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A comparison of resource depletion for the four models is presented in figure 14.

Figure 14: Model comparison: Magnitude of the resource depletion as a function of the distance from equilibrium for each of the four models studied in this paper. The fraction of initial capital with respect to equilibrium capital is plotted along the horizontal axis, and the fraction of resource at maximum depletion with respect to equilibrium resource along the vertical axis. Solid line: the basic model of section 2 with , =1; Dots: the model of section 3 with a carrying capacity (, , ); Dashes: the price model section 4 (, , , , and ); Dash dot: the minimum consumption model section 5 (, , , , , and ).

The main emphasis of the present paper has been to interpret resource exhaustion as a dynamical depletion following an exploitation overshoot due to capital (and labor) inertia. We have seen that this qualitative result does not seem to depend upon the degree of simplification of the model since introduction of a carrying capacity, price dynamics and minimal consumption does not destroy the observed depletion. Only one of these features was added at a time in order to simplify algebra and allow simple interpretation. But extra computer simulations done with all three features added gave the same qualitative results: resource depletion is still observed for many parameter choices although adding up several features often diminish the depletion effect with respect to equivalent simulations done with only one feature.

Other refinements can still be introduced, such as endowing agents with more rational behavior, but we suspect that as long as perfect rationality is not assumed, oscillations and subsequent depletion are generic dynamical properties of the dynamics of the ecological-economical system, i.e. that they do not depend upon the details of the model. We will summarize here those results which should be robust against new refinements of the models, and which should probably be true also for a number of real systems. Of course this quest for generic properties still has to be checked by more modeling and by comparison with data from observations.

We noted in section 2 that other production functions can be considered and that, in some cases, the equilibrium could become unstable. But even in those cases the fundamental result of this study is still valid: exploitation of renewable resources with open access yields significant depletion following overshoot when resource replenishment is fast with respect to capital depreciation.

Previous dynamical models were based on fast readjustment of effort in proportion to the difference between cost and benefit [Clark, 1990,Opsomer and Conrad, 1994]. Our model takes into account the fact that readjustment by capital variations has a lot of inertia, which is reflected by the capital depreciation rate. Of course this inertia is responsible for the overshoots and depletions. The first type of fast readjustment of effort would correspond to some kind of perfect economic rationality, including perfect information, but human organizations are more often driven by inertia than by perfect rationality, which explains our choice for the dynamics of the economic variable. Let us note here that a limited rationality in capital investment which would avoid reallocation of capital when profit is small would not prevent resource depletion because the most profitable investment is being done when the resource is above its equilibrium value. When investors would realize that they are loosing money, capital inertia would prevent fast enough readjustment of the effort: a readjustment rate fast with respect to oscillation frequency, which is of order (eq. 13), would be needed instead of the current adjustment rate of capital. The same qualitative argument applies to labor readjustments: they could prevent resource depletion if they would occur faster than , which could only be expected if some other sector of the economy were growing.

We will also discuss in this section some assessment of various government (or other institutional) intervention tools.

One of the simple results obtained in section 2 is that the oscillatory character of the dynamics is not changed by the production coefficients and . In other words, technological and marketing improvement won't save us from overshoots and depletion, but can increase the level of the renewable resource at equilibrium. The same conclusion also applies to some management policies such as shortened seasons and some gear restrictions: they only change the production coefficients.

We have seen that decreasing the capital depreciation rate, , and thus , amplifies the oscillations and level of resource depletion. In the case where government intervention consists of supporting investment by providing loans with low interest rates, is probably decreased and the instability increased. This is one example of a situation where government support of the industry, or of the fishermen, could be detrimental to the fish stock, and thus to the industry in the long term [CEE, 1991].

We have seen in section 3 that the oscillations and resource depletion are strongly attenuated when the fishing effort is such that the harvesting equilibrium is in the neighborhood of the carrying capacity . But, of course, the effect is only important when the fishing effort is small. If the harvesting equilibrium is far below the carrying capacity, the attenuation effect is quite reduced and oscillation amplitude is restored.

Market mechanisms and price variations can reduce resource depletion, but we have seen that this reduction is limited to a factor of two in the real part of the time constant, see eqn. 37. This prediction is made for a rather simple model, however our assumption that the price decreases inversely proportional to the production is probably an extreme since it corresponds to an income that saturates at high productions. We expect instead that prices drop less rapidly in normal situations, and thus that the reduction in resource depletion should be less important than the one predicted in section 4. The reduction in resource depletion by the market only occurs when the demand is sufficient to avoid an external support of prices at production. In this context, institutional support prevents the reduction of resource depletion by market mechanisms and might reduce the ecological sustainability of the system in the absence of any complementary measure.

Societies have introduced financial mechanisms which allow them to maintain the personal income of producers even when natural production is plummeting. The effect of maintaining a minimum consumption level has been investigated in section 5. Once more the dynamics shows that the system sustainability is decreased when a minimum consumption is maintained.

The above results converge in pointing out the fact that external intervention has to balance the short term interests of labour and capital with their long term interests when the renewable resource is in danger of being depleted.


We thank J. Roughgarden for communicating us his preprint prior to publication and A. T. Charles for a number of useful comments on the manuscript. The Laboratoire de Physique Statistique is associated with CNRS (URA 1306) ,Ecole Normale Supérieure, and universités Paris 6 et Paris 7 and we acknowledge financial support from the John D. and Catherine T. Mac Arthur Foundation, through the New Century Fund, the sponsor of Project 2050, and from the European Science Foundation (Network "Dynamics of Complex Systems in Biosciences").

next up previous
Next: Appendix Up: Influence of capital Previous: Simulation results

Fri Feb 7 13:18:37 GMT+0100 1997