c)− bR0 − λ = 0,
and
[R1] :
(a− c)− bR1
1+ ρ
− λ = 0;
combining these two
(1+ ρ) ((a− c)− bR0) = (a− c)− bR1
and using the cumulative extraction constraint
(1+ ρ) ((a− c)− bR0) = (a− c)− b (120− R0)
b (120) + ρ (a− c) = (2+ ρ) bR0
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Environmental and Natural Resource Economics · ISCTE-IUL, Fall 2017 Workout No.4
120+ 0.1 (102)
2.1
= 62 = R0
120− 62 = 58 = R1
(b) What is the resource price in each period?
R0 = 62→ P0 = 107− 62 = 45
R1 = 58→ P1 = 107− 58 = 49
(c) Show that the path of the resource price found in (b) net of extraction costs follows Hotelling’s
rule
According to Hotelling’s rule, the price net of the marginal extraction cost should grow at the
discount rate
∆% (P− c) = ρ
Plugging in the resource prices fond in (b)
∆% (P− c) = (P1 − c)− (P0 − c)
P0 − c =
(49− 5)− (45− 5)
45− 5 = 0.1
which is exactly equal to the discount rate ρ
Exercise 3 - Nonrenewable resources
(PMMC 15 Ex. 6)
Discuss, with diagrams, the consequences of the discovery of North Sea oil for
(a) the price and output levels for the oil market;
(b) the date of exhaustion of oil reserves.
A sudden increase in the resource stock will cause a parallel shift of the extraction schedule down
and to the left (ie., expansion of the extraction schedule). The area below the extraction schedule
((R, t)-space) is equal to S0
a) At any given date t < T extraction R will be higher; consequently, at any given date t < T the
resource price P will be lower
b) The exhaustion date will increase from T to T′ (i.e., it will take longer to fully deplete the re-
source)
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Environmental and Natural Resource Economics · ISCTE-IUL, Fall 2017 Workout No.4
Figure 1: Effect of ↑ S0
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Environmental and Natural Resource Economics · ISCTE-IUL, Fall 2017 Workout No.4
Exercise 4 - Renewable resources
(PMMC 17 Ex. 2-extended)
Suppose a simple bioeconomic model (that is, biological and economic) of an open-access fishery
in which resource growth is logistic and given by
G (S) = g
(
1− S
SMAX
)
S
Furthermore, the total harvest H depends positively on the harvest effort E and the available stock S.
The total harvest is given by
H = eES
The agents are price takers and that they sell their harvest at P per unit, while the marginal cost of
harvesting effort is constant and equal to w. Therefore, the fishery’stotal net benefits are
NB = PH − wE
Finally, the open-access entails that over time the total harvesting effort evolves according to the
following law of motion
E˙ = d (B− C)
(a) Show that the bioeconomic steady state equilibrium of this model is given by the following two
conditions
g
(
1− S
SMAX
)
S− eES = 0 ∗
and
PeES− wE = 0 ∗ ∗
The first equilibrium condition of the bioeconomic model is S˙ = 0, which simply is G (S) = H. Using
the harvest production function (H = eES) this equilibrium condition reads:
G (S) = g
(
1− S
SMAX
)
S = eES = H ∗
The second equilibrium condition comes from the economic sub-model and entails that the har-
vesting effort should remain constant over time (E˙ = 0) which occurs whenever net benefits are zero
(i.e., PH−wE = 0). Using the harvest production function (H = eES) this equilibrium condition reads:
PeES−wE = 0 ∗ ∗
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Environmental and Natural Resource Economics · ISCTE-IUL, Fall 2017 Workout No.4
(b) Demonstrate that the equilibrium fishing effort and equilibrium stock can be written as
EOA =
g
e
(
1− w
PeSMAX
)
and SOA =
w
Pe
Solving for S in the second equilibrium condition ∗∗
SOA =
w
Pe
Plugging this into the first equilibrium condition ∗:
g
(
1− w
PeSMAX
)
S = eES
and solving for E
EOA =
g
e
(
1− w
PeSMAX
)
(c) Using these expressions, show what happens to fishing effort and the stock size as the ‘cost-
price ratio’ w/P changes. In particular, what happens to effort as this ratio becomes very large?
Explain your results intuitively
SOA =
w
Pe
→ ∂SOA
∂w/P
> 0
and
EOA =
g
e
(
1− w
PeSMAX
)
→ ∂EOA
∂w/P
< 0
A higher relative cost of harvesting effort will lead to lower harvesting effort in equilibrium. Intu-
itively, a higher cost, will reduce the incentives to enter (everything else constant net benefits will be
lower). Lower incentives to enter lead to overall lower aggregate harvesting effort in equilibrium.
In turn, the lower harvesting entails a higher preservation of the resource and thus, in the steady
state equilibrium the stock of fish will be higher.
If w/P is sufficiently large, that is
w
P
≥ eSMAX
the effort drops to 0 as it is not economically viable to exploit the resource: NB < 0 for any E > 0
even if S = SMAX. Then EOA = 0, and SOA = SMAX.
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