Feedback stochastic decision rule for commodity stabilization : An application of control theory to the world cocoa markets

The feedback stochastic decision rule is developed for potential buffer stock agencies in cocoa. This rule prescribes the values of policy variables as a linear function of actually observed endogenous variables with time-varying coefficients which reflect the social welfare and is derived by, first, solving the deterministic non-linear optimization problem, and, second, by tracking the deviations of endogenous variables from their deterministic optimal paths. The welfare function is chosen as a weighted sum of the three eminently desirable objectives, export earnings, price stability and earnings stability. Instead of arbitrarily fixing the weights, we vary them within certain ranges so that policy-makers could be presented with a set of "possibility trade-offs" among our three objectives. The weights on price stability and earnings stability are inferred from simulation results. All of these results should be considered demonstrative rather than prescriptive. We also develop new solution algorithms for the deterministic control based on Jacobson\´s and Mayne\´s (1970) first-order DDP which allow for both non-negativity constraints and implicit dynamic systems and apply these to solve the 20-state variable, thirty three period non-linear model of the world cocoa markets developed by Goreux (1972). Since our algorithms converge slowly near the optimum, we employ the convergence criterion that computation is halted when no further improvement in the welfare functional can be made with the first-order DDP. As far as our experiments are concerned, the welfare gain vary from 17 to 20 percent of the welfare function computed before control. While the solution for the case of quadratic welfare and linear dynamic model itself requires no more than multiplying and adding matrices, it is a formidable work to calculate all second-order partial derivatives of the Hamiltonian. For practical implementation, therefore, it would be desirable to calculate all partial derivatives b- computer techniques.