Effect of Uncertainty on Optimal Control Policies

Abstract

IN THIS PAPER, I will provide a method for ascertaining the optimal control policy and the associated welfare cost in a control problem where the welfare cost is quadratic and the econometric model used is linear and the values of its parameters are uncertain. In previous papers, Chow [5,7,8], I have treated the control problem with quadratic welfare and linear model under the assumption that the parameters in the model are given for certain. It would be interesting to relax the assumption of certainty of the parameters, and to examine the effects on the optimal control equations and the associated welfare cost. As it has been generally recognized, one important use of econometric models is in the design of optimal quantitative economic policies. However, because our knowledge of the economic system is imperfect, one might be led to argue against the use of existing econometric models for policy purposes-although a Bayesian would not take this position but would rather find an optimal way to utilize his imperfect knowledge. One problem which I have tried to study in the application of econometric knowledge to policy decisions is the measurement of the possible advantage of an optimal policy based on an econometric model over a policy of maintaining constant rates of change for the instruments, under the assumption that the parameters of the model are known for certain. Calculations using a simple marco-econometric model presented in Chow [8] have indicated that the welfare costs for the latter policy can be about 40 to 80 per cent higher than for the optimal policy based on an econometric model when the model parameters are assumed to be known constants. A natural second problem is to measure the gain from optimal control when knowledge of the model parameters is uncertain. This and other problems of economic policy can be studied by methods of this paper. When we assume that the parameters of a linear econometric model are uncertain we can take one of two approaches in deriving the optimal control policies. The first is to assume a given joint density for these parameters which is available at the beginning of the planning horizon and which is not to be modified while the economic process is being controlled. The second is to allow for continuous modification of the joint density of the unknown parameters as more observations become available to the policy maker. To derive optimal control policies from the second approach is more difficult because policies applied to the early periods affect not only the performance of the economy