ll!""l‘l'llllll'llllllllllll“ LIBRARY 5-H???" Michigan State. University This is to certify that the thesis entitled MULTIPRODUCTS PRODUCTION RELATIONS IN MANUFACTURING PLANTS: AN EXPLORATORY STUDY ON SIX SELECTIVE MANUFACTURING ACTIVITIES IN KOREA presented by Seung Yoon Rhee has been accepted towards fulfillment of the requirements for PILD. degree in Economics fl/éfi W K? Major pryéssor Date OCtObEY‘ 12. 1978 0-7639 © 1978 SEUNG YOON RHEE ALL RI GHTS RESERVED MULTIPRODUCTS PRODUCTION RELATIONS IN MANUFACTURING PLANTS: AN EXPLORATORY STUDY ON SIX SELECTIVE MANUFACTURING ACTIVITIES IN KOREA By Seung Yoon Rhee A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1978 ABSTRACT MULTIPRODUCTS PRODUCTION RELATIONS IN MANUFACTURING PLANTS: AN EXPLORATORY STUDY ON SIX SELECTIVE MANUFACTURING ACTIVITIES IN KOREA By Seung Yoon Rhee The subject of a production technology is one of the areas of economics where the gap between theoretical formulations and empirical knowledges is still quite wide. Furthermore there have been only a few theoretical and empirical studies on the multi- input multi-output production technology until recently. The purposes of this study are (i) to understand the theory of a multi-input multi-output (and uni-output) production tech- nology, (ii) to investigate the workability of the multi-input multi-output production technology for a cross-section data system of the Korean Manufacturing Census, (iii) to find some knowledges on the first and second order properties of a production technology, estimated by the translog approach at an establishment level in unnufacturing activities, and (iv) to collect information on the usefulness of the Korean Manufacturing Census System which is quite a common type of data system in most other countries. Seung Yoon Rhee A brief review on theoretical formulations of a multi-input uni-output production technology and its extension to a multi-input mmlti-output technology are followed by another summary on eco- nometric backgrounds in this empirical estimation of a production technology. Considerations on exclusion rules in sampling establish— rents, selection of specific industries to be studied, and quality of sample data are followed by preliminary investigations on the industrial characteristics in terms of factor products, factor use ratios, factor prices, factor shares and their variations in sample establishments by size and by industry. Results on the first and second order properties in the estimated technology suggest, firstly, a strong objection on the conventional Cobb-Douglas form from denial of the self-duality between the translog production and cost functions and from rejections in the null-hypotheses test of the second order parameter estimates. Secondly, the conventional value added approach in production studies should be reevaluated, not only from the wide variations of the value added ratio to gross output across industries but also from the seemingly, unitary substi- tution elasticities of raw materials with respect to other factor 1nPuts. Thirdly, many interesting results from the second order Draperties of the estimated production technology are found in terms of the direct substitution elasticities, the Allen-Uzawa partial substitution elasticities, the McFadden shadow partial Seung Yoon Rhee substitution elasticities and the demand elasticities with respect to price changes over input and output bundles. Examples of speci- fic findings are the supplementary substitution relation between two heterogeneous labor inputs, the inconclusive but close-to-unity substitution elasticity between labor and capital inputs, etc. Fourthly, further results are found from the supplementary works, such that the workability of the production theory becomes weaker for the small-sized establishments, such that the inclusion of sample establishments with no capital inputs results in the seemingly, unitary substitution elasticity between labor and capital inputs, and such that gains from alternative explanatory variables of different quality are negligible in the Korean census data. Finally, we have learned something from our investigations, not the least of which is that just "more data" will not do. If we persist in asking rather complicated questions, we shall need much better and more relevant figures before we can hope to answer them previously. To my parents, Sung Rae Kim and Young Jin Rhee 11° ACKNOWLEDGMENTS Sincere appreciation I owe to the chairman of my committee, Professor Anthony Koo for his guidance and encouragements for me as a human being. Appreciation is also extended to the other member of the committee from Michigan State University, Professor Glenn L. Johnson, Professor Robert L. Gustafson and Professor Norman P. Obst for their interest in me and their support for the completion of my Ph.D. program in Economics. As I complete my graduate training, I am especially grateful to Dr. Mahn Je Kim, the president of the Korea Development Institute who has officially supported me for this education opportunity, and also to Professor Ray Byron of the Australian National Uni— versity, who initially suggested me this research project and has continuously encouraged me to the last moment. More personally, a heartfelt thanks, love and affection are offered to my parents and to my brothers and sisters, Seung Su Rhee, Eun Hua, Eun Young, Seung Keun, Kyoung Mee, Eun Suck and Hwa Yoon. Most importantly my love and affection to my wife, Myung Hee, my son,Dong Joo, my daughter, Sun Young and the third child who will come out to this world sometime next month, who have sacrificed so much particularly during the last three years when I have been away from the campus and learned about the hard reality in social life as well as the hardships in the life of an economist. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES . LIST OF APPENDICES . INTRODUCTION . Chapter Part A. 1. Sam: 3:3: 3>Lo Bola dd II. .>>>> NNNN hum—- Part B. II. WW CD 0 O 0 NM N (JON -‘ . UNIPRODUCT PRODUCTION THEORY . What is a Production Function Specification of a Technology. Axiomization of Production Structure . . . . . . Functional Forms in Trend. . The Transcendental Logarithmic Function MULTIPRODUCT PRODUCTION THEORY Introduction Specification of a Technology Functional Forms in Trend. . The Translog Generalization to the Multiproduct Situation . . . . INTRODUCTION DATA, MEASUREMENT PROBLEMS AND SAMPLE PROPERTIES General Descriptions on the “Census of Korea Mining and Manufacturing, 1973" Variables in the Record Derived Variables iv Page vi ix 10 27 46 46 61 68 85 88 88 96 Chapter 8.2.4. Selection and Exclusion Rules in the Sample Establishments . . 8.2.5. Industry Classifications and the Selection of Specific Industries to be Studied 8.2.6. Some Characteristics of the Industries Selected 8.2.7. Quality of Data: Some General Considerations 111. THEORETICAL AND STATISTICAL BACKGROUNDS IN THE EMPIRICAL ESTIMATION OF A PRODUCTION TECHNOLOGY . 8.3.1 Introduction . 8.3.2 Choice of the Estimating Equations . 8.3.3 Error Specification and its Properties 8.3.4 Estimation Methods . . . 8.3.5 Monte Carlo Experiments IV. EMPIRICAL ESTIMATION OF THE TRANSCENDENTAL LOGARITHMIC PRODUCTION FUNCTION AND THE TRANSCENDENTAL LOGARITHMIC COST FUNCTION 8.4.1. Introduction . 8.4.2. Main Results of the Empirical Estimation . 8.4.3. Supplementary Results . V. CONCLUSIONS AND RECOMMENDATIONS . APPENDICES BIBLIOGRAPHY . Page 108 111 121 132 139 139 141 146 150 164 181 181 275 296 309 347 Table I-1. II-1. II-Z. II-3. II-4. II-5. III-1. III-2. III-3. III-4. IV-1. V-l. V-2. V-3. V-4. VI-1. LIST OF TABLES Number of the Excluded Establishments Industries Selected in the Study . Major Commodities Identified Composition of Major Commodities Identified Distribution of Establishments Producing Multiproducts . . . . The Size Distribution of Establishments Factor Products by Industry Factor Use Ratios by Industry . Factor Prices by Industry Factor Shares by Industry Measurement Deviations (%) by Industry . Mean and Standard Errors of Translog Estimates of CES PargBeters: Base (3, 8) and with (l.O x 10’ ) . . . . . . . . Mean and Standard Errors of Translog Estimates of 555 Parametergb Base (3, 8) and with (1.0 X lO' ) . . . . . . . . . , 1.0 X 10' Mean and Standard Errors of Translog Estimates of CES Parameters: Base (8, I8;, (6, l0), (4, lOL (2, 10), and with (1.0 X 10' Estimated Elasticities of Substitution in the Monte Carlo Experiments: Base (8, 10)70§6, l0), (4, l0) and (2, l0) and with (1.0 X 10‘ Parameter Estimates for the Translog Functions-- Canning Industry vi Page llO ll3 ll5 116 118 120 122 124 126 129 135 173 175 177 179 184 Table VII-1. VII-2. VII-3. VIII-1. VIII-2. VIII-3-1. VIII—3-2. IX-1-1. IX-1-2. Parameter Estimates for the Translog Functions-— Leather Footwear Industry . Parameter Estimates for the Translog Functions-- Screw Products Industry Parameter Estimates for the Translog Functions-- Manufacture of Knitted Underwear Parameter Estimates for the Translog Functions-- Manufacture of Briquettes Parameter Estimates for the Translog Functions-- Molding Industry R2 and Relative Size of the Error Sum of Squares-- Canning Industry R2 and Relative Size of the Error Sum of Squares-- Leather Footwear and Screw Products Industries R2 and Relative Size of the Error Sum of Squares-- Manufacture of Knitted Underwear, Briquettes, and Molding Industry . . . . . Likelihood Ratio Test for Alternative Restrictions . . . . . F-Test for Alternative Restrictions . t-Test for each Restriction Imposed--Canning, Leather Footwear and Screw Products Industries . . . . . t-Test for each Restriction Imposed--Manu- factures of Knitted Underwear, Briquettes and Molding Industry . Number of Establishments, Not Satisfying Monotonicity Conditions--Multiproducts Industries . . . . . . . Number of Establishments, Not Satisfying Monotonicity Conditions--Uniproduct Industries . . . . . . . vii Page 192 198 206 208 210 215 216 218 222 225 228 230 234 235 Table IX-2-1. IX-2-2. IX-2-3. IX-2-4. X-1-1. X-1-2. XI-Z. XI-3. XII-1. XII-2. Eigenvalues of the Hessian Matrix--Multi- products Industries Eigenvalues of the Hessian Matrix--Uniproduct Industries . . . . . . . Average Prices of Outputs in the Uniproduct and the Multiproduct Firms . . Age Structure of Uniproduct and Multiproduct Firms Estimated Elasticities of Substitution-- Canning Industry . . . . Estimated Elasticities of Substitution-- Leather Footwear Industry . Estimated Elasticities of Substitution-- Screw Products Industry Estimated Elasticities of Substitution-- Manufacture of Knitted Underwear Estimated Elasticity of Substitution-- Manufacture of Briquettes . Estimated Elasticities of Substitution-- Molding Industry . . Share Elasticities with Respect to the Own Quantity Changes and the Own Price Changes . The Weighted Average of the Quasi- R2' 5 by Firm Size . . . . . Likelihood Ratio Test Statistics by Firm Size Scale Effects in Output Elasticities Comparison of the Weighted Average of the Quasi-Rz's Between the Inclusive and the Exclusive Cases . Factor Shares Estimated in the Inclusive Case viii Page 237 238 243 246 255 256 257 258 259 260 273 278 280 283 288 290 Figure II. III. IV. LIST OF FIGURES Depreciation Rate of Capital Stock . Estimated Transformation Curve Possible Transformation Curve with Heterogeneous Output . . . . Possible Transformation Curve with Technical Change ix Page 93 240 241 244 LIST OF APPENDICES Appendix Page A-I. Derivation of the Share Equations System in the Multiproduct Transformation Function . . . . . 3ll A-II. Intrinsic Property of the Share Equations System . . 3l4 B-I. Tables on Factor Prices by the Size of Establishments . . . . . . . . . . . . . 3l7 8-II. Tables on Factor Shares (%) by the Size of Establishments . . . . . . . . 324 B-III. Tables on Partial Elasticities of Factor Substitutions and Demand Elasticities with Respect to Price Changes, Estimated by the Establishment Size 331 B-IV. Tables on Partial Elasticities of Factor Substitutions Estimated in the Inclusive Case 337 B-V. Tables on Comparisions of Partial Elasticities of Factor Substitutions, Estimated in the Four Alternative Cases of Different Quality in the Explanatory Variables. 340 INTRODUCTION Whether or not there exists a stable functional relationship between inputs and output(s) in a production activity has long been a subject of economic inquiry. In economic theory the production function is a mathematical statement relating quantitatively the purely technological relationship between the output(s) of a process and the inputs of the factors of production, the chief purpose of which is to understand and explain the reality of a production activity, owing to several useful characteristics of a production function in economic analysis. Many efforts have been expended by economists in developing and refining the theory of a production, and in formulating and estimating the model of a production technology. For example, great efforts on the functional form of a production function, relating directly to the fitting of econometric production function, have been followed by the Cobb-Douglas production function as a simple form toward more complicated functions, such as the CES (constant elasticity of substitution) function, the VES (variable elasticity of substitution as various generalizations of the CES) function, and the translog (transcendental logarithmic) function. In particular, the formulation of the translog production function does not require any a priori assumptions on the functional form to be investigated in empirical works, distinctively different from other functions, such as the CES function requiring a_pgjgrj_ assumption of the constant elasticity of factor substitution. Most of the earlier studies of a production technology have been based either on a rather simple specification of the production function (such as the CD and the CES) to be estimated, or on highly aggregative data for the estimation of aggregate production function at a certain macro (or sector) level. Very few studies have actually dealt with data at the plant level and most of the previous micro production functions have been esti- mated for selected farming activities,'for the electricity generating industry, and for the railroad sector, where abundant plant data had become available through the operation of regulatory agencies. There have been only a few econometric studies of production based on individual plant data in manufacturing.2 In particular, most of the previous works on the production