In solving the dynamic fisheries management problem involving time, the modern optimal control theory formally known as the calculus of variations is used. The objective of the intertemporal analysis is to arrive at the optimal time path that will maximize the economic rent of the fishing industry managed by the fishermen as the operating firms. In economics, information on fisheries population is rarely available, as such the optimal time path for the management of the stock is therefore inaccurate. In view of this shortcoming, fisheries economic studies tend to divert attention to alternatively predict the future trend in the utilization of essential policy variables like the rate of harvest, level of effort application, and the pressure on the fisheries population arising from the current level of resource utilization. From current information the future direction is deduced and a sound management policy is recommended to the fisheries authority. One of these fisheries management policies in economics relates to the issue of sustaining the resource for the future.

In the next section, we will review the basic but useful intertemporal fisheries model developed by Clark [1977], and the theoretical aspects of this intertemporal model will be subsequently highlighted. The application of steady state and intertemporal fisheries models will follow in the light of sustainable resource management if the given fisheries were to be managed under open access, steady state, and the intertemporal limited entry.

**5.1 Intertemporal fisheries model** The widely discussed intertemporal fisheries model is the one developed by the mathematician Clark [1976]. In this theoretical development, fisheries stock is treated as synonymous with capital. Clark utilizes the normal catch effort and stock equation in the form of a Cobb–Douglas production function

- ((7.1))

- ((7.2))

where γ is a constant term, and *a* and *b* are parameters of the variable factors for fishing efforts and fish stock, respectively. The rest of the notations are defined as before. Assuming that *a*= 1, the economic rent equation becomes

- (8)

where π is the economic rent in terms of monetary value per season, *p* is the per unit price of fisheries harvest, and *c*(*x*(*t*)) =*c*/*F*(*x*(*t*)) is the cost per unit of fishing operation, which is a decreasing function of the fish stock. Whenever time is noted in the variable it means that it is a function of time. The objective of the intertemporal problem is to maximize the present value of the stream of the future economic rents within the range of time denoted by the integral that is from the initial time 0 to the terminal time ∞. This optimization objective is constrained by the per time stock-growth equation. Presented in mathematical expression the intertemporal fisheries objective is to

- (9)

subject to

where, δ is the discount rate and *e*^{−δt} is the discounting factor for a continuous problem for *t* number of years time horizon. The stock-growth equation is often referred to the equation of motion and *x*(*t*) is the control variable without which there is no need for intertemporal optimization. The objective is to find the optimal time path for the control variable fish stock that will maximize economic rent overtime and conforms to the aim of sustaining resource utilization into the future. The intertemporal maximization result proven by Clark [1976, 1977] is similar to that of golden rule basic equilibrium condition of the capital theory:

- (10)

*F*′(*x*) is the marginal growth rate of the fishery population and *c*′(*x*) is the marginal cost of harvesting the stock. When the marginal cost of harvesting fisheries stock is zero the discount rate equation would reduce to δ=*F*′(*x*), identical to short run demand for and supply of investment funds under the capital theory. The case of fisheries is somewhat different from the capital theory because of the stock effect represented by the second term of the right-hand side of the equation. Clark assures that this stock effect is negative because as the fisheries stocks decline due to the fishing harvest the extraction cost eventually raises. In other words, it needs higher operation cost to capture a smaller stock of fisheries.

In our intertemporal fisheries formulation we stick to the original economic rent equation represented by the difference between the normal definition of total revenue and the total cost of the fishing undertaking. We further assume that this fishing undertaking is operational under the government-managed firm like its authorized fisheries agency, with the objective of maximizing the economic rent function of the firm under a given simplified market condition of perfect competition. Under this last assumption the firm's output price and cost of harvesting the stock are held as constants. The objective function written in the Hamiltonian expression is given in equation (13) and solutions to this intertemporal fisheries problem can be found using the maximum conditions of optimum control theory.

- (11)

The partial differentiation of the Hamiltonian equation with respect to effort yields the simplified result of the optimum utilization of fishing effort in equation (12).

