The diagram of our simplest model is presented in figure 1.
Figure 1: Diagram of the different processes involved in
the dynamical model.
Resources, N, are renewed at a constant proportional rate r. N can be interpreted as a fish population, or better as the fish total mass. Resources are depleted because of harvesting by the fishermen. In this model the harvest obeys a generalized Cobb-Douglas expression:
where P, the production function, has decreasing returns in terms of the labor force, L, and the invested capital, K. This simple Cobb-Douglas expression, with half powers on K and L and a unitary power on N, was chosen to allow simple algebraic computations rather than to be realistic. Still, proportionality of production to N is a common assumption of many fisheries models ([Clark, 1990] and [Yoshimoto and Clarke, 1993] for instance). It corresponds to the hypothesis of a constant catch per unit effort. In the present model the effort is represented by the square root term in K and L. This term gives a constant return to scale and perfect substitution between K and L. We discuss in section 2.4 possible generalizations of equation 1 and some consequences specific to the present model.
We further distinguish between , the size of the harvest in monetary units, and , the size in number of fish, by multiplying P by coefficients and for, respectively, the size of the harvest in monetary units and in number of fish. can be thought of as a price.
The labor force is assumed to be constant in time. Let be the capital depreciation rate, i.e. the inverse mean life time of the boats represented by the capital, K. can be thought of as the profit from the fishery, coming from the production in monetary units after compensation for capital depreciation. Our main assumption in this section is that a constant fraction of the profit, , is spent on consumption. The rate of consumption, C, is assumed to be
The profits from the harvest are then either consumed by the fishermen or reinvested into new capital in order to increase future yields. In some sense, investment and consumption do not correspond to any optimization decided by fishermen upon some economic rationality, but simply to what they can invest or consume. This fixed reallocation ratio, independent of the size of the profits, of course differs from the usual assumptions of equilibrium economics, in which capital and labor instantly quit the activity when profits plummet, but a similar fixed reallocation ratio was observed during the twentieth century by A. Bowley in England and P. Douglas in the United States and is known in evolutionary economics as Bowley's law ([Samuelson, 1993]). Bowley's law is taken here as a simple first approximation to test the influence of capital inertia.
The following set of equations describe the coupled resource and capital dynamics:
In this simple form the model is reminiscent of a Volterra-Lotka predator prey model, where the capital plays the role of the predator and the resource that of the prey population. It also resemble some of the two differential equation models early proposed by V. L. Smith (1968), although Smith's differential equation for capital is based on different assumptions. But the main difference between Smith's approach and the present study is that Smith focuses on equilibrium while we are interested in resource depletion which here appears as a dynamical transient process (see further).