function 1In addition to early empirical efforts by Senator Paul Douglas with the Cobb-Douglas function, agricultural economists did considerable empirical work with the Cobb-Douglas, Mitscherlich, and several other functions, notable among these efforts are Tintner, Brownlee, Heady and Johnson. This line of work seems to continue to date. 2Recent examples are Krishna's (1967) study of combined cross-section time series data for three manufacturing industries in the U.S., Hodgins' (1968) study of economies of scale in Canadian manufacturing, Eisner's (1967) study based on data for individual companies rather than plants, and recently Griliches and Ringstadt's (1971) study for the rather higher level of industry (3-digit ISIC classification level) with simple functional forms such as the CD and the CES functions. have focused either on the possibilities of substitution between factors (mostly two factors) of production to achieve one given output, or on the possibilities of transformation between products (also mostly two) of production, paying no attention to the input side. There have been only a very few econometric studies of the production possibility frontier with more than two inputs and one output.3 The present study purports (i) to understand the theory of a multiproduct production function as an extension of a uniproduct production theory, (ii) to investigate the workability of multi- product joint production theory, using the Korean manufacturing census data, (iii) to find some knowledges on the parameters of a production function in a multiproduct establishment which is closer to the reality of most manufacturing activities, and finally (iv) to collect informations on the usefulness of the Korean manufacturing census data which is quite a common type of data system in most of the other countries also. More specifically this study investigates the production technology of an industry at the micro establishment level, via estimation of the translog production function and also the translog cost function with five distinct inputs and more than (or equal to) 3Recent examples are Christensen, Jorgenson, and Lau's (1973) works of the transcendental logarithmic production frontiers with two inputs and two outputs for the U.S. economy during 1929- ~1969, and Brown, Caves, and Christensen's (1975) of the translog cost function for the U.S. railroad industry. two outputs. Our study is very much conditioned by the availability of a particular body of data: the 1973 Census of Mining and Manu- facturing_in Korea. These data have several important advantages, not the least of which is their accessibility for research purposes. In addition, their comprehensiveness and the potentially large .number of observations may allow the testing of much more detailed hypothesis about the structure of production activities than was hereto possible. On the other hand, these data have also serious limitations. Some of the data have turned out not to be as good as zanticipated. But more importantly, our study is limited to only one year, 1973, and these data only to those items for which questions were asked in the Census. One of the main shortcomings is the poorness of data on the capital stocks and on the character- istic (age, sex, skill) of the labor force in the various estab- lishments. In particular, the absence of time series observations in the study makes impossible the construction and estimation of a complete production system inclusive of technical changes. Since we can not afford in this study to cover all of the manufacturings, only six sepecific industries are selected .randomly, the half of which produce multiproducts and the other half of which do a uniproduct. The plan of this study is as follows: the first part is devoted to a brief review of the theory of a production function, mainly focusing on various efforts in the formulation of the functional form both for the uniproduct (Chapter I) and the multi- product production function (Chapter II). The second part consists of the very ingredients of the current empirical investigation. Following the introduction of Chapter I of part 8, the data and measurement problems are dis— cussed in Chapter II, in terms of the variables in the record of the Census of Mining and Manufacturing in Korea, 1973 and of the derived variables adapted by this empirical investigations. Also the selection rules of specific industries to be studied and the exclusion rules of sample establishments included in the esti- mation are followed by the sample properties by industry and the general quality of data used. Chapter 111 contains the theoretical and statistical back- grounds in the empirical estimations, such as the choice of the estimating equations, the error specification, the estimation method and some Monte Carlo Experiments for the validity of the estimation methods adopted in this study. In Chapter IV, the empirical estimates of the production technology for the six selective industries are presented and tested. Also several alterative investigations are covered, such as those of alternative restrictions, separate results by different establishment size, those of alternative exclusion rules in sampling establishments, and those of alternative variables of different quality. The final chapter summarizes the main findings of this empirical study and suggests further efforts to be done necessarily to understand the reality of the micro production technology in the Korean manufacturings. PART A CHAPTER I. UNIPRODUCT PRODUCTION THEORY A.1.l. What is a Production Function A production function is a complex analytical tool which describes the maximum output that can be obtained from a given set 111: of inputs in the existing state of technological knowledge. can also be regarded as the technical relationship between the maximum quantity of output and the volume of inputs required to produce it, and as the technical relationships between inputs themselves. The parameters of the production function thus con- ceived represent the features of the technology according to which a given set of inputs is transformed into a certain output. In general four useful characteristics of a production function have been discussed in economic analysis. IThe engineers of the firm are concerned not only with inputs and outputs but with the properties of the energy sources and other factors of production required to transform materials, such as the feed mechanism of certain equipments, etc. An engineering produc- tion function can be transformed into an economists' production function so as to provide for it a physical-technical foundation, by leaving out some non-relevant information. The production func- tion is fabricated by the economist and it is probably foreign to the engineering and business world, because it is not directly measurable. The abstractness of the production function concept is precisely its source of value; it produce highly useful and verifi- able hypothesis and it enables economists to analyze a wide variety of problems. See H. Chenery (1953) and W. Salter. For a survey of the literature in which production functions are derived from engineering data, see A. A. Walters (1963; pp. ll-l4), and R. Dorfman, P. Samuelson, and R. N. Solow (1953; pp. 130). 8 They are the efficiency of the technology, the degree of economies of scale that are technically determined, the factor intensity of the technology, and the ease with which one input is substituted for another. Abstract technology2 is followed by the additional uses that is to be made of the production function. Firstly production functions can be used in measuring technical changes.3 Secondly, the relationships between production func- tions and isoquants can be broadened and the production function can be used to derive a more general description of technology. For instance, one can obtain the input requirement set of which the isoquants might be considered as the efficient sets. This more abstract and general view has some important analytical attributes and may yield some good returns in empirical work. It has been supported by a great deal of refinement on the axiomization of technology in the last decade. It deals also with the dual relationships between technology and cost functions, more generally, profit functions. The cost function has an important analytical value and in some circumstances some potential promise for 4 empirical work. The next question of great importance both in 2M. Brown call these four characteristics of a production function, taken together, an abstract technology. See M. Brown, 93_ the Theory and Measurement of Technological Change, Cambridge Press, 1968, pp. 12. 3Solow (1967) denoted the major part of his survey to discuss this issue. Some other works on this subject have appeared since. In spite of its importance, the subject is not discussed here. 4For instance, the allocation of costs into multiproducts can be dealt with the estimation of a joint cost function. See R. Brown, D. Caves, and L. Christensen (1975). 10 theory and in practice is that of aggregating quantities and prices. The question is when there is a natural way for such aggregation. This question is associated with the notion of the separability5 of a function, which is of prime importance by itself in empirical analysis and is closely related to our study. Various specifications of a production technology in the axiomization of production structure are summarized in terms of the input requirement set, the production function, the cost function, and the duality relationships among them, in the following section. Next, the functional forms in historical trend of most empirical researches are reviewed in terms of a quadratic form and of a combination of several subfunctions. Finally the trans- centdental logarithmic functions are defined for a production function and for a cost function, and its properties are investi- gated in relation to the theory of production and to its empirical analysis. A.l.2. Specification of a Technology: AAxiomization of ProdUction Structureb 2.1. The Input Requirement Set For the production of output y of a particular product, we need at most n factors. Let x_be a vector of inputs of these n 5"Separability" is discussed in later sections in detail. See the subsection 4.1.2. in this chapter. 6Let x, yeRn and i=1, . . . n. Then for every i;x>y=e xi>y1~,_x_3_y=>x1_>_yiandxfy,andxgyaxigyi. The inner product 15 denoted as x y, The positive and non-negative orthants of Rn are denoted by Di and On respectively. 11 factors. The technology specifies the various way of producing y, namely X(y) = {_;x_can yield y} (l) The properties of X(y) are assumed to be: (a) Location--X(y) is a non-empty subset of the non-negative orthant Rn denoted by On. It is possible that some factors will not be utilized, but the only output that can be obtained with no inputs at all is the zero output, that is, X(O) = On and if y > O =>O t X(y). (b) Closure---X(y) is assumed to be closed. That is, if a sequence of points {5?} in X(y) converges, the limiting point also belong to X(y), meaning that X(y) contains all its limiting points. (c) Monotonicity---If a given output can be produced by the input--mix x_it can also be produced by a larger input. Similarly, the inputs required to produce a given output can certainly produce a smaller output. (d) Convexity---X(y) is convex. 2.2. The Production Function Using the notion of the input requirement set, the production function can be defined by: f(x_) = "f," {y = 2: e X(y)} (2) When X(y) has the four properties defined above, f(x) has the following properties: (a) Domain--f(x) is a real-valued function of x_defined for every x c On and it is finite if.x is finite. (b) Monotonicity--an increase in inputs cannot decrease production: 12 3535' =>f(x) _>_ f(x') (c) Continuity-"f(x) is continuous. (d) Concavity-—-f(x) is quasiconcave over On. 2.3. The Duality between the Production Functions and the Input Requirement Sets The production function was derived from the input require- ment set. It is possible to assume a production function f(x) with the four properties above and to derive from it: X*(y) = {5: f(a) 3y. 3: can} (3) It turns out that x*(y) possesses the corresponding four properties of an input requirement set. Furthermore, if X*(y) is used in (2) to derive a production function, say f*(x), then f* = f. Similarly, if we start with X(y) to derive f(x) and then in turn use of f in (3) to derive x*(y) then X*(y) = X(y). Thus, there is a full duality between the input requirement set and the production function.7 2.4. The Cost Function In general, economic models involving production need rules ,of behavior, in addition to the production function. In the micro analysis the criterion is profit maximization. The selection of the optimal output can be done in stages, first selecting the 7This is discussed in detail by Diwert, w. E. (1971), "An Application of the Shepard Duality Theorem: A Generalized Leontief Production Function," Journal of Political Economy, 79: 481-507. 13 input mix which minimizes cost for any output y 6 Y and then selecting that y which maximizes profit. The cost minimization for all p_e 93 and y 6 Y is described by: C(y, P) = min { px_: 5.5 X(y)} (4) where p_is the vector of factor prices and C (y,p) is the cost function. If X(y) possesses the properties defined earlier, then C (y, E) has the following properties:8 (a) Domain--C (y, p) is a positive real-valued function defined for all positive prices 2 and all positive producible outputs. (b) Monotonicity--C (y, p) is a non-decreasing function in output and in prices. (c) Continuity--C (y, p) is continuous in y and in p, (d) Concavity-- C (y, p) is concave function in p, (e) Homogeneity--C (y, p) is homogeneous of the first degree in prices. 2.5. The Duality between the Sost Function and the technology Instead of deriving the cost function from the input requirement set according to (4) it is possible to postulate a cost function with the assumed four properties and to define the following set: BShephard (1953), Uzawa (1964), McFadden (1966). 9McFadden (1966), Diwert (1969). 14 X°(y) = {x,: po3_C (y, p) for every p_c 95 and x_e On} (5) The set X(y) has the correSponding four properties. Further, we can use (5) in (4) to derive a cost function that will be identical to that used in (5). Alternatively, if we use (4) for C (y, p) in (5), then X°(y) is identical to X(y) in (4). This is the duality between a cost function and an input requirement set. But in view of the duality between input requirement sets and production functions, there also exists a duality between the cost and the production functions. 