Optimal Stabilization Policies for Deterministic and Stochastic Linear Economic Systems

The analysis of different stabilization policies has been the subject of detailed investigations by economists for a number of years. Research in this area originated with early work by Phillips [11, 12] who proposed a number of stabilization policies and considered the stability properties of the system when these policies were implemented. In particular he showed how different policies could generate undesired oscillations and even instability, in economic activity. However, his analysis was purely descriptive in the sense that while the alternative policies he considered were plausible, they were not derived from any optimizing behaviour. Since that time attention has shifted to more normative questions and to the study of optimal stabilization policies. One of the first authors to consider this aspect was Holt [7], who assumed a quadratic criterion functional and obtained linear decision rules of the type first derived by Simon [15] and Theil [16]. Recent advances in control theory have led to the development of new, more convenient techniques, directly applicable to this problem (see e.g. Kalman [8] Pontryagin [13]). As a result of these various developments it is now known that assuming a quadratic utility functional and a linear model it is possible to obtain the optimal policy as a linear feedback law, which is a particularly convenient form in which to obtain the solution. The present paper has two objectives. The first is to apply these techniques to the stabilization of the deterministic multiplier-accelerator model. This problem has been recently studied by Sengupta [14], although he appears to have incorrectly applied the basic theorem which yields the feedback law. The solution is given for the general multiplieraccelerator model where the investment function is generated by the flexible accelerator. While the general nature of the optimum solution can be readily determined, it is extremely difficult to investigate any particulars and to perform any comparative dynamics in the general case. However, the solution can be explicitly obtained for the instantaneous accelerator and accordingly this case is studied in some detail. The second aim of this paper is to introduce stochastic elements into the model. Recently, considerable progress has been made in stochastic control theory (see Wonham [18-21]). It has been shown that the quadratic utility functional approach can be readily extended to cover various kinds of stochastic elements and turns out to give similar linear feedback laws to those for the deterministic case. Some work has already been devoted to introducing stochastic elements into stabilization theory. Brainard [3] introduces them into a static model; Holt [7] includes additive disturbances in his dynamic model; Henderson and Turnovsky [6] introduce stochastic parameters in a somewhat specialized dynamic