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Stochastic control for economic models
0005 1098/83$3.00 + 0.00 Pergamon Press Ltd. ~', 1983InternationalFederation of AutomaticControl.
Automatica, Vol. 19, No. 4, p. 455, 1983 Printed in Great Britain.
Book Review
Stochastic Control for Economic Models* David Kendrick verbal presentation of the underlying assumptions and limitations. For example, and as might be expected from the control literature, the schemes presented in Parts It and III essentially imbed the stochastic aspects into the mean and covariance structures of the states and model parameters. The rational expectations assumption is also made. The projections of the means/covariances are studied in detail in Sections 7.3 and 9.6. In general however, the decision maker takes into account the whole distribution of future uncertain elements if known, and not only the expected values as assumed. The fact is that economic behaviour is as much influenced by changing degrees of uncertainty, as by changes in expectations in the narrow sense. The anticipated states X,+~ in economy should be represented n o t only by a probability distribution, but by a full set of distributions, in principle one for each decision-maker. This would reduce the concentration on expectations, and reduce the neglec! of uncertainty, and of a preference structure in this uncertainty. Another related issue is the fact that economic decision makers often pursue discretionary policies in a stochastic environment, which means that the deviations between anticipations and realizations are not purely random, i.e. random normal, at least for some decision-makers. It is hoped that by stimulating a wider use of some of the available techniques, the m o n o g r a p h will help clarify the outstanding issue of whether the ~eal variables of an economy can or cannot be influenced through economic policy in a stochastic environment, e.g. when expectations serve as anticipations, and also when these are different as often seen. It is a little regretful that other approaches to adaptive control have not been surveyed, even if the reader of the m o n o g r a p h would be referred to a wider use of other reference documents. Especially the relation to stochastic filtering (for prediction, estimation, approximation, and modelling) is neglected. Works by Dungan, Fair, Aoki, Chow and others should have been surveyed. The formal derivations are carefully written in terms of notation, equations, and calculations. Dynamic programming is extensively covered in Section 2.2, Chapters 6 and 7, and approximation procedures are presented in a clear way (e.g. Section 10.3). As indicated, the analytical appendices are very extensive, and there is an index. The bibliography might have been supplemented by more diversified references. In conclusion, this m o n o g r a p h will stimulate experiments with some techniques for stochastic control of economic models, while inciting other authors to further comparisons and advances. It is a recommended reading for control engineers, wishing a flavor of the difficulties of stochastic economic models, and for advanced econometrics students and researchers.
Reviewer: L. F. P A U Ecole Nationale Sup6rieure Des T616communications, 46 Rue Barrault, F-75634 Paris, Cedex 13, France.
IMPORTANT developments in economic theory and applications in recent years have been the progress of economic analysis under uncertainty and the increasing degree of sophistication in the modelling of the formation of expectations and their impact on policy rules. The models for most economic systems, assuming one single decision maker for simplification, have evolutions governed by equations of the type
f, tx,, a,, z,, u,,{xo, x~ ..... x , - ~ l , X~+2 .... : , 3 = 0 (1) where X, is the endogeneous variables at time t; Z,, exogeneous variables at time t; a,, parameters of the model estimated at time t; U , instrument variables (values of controls) as determined by policy rules (control law); ~, unobservable random variables; X,+ ~, anticipated value of X,+~ at time t. Both types of advances as earlier mentioned have raised important questions concerning the possibilities of using stochastic control theory for the formulation of economic policy rules, as well as the effectiveness and feasibility of such policies. Much of this work has been centred on validations of, or contradictions to the so-called rational expectations model, in which the anticipated values )f,+i are taken equal to Et(X,+~), i = 1,2 . . . . . expected value at time t over c,,e,+ 1. . . . . as derived from the model (1). In his carefully written monograph, David Kendrick and his participating co-authors, have essentially tried to transfer to the economic community some specific algorithms and theories from optimal control, stochastic control, and dual c o n t r o l : Part 1 covers basic notions and equations of quadratic linear control problems, numerical methods, and an example; Part II deals with stochastic control with additive error terms in the state equation for X,, supplemented by some interesting specific results on multiplicative disturbances; Part III introduces adaptive control or dual control, with deterministic, cautionary, and probing terms in the