2.6. Implications of the Production Theory for Empirical Analysis The axiomization of the production structure has been refined in various ways but such refinement seems to have had, so far, little effect on the implications to be drawn for empirical analysis.10 For the empirical analysis also, the technology of the economy can be measured either in terms of the input requirement set or in terms of the production function or in terms of the cost function and from any of the three we can derive the other two. However, most empirical works are concerned with production functions. There is hardly any work on the direct measurement 10See also Lau's recent works: Lau (1976). A revision of "Some Applications of Profit Functions," Memorandum 86A and 688, Center for Research in Economic Growth, Stanford University, November 1969. 15 of the input requirement set, except that of Hanoch and Rothschild (1972) which attempts to set the ground for such an empirical analysis. More hope has been expressed in the literature on the appropriateness of the cost function to empirical analysis and some work has been done by Nerlove (1963) and Brown, Caves and Christensen (1975). From the point of view of empirical analysis, the properties of the production function in the previous section, 2.1., impose rather little. Any function used in such analysis assumes much more. In fact, the very notion of representing the production function by a given algebraic form is rather restrictive and very ‘1 This point is of prime likely can only yield an approximation. importance and has several repercussions. The algebriac formulation is essential for empirical analysis. However, nowhere is it stated that there should be one algebraic form which will give a good approximation for the whole domain. Yet, implicit or explicit in many works is the idea that the particular function should describe the process of production near the origin as well as for outputs which are many times the observed quantities. Therefore, it is suggested here that a particular function used in an empirical analysis should maintain the usual properties assumed only in the neighborhood of the observations. The 1]This is discussed in detail in the following section, the function form of a production function. 16 relevance of this observation will become evident in subsequent discussions.12 A.l.3. Functional Forms in Trend 3.1. Choice of the Functional Form The choice of functional form should depend on two properties. First, the functional form should be capable of representing a wide range of technologies in order to minimize the a_prigr_assumptions imposed on the estimating equations. Second, it should be tracta- ble within the assumption of the model. That is, the estimating equations should be simple enough to carry out the estimation with minimal computational burden and with ease of interpretation. In reality any choice is a compromise between these two objectives and such a choice must be based upon value judgements in general. It is with respect to the first criterion that simple functional forms of production and cost functions such as the Cobb-Douglas (CD). Leontief (L), and constant elasticity of substitution (CES) forms are dominated by more general forms such as the flexible functional forms to be illustrated below. For example, it is well known that the CD production function has a Hicks--Allen elasticity of substitution (AES) which is unity for all input pairs, and under cost minimization implies that factor 121" particular, our empirical work, using the translog approach, emphasizes it. 17 shares are constant. Since the substitution elasticities are measures of curvature around an isoquant it is evident that the shape of the isoquant is severely restricted by assuming a CD function. This is highlighted by the implication that factor shares are constant. The apparent success of the CD function in applied work seems to be due to two reasons. First, using aggre- gated time series the direct estimation of the CD function is reasonable since the substitution effects are not well identified by highly collinear data. Second, Fisher (1969) argues that the constancy of factor shares of labour and capital in aggregate data fits the co hypothesis.]3 The CES function permits the AES to deviate from unity but does require it to be constant by construction. The CES function thus generalizes the CD and L functions which assume a common constant elasticity of substitution which is unity or zero respectively. But the CES function is restrictive in the nature of the type of substitution permitted. In particular, all factors are equally substitutable with each other, a restriction which has no theoretical justification but which simplified the empirical work considerably. If there are just 2 inputs then this restriction may not be very hard to accept since it simply means that a single cross elasticity of substitution is constant. But to extend this constancy to a multifactor technology and assume that the AES 13But this does not necessarily mean that there exists an aggregate CD production (or cost) function but that we have yet to explain this constancy. 18 between electricity and machines is the same as between unskilled labour and materials, for example, seems to be unreasonable until verified empirically. Even with two inputs one may not wish to assume that the AES remains constant around the isoquant. We conclude therefore that g_prjg§i the CES function is restrictive in that it restricts the elasticities of substitution (a) to be constant, and (b) to be the same constant for every pair of 14 Such restrictions should be tested not imposed a_priori. inputs. Given the a_prjgri_presumption against the CES form, there has been considerable effort made to obtain less restrictive functional forms. One obvious approach is to make the common AES, o, a function of some variable such as the level of output or the factor ratio or factor share, etc. Such generalization have been called Variable Elasticity of Substitution (VES) functions and have been discussed by Revankar (1971), Lu and Fletcher (1968), Sato and Beckmann (1968) and Lovell (1973). A recent spurt of functional forms owes its origin to Diwert (1971) who generated a functional form that is linear in parameters and which provides a second order approximation to any arbitrary twice differentiable function. This Generalized 14Another feature of the CES function is that it is additive in terms of input combinations. This special form of separability is not independent of the constancy of the AES, since Berndt and Christensen (1973c) have shown that a certain separability has something to do with a certain set of AES equal. See also Russell 9 5 . 19 Leontief (GL) functional form15 was quickly followed by the Transcendental Logarithmic (translog) functional form developed by Christensen, Jorgenson and Lau (1971), and Sargan (1971). The Generalized Cobb-Douglas (GCD) developed by Diwert (1973) and a generalization of GL by Denny (1974) and Kadiyala (1972). In addition Diwert (1973) has developed functional forms for special functions such as revenue and variable profit functions as well as indirect utility functions and indirect production functions. Also recently Lau (1976) has developed a profit function. 3.2. Functional Forms in a Quadratic Form One interesting aspect in the formulation of a production function can be described in a quadratic form which accommodates various functions when properly interpreted. The form is Y0 = [111][0‘0 159i][1]= 0.0 TX'ETX'BX (6) 458 B x The functions to be reviewed use the following transformation: .=o- T1 . yi Xil , pi # O (7) T2 : y, = 1n xi, pi = O 15It is called so because when used as a cost function it yields the Leontief cost function as a Special case. 20 where X 0 output x. the i-th input 1 (a) CD-like (Cobb-Douglas production functionk This function is obtained from (6) by imposing (6) (B E O) 0 T2 (8) By this notation it is meant that the CD function is obtained from (6) imposing on the function 8 E O and the variables are obtained by a logarithmic transformation (T2). The result is, for the production technology of one output and of five inputs, 5 1n x0 = o0 + 121 ai 1n xi (9) (b) CES-like (constant elasticities of substitution): This function is obtained from (6) by imposing (6) n (8 E 0) 0 T1 [lipj = p}, j = 1, . . . 5 (10) By this notation it is meant that the CES function is obtained from (6) imposing on the function 8 E O and the variables are obtained by a power transformation (T1) of a constant exponent (p). The result is 5 xg= z a. x? (11) 21 (c) CRES-like (constant ratios of elasticities of substitution); The function is obtained from (6) by imposing (6) n (8 E O) n T; (12) The CRES function, developed by Mukerji (1963) and Gorman (1965) is a generalization of the CES-like function, the functional form of which is 5 X8= Z 01.x.pi (13) (d) CRESH (homothetic or homogeneous CRES): Hanoch (1971) defined and analysed a functional form for a one-output, many factors production function, which is homothetic (or homogeneous) and exhibits CRES.‘ Its functional form is 5 . . 1= 2: a. (:1)‘31 (14) _a _a O In the CRESH function its AES (Allen-Uzawa partial elasticities of substitution) vary along isoquants and differ as between pairs of factors, but the AES stand in fixed ratios everywhere, while the CRES function; however, is not homogeneous or homothetic, so that individual ES vary with output as well as factor combinations, the expansion lines (for given factor prices) being curved in a predetermined way. 22 (e) GL-like (generalized Leontief function): The GL function, developed by Diwert (1971), is obtained from (6) by imposing (6)n(ao=0)n(gEO)nT1n{pj=1s} (15) Hence the result is 5 5 x = Z Z B-o MIT (77 (16) ° i=1j=113 ‘ 3 (f) TL-like (Transcendental Logarithmic function): The TL function is develOped by Christensen, Jorgenson and Lau (1972). The function is obtained from (6) by imposing (6) n T2 (17) Hence its function form of a production technology of one output and of five inputs is 5 5 5 X0 = a0 + 1:] “i 1n xi + 121 jil Bij 1n xi 1n xj (18) The CD function captures two important properties of a production function: monotonicity and concavity. It does so with a small number of parameters. This, in addition to the other two plausible reasons explained in the previous section, may explain its dominance of the field for so many years. The work by Arrow, Chenery, Minhas and Solow (1961) added a new dimension to the analysis: the ease of factor substitution. 23 The generalization of this measure to the case of more than two inputs, and the implications of such generalization for the form of the production function have been widely discussed. The functions listed in items (b), (c) and (d) represent the results of this discussion. The constraint (8 a O) on (6) implies that the production function is strongly separable between inputs. This property constitutes a strong constraint which simplifies considerably the empirical work. As indicated by Mundlak and Razin (1971) sepa- rability has been imposed without being tested and that raises a question with respect to the proper use of this assumption. It is against such a background that the translog function approach broadens the scope of analysis.16 We have thus singled out three major stages in the develop- ment of algebraic formulation of the production functions: (1) Cobb-Douglas, (2) broadening the scape for factor substitution, and (3) submitting separability to empirical test. 3.3. Functional Forms in a Combination of Subfunctions Another aspect in the formulation of a production function can be reviewed as a combination of several subfunctions which 16See Christensen, Jorgenson and Lau (1972). The translog function was also discussed by Griliches and Ringstad (1971) and Sargan (1971), but with no particular emphasis on separability. 24 accommodates various functions when properly interpreted. The form is f(x) = 9(5) * h(§) (19) where f(x) is the production function, 9(5) and h(x) are two arbitrary functions and * is an arbitrary operator, such as addition, multiplication, or an exponent. This approach provides a convenient framework for classifying functions which do not fall within the general quadratic form of the previous section. (a) VES function: Revankar (1971) suggested the following function in order to make the elasticity of substitution a linear function of the capital-labor ratio: y = “o X?‘ (x2 + 11x11“ (20) If we let 9(5) = 6.0x?l and h(x) = (x2 + Y1x1)p then we can write this function as y = 9(§)h(5) (21) That is simply the product of the CD form and the CES form. (b) Constant marginal share: This function was suggested by Bruno (1968); y = aOX‘i‘lx‘é‘Z - 1x2 2 9(5) + Mg) (22) Again 9(5) has the CD form where h(x) is linear. 25 (c) Transcendental production function: Halter, Carter and Hocking (1957) use a function y = aoxglxgz eYixi + Y2X2 (23) which can be immediately decomposed into y = 9(5)h(2s) (24) where 9(5) is a CD and h(;) = eY1X1 + szz The same procedure can be followed with more than two subfunctions. Having decomposed a particular algebraic form into its components it is then possible to trace the origins of particular properties and search for ways to achieve the same property with as few parameters as possible. 3.4. Functional Form Flexible in Prices Attentions on functional forms which are flexible in the sense of providing second order approximations in input prices to an arbitrary continuously differentiable cost function, have been paid since it may be unlikely that the production function approach will be useful at the industry level of disaggregation. Diwert (1971) generated a functional form, called the Generalized Leontief (GL) function, of 5 5 C(w) = Z 2 bij lig’ MW; , where bij = bji (25) i=1 j=l 26 to describe a wild range of substitution possibilities for a multi- input technology. Diwert (1973) also suggested another generali- zation of Cobb-Douglas form, called the Generalized Cobb-Douglas function, of 5 5 ln C(w) = bO + iil jil bij 1n (wi + wj) (26) where bij = b.. J, and Z Z bi' = l. 133 Following to Diwert, Christensen et a1. (1971) developed the transcendental logarithmic or translog (TL) form of 5 5 5 boo + if] boi 1n wi + ifl jEl bij 1n wi ln "j (27) 1n C(w) where bij = bji’ E bio = l and g bij = 0, i=1, . . . 