Maximizing the Present Value of Resource Rent in a Gordon-Schaefer Model

​
unit cost of effort
0.4
unit price of harvest
1.5
discount rate δ
0.1
The classical Gordon–Schaefer model presents equilibrium revenue (
TR
) and cost (
TC)
, including opportunity costs of labor and capital, in a fishery where the fish population growth follows a logistic function. Unit price of harvest and unit cost of fishing effort are assumed to be constants. In this case, the open access solution without restrictions (
OA
) is found when
TR=TC
and no rent (abnormal profit,
Π=TR-TC
) is obtained. Abnormal profit (here resource rent) is maximized when
TR'(X)=TC'(X)
(maximum economic yield,
MEY
). Discounted future flow of equilibrium rent is maximized when
Π'(X)/δ=π
, where
π
is the unit rent of harvest and
δ
is the discount rate. This situation is referred to as the optimal solution (
OPT
), maximizing the present value of all future resource rent. The open access solution and
MEY
equilibriums are found to be special cases of the optimal solution, when
δ=∞
and
δ=0
, respectively.

Details

The basic Gordon–Schaefer model includes the following:
Surplus growth of the fish stock population (logistic growth):
f(X)=rX(1-X/K)
X
: fish stock biomass
r
: intrinsic growth rate
K
: environmental capacity level in terms of stock biomass
Assume that the fish harvest (
H
) is linear in stock biomass (
X
) and fishing effort (
E
):
H=qEX
.
q
: catchability coefficient
The equilibrium catch is found at the stock biomass value
X
where
H=f(X)
,
X=0
, or
X=K(1-qE/r)
.
Assume further a constant unit price of harvest,
p
, and a constant unit cost of effort,
c
. Total revenue (
TR
) is then
TR=pH
and total cost (
TC
) is
TC=cE
.
Assume
c
includes all opportunity costs, reflecting the normal profit in perfect markets. Abnormal profit (rent) is then
Π=TR-TC
,
which in equilibrium (
H=f(X)
) could be written as a function of
X
as
Π(X)=(p-cX/q)f(X)
.
Denote the unit rent of harvest by
π
; then
π(X)=p-cX/q
.
The optimal equilibrium solution (maximizing the present value of future harvests in equilibrium) is obtained when the short-term loss of not fishing one unit more (
π(X)
) equals the long-term discounted benefit related to this unit being included in the future stock (
Π'(X)/δ
). See Clark (1976) for further details.
C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, New York: Wiley–Interscience, 1976.
​

Permanent Citation

Arne Eide
​
​"Maximizing the Present Value of Resource Rent in a Gordon-Schaefer Model"​
​http://demonstrations.wolfram.com/MaximizingThePresentValueOfResourceRentInAGordonSchaeferMode/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011