expression of the cost-to-go, and the dual role of control and learning assigned to the instrument variables. The main strength of the book is the good progression in the complexity of the algorithms presented, and the fact of using several times in their context two case models: Chapters 7 and 12 o n the o n e hand, and Chapters 4, 10 and 12 on the other. One interesting issue for the experienced reader is whether this book has been able to integrate the two aspects of economic analysis and model sophistication as mentioned in the introduction of this review. More specifically, many problems in the field of stochastic control of economic models are about what the anticipations )(,+~, i = 1,2 . . . . . should mean and how they should be represented. The engineering literature only provides little help in this respect. The fear one m a n y have as a reviewer is that this book, because of its stated goal of 'information translation for economists', provides too little analysis of these basic economic questions. One may also worry a little bit about the lengthy development warranted to some specific algorithms, to the expense of many others or of the comparison hereof. As might be expected, the economist may be puzzled by the extensive analytical derivations or explanations (appendices run from page 143 to 227), in contrast to a more fundamental and
About the reviewer L. F. Pau received his M.S., M.A., Ph.D. and D.Sc (Doctorat d'Etat) degrees from Paris University. After serving as an assistant, then associate professor at the Technical University of Denmark, he has since 1974 been a professor at ENS T616communications, Paris, and department head since 1977. During 1976-1977, he was visiting associate professor at the Department of Electrical Engineering and Computer Science, M.I.T., and 1980 1982 he was professorial lecturer at the Department of Computer Science, University of Maryland. His main areas of interest are: economic modelling and control, applications of game theory, failure diagnosis, and pattern recognition. For the 1981-1984 term, he serves as chairman of the economic and management systems committee of IFAC.
* Stochastic Control Jbr Economic Models, by D. Kendrick. Published by McGraw-Hill, New York (1981). 242 pp., US$39.50.
455
Automatica, Vol. 19, No. 4, p. 455, 1983 Printed in Great Britain.
Book Review
Stochastic Control for Economic Models* David Kendrick verbal presentation of the underlying assumptions and limitations. For example, and as might be expected from the control literature, the schemes presented in Parts It and III essentially imbed the stochastic aspects into the mean and covariance structures of the states and model parameters. The rational expectations assumption is also made. The projections of the means/covariances are studied in detail in Sections 7.3 and 9.6. In general however, the decision maker takes into account the whole distribution of future uncertain elements if known, and not only the expected values as assumed. The fact is that economic behaviour is as much influenced by changing degrees of uncertainty, as by changes in expectations in the narrow sense. The anticipated states X,+~ in economy should be represented n o t only by a probability distribution, but by a full set of distributions, in principle one for each decision-maker. This would reduce the concentration on expectations, and reduce the neglec! of uncertainty, and of a preference structure in this uncertainty. Another related issue is the fact that economic decision makers often pursue discretionary policies in a stochastic environment, which means that the deviations between anticipations and realizations are not purely random, i.e. random normal, at least for some decision-makers. It is hoped that by stimulating a wider use of some of the available techniques, the m o n o g r a p h will help clarify the outstanding issue of whether the ~eal variables of an economy can or cannot be influenced through economic policy in a stochastic environment, e.g. when expectations serve as anticipations, and also when these are different as often seen. It is a little regretful that other approaches to adaptive control have not been surveyed, even if the reader of the m o n o g r a p h would be referred to a wider use of other reference documents. Especially the relation to stochastic filtering (for prediction, estimation, approximation, and modelling) is neglected. Works by Dungan, Fair, Aoki, Chow and others should have been surveyed. The formal derivations are carefully written in terms of notation, equations, and calculations. Dynamic programming is extensively covered in Section 2.2, Chapters 6 and 7, and approximation procedures are presented in a clear way (e.g. Section 10.3). As indicated, the analytical appendices are very extensive, and there is an index. The bibliography might have been supplemented by more diversified references. In conclusion, this m o n o g r a p h will stimulate experiments with some techniques for stochastic control of economic models, while inciting other authors to further comparisons and advances. It is a recommended reading for control engineers, wishing a flavor of the difficulties of stochastic economic models, and for advanced econometrics students and researchers.