5. Each unit cost function is linear homogeneous in prices as theory requires. Diwert has shown that GL and GCD are decreasing concave functions if the bij are non-negative while BL is positive if some bij > o as well and GCD is positive if b0 > m. Under certain parameter restrictions the CL and GCD functional forms satisfy all of the conditions required of a cost function. The TL 27 form satisfies all these conditions globally only if bij = O in which case it reduces to a Cobb-Douglas function. A more general functional form permitting a wider range of special cases have been provided also by Denny (1972) and Kadiyala (1972); C(w) = {z z bij "i/2 wg/zil/Y (28) where bij = bji This reduces to CL when r - l and to the CES form when bij = O, i f j which in turn reduces to the CD and L forms as limiting cases. A.l.4. The Transcendental Logarithmic Function 4.1. The Translog Production Function 4.1.1. Introduction A new class of production function, named the "Transcendental Logarithmic Production Function," or more briefly the translog production function is defined by the following form: 5 5 , 5 15(2 y..lnx.) v = do n x?) n x, 5:1 ‘3 J (29) i=1 i=1 where V = quantity of output X = quantity of the i-th input and Yij = in for i, J = 1, . . . 5 28 Equivalentely, the translog function may be written as 5 5 5 1n V = 1n a0 + Z oi 1n xi + k 2 X y.. 1n x. 1n x. (30) i=1 i=1 j=1 ‘3 l J The translog function has many desirable features in both theoretical and empirical applications. In particular, it reduces to the CES and the CD functions as special cases--the former as a second-order approximation. It also reduces to most of the CES-like functions as special cases with appropriate restrictions, such as the Uzawa generalization of the CES production function (1962), the McFadden generalization of the CES function (1963), the Mukerji Generalized SMAC Function (1963), the Sato Two-Level CES production function (1967), the Hildebrand—Liu generalization (1965), the McCarthy generalization (1965), and the Transcendental Generali- zation of Halter, Carter, and Hocking (1957), etc.17 17The Uzawa generalization of the m-factor C.E.S. function is given by 5 : where v = Yo 521 25 z .19 xssw --2; , z .19.] Z = v ieNS 1 1 ieNS S S and N is the set of indices of inputs in set 5. Uzawa (1962) has proved that the above function completely characterizes the class of homogeneous m-factor production function with constant Allen- Uzawa partial elasticities of substitutions (AES). McFadden (1963) has derived the class of homogeneous m-factor production functions which possess constant direct elasticities of substitutions (DES)-- the block additive linear homogeneous functions. The McFadden generalization of the C.E.S. production function is defined by S 2 B = 1 = Ki '9 1 Y 8s ieiINsi‘T) ’ 5:] s s=l 29 The translog production function provides a second order approximation to any arbitrary production function for values of inputs near unity. The Mukerji generalized SMAC Function is shown in the equation (13) of the previous section 3.2. in this chapter. The Sato Two-Level C.E.S. production function is given by s -1 = -o 9 V 521 as ZS , where 1 Zs = [2 81‘s) (X§S))'ps:l 95. as. 81(5) > o, —1 < 9, 95 < °° ieNs as= ZB§5)=1 1 ieNS "MM 5 Here V is a C.E.S. function in {Z} and 25, in turn, is a C.E.S. function in {X 5 }. Hence V is a "two-level" C.E.S. function in {x}. The last three generalizations are of the C.E.S. production function in the two-factor case. Hence the Hildebrand-Liu generalization has the form of - -9 V = Y EK") + (M) n (§)'°(Hp) L'] P , and the McCarthy generalization is given by -.E = “o -n n-p -o p v YEK +62K L +63L] , and the Transcendental generalization has the form of v = Y Kan Lil-alu e8(K/L) 30 L81: V = FEX), X2, X3, X4, X5] (3]) be written as ln V = G[ln x1, ln x2, 1n x3, 1n x4, 1n x5] (32) where V is an arbitrary production function. Expanding G in a Taylor's series expansion in 1n xés around xi = 1 (or 1n xi = O), i = 1, . . . 5, we have 1n v = G[O] + z a?“ I 1" X1 _' 1=l n x1 ln X = [QJ 5 5 2 a G I 1n X. l X. +3522 1 J i-l 5:1 31n xi 5Tn xJ 1n x = [OJ + the high-order terms, (33) where x and 1n.x represent the vector of xi's and 1n xi's respectively and [DJ is a vector of zeros. A comparison of Equation (30) with (33) indicates that we may set ln a0 = G[_O_] _ as = “i - aln X, i ’ 1 1’ 5 lna=lm 32 G | . 1.1=1. 5 Yij = §1n xi STTn xj 31 Hence the translog function provides a second order approximation to any arbitrary production function around 1n x_= [0]. 4.1.2. Properties of the TranslogfiProduction Function (1) Monotonicity condition. A neoclassical production function should be increasing in all its arguments, i.e., 8V 3 Xi __>_O,i=l,...5, at least in the region of observed operation. This implies that ainv_f_1_av ._ m-v3x1;0,1-1,...5, because of the strict positivity constraints on V and X1. Hence the monotonicity constraint becomes Yij ln X.:; 0, i = l, . . .5 (35) (2) Convexity condition. In addition to the monotonicity property, a neoclassical production function must also be concave--i.e. it exhibits de- creasing returns to scale. Hence [Fij] must be negative semi- definite. A necessary condition is that Fii=é O or 2 §__JL-=;_!. (@.lflLJL._ 1) 3.1!L1L.+.Y 3 2 x 2 a 1n x. a 1n x. xi .i 1 1 1.1. __<__O (36) 32 These must be satisfied in particular at the point of approximation. Hence “g“;;§ = V [(01 ' 1) 01 + Yii] ; 0 (37) lna=£91 A set of sufficient conditions given mononicity is 1:; “i=3 0; y.. i 0. Moreover 11 32" V aan aan :I '_____—'= + Y-- (38) 8x1 axj Xi xj [:3 ln xi aln x3. 13 and also _ai_V_ l = . .= 3X1 3Xj V [(11 aj +Y1j]’ 1 M J 1, . . . 5 lnx_=[QJ (39) We note that if V is concave at ln x_= [9), then by a continuity argument it can be shown that V is locally concave in a neighbor- hood of 1n x.= [93. This local concavity does not rule out the existence of uneconomic or convex regions and especially increasing returns to scale in certain ranges of inputs. A necessary and sufficient condition for local concavity at 1n x.= [9] is that the matrix 1" ‘1 (01-1101 + Y11 . . . . azas + Yis F.. = [‘3 ln_>_<_=[9] (40) Q L_ 0501 + Y5: ~ - - ~ (05'1)Gs + Yss 33 is negative semi-definite, which in turn requires that all the principal minors be negative semi-definite, or equivalently all the characteristic values of the matrix are non-positive. (3) Homogeneity conditions. For homogeneity of degree k of the translog production function we require that 1n V[Xx) = 1n v + k 1n A. This implies the following set of necessary and sufficient conditions on the translog function; 5 Z a. = k (41) Yis=0,i=1900059 (42) Y1..=0,j=1,...5, (43) i.e. the row sums and column sums of [Ylj] are identically zero. (4) Separability conditions. To define separability among inputs, first denote the set of n inputs by N = {i, . . . n}. A partition S of N is given by {N1, . . . NS} where N = N1U'N2. . .U Nsand NrnN 3F 3 x1 = fi, etc. A basic condition to which we refer tis empty for r f t. Let below is the independence of the marginal rate of substitution of pairs of inputs from another input: 34 f. a(fi)/axk=o (44) J We say that F is strongly separable ($5) with respect to the partition 5 if (44) exists for all ieNr, jeNt and ktNrU Nt' The function is weakly separable (WS) with respect to the partition 5 if (44) exists for all i, j e Nr and k t Nr.]8 8y differentiation we immediately obtain that (44) is equivalent to f5 fik - f1 fjk = O (45) The condition for inputs i and j to be functionally separable from input k is that the first and second derivatives 19 20 of F satisfy, for the translog function, 18Goldman and Uzawa (1964) showed that a function f(x) is SS with respect to the partition S (s > 2) if and only if f(x) = F [E ft (x?)] where F is monotone increasing and ft(xF) is a function of x. The function is WS if and only if it is of the form; f(x) = erg‘u‘). . . . 95(1511 Also Lau (1972) showed that the cost function is WS(SS) with respect to the partition S in input prices and in input quantities if and only if f(x) is homothetic. And Berndt and Christensen (1972) related separability to AES. 19For weak separability this condition must hold for inputs i and j in one subset and input k in another subset. For strong separability this condition must hold in addition for inputs i,j, and k all in distinct subsets. See Berndt and Christensen (1973b) far a summary discussion of separability conditions. 20This is derived by differentiating (31) and (32), and substituting into (45). 35 5 5 ij (a i + £2] Yil 1n x2) - Yik (aj + £2] ng 1n x2) = 0 (46) The set of conditions necessary and sufficient for inputs i and j to be globally separable from k are that21 0‘i ij ’ “j Yik = 0' Y12 ij - ng Y-ik = 09 isjsks’e =19 - o - 5 (47) When ij and yjfl are nonzero, we can divide by these parameters and alternatively write the separability conditions as ——-:-=———-—.--, 2:1,...5 (48) 4.1.3. Elasticity of Substitution There exists a transcendental logarithmic production function of 5 inputs which attains both a given arbitrary set of "Direct Elast1c1ties of Subst1tut1on {Grs r, s = l, . . . 5; ars = asr} and a given set of "Allen-Uzawa Partial Elasticities of Substi- tut1on {or r,s = l, . . . 5; or = Osr}’ at given quant1t1es s s 22 of output and inputs. 21For the derivation of this condition, see Berndt and Christensen (1973a). 22About the proofs of these statements, see Jorgenson, Christensen, and Lau (1971), Part II: The Transcendental Logarithmic Product1on Function, pp. 21-57. 36 The Direct Elasticity of Substitution (DES) between inputs r and s is defined as23 Fr FS(FF X? + F5 XS) = - (49) 6 PS 2 _ 2 XY‘ xS(FY‘Y‘ FS 2F? Fs FY‘S + FSS FY‘ ) where Fr's and Frs's are the first and second partial derivatives respectively. For the translog function, " ( 5 ) F =— a + 2 Y01n X. r xr r i=1 r1 1 V 5 5 Frr = ia-‘Ivrr + (or + .5 yri ln xi - 1)(ar + -E Yri 1n xi)] (50) r 1-1 1'] V [ ( 5 )( 5 )1 F = y + a + 2 y . ln X. a + 2 y . 1n X. rs xr xS rs r i=1 r1 1 5 i=1 s1 1 Hence the D.E.S. is given by, in terms of the parameters of the translog function, 6 = _ MrMs(Mr+Ms) PS 2 2_ _ 2 2_ M5(Yrr+Mr Mr) 2MrMs(Yrs+MrMs)+Mr(Yss+Ms Ms) 23See Allen (1938), pp. 340-345, 503-505, Frisch (1959), and McFadden (1963). 37 where M = a + _ a 1n F i a 1n X, i IIMU'I . 1 Yij 1n xj, i=1, . . . 5 (51) J The Allen-Uzawa Partial Elasticities of Substitution (AES) is defined as24 | rs rs (52) where |F|= 0 F1 000 F5 r, F11 F15 F5 F51 ° ' ' F55 and lFrsl is the cofactor of Frs in |F|. and the A.E.S. can be again expressed, in terms of the parameters of the translog function, as rsl (53) rs IGI IG O 24See Allen (1938) and Uzawa (1962). 38 where [G] is the determinant of G: 0 M1 0000 M5 M1 Y11+M12'M1 ° ° ' ' Y15+M1Ms Ms Y51+M5M1 ° ° ° ° YssTMsz'M5_j L. and lGrsi 15 the cofactor Grs 1n G. The formulae for the D.E.S. and the A.E.S. are functions only of the Mi and the Yij‘ Since the regressors are logarithmic, estimates of the y. are independent of units of measurement. The 11' fitted values Mi are also invariant to scaling of the regressors. Therefore, the estimates of the o are independent of units of rs measurement. In general, neither the DES nor the AES of the translog function is constant for all quantities of inputs--and hence indirectly, for all prices of input--as one can readily verify by computing equations (51) and (53) for the translog function. Hence, the translog function exhibits the property of variable DES and AES. Actually this is to be expected in view of the theorems of McFadden (1963), Uzawa (1962) and Gorman (1965), which characterize completely the various highly restrictive classes of 39 functional forms which exhibit the property of constancy of various definitions of elasticities of substitution.25 4.1.4. Profit Maximization Let Po price of the output, P* = price of a unit of input i, i=1 . . . 5, * .- Pi = P10 /Po, 1-] o o o 5, 1 Po V V ’ Then the usual marginal conditions for profit maximization can be written as aln V P X. ___= 1 1 = M1 (54) 31h Xi V where Mi is the ratio of expenditure on input i to total sales. Equation (54) results in the following system of share equations. 1 Yij 1n xj, i=1, . . . 5 (55) 3 ll Q + II M01 Equation (55) is linear in parameters and in addition, there are equality restrictions from the homogeneity conditions across the individual equations corresponding to the Yij'5° 25See McFadden (1963) and Mundlak (1968). 40 Other important properties of the translog production function in connection with the technical change are omitted from this discussion, because they are beyond the scope of our studies. Also further discussions on the translog function re- garding empirical implementation and its advantages and dis- advantages are postponed to the next chapter, the section 4.3. ' 4.2. The Translog Cost Function 4.2.1. Introduction A convenient functional form for the unit cost function is the transcendental logarithmic (or translog) cost function,26 5 5 5 ( 1nC=o+Za.an.+Z ZB..1nW.an. 56) o 1:] 1 1 i=1 j=l 13 l J where C is the production cost and W, is the i-th input price. Also the translog form provides a second order approximation to an arbitrary twice continuously differentiable unit cost function. 4.2.2. Properties of the Translog Cost Function (1) Monotonicity condition 26Dual to the production function is a cost function, C* = J(y, W1, W2, W3, W6, W5) where C* is total cost of production, y is aggregate output, and W1 is the i-th input price. If the production function is a positive, nondecreasing, positively linear homogeneous, concave function, then the cost function can be written C* = y-C(W1, W2, W3, Wt, W5) where C is a unit cost function satisfying the same regularity conditions (Diwert, 1973). See also Christensen, Jorgenson and Lau (1973) on the definition of the price possibility frontier under constant returns to scale, by duality in the theory of production. 41 The cost function must be an increasing function of the input prices. In terms of the parameters of the translog cost function, this implies, 3 1" C - + 5 1 w 0 '—1 5 57 m-OL. _B..n.__>_,1-,... () (2) Concavity condition The cost function must be concave in the input prices. 32c (w) This implies that the matrix must be negative- definite within the range of input prices observed, or equivalently all the characteristic values of the matrix are non-positive. (3) Homogeneity condition It is also well known that the cost function for a cost- minimizing firm must be homogeneous of degree one in the input prices. Hence, ll MU'I Q II ._n 0 d. d d and (58) 0, i=1, . . . 