The Dynamics of Spatial Pollution - the Case of Phosphorus Runoff from Agricultural Land

Negative production externalities have been the concern of many economic studies. In the case where the pollutants accumulate over time these studies are conducted within a dynamic framework. Besides the intertemporal aspect, the heterogeneity of space may be an important consideration for the economic analysis of negative production externalities. Traditionally, economists think of space in the form of a distance measure. However, two locations next to each other and at an equal distance from a receiving medium, may demonstrate completely different vulnerability to negative agricultural production externalities. Thus, we must not only consider site characteristics in a land classification system, but must also consider the characteristics of the receiving medium to depict the environmental vulnerability of locations. We base our analysis on the full-information approach by employing a land classification system and considering the use of improved monitoring technology to account for heterogeneity in residue (pollutant) production. For the sake of concreteness of the analysis we consider the example of Phosphorus runoff into a lake, but also discuss our results in the context of Nitrate leaching. We assume that a social planner for the watershed of a lake exists and that she/he wants to maximize the net benefits resulting from agricultural production taking account of the monetary damage due to pollution of the lake. To obtain an analytical solution more easily, we propose the framework of a two stage optimal control problem. In the first stage we solve the spatial problem by determining the optimal trajectories of the control variables in the dimension of space given by the range of the land classification system. The solution of the social planner's decision problem in the first stage is depicted by the value function which is defined as the value of the maximization of the net benefit from agricultural production allocated over space subject to the law of motion and other conditions appropriate for the problem. Thus, the value function, evaluated along the optimal path of the control variables, reflects the value of the solution for the optimal control problem of the first stage (spatial problem) given some parameters. In the second stage we solve for the optimal intertemporal allocation of the already optimized spatial allocation. For this purpose the value function is used as the benefit function. In the policy analysis, we address questions related to zone taxes, zone permits and zone standards. Most importantly, however, we present a modeling approach which consists of the simultaneous solution of the micro level (farm) and macro level (aggregate supply and demand) production over space and time. In particular, it allows us to derive relationships between long-run and short-run supply and input demand functions