Table Of ContentOptimal Control Methods
for Linear Discrete-Time
Economic Systems
Yasuo Murata
Optimal Control Methods
for Linear Discrete-Time
Economic Systems
Springer-Verlag
New York Heidelberg Berlin
Yasuo Murata
Nagoya City University
Faculty of Economics
Mizuhocho Mizuhoku
Nagoya, 467
Japan
AMS Classifications: 90A19, 90A20
Library of Congress Cataloging in Publication Data
Murata, Yasuo, 1931-
Optimal control methods for linear discrete-time
economic systems.
Includes bibliographies and index.
I. Economic policy-Mathematical models.
2. Economic stabilization-Mathematical models.
3. Control theory. I. Title.
HD75.5.M87 1982 338.9'00724 82-19137
With 2 Figures.
© 1982 by Springer-Verlag New York Inc.
Softcover reprint of the hartdcover I st edition 1982
All rights reserved. No part of this book may be translated or reproduced in any
form without written permission from Springer-Verlag, 175 Fifth Avenue, New
York, New York 10010, U.S.A.
Typeset by Computype, Inc., St. Paul, MN.
9 8 7 6 543 2 I
ISBN-13: 978-1-4612-5739-4 e-ISBN-13: 978-1-4612-5737-0
001: 10.1007/978-1-4612-5737-0
To Hiroko and Akiko
Preface
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.
Sections 3.5 and 5.2), we are concerned in Chapter 5 with stochastic control
methods in three types of uncertain environments and with the certainty
equivalence principle. Existing macroeconomic applications of our control
viii Preface
rules are examined in Section 5.5. Our main application, an economlC
stabilization problem, is found in Chapter 6.
Alternative routes for the journey through this book are shown in the
following flow chart, where the reader will find the convenient short cut:
Ch. 1 ~ Ch. 2 ~ Ch. 5 ~ Ch. 6. Whether the short or long course is taken,
we hope that those readers with intermediate knowledges of economics,
mathematics, and statistics will find few obstacles.
Ch.l ( Instrument) Ch.6 ( Macroeconomic)
Instability Stabilization
J., t
Ch.2 ( Deterministic) Ch.5 ( Stochastic)
~
Control Control
J., t
Ch.3 (Observers) Ch.4 (Filters)
~
The author appreciates the suggestions concerning related literature
made by Associate Professor Shin-ichi Kamiyama, my colleague. I also
express my thanks to Miss Y oshie Mizutori for her careful typing of my
cumbersome manuscripts and Miss Mariko Akiyama for her efficient
programming of computations.
Nagoya, Japan Y ASUO MURATA
July, 1982
Contents
CHAPTER I
Macroeconomic Policies and Instrument Instability
1.1 A Keynesian Economy under the Government Budget Constraint
1.2 Optimal Policies by Means of Two Instruments 4
1.3 A Combination Policy 8
1.4 The Case Allowing Instrument Cost \0
1.5 Formulation of a Standard Control System 12
REFERENCES 14
CHAPTER 2
Optimal Control of Linear Discrete-Time Systems 15
2.1 Fundamentals of Discrete-Time Control 15
2.2 Controllers for One-Period Lag Equation Systems 24
2.3 Solving Discrete Riccati Equations 33
2.4 Application to Control of a Dynamic Leontief System 39
2.5 Controller for a Dynamic Inequality System 42
REFERENCES 45
CHAPTER 3
Observers for Linear Discrete-Time Systems 47
3.1 Preliminaries to Discrete-Time Observers 47
3.2 Luenberger Observers for Discrete-Time Systems 53
3.3 Cost Performance of Optimal Control Incorporating Observers 60
3.4 Recursive Minimum-Cost Observer 63
3.5 Separation of Observer and Controller 70
REFERENCES 75
x Contents
CHAPTER 4
Filters for Linear Stochastic Discrete-Time Systems 77
4.1 Preliminary Least-Squares Estimators 77
4.2 Kalman Predictor and Filter 84
4.3 Minimal-Order Observer-Estimators: Existence 92
4.4 Minimal-Order Observer-Estimators: Computation 97
4.5 Economic Applications of Kalman Filtering Methods 105
REFERENCES 110
CHAPTER 5
Optimal Control of Linear Stochastic Discrete-Time Systems III
5.1 Controllers for Linear Systems with Additive Disturbances III
5.2 Controller in an Imperfect Information Case 119
5.3 Controllers for Linear Systems with Stochastic Coefficients 127