- ((12.1))

- ((12.2))

The result in equation (12.2) shows that the marginal user cost, μ(*t*) =*e*^{−δt}λ(*t*), is defined as the difference between price and the marginal cost of fishing effort. The concept of marginal user cost coined by J. M. Keynes (see Clark[1976, p. 103]) is used interchangeable with the scarcity rent concept used to explain λ(*t*) in resource economics. In our fisheries intertemporal problem the marginal user cost is the current scarcity rent value (marginal user cost multiplied by the quantity of catch) of the fisheries resource after discounting.

On the other hand, the same result when translated into the marginal physical product concept as often used by agricultural production economist reduces to the ratio of per unit effort to the difference of catch price and marginal user cost. Without time dimension, μ(*t*) is absent and the optimum result for effort would be reduced to the familiar steady-state condition.

Because the value of marginal user cost of the *in situ* fisheries is positive, the input–output price ratio for the intertemporal optimization should be higher than the steady-state optimization case, that is *c*/*(p*–μ(*t*)) > *c*/*p*. Evidently this outcome tends to support the fact that effort application under the intertemporal optimization would be smaller compared to the steady-state optimization. Therefore, the pressure on resource utilization is relatively milder under the intertemporal case and thus more appealing to resource sustainability. With a lower fishing effort applied to the fisheries, it is likely that the impact in terms of damage inflicted on the environment and fisheries habitat is also minimal. As a result fisheries population tends to breed favorably under the intertemporal management condition.

For this to be really achievable it must be conditioned such that the assignment of the property right in fishing jurisdiction is made possible and that enforcement to refrain unauthorized users is effectively implemented. Private and social cost involved in the assignment of property rights and policing of laws and regulations should fall short of the potential benefits generated from fishing operation.

Partial differentiation of the Hamiltonian expression with respect to stock requires the use of the Pontaryagin maximum principle, that is *d*λ/*dt*=−∂H/∂*x*(*t*) (see Nagatani [1981, pp. 15–17]). This condition can be derived and simplified further based on our earlier reference to the definition of the discounted marginal user cost λ(*t*) =*e*^{−}^{δt}μ(*t*), which reduces to the equilibrium condition for fisheries stock as:

- (13)

Using the above equilibrium intertemporal condition yields the following desired optimal time path for fisheries stock:

- ((13.1))

Expressed in terms of the rate of change of the marginal user cost over time equation (13.1) becomes

- ((13.2))

When price equals the marginal user cost (*p*=μ), the growth rate of marginal user cost over time should be equal to the discount rate minus the rate of growth of stock effect. For the case in which price is greater or smaller than the marginal user cost, the growth rate of marginal user cost is greater or smaller than the discount rate depending on the magnitudes of catch- and growth-stock effects. The above intertemporal optimum time path result also shows that the growth rate of marginal user cost would be equivalent to the discount rate in the absence of catch- and growth-stock effects on the fisheries. In the case of the undisturbed fisheries whereby the impact of fishing mortality on stock is minimal, the change in the marginal user cost over time should be equal to the discount rate minus the rate of growth of the fisheries stock.

Letting ∂μ/∂*t*= 0 and μ= 1, an equilibrium steady state condition in output space representing per unit cost and price on the *y*-axis and quantity of harvest on the *x*-axis could be derived from equation (13.2), which is equal to

- ((13.3))

where *MC*_{x}=δ/*h*(*x*), the marginal cost of stock extraction, *g*(*x*) is the stock growth rate, and *h*(*x*) is the stock harvest rate. An interpretation of the above equality is that when *g*(*x*) =*h*(*x*) the equilibrium is achieved as in the case of the perfect competition steady-state fisheries management. For *h*(*x*) > 0 and *g*(*x*) > *h*(*x*), the marginal cost curve or the supply function for the fishing industry shifts to the right, which means that at the same existing price more quantity of fisheries can be caught. On the other hand, when *g*(*x*) < *h*(*x*) the equilibrium condition requires that price be equated to marginal cost of stock extraction plus some percentage of marginal user cost represented by 1. This scenario seems to support a smaller level of catch because the marginal cost curve with the marginal user cost for the dynamic equilibrium is higher (above) the marginal cost under the static optimization.

In fact we may incorporate the effect of the environmental degradation as the result of resource extraction, which requires a higher marginal cost curve by the magnitude of the marginal environmental cost and a much lower quantity of harvest for an equilibrium condition. By this reasoning, fish population and fish habitat exposed to constant exploitation is most likely to exhibit higher probability of environmental and ecosystem damages.