Details

The basic Gordon–Schaefer model includes the following:

Surplus growth of the fish stock population (logistic growth):

: fish stock biomass

: intrinsic growth rate

: environmental capacity level in terms of stock biomass

Assume that the fish harvest () is linear in stock biomass () and fishing effort ():

.

: catchability coefficient

The equilibrium catch is found at the stock biomass value where , , or .

Assume further a constant unit price of harvest, , and a constant unit cost of effort, . Total revenue () is then

.

Assume includes all opportunity costs, reflecting the normal profit in perfect markets. Abnormal profit (rent) is then

,

which in equilibrium () could be written as a function of as

.

Denote the unit rent of harvest by ; then

.

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 () equals the long-term discounted benefit related to this unit being included in the future stock (). See Clark (1976) for further details.

C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, New York: Wiley–Interscience, 1976.