Natural Resource Economics
Academic year: 2016-2017
Prof. Luca Salvatici
[email protected]
Lesson 22
Renewable resources: Schaefer
model
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Monday Novembre 28th
Andrew Marincola
The dynamics of deforestation and
reforestation in a developing economy
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Genuine Saving and GDP
Presentation by Francesca Insabato: November
30th
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Outline
• Schaefer model: Schaefer, M. B. 1954.
“Some aspects of the dynamics of
population important to the management
of commercial marine fisheries”. Bulletin of
the
Inter-American
Tropical
Tuna
Commission, 1, pp. 25-56
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Harvesting
• y(t) = harvested/culled quantity in each
period (control variable)
• How does the state equation change?
.
• Hp:
y0
• 3 cases: y < max f(x)
y > max f(x)
y = max f(x)
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Graph: y < max f(x)
dX/dt
y
X
X1
X2
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Schaefer model: harvest
production function
1. The harvest will depend on the amount of resources devoted to
fishing. In the case of marine fishing, these include the number of
boats deployed and their efficiency, the number of days when
fishing is undertaken and so on. For simplicity, assume that all the
different dimensions of harvesting activity can be aggregated into
one magnitude called effort, E.
2. Except for schooling fisheries, it is probable that the harvest will
depend on the size of the resource stock. Other things being
equal, the larger the stock the greater the harvest for any given
level of effort. Hence, abstracting from other determinants of
harvest size, including random influences, we may take harvest to
depend upon the effort applied and the stock size.
==> y (x) = E x q
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Schaefer model: steady state
• State equation:
x
x  ax(1 
)  qEx
K
.
qE

)
a
 x1  K (1 
x( a  qE 
x)  0  
a
K

x2  0
• If E < a/q, steady-state fished quantity:
qE
Y  qE ( x1)  qEK (1 
)
a
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Schaefer model: graph (f(x), x)
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Schaefer model: Y(E)
qK 2
 qE 
x1  K 1    Y  qKE 
E
a 
a

2
Intercept with the horizontal axis and EMSY:
qE
a
Y  0 1
0 E 
a
q
dY
q 2 K dY
aqK
a
 qK  2 E
, 0 E  2 
dE
a dE
2q K 2q
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Schaefer model: graph (Y, E)
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Schaefer model: depensation
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Schaefer model: critical depensation
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Gordon-Schaefer model: revenue
• Constant price (p) => total revenue (TR) equal to:
TR(E) = pY(E)
• This is not a «generic» total revenue (py), rather
the «sustainable» total revenue (pY)
• Decision variable is effort (E) rather than harvest
rate (y)
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Gordon-Schaefer model: costs
• The total cost of harvesting, C, depends on the
amount of effort being expended. For simplicity,
harvesting costs are taken to be a linear function
of effort: Total costs (TC): TC(E) = cE
where c is the cost per unit of harvesting effort,
taken to be a constant.
=>
y
TC ( x, y )  c
qx
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Gordon-Schaefer model: rent
dissipation
Rent: TR - TC
Free access ==> TR - TC = 0
• E > E* ==> TC > TR
• E < E* ==> TC < TR
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Gordon-Schaefer model: graph
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