 # Maximizing the Present Value of Resource Rent in a Gordon-Schaefer Model Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The classical Gordon–Schaefer model presents equilibrium revenue ( ) and cost ( , 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 ( ) is found when and no rent (abnormal profit, ) is obtained. Abnormal profit (here resource rent) is maximized when (maximum economic yield, ). Discounted future flow of equilibrium rent is maximized when , where is the unit rent of harvest and is the discount rate. This situation is referred to as the optimal solution ( ), maximizing the present value of all future resource rent. The open access solution and equilibriums are found to be special cases of the optimal solution, when and , respectively.

Contributed by: Arne Eide (March 2011)
Open content licensed under CC BY-NC-SA

## Snapshots   ## 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.

## Permanent Citation

Arne Eide

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send