5 C4. II M 01 cad “u: do Ll. ll (4) Separability condition Similarly to the case of the translog production function the separability conditions for inputs i and j to be functionally 42 separable from the input k is that the first and second derivatives of C satisfy, for the translog cost function,27 5 5 83°15‘11" ElYi‘e 1n HZ) - lik(“i+,§,% 1n 14,) = 0 (59) 4.2.3. Elasticity of Substitution A dualistic concept in the cost function to the direct elasticity of substitution (D.E.S.) in the production function can be defined by applying the two-factor elasticity of substitution formula to each pair of factors, holding fixed the imputed prices of the remaining factors and the imputed total cost. McFadden (1963) named it the shadow partial elasticity of substitution (S.E.S.). The S.E.S. can be defined in terms of the cost function C = C(y,W) of the producer, which specifies the minimum imputed cost C of producing the output y with the according price vector E = (1'41, ”2, ”3, Nu, W5) SUCh as 2 2 - (cfixci ) + mom/ci ca.) - (c../c. ) 6 * _ 33 J ‘3 (1/w, 0,) + (1/chj) where C = §—£—- and C = ———§:9-——- are evaluated at (y W) i 3 W, ij 3 W, a ”j ’ — 27See the equation (44) through (48) of the section 2.1.2. in this chapter. 43 Hence the S.E.S. is defined as, in terms of the parameters of the translog cost function, 22_ _ 22- -fMj (Mi MiTYii) ZMiMj(MiMj+Yij)+Mi(Mj Mj+Yij)] * 6.. = 13 MiMj(Mi+Mj) On the substitution possibilities among inputs implied by the translog cost function, Uzawa (1962) demonstrated that elasticities of substitution (AES) could be computed directly from the cost function and its derivatives. The formula28 is 0.. = (62) = = 2 where Ci aC/aWi and Cij a C/awiawj. For the translog cost function this becomes 28The A.E.S. formula is shown in (47) as £355 15.1 0" = '1F‘1f“" "" ‘3 ii IFI where y = F(X1, X2, X3, Xu, X5). Also we have C. = a£—-= X. and 3X. 1 awi 1 cij = EWIT' from the cost function C = C(Wl, W2, W3, Wu, W5) where -§wx H _c 3x1.__J_c1..C COD-if1 i i' 9"” C’i.i"x1..xj 3113's1 03. ° 71.1 + Mi ".1 0.. = , i f j 1.1 (53) 111 + "i(Mi") 011 - M 2 where the Mi's are fitted values of the cost share equations. The elasticities of demand with respect to price changes are closely related to the AES:29 114*”1' "a . . nij-MJ-Cij- M. 91fJ 1 (64) Y11 + Mi(Mi-]) nii‘MiGii= M 4.2.4. Cost Minimization The system of share equations are also obtained by logarithmic differentiation of the unit cost function,30 29See also Uzawa (1962) and Brown, Caves and Christensen (1975), p. 26. 30In the total cost function of C* = y-C(W1, W2, W3, W“, W5), aC/BW. = Xi/y, where X1 is the cost minimizing quantity of the i-th input. Since the cost function is linear homogeneous in prices, 5 , C* = z WiXi by Euler's theorem. Therefore, C = z WiXi/y. i=1 i= From these relations, we can get, 1 45 . 5 = ____.__.= M = a. + .2 8.. 1n W., (65) where Mi is the cost share of the i-th input. 31nNi 31111. C Y C Y 5 X W.X 1:] 11 wix1 = 5 =M‘i 2 W X i CHAPTER II. MULTIPRODUCT PRODUCTION THEORY A.2.1. Introduction Economic speculations on the behavior of multiproduct firms can be traced to Pigou (1932), and Robinson (1933), and more recently, Reder (1941), Gordon (1948) and Bailey (1954). These analyses focussed on the selling behaviors (revenue side) of a multiproduct firm, viewing its pricing process as an extended application of the Pigou-Robinson theory of price discrimination. These were followed by those of Hicks (1929), Dorfman (1951) and Ferguson (1971), who adopted the conventional marginalists' method to analyze the profit--maximization behavior of a firm that produces more than one product by means of several variable inputs and occasionally of fixed inputs. Until the study of Samuelson's singularity theorem for non-joint production was published in 1966, there had been no extensive studies on the specification of a production technology for a multiproduct firm. Samuelson (1966) established the necessary condition for the production possibility frontier not to involve joint production, and the work has been extended by Hirota and Kuga (1971) and Burmeister and Turnovsky (1971). Burmeister and Turnovsky studied the case where commodities are 46 47 partitioned into joint groups and obtained necessary conditions for such to occur in terms of the second derivatives of the production possibility frontier. They assumed rather a simple type of partition of commodities into non-overlapping groups. More extensive and useful concepts and analysis in commodity structure of group formation, in general cases followed by Kuga (1973). Kuga extended the concept of Hirota-Kuga's "intrinsic non- jointness" to that of "marginal non-jointness," where it is not advantageous for the producer as a whole to change its output level infinitesimally, at the going factor--and commodity--prices, from suitable changes in the factor input, but without requiring any change in the output level of other commodities. He also used the concepts of a "weak joint group" where a commodity may enter joint relations with more than one joint group in which a number of joint groups are formed, not necessarily of the non- overlapping type, and of a "strong joint group" which roughly corresponds to the non-overlapping joint group of Burmeister and Turnovsky. Together with the problem of jointness in the theory of 1 between inputs the multiproduct firm, the concept of separability and outputs has become focussed on in the specification of the multi-input, multi-output production technology. Recent progress in the specification of the multiproduct production technology has been achieved in two distinctive 1The definition of separability is already discussed in the previous chapter. See the section 4.1.2. Chapter I. 48 directions. The first approach deals with the production possi- bility frontier, originally proposed by Mundlak (1963) under certain restrictive assumptions. The second contribution focusses on the further developments and various applications of duality theory.2 The applications of duality theory in the theory of the multiproduct firm have been elaborated mainly in the two different ways. The dual relationship between the transformation function and the profit function have been speculated by McFadden (1966), Diwert (1973), GHd Lau (1972, 1976). Christensen, Jorgenson and Lau (1971, 1973) have also made an empirical application to the U.S. economy. In addition, Hall (1973) has approached the problem from the point of view of the dual relationship between the transformation function and the joint cost function, using a generalization of the Generalized Leontief cost function due to Diwert (1971). Also recently Brown, Caves and Christensen (1975) have made an empirical application of a joint cost function to the U.S. railroad industry. The basic duality concepts which underly all these studies may be traced back to the pioneering work of Shephard (1953). A.2.2. Specification of a Technology 2.1. The Factor Requirement Function In the specification of a technology with a multiproduct production process, the simplist procedure is to appregate inputs 2A very extensive survey on this topic was done by W. E. Diwert (1972). 49 of various factors. Suppose we have a single input x which can be used to produce various conbinations of three outputs, Y_= (Y1, Y2, Y3). The technology of multiple outputs, single input firm may be sum- marized by a factor requirements function g(Y) which gives the minimal amount of input x required to produce the vector of out- puts _Y_. The properties of 9(1) are assumed to be,3 (a) Domain--g(Y) O and is a real valued function defined for Y“: Q with 9(9) T 9(1) > 0 if Yh3_9_(b) Closure--if Yn 3_Q_and lim 1 Yn n—m + m, then lim 9(Yn) = + w, (c) Monotonicity--g(Y) is a nondecreasing function, new (d) Convexity-- 9(1) is a quasiconvex function, and (e) Continuity-- 9(1) is continuous from below, i.e., for every 0.: O, the set {i,: 9(1)‘§_a} is closed. Condition (a) states that zero input produces only zero output and that a positive amount of input is required in order to produce a positive amount of any output. Condition (b) states that an infinite amount of input is required to produce an infinite amount of any output. Condition (c) states that if more output is produced, then the minimum amount of input needed will not decrease. Condition (d) is a generalization of the classical condition of increasing marginal rate of substitution between products.4 3The generalized C.E.S. form of the function was first introduced by Powell and Gruen (1968). Also their properties are well clarified again by Diwert (1974). 4On this classical condition, see Hicks (1946), p. 87. 50 Condition (e) is a weak mathematical regularity condition. If 9(1) is a continuous function, then (e) will be satisfied. But the difficulty in this simplist procedure of aggregating various factors into one input is that the aggregate input require- ment function is not a single valued function, and its parameters depend on the composition of inputs, which in turn depends on, among other things, the prices in question. This difficulty can be avoided by working on a lower level of aggregation where inputs are not combined. 2.2. The Transformation Function A well behaved technology can be described equally well in terms of relations between prices, or relations between quantities and prices, as long as markets are competitive and profits are maximized. The basic relation among quantities for our purposes is the transformation function t(Y, X) 3_O if [_can be produced with X. We assume that t(Y, X) is defined and continuous for all non-negative Y_and X_and that it is decreasing in Y_and increasing in.X. Alternatively speaking on the transformation function, the production function is also defined by:5 5For a multiple-input, multi-output firm, there is no natural numeraire commodity, such as the single output, to define the production function representation of technology. Following Jorgenson and Lau (1974), the convention of choosing as the left- hand-side variable for production function a variable input which is nonproducible, is adopted here. See also Lau (1976), pp. 52-53. 51 L = t(Y, X) (l) the minimum value of L for given values of‘Y and X such that the production plan (Y,X, - L) is feasible, where L is quantity of the left hand-side variable and nonproducible net input, Y is the vector of net outputs, and X is the vector of net inputs. It is assumed that t(Y,X) possesses certain properties, which parallel similar properties of the single output case. (a) Domain--t is a finite, nonnegative, real-valued function defined on R4" X R;m, where R," denotes the closed nonnegative orthants of Rn for n outputs and R;m denotes the closed nonpositive orthants of Rn for m inputs. (b) Continuit --t is continuous on §;n X R;m. (c) Monotonicity--t is nondecreasing on §;n X R;m and strictly increasing on R+n X R_m where R+n denotes the interior of the nonnegative orthant of Rn and RT denotes the interior of the nonpositive orthant of Rn' (d) Convexity-~t is convex on R" X Rm and locally strongly convex on RE X RT. «1» .. (e) Twice differentiability--t is twice continuously differ- entiable on R2 X RT. (f) Boundedness-- t(>.Y_, xx) (D lim A+w X for every 1, x e 8+" x 1C“, 1, _x a! g. Alternatively saying, _v_ is finite for all finite X_and X_is finite for all finite 1, Also Y_ becomes unbounded for unbounded X, 52 2.2. The Cost Function and the Profit Function The cost function is defined by the relations among quanti- ties and prices, i.e. the function C(W,Y) giving the minimum cost at which outputs Y can be produced when factor prices are H, We assume that C(W,Y) is defined for all positive Y_and N, that it is a continuous, nondecreasing function in [_and W, and that it is homogeneous of the first degree in W, Parallel to the definition of the cost function, the profit function is also defined by: II(P1, _Y_z) s 111an {£111 : 13(11. 12) = 0} (2) 1 the maximization of a linear function PAY) over the set of 1) such that t(Y}, X2) = D where Y; is a vector of choice variables, 2, is the corresponding price vector and Y; is a vector of fixed variables. Here the profit function 8(3), 11) is dual to the transformation function t(!}, 13) in the sense that each may be completely derived from knowledge of the other. Certain regularity conditions are required, of course, leading to different duality theorems.6 In particular, different theorems apply depending upon the nature of Y; and 1;. If 1; refers to a set of inputs then we refer to “(31, X?) as (the negative of) a cost function which may be a 6For a sampling of the literature on duality see Shephard ($323, 1970), Uzawa (1964), Diewert (1971, 1974) and Lau (1969, 53 total cost function if Y; refers to outputs only or a variable cost function if 13 includes some fixed factors. If X; refers to fixed inputs only then 8(3), X3) is called a variable profit or gross profit function and if 11 does not exist then it is called a profit function. If 12 includes only primary inputs H is a value added function and if X; includes only outputs it is called a revenue function. The normalized profit function is given by7 11(3, W)=$u§{f_'l+fl'l-Xt(}_, )9 :1, XeRnXRm} (3) where £_and W_are respectively the normalized prices of [_and X in terms of L, the numeraire commodity nonproducible in the production function. The corresponding properties of this function are; (a) Domain-~H is a finite, positive, real-valued function defined on RT X RT. (b) Continuity--H is continuous on RT X RT. (c) Monotonicity--n is strictly increasing in E_and strictly decreasing in W,on RT X RT. (d) Convexity--H is locally strongly convex on RT X RT. (e) TWice Differentiability--H is twice continuously differentiable on RT X RT. (f) Boundedness-- 1103. it!) n lim ———-——— = 0°, for every _P_, W e R+ X RT. Xe» X H is finite for all finite E_and W, 7This specification is due to Lau (1976), pp. 54-55. 54 The importance of the theory for our purposes is that, under certain regularity conditions, H(E}, 1;) and t(ly, lg) O are two equivalent representations of the technology. First we provide here an explicit statement of duality between the cost function and its underlying technology.8 2.4. Shephard-Uzawa-McFadden Duagity Theorem for the Cost Functions Suppose the transformation function t(Y, X) has a strictly convex input structure; that is, the input requirement set X(Y) = 10 {X|t(Y,X) 3 O} is closed and strictly convex. Then there is a unique cost function C(Y,W), differentiable in W, defined by C(Y,W) = min {WX}. (4) XeX(Y) Further, C(Y, W) is positive, linear, homogeneous, non- 11 decreasing, and concave in the factor prices, W. Finally it obeys Shepard's Lemma (1953), t(v.3—9-§-,‘$—ifl)) = 0. (5) 8On the duality between the transformation function and the profit function, see Lau (1976). 