Reviewer: L. F. P A U Ecole Nationale Sup6rieure Des T616communications, 46 Rue Barrault, F-75634 Paris, Cedex 13, France.
IMPORTANT developments in economic theory and applications in recent years have been the progress of economic analysis under uncertainty and the increasing degree of sophistication in the modelling of the formation of expectations and their impact on policy rules. The models for most economic systems, assuming one single decision maker for simplification, have evolutions governed by equations of the type
f, tx,, a,, z,, u,,{xo, x~ ..... x , - ~ l , X~+2 .... : , 3 = 0 (1) where X, is the endogeneous variables at time t; Z,, exogeneous variables at time t; a,, parameters of the model estimated at time t; U , instrument variables (values of controls) as determined by policy rules (control law); ~, unobservable random variables; X,+ ~, anticipated value of X,+~ at time t. Both types of advances as earlier mentioned have raised important questions concerning the possibilities of using stochastic control theory for the formulation of economic policy rules, as well as the effectiveness and feasibility of such policies. Much of this work has been centred on validations of, or contradictions to the so-called rational expectations model, in which the anticipated values )f,+i are taken equal to Et(X,+~), i = 1,2 . . . . . expected value at time t over c,,e,+ 1. . . . . as derived from the model (1). In his carefully written monograph, David Kendrick and his participating co-authors, have essentially tried to transfer to the economic community some specific algorithms and theories from optimal control, stochastic control, and dual c o n t r o l : Part 1 covers basic notions and equations of quadratic linear control problems, numerical methods, and an example; Part II deals with stochastic control with additive error terms in the state equation for X,, supplemented by some interesting specific results on multiplicative disturbances; Part III introduces adaptive control or dual control, with deterministic, cautionary, and probing terms in the expression of the cost-to-go, and the dual role of control and learning assigned to the instrument variables. The main strength of the book is the good progression in the complexity of the algorithms presented, and the fact of using several times in their context two case models: Chapters 7 and 12 o n the o n e hand, and Chapters 4, 10 and 12 on the other. One interesting issue for the experienced reader is whether this book has been able to integrate the two aspects of economic analysis and model sophistication as mentioned in the introduction of this review. More specifically, many problems in the field of stochastic control of economic models are about what the anticipations )(,+~, i = 1,2 . . . . . should mean and how they should be represented. The engineering literature only provides little help in this respect. The fear one m a n y have as a reviewer is that this book, because of its stated goal of 'information translation for economists', provides too little analysis of these basic economic questions. One may also worry a little bit about the lengthy development warranted to some specific algorithms, to the expense of many others or of the comparison hereof. As might be expected, the economist may be puzzled by the extensive analytical derivations or explanations (appendices run from page 143 to 227), in contrast to a more fundamental and
About the reviewer L. F. Pau received his M.S., M.A., Ph.D. and D.Sc (Doctorat d'Etat) degrees from Paris University. After serving as an assistant, then associate professor at the Technical University of Denmark, he has since 1974 been a professor at ENS T616communications, Paris, and department head since 1977. During 1976-1977, he was visiting associate professor at the Department of Electrical Engineering and Computer Science, M.I.T., and 1980 1982 he was professorial lecturer at the Department of Computer Science, University of Maryland. His main areas of interest are: economic modelling and control, applications of game theory, failure diagnosis, and pattern recognition. For the 1981-1984 term, he serves as chairman of the economic and management systems committee of IFAC.
* Stochastic Control Jbr Economic Models, by D. Kendrick. Published by McGraw-Hill, New York (1981). 242 pp., US$39.50.
455