حل مشکلات مدیریت زیست محیطی با استفاده از برنامه ریزی پویا

We study an ecological management problem where the interaction of the maximizing-welfare dynamic decision and the dynamics of the ecosystem admits multiple equilibria. We follow the example of a shallow lake by Brock, etc. Low loading preserves resilience of the ecosystem while high loading may lead to the deterioration of the ecosystem. We consider instruments of a regulatory agency that may help to maintain and enhance resilience by enlarging the domain of attraction of the low-pollution equilibrium. The global dynamics of all our model variants, without and with tax rates, are analytically studied by the Hamilton–Jacobi–Bellman method and numerically solved through dynamic programming.

Research in the last decade has shown that the relationship between the environment and economic activity is a rather complex one. Recent studies on ecological management problems reveal that the interaction of human activity and the ecological dynamics generates intricate dynamics that we need to understand in order to undertake welfare analysis and to make policy decisions. To demonstrate the existence of such a complexity, researchers often have focused on a lake management problem. The lake model can be viewed as a symbolic model. Here, the problem of the behavior of a shallow lake that is subject to pollution by phosphorous is studied. The model employed here reflects the characteristics of many ecological systems. They imply the integration of two aspects: the dynamic decision problem maximizing welfare and the dynamics of the ecosystem. This often generates properties of complex dynamics such as multiple equilibria, history dependence, thresholds between multiple attractors and a discontinuous policy function. Carpenter et al. (1999b) and Brock and Starrett (1999) propose a deterministic version of an optimal management problem for ecosystems where there is a phosphorous loading into the lake due to economic activities. This affects the stock of phosphorous in the lake. When the stock of phosphorous becomes too high, internal positive feedback mechanisms start to impair the ecosystem’s ability to absorb and biodegrade the loading. Welfare is defined as sum of utilities from loading rate and disamenities from degraded lake quality proportional to the level of the stock variable. The management can measure the stock and can control the loading as a function of the stock. The management chooses these loadings to maximize welfare of the conflicting interests of polluters and lake users. The model has one state variable, and its dynamics is simplified as a one-dimensional ordinary differential equation. The work by Carpenter et al., 1999a and Carpenter et al., 1999b is extended in Ludwig et al. (2002) to give a more precise account of the internal loading due to changing levels of dissolved phosphorous in the lake sediment in a stochastic environment. Using two state variables, the phosphorous in the water and the phosphorous in the lake sediment, the optimal loading is derived as a function of those two state variables. They argue that simple policies that neglect dynamics of the phosphorous in the sediments are inadequate unless the time horizon is short and the dynamics are slow. They also mention that a stochastic model is essential if there are substantial random fluctuations in loading. It is important particularly when the optimal solution to the system shows history dependence where the lake possibly flips between two attractors due to the disturbance. Thus, they argue that the management problem should be described as a stochastic two-dimensional state variable problem. Another stream of literature to analyze this problem uses a game theoretic approach with N communities as in Mäler et al. (1999) and Dechert and Brock (2000). They focus on the Nash equilibria solutions to the dynamic lake game. Dechert and Brock (2000) use discrete dynamic programming to obtain a numerical solution and show that the social optimum loading is always less than the total loading in the dynamic game. As N becomes large, the divergence between the dynamic game solution and the social optimum becomes extreme and complex dynamics can appear in between. Mäler et al. provide the optimal management solution in the same context but investigate whether it is possible to induce an optimal management in case of common use of the lake by means of a tax rate. Their results shows that, for a small number of communities, a constant tax on phosphorous loading can induce optimal behavior in the long run, but for a large number, it depends again on the history of the lake. Perrings (1999) suggests that in the shallow lake example, an aim of decision maker should be to maintain the system in a desirable state and to assure the sustainability of desirable states by assuring the resilience of the system in those states since there are significant costs to being trapped in a locally stable eutrophic state. As Carpenter et al. (1999b) show, eutrophication causes the loss of the potential benefits of fresh water, including consumption of water by the surrounding population, irrigation, industrial uses and recreation. Once a lake reaches an eutrophic state, it is very costly to reduce the phosphorous level and restore clean water. However, note that disamenities from degraded lake quality is already taken into account in our optimization problem. It means that the optimal management does not necessarily choose the path to the oligotrophic state. It happens when the society does not care much about lake quality and when the path going to the eutrophic state generates a higher value than the path going to the oligotrophic state for a given initial phosphorous state. However, we still could argue that the eutrophic state is not desirable and there is enough reason for a neutral institution, for example, a public agency, to intervene to restore a resilient oligotrophic state. There are several reasons that one should intervene into the system dynamics. First, human knowledge on the dynamic mechanisms of the ecosystem is still not complete. We may overlook further negative externalities from the lake locked into eutrophic state, such as the unexpected loss of diversity of species in a food chain. The collapse of a food chain that starts from the eutrophic state of the lake may cause an irreparable crisis. Our knowledge of those facts is rather poor. There is also a problem of incorporating the dynamics of each species and their interaction into our model. Its theoretical treatment is hopeless due to the complexity of the interaction effects. Those unexpected negative externalities, which we neglect in the model may give a rationale for intervention. Another reason is that the optimization is done only from the viewpoint of the current generation in the sense of their utility and subjective discount rate. However, it is impossible to know the future generation’s preference. We also know, as aforementioned, that once a lake reaches an eutrophic state, it can be very costly to restore clean water, and future generations will have to pay a high cost for restoring such a lake when they prefer an oligotrophic state. It seems that for the decision maker, it is fundamental to maintain the desirable function of the ecosystem where the ecosystem keeps the capability to absorb and biodegrade loading. The ecosystem should also be a resilient one about the oligotrophic state. In this paper, we go back to the simplest deterministic version by Carpenter et al. (1999b) and Brock and Starrett (1999). They detect the existence of multiple equilibria. When multiple equilibria arise, our interest will be to study the global dynamics and its policy implication. It has been recognized by Brock and Starrett that one could obtain three different scenarios on global dynamics in the simplest variant of the model with three equilibria: (1) Lower steady state (LSS)—the stable manifold associated with the low pollution equilibrium is dominant for any given initial condition, (2) Skiba—there is a threshold (Skiba point) that separates different domains of attraction and (3) Upper steady state (USS)—the stable manifold associated with the high pollution equilibrium is dominant for any given initial condition. We study those three scenarios analytically by using the Hamilton–Jacobi–Bellman (HJB) equation and numerically by dynamic programming that permits us to detect a threshold, if it exists, and to derive a policy function which may be discontinuous. For the purpose of an ecological policy, we are in particular interested in the scenario (1), the situation for which for any starting point the optimal policy will converge on the stable manifold to the low pollution equilibrium and make the oligotrophic state a global attractor. We will explore the possibility of enlarging its domain of attraction by introducing a tax rate on the phosphorous loading whereby the tax rate will depend on the stock of phosphorous. Such a tax rate will make the lake dynamics converge to the low pollution equilibrium. Overall, our interest is whether we can create a resilient system with oligotrophic optimal state by a tax rate on loading imposed by the manager of the lake or some regulatory agency. Our definition of desirable resilient system is where we obtain the scenario LSS; the stable manifold associated with the low pollution equilibrium is dominant for any feasible initial state. Our tax rate that is increasing in the stock of phosphorous is successful in the sense that we indeed can convert the scenarios of both Skiba and USS into a scenario of LSS. If the decision maker is successful to keep the latter scenario, for any initial state of phosphorous, the path to the oligotrophic state will generate the maximum value, and it will be actually chosen by the decision maker. Since the history dependent property disappears and the size of the stability domain corresponding to oligotrophic attractor has been enlarged, the system may be able to absorb and biodegrade loading to a certain degree even if external shocks occur to the system. Finally, we will extend the treatment of problem and explore optimal tax schemes where we allow the tax revenue to be used for purification efforts by the public agency to clean up the lake. We will show that in a model with multiple equilibria and Skiba points, “best tax schemes” (tax schemes that reduce welfare least) will be state dependent. The remainder of the paper is organized as follows. Section 2 presents the model. Section 3 studies the problem by using the HJB equation and shows the analytical procedure. Section 4 introduces a tax rate into the model and applies again the HJB equation to this new problem. Section 5 presents results using dynamic programming to compute the global dynamics of the model without and with tax rate. Section 6 explores optimal tax schemes. Section 7 concludes the paper.