9We already speculated its correspondence in the case of a uniproduct technology earlier. See Shepard (1953, 1970), Uzawa (1964), and McFadden (1973). 10This rules out the case of factors that are perfect substitutes or perfect complements. See McFadden (1973). 11The concavity of the cost function does not follow from the convexity of the technology. All cost functions are concave, ir- respective of the characteristics of the underlying technology. 55 that is, the vector of cost-minimizing factor inputs is equal to the vector of derivatives of the cost function with respect to the factor prices.12 Also when the transformation function t(Y, X) is differentiable in outputs, Y, the following condition holds: a C(Y,W) / 3Y1 a t(Y,X) /avi = 9 (6) 3 C(Y,") / an 3 t(Y,X) /3Yj that is, the ratio of the marginal costs of two goods is equal to the marginal rate of transformation between them. Thus the production possibility frontier is tangent to the isocost surface at the point where production takes place. 2.2. Homogeneity and Almost Homogeneity In the case of multi-input, multi-output transformation functions, the concept of homogeneity is somewhat imprecise unless it is homogeneity of degree one. Intuitively, one would like to say that a transformation function is homogeneous of degree k, if when all inputs are increased by some proportion A all outputs are increased by the proportion Ak. Furthermore the concept of "almost homogeneity" has been introduced to facilitate the analysis of the technology with multi-input and multi-output. A function (Y, X) where Y and X are vectors, is almost homogeneous of degrees k1, k2 12The proof of this theorem is given by McFadden (1973)- 56 and k3, respectively, if and only if t(Xk’Y, XkZX) = Xk3t(Y,X) for every scalar X > 0.13 It is straightforward to see that the transformation function exhibits constant returns to scale if and only if C(Y,W) is homogeneous of degree one in Y. The transformation function is homogeneous of degree one if t(XY, AX) = t(Y,X) = o (7) This implies C(XY, W) = min 2 Wi X Xi = X min 2 WiXi = XC(Y,W) (8) Similarly (8) implies (7). , Further meaningful analysis on homogeneity and almost homogeneity among the transformation function, the cost function and the profit function, are beyond the scope of the current study and omitted here.14 2.6. Separability Between Inputs and Outputs Most studies of the structure of production utilize a single variable to represent output, no matter how diverse its 13It is clear that k1, kg and k3 are in general unique. This more general representation is used to allow the possibility of some k1 being equal to zero identically. 14For a sampling of the literature on homogeneity in relation to the transformation function, the cost function, the profit func- tion and the revenue function, see Lau (1972, 1976;, Diwert (1974). Brown, Caves and Christensen (1975), and Hall (1973 57 actual components. The question here is whether there exists an unambiguous measure of output which is valid independent of the relative factor intensities, i.e. whether the transformation function can be written as t(Y,X) = t[H(Y), X] = H(Y) * G(X) = 0 (9) where H(Y) and G(X) are scalar functions of the Y and X vectors respectively, and * is any arbitrary operator such as addition, multiplication or an exponent, etc. Thus the existence of an output index H(Y) implies the existence of an input index G(X). The existence of these indexes is equivalent to t(Y, X) being separable in outputs and inputs. Lau (1969) has proved many useful theorems relating the properties of transformation functions and profit functions. He noted that revenue and cost functions can be regarded as special cases of the profit function with inputs and outputs fixed. Thus his Theorem is directly applicable in the present context.15 Theorem 1 (Lau): t(Y,X) = t[H(Y), X] = O (10) if and only if C(Y,W) = C[H(Y), W]. 15Han (1973) and Burgess (1974) also demonstrated similar theorems on separability between inputs and outputs, confining most of their attention to the case of constant returns to scale--a very reasonable specification for the analysis of aggregate data, but less reasonable for the analysis of microeconomic data. 58 This intuitively appealing result says that the transformation function is separable in outputs and inputs if and only if the joint cost function is weakly separable in outputs.16 Another result can be adapted from Lau (1969, ) to illustrate the case where both the input and output indexes exist. Theorem 2 (Lau): C(Y,W) H(Y) F(W) H(Y) + G(X) = 0 (11) if and only if t(Y,X) and G(x) is homothetic. Thus strong separability of outputs and factor prices in the joint cost function is equivalent to separability of the trans- formation function with the input index being homothetic. 2.7. Non-Jointness in Production The problem of non-jointness has been investigated by Samuelson (1966) and Kuga (1973) who derived necessary and sufficient conditions for a production function to represent a non-joint technology, using the transformation function. A production function of five inputs and three outputs L = t(Y,X) is said to be non-joint in inputs if there exist individual production functions lasince the cost function is an implicit function, separa- bility in outputs does not imply separability in factor prices. Hence "weak" separability must be distinguished from "strong" separability. See Berndt and Christensen (1973b) for a recent discussion of "weak" and “strong" separability. 59 Y]. = fi(xi]’ XiZS Xisg X14, X15), 1:], o o o 3 (12) such that t(Y,X) = 0, if and only if Yi = f1 and 1 ij = Xja J=1, . . .5 "MOD 1 and the inputs are so allocated amongst the industries that the output of no one industry may be increased without decreasing the output of some one industry and no one input may be decreased without increasing another input. It is said to be non-joint in outputs if there exist individual factor requirements functions X. 1 91(Y113 YiZ’ YT3)’ i=1, - - - 5 (13) such that t(Y,X) = 0, if and only if, Xi = 9i and 1Y1.J.=YJ.,j=l, . . . 3, "NOT 1 and the outputs are so allocated that no input may be diminished without increasing the input of some one joint production process. To show that a technology is nonjoint, we must exhibit the individual functions f1 = 91 and show that they meet both of these 17 requirements. Although we have the natural definition of fir 17Note that nonjointness requires only that the fi exist as functions: there is no requirement that there be physically separate processes producing the various outputs, Y1. Thus the observation that more than one output is produced in the same plant is not sufficient to rule out nonjointness. 60 nonjointness, there is no obvious way to translate this definition into an econometric restriction that can be imposed on a more general specification of the technology. Since necessary and sufficient conditions for a transformation function to represent a non-joint technology are not particularly helpful in the specification of functional forms for econometric analysis, Hall (1972) has approached the problem using the joint cost function and Lau (1972) has ap- proached the problem using the profit function. However, the details of this problem seem to be beyond the scope of the current study and here we briefly review an alternative characterization of non-jointness in terms of the joint cost function, suggested by Hall Theorem 1. (Hall): A necessary and sufficient condition for nonjointness is that the total cost of producing all outputs be the sum of the costs of producing each separately: C(Y, w) = y1 o(])(w) + ...... + Yn 6(")(w) (14) where ¢(i)(W) is the cost of producing a unit of output i. If the technology is nonjoint, the marginal cost of each output is independent of the level of any output. Lastly in the case of the separability of technology, the ratios of the marginal costs depend only on the output mix, while with nonjointness, marginal costs are independent of the output mix. This subject that the overlap between the two restrictions is very small. Hall (1973) proved the following theorem: 61 Theorem 2 (Hall): No multiple-output technology with constant returns to scale can be both separable and nonjoint. That is, the individual production functions in such a technology are identical except for a scalar multiple, implying that there is effectively only a single kind of output.18 Hence nontrivial separable technologies are inherently joint, and their use in empirical work forecloses investigation of the hypothesis of nonjointness. A.2.3. Functional Forms in Trend 3.1. Functional Forms of the Production Possibility Frontier In the specification of a technology with a multiproduct production process, the simplist procedure is to aggregate outputs of various products. The difficulty in such an approach is that the aggregate production function is not a single valued function, and its parameters depend on the composition of output, which in turn depends on, among other things, the prices in question. This difficulty can be avoided by working on a lower level of aggregation where outputs are not combined. Recent progress in the specification of this multiproduct production technology has been achieved in two distinct directions. 18See Hall (1973) on the proof of this impossibility theorem for separable nonjoint technologies, pp. 885-886. 62 The first approach deals with the production possibility frontier-- under certain restrictive assumptions--and the second contribution focusses on the further developments and various applications of the duality theory. The first approach was originally proposed by Mundlak (1963). He suggested the estimation of the production possibility frontier giving an implicit relation between a vector of outputs, say Y and a vector of total inputs, say X. In general, a production possibility frontier can be defined in terms of a transformation function: t(Y, X) = O (15) In the absence of further restrictions, this formulation of a technology permits arbitrary kinds of interaction between total factor intensities and the trade-off between various types of output. Mundlak introduced a substantive restriction on the form of the transformation function: he assumed that it can be written in the following way: i.e. W. X) = H(Y) - G(X) = 0 (16) Specifically he suggested a transcendental function which forms a generalization of a Cobb-Douglas production function; t(Y,X) = Y101Y202931Y1+82Y2 _ X1Y1X2Y2e51X1+62x2 = 0 (17) 01 O 63 This separability restriction between inputs and outputs has a number of implications. First, separability almost always means that outputs are produced jointly. The only case in which the production structure of a multiproduct firm can be portrayed by separate production functions for each kind of output is the case where all the production functions are identical. Second, separability implies that output price ratios or marginal rates of transformation are independent of factor intensities or factor prices. This rather undesirable property makes it apparent that a specification of joint production with the separability constraint is no more general, in at least this crucial respect, than the specification of a uniproduct technology. Following Mundlak's transcendental multiproduct production function, a generalization of a Cobb-Douglas production function, Powell and Gruen (1968) derived the family of constant elasticity of transformation (CET) production possibility schedules which turn out to be algebraically identical to the CES isoquants, apart from one difference of sign determining their concavity. Measuring the basic shape of the frontier of production possibilities by the elasticity of transformation between products 1 and 2 as follows: 31.1.) 3Y2 dig-if) ( ii“) Y1 d(-7-2- ( T12 ’ 64 his derived CET function has the functional form of Y (I‘k) 1 + A 129'") = B(_l-k) (19) where k =-¥L— , A = Ck T12 > O, and B and C are constants. This expression is nothing but a general mathematical expression of an ellipse. They further showed that a given, constant elasticity of transformation is compatible with product-neutral and product- biased shifts in the location of the frontier, pointing out that the CET model therefore is of potential value in the analysis of technical change. Further extension of the CET functional form was done again by Mundlak and Razin (1971), applying a n-factor generalization of the CES function of the form presented by Sato (1967) into the specification of a multiproduct technology, called a nested multi- stage multiproduct production functions. "Let there be A products. The output of the a product is denoted by Ad. In the first stage of aggregation (stage a), the A products are grouped into 8 disjoint and exhaustive groups. A function b is defined on each of these 8 groups. The 8 function bB are grouped into I disjoint and exhaustive groups, and new functions cY are defined on these groups. This process continues until a final aggregate function results. So we get l/o stage Alpha b = [: 2 A apé] 8, B=l, . . . 8 (20) , v=l, . . . r (21) 91 stage Beta c = )3 B b Y BECY I.19 But their main contributions in the production studies concern the problem of index numbers in terms of an appropriate aggregation scheme in the measurement of technical change. Hence either single- or multi-stage multiproduct production functions are still conditioned by a severe restriction, i.e. separability, as in the previous Mundlak specification. Also a functional form without explicit separability between inputs and outputs was suggested by Mundlak and Razin (1971). A simple representation for the two-output two-input case is o o 1/0 _ 6 6 1/6 [a] Y1 + 02(k)Y2 J - [31Xl + 82X2 ] (22) where k = X1/X2. The right-hand side of (22) is the usual C.E.S.-- like formulation for the factor side and the left-hand side is a similar formulation for the product side. But since “2 is written as a function of the factor ratio k, the dependence of the trans- formation curve on the factor ratio is explicitly introduced, rejecting an explicit separability between inputs and outputs. But the most comprehensive and general representation of the production possibility frontier, not restricted by any g_priori 19See Mundlak and Razin (1971), pp. 493-494. 