Optimal Control Methods for Linear Discrete-Time Economic Systems

As our title reveals, we focus on optimal control methods and applications relevant to linear dynamic economic systems in discrete-time variables. We deal only with discrete cases simply because economic data are available in discrete forms, hence realistic economic policies should be established in discrete-time structures. Though many books have been written on optimal control in engineering, we see few on discrete-type optimal control. More­ over, since economic models take slightly different forms than do engineer­ ing ones, we need a comprehensive, self-contained treatment of linear optimal control applicable to discrete-time economic systems. The present work is intended to fill this need from the standpoint of contemporary macroeconomic stabilization. The work is organized as follows. In Chapter 1 we demonstrate instru­ ment instability in an economic stabilization problem and thereby establish the motivation for our departure into the optimal control world. Chapter 2 provides fundamental concepts and propositions for controlling linear deterministic discrete-time systems, together with some economic applica­ tions and numerical methods. Our optimal control rules are in the form of feedback from known state variables of the preceding period. When state variables are not observable or are accessible only with observation errors, we must obtain appropriate proxies for these variables, which are called "observers" in deterministic cases or "filters" in stochastic circumstances. In Chapters 3 and 4, respectively, Luenberger observers and Kalman filters are discussed, developed, and applied in various directions. Noticing that a separation principle lies between observer (or filter) and controller (cf.

Table of Contents:
1 Macroeconomic Policies and Instrument Instability.- 1.1 A Keynesian Economy under the Government Budget Constraint.- 1.2 Optimal Policies by Means of Two Instruments.- 1.3 A Combination Policy.- 1.4 The Case Allowing Instrument Cost.- 1.5 Formulation of a Standard Control System.- References.- 2 Optimal Control of Linear Discrete-Time Systems.- 2.1 Fundamentals of Discrete-Time Control.- 2.2 Controllers for One-Period Lag Equation Systems.- 2.3 Solving Discrete Riccati Equations.- 2.4 Application to Control of a Dynamic Leontief System.- 2.5 Controller for a Dynamic Inequality System.- References.- 3 Observers for Linear Discrete-Time Systems.- 3.1 Preliminaries to Discrete-Time Observers.- 3.2 Luenberger Observers for Discrete-Time Systems.- 3.3 Cost Performance of Optimal Control Incorporating Observers.- 3.4 Recursive Minimum-Cost Observer.- 3.5 Separation of Observer and Controller.- References.- 4 Filters for Linear Stochastic Discrete-Time Systems.- 4.1 Preliminary Least-Squares Estimators.- 4.2 Kaiman Predictor and Filter.- 4.3 Minimal-Order Observer-Estimators: Existence.- 4.4 Minimal-Order Observer-Estimators: Computation.- 4.5 Economic Applications of Kaiman Filtering Methods.- References.- 5 Optimal Control of Linear Stochastic Discrete-Time Systems.- 5.1 Controllers for Linear Systems with Additive Disturbances.- 5.2 Controller in an Imperfect Information Case.- 5.3 Controllers for Linear Systems with Stochastic Coefficients.- 5.4 Certainty Equivalence in Stochastic Systems Control by Theil.- 5.5 Optimal Control of Macroeconomic Systems 148 References.- References.- 6 Stabilization of Economic Systems under the Government Budget Constraint.- 6.1 Dynamic Process of a Government-Budget Constrained Economy.- 6.2 Instability of an Economy with Government Budget Deficits.- 6.3 Instability of an Economy with Government Budget Deficits and Keynesian Policy Assignment Continuous-Time Case).- 6.4 Optimal Control of Economic Systems with Bond-Financed Budget Deficits.- 6.5 Optimal Control of an Open Economy with Bond-Financed Budget Deficits.- References.- Appendix Differentials of Matrix Traces.- Author Index.