Modelling producer decisions on land use in a spatial continuum

The paper describes how stochastic optimization techniques can be used to model profit maximizing producer behaviour in a spatial continuum. The main methodological issues to be addressed are, first, that the representation of optimal allocations in a spatial continuum naturally leads to models that contain integrals over space, and the second that the resulting model tends to have a multi-level structure, i.e. requires solving nested optimization problems because it should combine the profit maximization by individual producers with market clearing at regional level. We specify four regional models that may illustrate the approach. The first determines the optimal output level for factories that emit pollutants which reduce the crop output of neighboring farmers. The main issue is to compute the associated level of compensation to be paid by the factories to the farmers. The second model deals with optimal zoning. It computes the optimal crop routing for farmers who can choose to sell their crop to factories situated at given locations. This is an optimization problem in functional space, which can be reformulated as a dual stochastic optimization problem. In the third model, the farmer has the possibility of routing his crop along different roads or distribution nodes to the various factories for processing. It can describe the optimal choice of distribution centres at given locations, around plants or cities, and produces optimal boundaries for the zones that supply to or buy from these centres. The fourth model deals with the problem optimal land consolidation, distinguishes between consolidation processes with and without sidepayments. To each of these four models we associate a decentralised, stochastic quasigradient (SQG-)procedure for attaining the (global) optimum, which has a natural interpretation as a device for decentralized adaptive planning.