66 assumptions such as separability, homogeneity, etc. was recently developed by Christensen, Jorgenson and Lau (1973). Their trans- cendental logarithmic production frontier is represented by a function that is quadratic in the logarithms of the quantities of inputs and outputs.20 This function provides a local second-order approximation to any production frontier. The resulting frontiers permit a greater variety of substitution and transformation patterns than based on the CET-CES. Its functional form of five inputs and three outputs can be shown as follows: 5 3 1nF=oL+Zo¢.1nX.+ZB.1nY. ° i=1 1 1 i=1 3 3 5 5 5 3 + .. 1 X. l . + .. l X. 1 . iEl jEl 713 n 1 n XJ iEl jfl ETJ n 1 n YJ 3 3 + )3 Z 6.. 1n Y. ln Y. (23) i=1 i=1 ‘3 l 3 where F = t(Y, X) = O. (24) 20They also developed the specification of a price possibility frontier, based on a complete model of production with a production possibility frontier and with necessary conditions for producer equilibrium under constant returns to scale with the existence of prices consistent with zero profits. 67 3.2. Functional Forms Flexible in Prices The second approach to the specification of production process with several kinds of outputs was contributed by Diwert (1971) who generated a functional form which is linear in parameters and which provides a second-order approximation to any arbitrary twice differentiable function. His Generalized Leontief (GL) functional form was quickly followed by the translog (TL) func- tional form of the price possibility frontier developed by Christensen, Jorgenson and Lau (1971) and Sargan (1971) for the multi-input, multi-output technology. 5 3 1n H(H, P) = do + .E ai ln Wi + .E Bj 1n Pj 1-1 j—l 5 5 5 3 +2 Zy..an.an.+z Ze..an.lnP. 1:] i=1 13 1 i=1 j:] 13 1 J (25) 3 3 + .. l P. 1 P. 1:13;613 n ‘ n 3 where ”i = price of the i-th input Pj = price of the j-th output Also a Hybrid Diwert joint cost function (HD) was defined by Hall (1973), the functional form of which is, 68 5 5 3 3 1,1: = z )3 z: .. . . c( ) i=1 2=1 i=1 k2] Aka/YJ (IO/w1 "2 (26) And Jorgenson et a1. (1970) also defined a translog cost function (TC), later extended more by Brown et a1. (1975). The TC form is, 5 3 Y, = + 2 . l w. + . l . ln C( W) a0 i=1 011 n 1 jg] BJ n YJ 5 5 5 3 + z z 7.. 1n W. 1n W. + X Z 5.. 1n w. ln Y. 1:] j=1 13 1 i=1 j:] 13 1 J (27) 3 3 + Z Z 6.. 1n Y. 1n Y. i=1 i=1 ‘3 l J where Yij = in and 6ik = ski A.2.4. The Translog Generalization to the Multiproduct Situation 4.1. Extension of the Translog Transformation Function to Multiple Outputs and Multiple Inputs 4.1.1. Introduction The translog function has an added advantage that it can be generalized to the case of multiple outputs in a straight forward manner. The general transformation function for a multi-output and 21 multi-input technology may be written as F(Y, X) = l, where Y and 2lIt is clear that F is unique only up to a monotonic transfbrmation f such that f(1) = l. 69 X are vectors of outputs and inputs respectively. As usual, we approximate 1n F by a second order Taylor series expansion in 1n X and 1n Y. Thus the translog transformation function for 3 outputs and 5 inputs can be written; 5 3 lnF=01+Zoi.lnX.+2a.lnY. 5 5 5 3 +15 2 z y..ln X. lnX.+ Z Z e..ln X. 1n Y. (28) i=1 j=1 ‘3 1 i=1 j=1 lJ l J 3 3 + .. . . 5 1:1 3:1 513 ln Y1 1n YJ where 5ij = Gji , Eij = eji and Yij = in Similar to the single output case, the translog function provides a second order approximation to an arbitrary transformation function at a specified set of values of Y and X, particularly near unity. 4.1.2. Properties of the Translog Transformation Function (1) Monotonicity condition. The corresponding monotonicity conditions are, subject to the convention that 38; < 0, i=1, . . . 5, are 1 3 y..1nX.+ Z 1 ‘3 J j=1 $2 + IIMU'I .. . , '= , . . . 5 2 j e1JlnYJ;O1l (9) 70 5 3 Bi + .: Eji 1n Xj + .2 5ij 1n Yj=g O, 1=1, . . . 3 (30) J l J 1 In part1cular, ai i=0 , 1=1, 5 (31) 8 3.0 , i=1, 3 As in the single-output case these monotonicity conditions cannot be globally satisfied. Hence we have uneconomic regions for the transformation function.22 (2) Convexity condition Convexity conditions require, with our sign convention, that [Fij] be positive semi-definite. This implies, in particular, at X = Y = [1], that the following matrix be positive semi- definite,23 (01'1)01+Y11 ' ‘ ‘ 'alaSTYis I G181+€11 ° ‘ ' a183+€13 . . l . : a501+Y51 ' ' ' ' (05")05+Y55 0581+€51 ° ' ' a583+€53 8101+€11 ° ' ' ' Bla5+€15 (31'1)81+511. . . 8183+513 . . I . . 8301+€31 . . . . 8305+€35 I 8381+531 ° ° '(Ba-I)83+533 TL. 22 Moreover, the possibility of an input becoming an output or vice versa is allowed by the translog transformation function, the switch occurring when the monotonicity condition is reversed for that particular commodity. This gives a great deal of flexi- bility in the analysis of inherently joint production processes. 23All these conditions =may be tested for each observed value of Y and X and at X= (l) in empirical analysis. 71 (3) Homogeneity condition Imposition of the assumption of "almost homogeneity" of degree k of the transformation function leads to another set of restrictions, which in terms of the parameters of the translog function, implies 5 3 5 5 2 a.1n A + k z 81. ln i + is z 1: y..(lnx.1n).+1nx.1n)() i=1 ‘ i=1 i=1 i=1 ‘3 l J (32) 5 3 3 3 + Z Z e..(lnX.1nX+lnY.lnX)+klmc ccm .mmcox mo owpnanmm .ucmom m:P==m_a owsocoom .HH mmmcmm .cowpmowmwmmmpu xgumzucH ucmucmum mmcox "mumzom mama Pomum _mmpm xoppm mo mummzm Pmmum aoFFN mo mace: mmuymscwcm cmozcmvca cwuuwcx .opw.m3mguw .mum>wm mp3: ucm mu—om cmsoz com cmmzpoom cmsummb cos sow cmmzuoom conumwm cmwcppmem new game co mcpcceu muwscw vmccmu mmpampmmo> umccmo msmz Nomopnm _pNopmm mpmopnm Popoemm Fommpmm mama—mm Foompmm Noooemm poooemm Popeppm —o—mppm Nopmppm muou Nem_o-N ume N - »N_eoseou Lena: N - suaeoesoo cone: F - successou Lone: NoFNm ">Nemaozfi ozHoNoz _ - saweoesoo Noam: Foemm "mmhemacfimm No NN=NNmhmzozH whozooma 3mmum N - speeossou town: F - Numeerou tone: oocmm N>mpm=oza mmhmsozm wszze ee eememee Neemcee emece>e egg memes Ngexcez we geese: mshe A Nee.eoNv ANN.N v NNN.N v NN_.N v ANN.N v ANN.NNV ANN.¢NV Noe.NNv NNN.N V Ngemsece NNN N. N N N_ NN 4N Ne NN NNNNNOZ Noe.ee_v Noo.e V ANN._ V ANN._ v ANN.N v NNN.N v NNo.NNV ANN.NNV Nee.Nev NNNNNNNNNN NNN o e e NN eN NN NN NNN Neo.eeNv Nee.e V NON.N V NON.N v ANN.N v ANN.N N NNe.eNv ANN.NNN ANN.NNV Namzceece Na. 0 e N N eN Ne NN NN eNNNch ANN.NNNN ANN.N v ANN._ v ANN.N v ANN.N v ANN.N v ANN.NNN ANN._NV ANN.NNV Neeseoaa NNN a N N N N NN NN Ne sageN Nee.eeNv ANN.N v Nee._ v ANN._ v ANN.N v ANN.N v ANN.N V NNo.N_v Nee.NNv amazeooa NNN N N a N N _N NN NNN Legumes Nee.eeNv ANN.N v ANN.N V ANN.N v ANN.NNV Nee.eNv ANN.NNV ANN.N v ANN.NNV Neemsece NN N N N N. NN NN N NN checae urameecfi NNNON aco:-eeN NNN-NNN NNN-eeN NN-ee_ NN-NN NN-NN NN-NN N-N ameaxeoz No .oz N N N N e N N N NNNN .ANNNegueeNee cw emeeeeegee.NaeeE;NN_eNeNe me .ez Newezv Nuemscmppeepmm we :eNueewcumNo eNNm ezN--.m-HH m4m Nev «amaze mmesm an twee—eepeue .Nezeeemee; ace.— eN easemeos .NuNeeeee Nezeeemce; Ne eoeN>Ne Aeoeee espe> Nev Neeuee NNeNm Ne eeuepaepeuu .ueme;N__eeuNe :e em whee mcvxcez we geese: ouncese Npguees an» N: eeNNe_up:e NesNueNeee me guess: we» Np mospuegwee Novices we geese: ecu .opeEexo Lem .mcexsez Novices we geese: use an emep>Ne Aeoeee ospe> Nev venuee mecca Ne easemeeze .meewuepsoe eceeeeum meweeeemoccee ecu wee Nemeseeecee :_ ewes» ecu acumeecN gene :Nsupz Nueee:NN~eeume :N :e: cos.— e. easemeee mueeeece Levee» Ne Nemegese N—zueee use use «New» on» cw exegm Nuceevw mghe ’m ANN.N V NNe.N N Ne_.N V Ne¢._ V NNN._ N Nee.N V NN.N NN.N NN.N NN.N NN.N NN.N exeoeN NNNNNNN Na: ANN.N e NeN.N . NNe.N V ANN.e V ANN.N V NNN.N V NN.N NN.N NN.N NN.e NN.N NN.N eueeseNseN Nesea ANN.NNV ANN.NNV ANN.NNN ANN._NV NNN.N_V ANN.NNN NN.NN NN.NN NN.NN NN.N. NN.N. NN.NN NNa>NNNNNNN=Nse< ANN.eNV .Ne.N_V ANN.N V NNN.N V NN¢.N V NN_.NNV NN.N NN.N NN.N NN.N NN.N NN.N NuasNNNNoNe NNNNN N=N<> eN NNNNNNN NNN:-_N ANN.NNV NNe.eNv NNe.N V NNN.N v NeN.N N ANN.NNV NN.N NN.N NN.N NN.N NN.N NN.N exceeN NNNNNNN Naz ANN.N V NNN.N N ANN.N V .eN.e . ANN.N V NNN.NNN NN.N NN.N NN.N NN.N NN.N NN.N eeeusNNNNN Nazca NNN.Ne_V ANN.NNV NNe.eee N_e.¢Nv NNN..N. ANN.¢NV NN.NN NN.NN NN.NN NN.NN NN.NN NN.NN NaosNNNNNNNNNEeN ANN.NNV ANN.NNN NNN.N v NNN.N v NNe.N N ANN.¢NV NN.N. NN.NN NN.N NN.N NN.N NN.N. NNNNNNNNNNN NNNNNN NNeNe eN NeNNNNN =N_:-_ auamaec~ Neuueeewcm we Newsceeez Naeeeece Neezueem Nuumeee~ mueeeece Levee; nevepez eceueekeeez eeuumcx recon sesame; ocNeeeu Nxcumzecm .Neoz eee._ ”NNNNN .Neemseee Na Nauseoca NONUNN--..-N_N NNNNN 123 In the industrial comparison of average factor product, the relatively higher average products for labor and capital inputs are revealed in the canning industry, manufacture of briquettes, and the molding industry. On the other hand, the leather footwear industry and the screw product industry show relatively lower average products for both inputs. The manufacture of knitted underwear shows relatively lower product for labor input but higher product for capital input of horsepower (see 2.51 in the 4th column of the third row in Table 111-1). Average factor products with respect to the industry's value added reveals, in general, the similar phenomena as mentioned above. 6.2. Factor Use Ratios The factor use ratio is focused especially upon the relation- ships between capital and labor inputs by industry. Two proxies are chosen separately for the capital input, power equipment and net 18 capital stock. As shown in Table III-2, the highest capital-labor industries seems to be closely related to the low level of multiple correlation coefficients in the input share equations to be esti- mated. See the discussion on the size of error sums of squares in the next chapter. 18The dispersion of these factor use ratios in the distri- bution of the sample establishments shows the highest coefficient of variation, for example, 6.21 in the horsepower-total worker ratio in the canning industry, while the leather footwear industry also shows a high coefficient of variation, 5.32. On average all the factor use ratios evaluated here, except the horsepower-worker ratio in the screw products industry, have a degree of relative dispersion greater than unity. 124 .eez oco._ me m=Ne> one :N NP xeeum Neuweee umze .Aezv Nezeemmce; ooo.~ :N emcemeee N? 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NN.N NN.NNN NNNoNNNazeNaNNON Ngumeeeu NeupeeeNNm Ne Neezcmeez Nueeeece Neezueeu ngmeeeH Newuem cheNez eceueemeeez eeeumex zeNem Nesuee4 merceeo "acumeeeH L e.»gpm=ee~ an mewuem em: Neaeem--.~-HHH NNNwe ece :ewueeeece cw emu: N—ewgeues see we mpeeese emezecee use an eouezwe>e mw Nweweeoee see we mewse egww .Nuepee ooo.oco.w New easemees mw easemceu Neew we eewee egwe .ce: ooo.~ cw xeeum weuweee we: eee e: ooo.~ cw aeoseweee Nezee ..e.w .Nueeew neweeeemegcee use we eoew>we useew Neuweee :e magnum; as» we eeueweewee wee pee .eewu -wewwee pewgpm ewes» cw xeeum _euweee we: we eee aceseweee Nezee we Neewge use we: ege emeswe .emee zpwee cw easemeoe use emezwe .emee Npgaees cw mgexce: meweeeemeceee use we geese: use an eeew>we .mgexgez eswuecamwewsee Lew Newgewem eee mgexgez eswuegeee New meme: use he eeuepeeweue .Neewuew>ee egeeceum meweceemeseee ewes» use memenucegee cw ewes» eee asymeeew seem ewguws Necesgmwweeame we eowce Neueew we Nemegose men use exegm Neeamww egwe ANN.NNV NNN.NNNV ANN.N V NNN.N V NNN.NNNV ANN.NNV NN.NN. NN.N NN.N NN.NN NN.NN NN.NN NN.NNNNNNz gem ANN.NNNV ANN.NNV NNN.NNV ANN.NNV ANN.NNV ANN.N_V NN.NN NN.NN NN.NN NN.NN NN.NN NN.N NNNNN NNN.N V NNe.N V NNN.— V Nee.N V NNN._ V ANN.N V NN.N NN.N NN.N NN.N NN.N NN.N exuoem Neeweee Nez ANN.N V ANN.N V ANN.N V ANN.e V Nee.N V ANN.N V NN.N NN.N NN.N NN.N NN.N NN.N eeeaersaN Nazca ANN.N V ANN.N V NN¢.e V ANN.e V ANN.N V Nee.N V umNNN-=N= NN.N NN.N NN.N NN.N NN.N Ne._ \mosweeaemwewse< NN¢.e V ANN.N V NNN.e V NNe.e V ANN.N V ANN.e V NN.N NN.N NN.N NN.N NN.N NN.N eNNee-:N=\Ne>weeeeNe NNN.NNV NeN.NNV ANN.NNV NNN.¢NV ANN.NNV ANN.NNV NN.NN NN.NN NN.NN NN.NN NN.NN NN.NN amasweeaemw=N5e< ANN.N V NNN.N V NNN.N V ANN.N_V NN¢.N V ANN.NNV NN.NN NN.NN NN.NN NN.NN NN.NN NN.NN amasweeeoae Ngumeeew Neuueeewgm we Neezweeee Nueeeega Neezueem Nuumaeew Neueem aewepe: eceueeweeez eeuaweg :ewum segueee meweeeu "Newmaemw .Nco: eee.. NN.NNV eNgemseee Na Naewea NONUNN--.N-NNN NNN Neuew a .Neewuew>me eweeeeum meweeeemeegee egg wee memecucewee cw omega ece Ngumeeew :eee cwguwz Nueesgmwweeume we .emeaceeNee cw .Neeenm Neueew we Nemegm>e ecu wee czegm megemww mswe ANN.NNV ANN.NNV ANN.NNV ANN.NNV NNN.N_V NNN.NNV NN.NN NN.NN NN.NN NN.NN NN.NN NN.NN mecseaN NNNNNNN ANN.N V NNe.e_V NNN.N V ANN.N V NNN.N V ANN.N V NN.N NN.NN NN.N NN.NN NN.NN NN.N maNNNNNN ANN.NNV ANN.NNV NNN.N_V ANN.NNV ANN.N_V ANN.NNV NN.NN NN.NN NN.NN NN.NN NN.NN NN.NN meme: NNNNN N=N<> ow NeNNNNN INN: ANN.NNV ANN.NNV ANN.NNV ANN.N_V ANN.NNV ANN.NNV NN.NN NN.NN NN.NN NN.NN NN.NN NN.NN NNNNeNeNz zNN ANN.N V ANN.N V ANN.N V Nee.N V ANN.N V ANN.N V NN.N NN.N NN.N NN.N NN.N NN.N coNNNENNNON Nesa NNN.N_V ANN.NNV ANN.N_V ANN.N.V NNN._NV ANN.NNV NN.NN NN.NN NN.NN NN.NN NN.NN NN.NN Nemee< NN.NN Neeow ANN.NNV ANN.¢NV ANN.N_V ANN.NNV ANN.N_V ANN.NNV NN.NN NN.NN NN.NN NN.NN NN.NN NN.NN NetsemN Neewaee NNe.N V N_e.N V NN¢.N V NNN.N V ANN.N V ANN.N V NN.N NN.N NN.N NN.N NN.N NN.N NNNNNNNN ANN.N V ANN.N V ANN.N V ANN.NNV ANN.N V NN4.N V NN.NN NN.N NN.NN NN.NN NN.NN NN.NN meme: NguNeeew monumeewgm we Leezgeece Npeeeeee Neezpeem ngmeecw Neeesm Levee; mcwe—ez egepeeweeez emuuwcx 3eNem gecueee mcweceu Nxcumeecw 1r .NN NewceV NNNNNNeNN Na Neaeem Noeeea--.e-eee NNNNN 130 negligible, far less than 10% of total production, where the highest 7.76% is revealed in the molding industry, and the next 4.22% is in the screw products industry (see the fifth row in Table III-4). The share of raw materials in total production varies from industry to industry, the highest 71.48% in the manufacture of briquettes and the lowest 44.08% in the screw products industry. This implies a strong suspicion on one of the conventional hypothe- ses that the elasticity of substitution between value added and raw material is zero, raw materials being used in fixed proportion to output, i.e., M = ox, M is the amounts of raw materials used, X is the amounts of total production, and a is a fixed coefficient.2] This suspicion becomes more evident when the distribution of the average shares of raw materials used by firm size within each industry, are referred to (see Appendix B-II, Table II-l through Table II-6). That is, more significant variations in the average share of raw materials used in total production are revealed within each industry, as the size of establishments varies. The value added ratios, defined as the ratio of value added to gross output, consequently vary from industry to industry with a significant variation, even among these six manufacturings. The highest ratio of 0.517 is revealed in the screw products industry and the lowest of 0.271 is in the manufacture of briquettes. Conse- quently, this simple observation suggests that the hypotheses on 2ISee the previous discussions on the role of raw materials in the production theory in section 8.2.3. 