Table Of ContentLecture Notes in Economics and Mathematical Systems
For information about Vols. 1-49, please contact your bookseller or Springer-Verlag
Vol. 50: Unternehmensforschung Heute - Obersichtsvortrage der Vol. 76: G. Fandel, Optlmale Entscheidung bei mehrfacher Ziel
Zaricher Tagung von SVOR und DGU. September 1970. Heraus setzung. II, 121 Seiten. 1972.
gegeben von M. Beckmann. IV. 133 Seiten. 1971.
Vol. 77: A Auslender. Problemes de Minimax via I'Analyse Con
Vol. 51: Digitale Simulation. Herausgegeben von K. Bauknecht vexe et les Inegali"'s Variationelles: Theorie et Algorithmes. VII,
und W. Nef. IV. 207 Seiten. 1971. 132 pages. 1972.
Vol. 52: Invariant Imbedding. Proceedings 1970. Edited by R. E. Vol. 78: GI-Gesellschaft liir Informatlk e.V. 2. Jahrestagung, Karls
Bellman and E. D. Denman. IV. 148 pages. 1971. ruhe, 2.-4. Oktober 1972. Herausgegeben im Auftrag der Gesel'
schaft fUr Informatik von P. Deussen. XI, 576 Seiten. 1973.
Vol. 53: J. Rosenmaller. Kooperative Spiele und Markte. III, 152
Vol. 79: A. Berman. Cones, Matrices and Mathematical Program
Seiten. 1971.
ming. V, 96 pages. 1973.
Vol. 54: C. C. von Welzsacker. Steady State Capital Theory. III. Vol. 80: International Seminar on Trends in Mathematical Model
102 pages. 1971. ling, Venice. 13-18 December 1971. Edited by N. Hawkes. VI.
288 pages. 1973.
Vol. 55: P. A. V. B. Swamy, Statistical Inference in Random Coef
ficient Regression Models. VIII, 209 pages. 1971. Vol. 81: Advanced Course on Software Engineering. Edited by
F. L. Bauer. XII, 545 pages. 1973.
Vol. 56: Mohamed A. EI-Hodiri, Constrained Extrema. Introduction
to the Differentiable Case with Economic Applications. III, 130 Vol. 82: R. Saeks, Resolution Space, Operators and Systems. X,
pages. 1971. 267 pages. 1973.
Vol. 57: E. Freund, Zeitvariable MehrgroBensysteme. VIII, 160 Sei Vol. 83: NTG/GI-Gesellschaft liir Informatik, Nachrichtentech
ten. 1971. nische Gesellschaf!. Fachtagung ,Cognitive Verfahren und Sy
steme", Hamburg, 11.-13. April 1973. Herausgegeben im Auft rag
Vol. 58: P. B. Hagelschuer. Theorie der linearen Dekompositlon. der NTG/GI von Th. Einsele, W. Giloi und H.-H. Nagel. VIII, 373
VII. 191 Seiten. 1971. Seiten. 1973.
Vol. 59: J. A. Hanson, Growth in Open Economies. V, 128 pages. Vol. 84: A V. Balakrishnan. Stochastic Differential Systems I.
1971. Filtering and Control. A Function Space Approach. V, 252 pages.
1973.
Vol. 60: H. Hauptmann. Schatz- und Kontrolltheorle In stetlgen
dynamischen Wirtschaftsmodellen. V, 104 Seiten. 1971. Vol. 85.: T. Page, Economics of Involuntary Transfers: A Unified
Approach to Pollution and Congestion Externalities. XI, 159 pages.
Vol. 61 : K. H. F. Meyer, Wartesysteme mit variabler Bearbeitungs 1973.
rate. VII, 314 Seiten. 1971.
Vol. 86: Symposium on the Theory of Scheduling and its Applica
Vol. 62: W. Krelle u. G. Gabisch unter Mitarbeit von J Burger tions. Edited by S. E. Elmaghraby. VIII, 437 pages. 1973.
meister. Wachstumstheorie. VII, 223 Seiten. 1972.
Vol. 87: G. F. Newel', Approximate Stochastic Behavior of n-Server
Vol. 63: J Kohlas. Monte Carlo Simulation 1m Operations Re Service Systems with Large n. VII, 118 pages. 1973.
search. VI. 162 Seiten. 1972.
Vol. 88: H. Steckhan, GUterstrome in Netzen. VII, 134 Seiten.
Vol. 64: P. Gessner u. K. Spremann, Optimierung in Funktfnen 1973.
raumen. IV. 120 Selten. 1972. Vol. 89: J P. Wallace and A Sherret. Estimation of Product.
Attributes and Their Importarlces. V, 94 pages. 1973.
Vol. 65: W. Everling, Exercises in Computer Systems Analysis.