Using the Hamilton–Jacobi–Bellmann equation, we have studied an ecological management problem with multiple equilibria and different scenarios for the global dynamics. When such different scenarios in the multiple equilibria model arise, knowing the global value function is crucial to detect the unique optimal path and the global policy function. The global value function is used for path selection for given initial state. One path may generate a higher value than the other paths for any given initial state, yet those best paths may lead to low or high pollution attractors or a threshold may exist between two attractors and the optimal path will switch at the threshold—the Skiba point. For our model of a shallow lake we have explored the possibility of enlarging a resilient domain of an oligotrophic attractor. Introducing a state dependent tax rate on the phosphorous loading, we demonstrated that the goal of maintaining or enhancing a desirable resilient system could be achieved with state dependent tax rates. Such an intentional action by some regulatory agency should be justified since there exist significant costs to being trapped into an eutrophic state. There are negative externalities from the lake being locked into a eutrophic state. Those concerns not only immediate effects on the ecosystem but also future generations’ environmental conditions. Those externalities were not explicitly theoretically treated in order to avoid greater complexity. However, we can have an intuition, from our numerical example, Fig. 7, on how the future generation is affected by the current generation’s choice. Fig. 7 shows the value functions for different k s, the preferences for the lake state over the benefit from phosphorous loading. It is clear that the value loss from increasing k is larger for the higher level of phosphorous stock, x. It means that the future generation favoring a cleaner lake will suffer a large loss of value by attaining the oligotrophic state when the eutrophic state is already given as an initial state. Therefore, maintaining the resilient system with oligotrophic attractor might be a safe way to avoid the unfair treatment of different generations. It is also remarkable that the system is strongly resilient in the face of shocks, preventing a lake from flipping over easily, unlike the Skiba case which has the property of history dependence. We use taxation as an instrument of a regulatory intervention. It seems to work well for the purpose of restoring a resilient ecosystem. As a result, however, there exists a value loss after increasing the coefficient of the tax rate, γγ. Finally, we compare different tax schemes and environmental policies from the point of view of value loss. It is noteworthy that the computation of the value function can be usefully utilized for evaluation of tax schemes and environmental policies.