5.4 Certainty Equivalence in Stochastic Systems Control by Theil 139
5.5 Optimal Control of Macroeconomic Systems 148
REFERENCES 158
CHAPTER 6
Stabilization of Economic Systems under the Government
Budget Constraint 160
6.1 Dynamic Process of a Government-Budget Constrained Economy 160
6.2 Instability of an Economy with Government Budget Deficits 167
6.3 Instability of an Economy with Government Budget Deficits and
Keynesian Policy Assignment (Continuous-Time Case) 169
6.4 Optimal Control of Economic Systems with Bond-Financed
Budget Deficits 174
6.5 Optimal Control of an Open Economy with Bond-Financed
Budget Deficits 181
REFERENCES 190
ApPENDIX
Differentials of Matrix Traces 192
Author Index 197
Subject Index 201
CHAPTER 1
Macroeconomic Policies
and Instrument Instability
The typical macroeconomic stabilization problem presented in this chapter
will serve as a motivation for our later discussion of optimal control
methods. When, in a dynamic economic system, one tries to minimize the
deviation in national income from its target value, one often finds an
ever-increasing need to adjust policy instruments in order to offset the
effects of past policies. This phenomenon was first recognized as an
instrument instability by Holbrook (1972). We demonstrate this type of
instability in a two-target-two-instrument setting, as an extension of Cur
rie's article (1976) for a one-target-two-instrument setting. After presenting
a Keynesian IS-LM framework for stabilization in Section 1.1, we perform
optimal policies which result in the instrument instability (Section 1.2). To
avoid the instability, we first propose a combination policy (Section 1.3)
and then an alternative objective function involving an instrument cost
(Section 1.4). Finally, in Section 1.5, we modify the optimization problem
into a standard control problem so that we come naturally to the subse
quent chapters.
1.1. A Keynesian Economy under the Government
Budget Constraint
Currie (1976) presented optimal stabilization policies for one target (na
tional income) in an IS-LM economy with three alternative instruments
(money, government expenditure, and bonds). We shall extend his analysis
to a two-target setting (national income and price level). That is, our
objective is to minimize the expectation of a weighted sum of squared
2 1. Macroeconomic Policies and Instrument Instability
deviations of target variables from their equilibrium values, i.e.,
minE(y,2+ wp;), (1)
where y, and p, denote the deviations in real national income Y and price
level P in period t from their equilibrium values Y and ii, respectively; w
stands for some weight attached to p,2.
Our basic model is composed of Currie's equations «2), (3), and (4» and
price determination equation (5) in addition:
°
[Budget] 6.M, + R,-'6.B, + 'TP, Y, - P,G, - (I - 'T)B, = (2)
[IS] Y, = Y( G, ,R" M" B, ,uY') (3)
[LM] M,/P,=M(Y"R"B"um,) (4)
[Price] (5)
where
G real government expenditure,
M stock of high-powered money,
B interest payments on government bonds outstanding, in money
terms,
R nominal rate of interest,
'T tax rate,
Ui random disturbances, (i = y, m, p).
6. symbolizes a change in variable from one period to another.
Innovation is found in (5) where P is assumed to depend on Y and M in
the following manner:
y, = ap /a Y > 0,
The other equations are the ones expounded in Currie's paper; we briefly
explain the equations here and define new notations. (Note that, since
Currie assumes fixed price, his equations do not contain price P, as
compared with equations (2) and (3).) Equation (2) shows the government
budget constraint. In (3), which represents the IS curve, impact effects on Y
with respect to instruments are
(\',=aY/aG>o,
By the negative slope of the IS curve, we have
(\'2= -(ay/aR),s>O.
It follows from the LM curve (equation (4» that
/3, = aM /a Y = I/(a Y /aM)LM > 0,
/32 = - aM jaR = -1/(aR/aM)LM > 0,
and portfolio balance implies
/33=aM/aB >0.