VIII. 184 pages. 1972. Vol. 90: J-F. Richard, Posterior and Predictive Densities for
Simultaneous Equation Models. VI. 226 pages. 1973.
Vol. 66: F. Bauer. P. Garabedian and D. Korn. Supercritlcal Wing
Sections. V, 211 pages. 1972. Vol. 91: Th. Marschak and R. Selten. General Equilibrium with
Price-Making Firms. XI, 246 pages. 1974.
Vol. 67: I. V. Girsanov, Lectures on Mathematical Theory of Vol. 92: E. Dierker, Topological Methods in Walrasian Economics.
Extremum Problems. V. 136 pages. 1972. IV, 130 pages. 1974.
Vol. 68: J Loeckx. Computability and Decidability. An Introduction Vol. 93: 4th IFAC/IFIP International Conference on Digital Com
for Students of Computer SCience. VI. 76 pages. 1972. puter Applications to Process Control, Part I. ztlrich/Switzerland,
March 19-22, 1974. Edited by M. MansolIl' and W. Schaufelberger.
Vol. 69: S. Ashour, Sequencing Theory. V. 133 pages. 1972. XVIII. 544 pages. 1974.
Vol. 70: J. P. Brown, The Economic Effects of Floods. Investiga Vol. 94: 4th IFAC/IFIP International Conference on Digital Com
tions of a Stochastic Model of Ratlonallnvestmen!. Behavior in the puter Applications to Process Control, Part II. ZUrich/Switzeriand,
Face of Floods. V, 87 pages. 1972. March 19-22,1974. Edited by M. Mansour and W. Schaufelberger.
XVIII, 546 pages. 1974.
Vol. 71 : R. Henn und O. OPitz, Konsum-und Produktionstheorie II. Vol. 95: M. Zeleny, Linear Multiobjective Programming. X, 220 pages.
V, 134 Seiten. 1972. 1974.
Vol. 72: T. P. Bagchi and J. G. C. Templeton, Numerical Methods in Vol. 96: O. Moeschlin, Zur Theorie von Neumannscher Wachs
Markov Chains and Bulk Queues. XI. 89 pages. 1972. tumsmodelle. XI. 115 Seiten. 1974.
Vol. 73: H. Kiendl, Suboptimale Regier mit abschnlttweise Ii nearer Vol. 97: G. Schmidt, Ober die Stabliitat des einfachen Bedienungs
Struktur. VI, 146 Seiten. 1972. kanals. VII. 147 Seiten. 1974.
Vol. 98: Mathematical Methods in Queueing Theory. Proceedings
Vol. 74: F. Pokropp, Aggregation von Produktlonsfunktionen. VI,
1973. Edited by A. B. Clarke. VII. 374 pages. 1974.
107 Seiten. 1972.
Vol. 99: Production Theory. Edited by W. Eichhorn, R. Henn,
Vol. 75: GI-Gesellschaft fUr Informatik e.V. Berlcht Nr. 3. 1. Fach
O. Opitz. and R. W. Shephard. VIII, 388 pages. 1974.
tagung aber Programmiersprachen . Manchen, 9.-11. Marz 1971.
Herausgegeben 1m Auftrag der Gesellschaft liir Informatlk von H. Vol. 100: B. S. Duran and P. L. Odell, Cluster AnalYSIS. A Survey.
Langmaack und M. Paul. VII, 280 Seiten. 1972. VI. 137 pages. 1974.
continuation on page 97
Lecture Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
Economic Theory
212
Ryuzo Sato
Takayuki Nono
Invariance Principles
and the Structu re
of Technology
Spri nger-Verlag
Berlin Heidelberg New York Tokyo 1983
Editorial Board
H. Albach A. V. Balakrishnan M. Beckmann (Managing Editor)
p. Dhrymes, J. Green W. Hildenbrand W. Krelle (Managing Editor)
H. P. Kiinzi K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fOr Gesellschafts-und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Authors
Prof. Ryuzo Sato
Department of Economics
Brown University, Providence, RI 02912, USA
and
J.F. Kennedy School of Government
Harvard University
Cambridge, MA 02138, USA
Prof. Takayuki N6no
Department of Mathematics
Fukuoka University of Education
Munakata, Fukuoka 811-41, Japan
ISBN-13: 978-3-540-12008-7 e-ISBN-13: 978-3-642-45545-2
001: 10.1007/978-3-642-45545-2
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of the German Copyright Law where copies are made for other than private use, a fee is
payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983
2142/3140-543210
PREFACE
The theory of Lie groups has proven to be a most powerful
analytical tool in many areas of modern scientific endeavors. It
was only a few years ago that economists discovered the
usefulness of this approach in their study of the frontiers of
modern economic theory. These frontiers include the areas of
technical change and productivity, technology and preference,
economic conservation laws, comparative statics and integrability
conditions, index number problems, and the general theory of
observable market behavior (Sato [1980, 1981], No~ no [1971], Sato
and N~no [1983], Russell [1983]).1
In Nono [1971] and Sa to [1981, Chapter 4] the concept of
"G-neutral" (group neutral) technical change was first
introduced as a natural extension of the well-known concepts of
Hicks, Harrod, Solow and Sato-Beckmann-Rose neutrality. The
present monograph contains a further extension of the G-neutral
technical change to the case of non-constant-returns-to-scale
technology and to the case of multiple factor inputs. The
methodology of total productivity estimation by means of Lie
group transformations is also developed in this monograph.
