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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.