Department of Environmental Economics and Policy
University of California
Economics EEP101/ECON125, Spring 2003
Professor Zilberman
Homework Set #4
Due at the Lecture on April 15th, to Yanhong Jin
1
Dynamic System
Diminishing Island has a special tree specie. Its population
s
grows according to the following relationship:
g
(
s
) = 0
.
8
s

0
.
00001
s
2
. Cutting down a tree yields a constant marginal benefit of
10
. The total cost of
annual harvest depends on the size of its current stock (
s
) and the amount of harvest (
x
), i.e., the total cost
is given by
T C
=
200000
x
s
.
Note: the growth function is not appropriate for forestry economics (see detailed notes in Homework
homepage). The appropriate growth function is a function of time rather than stock. Dynamic man
agement of forestry is to determine when to cut down the entire stock. You may regard this problem
as a dynamic system for fishery.
1. What is the maximum carrying capacity of the stock?
Solution:
The maximum carrying capacity is achieved when
g
(
s
) = 0
and
dg
ds
<
0
.
g
(
s
) = 0
.
8
s

0
.
00001
s
2
= 0
→
s
1
= 80
,
000
and
s
2
= 0
(1)
dg
ds
= 0
.
8

0
.
00002
s <
0
→
s >
40
,
000
(2)
Thus, the maximum carrying capacity of the stock is
s
= 80
,
000
.
Note: The stock will decline as its size goes up due to the limited space and nutrients.
g
(
s
) = 0
implies that the stock either achieves its minimum or maximum; and
dg
ds
<
0
implies that the stock
goes down at the current value of
s
. Thus, we can tell that at the current value of
s
, the stock
achieves its maximum rather than minimum.
2. What are the sizes of the stock and the annual harvest at the maximum sustainable yield?
Solution:
Maximum sustainable yield is achieved when
dg
ds
= 0
.
dg
ds
= 0
.
8

0
.
00002
s
= 0
→
s
= 40
,
000
(3)
3. Discussion on equilibrium outcomes with different management agents
We are looking at the equilibrium at the steady state.
(a) The government believes that everyone should be allowed to use forest, and sells licenses for
$6 per tree to anyone willing to buy.
What are the sizes of stock and the annual harvest at
equilibrium?
Solution:
In this case, government charges a license fee (
L
= 6
per tree), and allows everyone
who are willing to pay to cut down trees. This is an open access situation in which profits are
ultimately driven to zero. Otherwise, more people will buy license and cut down trees. If a steady
state is achieved, the following conditions must hold:
zero profit condition:
revenue

cost
= 0
(4)
sustainability or steady state condition:
x
=
g
(
s
)
(5)
At steady state,
profit
=
T R

T C
= 10
x

(200000
x/s
+ 6
x
) = 4
x

200000
x/s
= 0
⇒
s
= 50
,
000
(6)
1
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=
g
(
s
) = 0
.
8
s

0
.
00001
s
2
= 15
,
000
(7)
Thus, at the equilibrium, the stock and the annual harvest are given by
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