We would like to express our sincere thanks to many
individuals notably to Professor M.J. Beckmann, Professor F.
Mimura, Professor G. Suzawa, T. Mitchell, K. Mino and P. Calem,
for their numerous contributions at various stages of this
work. We are also grateful to Marion Wathey for her usual superb
typing of this difficult manuscript.
Providence, R.I., U.S.A. and Tokyo, Japan
December 1982
Ryuzo Sato
Takayuki No~ no
FOOTNOTE
ISee T. Nono [1971], "Classification of Neutral Technical
Changes," Bulletin of Fukuoka university of Education, 1971.
R. Sato [1980], "The Impact of Technical Change on the
Holotheticity of production Function," Presented at the World
Congress of the Econometric Society, Toronto, 1975, published in
Review of Economic Studies, Vol. 47 (July 1980), pp. 767-776.
R. Sato [1981], The Theory of Technical Change and Economic
Invariance: Application of Lie Groups, Academic Press, New York,
1981.
R. Sato and T. Nono [1983], "Invariance Principle and 'G-Neutral'
Types of Technical Change," Technology, Organization and Economic
Structure: Essays in Honor of Professor Isamu Yamada, ed. by
Ryuzo Sato and Martin J. Beckmann, Lecture Notes in Economics and
Mathematical Systems, No. 210, Springer-Verlag, Berlin
Heidelberg, New York, 1983.
T. Russell [1983], "Notes on Exact Aggregation," Technology,
Organization and Economic Structure: Essays in Honor of
Professor Isamu Yamada, ed. by Ryuzo Sato and Martin J. Beckmann,
Lecture Notes in Economics and Mathematical Systems, No. 210,
Springer-verlag, Berlin-Heidelberg, New York, 1983.
TABLE OF CONTENTS
Chapter 1. Introduction 1
Chapter 2. Lie Group Methods and the Theory of
Estimating Total Productivity 6
I. Ho1otheticity and the Scale Effect 6
A. Lie Group Theory 6
B. Estimation Procedures 13
C. Estimation of the Scale Effect 17
II. The Lie O~erator Technique for Estimating
productiv~ty 19
III. The Effect of Technical Progress Represented
by New Forms of the Production Function 24
Chapter 3. Invariance Principle and "G-Neutra1" Types
of Technical Change 29
I. Introduction 29
II. "Neutral Types" of Technical Progress 30
III. "G-Neutra1" Types of Technical Change 34
IV. G-Neutra1 Technical Change Generated by
the One-Parameter Lie Subgroups of GP(2,R) 42
V. G3-Types of Neutral Technical Change 46
VI. Invariance of the Regularity Conditions
Under Techn~ca1 Change 48
Chapter 4. Analysis of Production Functions by "G
Neutral" Types of Techn~ca1 Change 53
I. Introduction and Summary 53
II. G-Neutra1 Technical Change 53
III. Symmetry Groups of Neutral Technical
Changes 57
IV. G3-Fami1y of Neutrality 63
V. Sato-Beckmann Types of Neutral Technical
Changes 68
Chapter 5. Neutrality of Inventions and the Structure
of Production Functions 72
I. Introduction and Summary 72
II. G-Neutra1 Technical Change 72
III. s*mmetry Groups of Neutral Technical
C anges 75
IV. Hicks-Harrod-So1ow Family of Neutral
Technical Change 87
References 90
Chapter 1. Introduction
In this book we accept the view that production processes
can be described in a meaningful way by a simple mathematical
production relationships. We will call production functions
"technologies" when the term can be used unambiguously. The
technologies encountered here will be of the one output variety
most common in the literature. Many will also contain two
inputs--presumably capital and labor--as is most common, but
we do not want to constrain ourselves to only two factor cases.
Technical progress plays a crucial part in the process of
economic growth, and its analysis occupies a central place in
contemporary growth models. In one general and widely used
approach, technology appears as a parameter of the neoclassical
production function. Our primary concern is the measurement of
technical progress and its relationship to the factor inputs.
Technical progress is the phenomenon by which fixed quantities
of inputs produce even greater quantities of output over an
extended period of time. This can be accomplished through an
improvement in the quality of machines or perhaps a better
educated labor force. An important question is what is the
relative contribution of each factor to the production gains
observed over time.
