Optimal management of an eco-system with an unknown threshold

There has been a extensive discussion in the ecology literature how an apparently rational economic management of an ecosystem can lead to repeated collapses. The standard model postulates that phosphorus concentrations evolve over time according two distinct regimes, depend on whether the current concentration is above or below a critical level. Once the phosphorus concentration exceeds a critical level, phosphorus recycling occurs, resulting in an additional discrete phosphorus intake in the next period. The optimal management balances the positive utility from the phosphorus-generating activity against the probability of lake eutrophication that is increasing in phosphorus concentration. Earlier studies assume that the manager only recognizes that there might be two different regimes, but never realizes that the current regime is depend on whether or not the critical level has been crossed. We assume that there is uncertainty about this critical level, but allow the manager to learn from past observations where this critical level is likely to be. There is an important difference: in the earlier version managers update their beliefs whether phosphorus recycling is present or not, but these beliefs are constrained to be independent of the rlation between the current phosphorus concentration and the critical level.. In or study the manager updates his or her belief about this critical level which yield a probability whether phosphorus recycling is present. The updating of beliefs between consecutive periods follows Bayes’ rule. Several noteworthy results emerge from the analysis. First, when the critical level is known to the manager, there is an interior region of critical phosphorus concentrations where it pays to reduce loadings to avoid crossing the threshold, but this is not the case for very large or very low critical values. Intuitively, if the threshold for recycling is very high, the probability of crossing it is very low, and hence it doesn’t pay to restrict the phosphorus-generating activity. Similarly, if the threshold is very low, phosphorus recycling will occur even if the pollution-generating activity is reduced, and it again doesn’t pay to do so. Only for interim values does it pay to reduce the pollutiongenerating activity. Second, uncertainty about the critical value, as measured by the variance in beliefs over the critical value, can either increase or decrease the optimal phosphorus concentration. This is a direct consequence of results obtained under perfect certainty. If the increase in variance places more probability mass on the lower or upper end of the support of critical values, a region where it doesn’t pay to reduce the pollution generating activity, the human phosphorus loading will increase. Similarly, if probability mass is shifted towards the portion of the support where it pays to restrict the economic activity, human-induced loadings might increase. Consequently, we don’t find any support of a precautionary principle where increases in uncertainty always result in more precautionary actions. The model can easily be generalized to other applications with possible thresholds, e.g., climate change. Third, once we allow the manager to update his or her beliefs and learn about the correct critical value, the system will collapse less frequently than under earlier management rules where the regulator does not realize that the two regimes are separated by the critical value. Hence, repeated collapses are not a direct consequence of economic optimization, but follow from the incorrect assumption that managers place on the environmental system.