Assume as usual that there are two productive factors,
capital K and labor L, and one output Y which is subject
to the following neoclassical production functions:
Y = F{K,L,t) (I)
where t denotes time, or alternatively, an index of technical
change. In this form, the role of technology is much too
general to permit a thorough-going analysis. It is essential
to specify the way in which technical progress enters the
production function. The usual procedure has been to formulate
certain hypotheses concerning the way in which technical progress
has affected certain important variables that are derived from
the production function. These variables include: (I) the
2
capital-output ratio; (2) the output per man; (3) the factor
proportions; (4) the marginal productivities; and (5) the marginal
rate of substitution. Thus one might postulate that technical
progress has affected anyone of these characteristics in a pre
determined way; for instance that it has left a certain variable
invariant. However, since these variables will depend not only
on technology but also on input proportions, it is necessary to
neutralize the effect of any changes in inputs. Thus one
arrives at one famous criterion of the so-called neutrality,
that technical progress is neutral--in the sense of Hicks--if
the marginal rate of sUbstitution is invariant under technical
change as long as the factor proportions are unchanging. By
contrast invention is called Harrod neutral whenever the
capital-output ratio is invariant as long as the interest rate
does not change.
The implications of the two types of technical progress
are well-known and indeed far reaching. If we intend to clarify
the nature of the specification of technical change, the
following three questions deserve close examination.
(I) Are there alternative ways of describing--and hopefully
of justifying theoretically--the known types of technical progress?
We might include among the known types purely capital augmenting
progress and also a combination of Hicks and Harrod neutrality,
the so-called factor augmenting technical progress.
(2) Are there any other economic variables which might be
considered to be invariant under technical change, such as the
elasticities of output with respect to an input or the
elasticity of factor substitution? Or are there any other
meaningful combinations of the usual variables considered
so far?
(3) As a result of alternative specifications, how many
pure types of technical progress can be distinguished, and what
is their functional form? An answer to a very special case is
given by Sato and Beckmann 11968] for Harrod and Hicks neutrality.
3
Another purpose in setting up alternative specifications of
technical change is, of course, to obtain hypotheses about
technical change which might be tested and (in all but a few
cases) refuted. The critical nature of technical progress
requires that a theoretical analysis be made of the principal
ways in which the functional relation between output, input,
and technical progress can be specified. Suppose one wishes to
analyze the long-run behavior of some crucial economic variable,
such as the return·to capital, the wage rate, or their ratios
in terms of other variables; then what variables should be
selected depends on the type of technical progress one has
postulated. For instance, if we assume Harrod neutrality, then
the major variable that would explain the return to capital
must be the capital-output ratio; moreover t which refers to
the state of technology should not be included among the
explanatory variables--by definition. Contrariwise if technical
progress is Hicks neutral, then the long-run behavior of the
ratio of marginal productivities, that is the marginal rate of
substitution, should be dependent only on the capital-labor
ratio and not on time. Suppose, however, that tests show that
the marginal rate of substitution is better correlated with
some other variable and/or with time, what should then be our
conclusion as to the way in which technology enters into the
production function? Partial answers to these questions are
given by Sato and Beckmann [1968], Rose [1968], Nono I197l],
and Sato [1981]. By prescribing the "neutral" and "invariant"
relationships among economic variables, they could infer
the properties of the underlying technologies consistent with
the given invariant relationships.
We model technical progress by allowing each effective
factor quantity to depend not only on the physical amount of
factor inputs, but also the level of technology and the other
factor quantities. This is only a slight extension of the
common factor augmenting technical progress notion. In the
two factor cases, the effective quantities of capital and labor
4
are given by the technical progress functions,
¢(K,L,t) (2a)
T:
{: =
~(K,L,t) (2b)
where t is some measure of the level of the technology or
technical progress and K and L are the real factor quantities.
The production function itself is then
y (3)
a function of the effectiveness quantities. If we impose
certain restrictions on the transformation of real capital and
real labor into effective capital and effective labor (equa
tions (2a) and (2b», the transformation can serve as a useful
device for studying the problem of inventions and technical
progress. Lie group theory provides the key to resolving
the problem. In earlier works by Sato [1980, 1981] and
Nono [1971], Lie group theory was extensively used to identify
the underlying structure of production functions generated by
certain invariant relationships.
The purpose of the present project is to pursue further the
application of the invariance principle in the theory of Lie
groups to the study of: (i) productivity estimation,
(ii) classification of technical change, and (iii) analysis of
the production technologies generated by the invariant
relationships. Chapter 2 presents the theory of estimating
total factor productivity from the point of view of Lie groups
and transformations. It will be shown that the invariance
principle provides a new tool for estimating the parameters of
a technical progress transformation. The effect of technical
progress can be represented by new forms of the production
function. Several examples are presented and some possible
applications are stated.