131 the elasticity of substitution between value added and raw materials (which are assumed either zero or infinite conventionslly in most production studies of the value added approach) should be a real question of empirical investigation. The shares of labor and capital also show quite a significant variation among our six industries. And in general, as noted above, the relatively higher capital returns are revealed in those industries of canning, briquettes, and molding, which show relatively higher factor products and have higher capital-labor ratios. 6.5. Summary In summary, the characteristics of the industries to be studied are such that the canning industry, the screw product industry, the manufacture of briquettes, and the molding industry show, relatively to the other three industries selected here, higher levels of factor products for labor and capital inputs and have higher capital-labor ratios. And relatively high level of factor remunerations for labor and capital inputs are paid up in these industries and also distributive shares in value added are more favorable for the capital input than for the labor inputs. One important observation on the relation between value added and raw materials is added. That is, there exist quite a significant variations in the cost share of raw materials by the size of establishments within an industry, where there are no sound evidence for assuming any a priori_role of raw materials in the study of production technology, based on the cross-section data. 132 8.2.7. Quality of Data: Some General Considerations In general, two major measurement problems plague more or less all empirical studies. First, the correspondence of the main variables used in the study to the presumably "correct" measures of these variables and second, the reliability of the information provided by the census about the components from which the variables were actually constructed. 7.1. Aggregation Most of our variables are still aggregated although correct 22 In the measurement depends very much on correct aggregation. aggregations of miscellaneous products and raw material inputs to form the respective quantity index, we used the conventional quantity indexing method which inevitably results in the so-called index number problem. However, these aggregations in this study exist not because of any theoretical or practical restrictions on constructing a multi-product production function or a "correct” aggregation but simply as a matter of convenience in empirical estimation with a large number of variables. Regarding remuneration of administratives including working proprietors and unpaid family workers, the use of average salary levels for the third category of labor may bias cost shares either upward or downward, presumably downward when we refer to the common 22But that is easier said than done, in most empirical works. 133 practices in most Korean businesses. Nevertheless, the degree of the bias and its direction cannot be stated a_prjgri_unless very specific evidence is available. The measure of operatives and administratives does not allow for the differences in the quality of labor among establishments. There is no information available about educational, occupational, or skill levels that would make it possible to adjust for such nonhomogeneous factors. There are also variations in efficiency of labor between regions and/or different firm sizes, but in the present study it is simply assumed that all these differentials are equally well reflected in the respective variations of wages and salaries. Conceptually, the replacement value, not the book value, of capital stock evaluated at market prices should be a good measure to use in our context, since it reflects both the quantity and quality of components. It does not, and should not, reflect the capitalized value of monopoly, location, or other sources of rent. Hence, we may escape Friedman's (1955) criticism against capital accounting measures which imply constant returns to scale 23 But this by capitalizing all rents into the value of capital. capital stock measures is still based on what an establishment owned, not used during the production process. In short, there were no data available to adjust for capital utilization and furthermore there were many establishments with no capital stock 23See Grilliches and Ringstadt (1971), pp. 59. 134 values available in the census, as shown in the previous section under exclusion rules. As an alternative to capital inputs, power equipment was measured and aggregated in a common physical unit (HP). One potential source of measurement error is that the capacity of power equipment is recorded by the horsepower figures shown on the labels or specifications neglecting any present physical and/ or economic obsolescence. Lastly, various fuels are aggregated into a rather perfect dimensional unit, say kilocalories of heating value, which allows comparison between ingredients of coal, oil and electricity.24 7.2. Modifications of Some Unclassified or Miscellaneous Factors There are several cost factors of which only amounts are available such as water purchased, payments for contract work on raw materials. and the cost of maintenance of the production facilities. In the current works, the costs of water are counted as a part of the cost of fuel consumption, the payments for contract work as raw material costs, and the maintenance costs as capital services.25 As shown in Table IV-l, although their portions are 24Here the only possible doubt may reside in the significant variations in the conversion coefficients for each type of energy. See footnote 11 in this chapter. 25These share adjustments seem to be rather appropriate, even if there may be some other alternative way to handle them more appropriately. 135 .meewuew>ee eweeeeem Nwegp eNe Nemesueegee cw emesp eee ngmeecw seem New NNV Nemeueeegee esp :w execm use Negemww ezwe useueo Nmecw NNN.NV NNN.NV NNN.NV NNN.NV NNN.NV NNN.NV e» V NN.N NN.N NN.N NN.N NN.N NN.N Nxcoz Neeeecoe + Nepez + Neweeemv . Npewgepez :em NNN.NV NNN.NV NNN.NV NNN.NV NNN.NV NN.NV N V NN.N NN.N NN.N NN.N NN.N NN.N Nee: Neeeeeoe eeEzmceu wee; NNN.¢NV NNN.NNNV NNN.NV NNN.NV NNN.NNV NNN.NNV N V NN.N NN.NN NN.N NN.N NN.N NN.NN eemmeeesa twee: NeeeN NNNNNNN emz NNN.NNV NNe.eNV NNN.NNV NNN.NNV NNN.NNV NN.NNV N V NN.N NN.N NN.N NN.N NN.N NN.N NNNNNNN Neeueo mmegw NNN.NV NNN.NV NN.NV NNN.NNV NNo.NV NN.NV N V NN.N NN.N NN.N NN.N NN.N NN.N asceseN NeNNe - . . Nueeueo mmecw NNN NV NNN.NV NNN.NV NNN.NV NeN.NV NNN.NV N V NN.N NN.N- NN.N NN.N NN.N NN.N NNNNNONN-=N-NN03 ngmeecN Nepueeewgm we geezeeeee meeeeece Neezueew NyemeeeN Neepeeu mewepez egaeeewecez empuwcx sewem NesueeN newceeu "NeumeecN meeecepweemwz e.xgamzecN an NNV Neewuew>eo Newsweemeezuu._i>u u4m5 l with its covariance matrix as, V(fi) = [c - CR'(RCR')'1 RC]. (24) asymptotically efficient. See also Kment (1971) on the same asymptotic properties of the two-stage Aitken estimator as the maximum likelihood estimator. 24.See Theil (1971), p. 308 and also see Wallace and Anderson. 155 In the actual estimation, a consistent estimation of 2, S, also is replaced for X in (23). 4.2. Some Properties in the System of the Share Equations 4.2.1. The Case of Identical Explanatory Variables As already noted in (17), we have the system of the share equations with the same explanatory variables. Therefore, the following relations hold; c =[x'12 911'1x1'1=[(1 e 1'112'1911119 x11" = [2'1 9 (X'X)]‘1 = [z 9 (2'11'11. and = [2 911'11’11191'112'1 9 I)Y = [1901)" XJY. (25) :1) That is, the fi vector becomes identical to the Ordinary Least Squares (OLS) estimates. 4.2.2. Restrictions Implied in the Estimates An intrinsic property of the share equations system, such that the sum of the factor's share is unity, ll MG) 01 II S. = 1.0 and 1.0 (26) i=1 i 156 implies the equivalent restrictions to linear homogeneity (to be discussed later) on the estimates of the system parameters such that:25 5 8 5 Z a. = -1 0, 2 B. = 1 0, 2 y.. = O 0, i=1 ' i=6 ' 1=1 '3 (27) 5 8 8 .. = . , .. = . , d .. = . . 1:1 613 O O iEG E13 0 0 an iE6 513 O 0 Here (26) implies that the disturbance terms in (15) must sum identically to zero. Thus the covariance matrix of the distur- bance terms must be singular, and the systems estimation method of the GLS will not be operational. This difficulty has generally been overcome by deleting one of the share equations from the estimation procedure and by iterating the so-called Zellner procedure, provided that the parameter estimates with converging iteration are independent of which share equation is deleted.26 But as already discussed in the previous section, 4.2.1., regarding the joint GLS estimation with identical explanatory 25The deviation of these conditions is shown in the Appendix A-II. 26Barten (1969, pp. 24-25) has shown that maximum likelihood parameter estimates of a system such as the one being considered are independent of which equation is deleted. Kmenta and Gilbert (1968) have shown that if one iterates the Zellner procedure, the parameter estimates (if they converge) will converge to maximum likelihood values. See Berndt and Christensen (1973) for further discussion of estimation procedures for translog share equations. 157 variables, the fact that the estimates of the joint GLS are equiva- lently identical to those of the OLS, allows us to get consistent estimates of the covariance matrix, S, from that of the OLS covari- ance matrix. This implies that the estimates of parameters in the unrestricted as well as the restricted cases are independent of the equation(s) deleted.27 Thus in the present study the two-stage Aitken estimation method is used in the place for the Iterative Zellner Efficient method in order to avoid heavy computational burden at the cost of probably negligible efficiency gains from the iterative procedure.28 4.3. Restrictions Considered In the empirical estimations, we want to test several hypotheses, such as the combination of homogeneity of degree 1 and symmetry conditions, and explicit separability between input and output. 4.3.1. Homogeneity Restrictions The sufficient conditions for homogeneity of degree 1 in the parameters of the share equations can be written as follows: 27Differently from our case, the Berndt and Christensen result (1973) is specific to systems with autoregressive errors. If the errors are not autocorrelated (as in our cross section data, there is no reason to assume the errors are autocorrelated), the results become invariant to the choice of equation deleted. 28No efficiency gains from the iterative procedure are found in the empirical results (see Chapter IV). 158 5 8 kg] Gk -1.0, kEG Bk - 1.0, 5 8 kE} Yik = 0.0, k§6 61k = 0.0, 1:], o o o 5’ (28) 5 8 k2] gjk = 0.0, kE6 Ojk = 0.0: 3:6, . . . 8; Thus the total number of restrictions become 18. But in the formulation of the matrix R, only the number of 12 restrictions are counted in the actual estimating equations system.29 4.3.2. Symmetry Restrictions The symmetry restrictions can be described straight forwardly: i,j = 73,. 1.3 = l. . 5. eij = gji, 1 = 1, . . . 5, j = 6, . . . 8, (29) and 5ij = Gji, 1,j - 6, 5 29Since one of the share equations among inputs and among outputs is deleted respectively in the actual estimating equations system, the first two restrictions on the o1 and 5i are not counted. Furthermore, among the rest four sets of conditions, each one of them is not counted. Hence, 6 restrictions as a total become eliminated in the actual estimation from two deleting equations. 159 The number of the restrictions here becomes 28, but only 15 restrictions are encountered in the matrix R.30 4.3.3. Strong Separability Between Ipputs and Outputs The strong separability conditions are equivalent to assuming that: 13 and (30) £3.1- 0.0 j = 6, . . . 8. The estimation of the system with this restriction can be done separately in the following joint GLS estimation without forming any restriction such as y = RH; Yij 1n XJ- s 1:], . . . 5s (3]) and 8 S. = s. + z 5.. 1n Y. , i=6, . . . 8. (32) 30When there are M inputs and N outputs, the total number of restrictions are counted by the following relation, i.e., %M(M-l) + 5N(N-l) + aMN. But in actual estimation with those deleting equations, it becomes k(M-l)(M-2) + $(N-l)(N-2) + (M-l)(N-l). Therefore in our case it is 15. This is because when we estimate the restricted model we only take (M-l) input equations and (N-l) output equations to estimate jointly. See the section 4.2.2. Also see Berndt & Christensen (1973) and Brown, Caves and Christensen (1975). 160 4.4.1. Likelihood Ratio Test max L The likelihood ratio, A = mgx , depends on the maximum 9 value of the likelihood function for the unrestricted (0) system and that of the likelihood function of the system subject to the restriction (w). The test statistic for each set of restrictions is based on minus twice the logarithm of the likelihood ratio -2 1n A = T[ln [Em] - 1n IZQI] (33) where T is the number of data points, IEN1 is the determinant of the restricted estimate of the covariance matrix, and IEQI is the determinant of the untrestricted estimate. Under the null hypothesis -2 1n A is distributed asymptotically as a chi-square with the degree of freedom equal to the number of restrictions being tested.31 4.4.2. F Test for Small Sample For the hypothesis y = Rn, where r and R have q rows, R has rank q, and H is the parameter vector of the share equations system, two X2 variates can be considered with the M+N-normal variates (Di). one being x2(q) if the null hypothesis is true:32 31Edward (1972) and Seber (1966). 3zThe test statistics is known as the Wald for testing linear restrictions on the coefficients of certain linear models (Wald 1943). 161 (y - Rfir mm" 9 11x1" Ru" (1 - 1211) (34) and the other being or - xfi)' (2" 91) (v - xii). (351 having (M+N)-T-K-(M+N) = 0 (say) degrees of freedom, where K is the number of the unknown parameters in each equation. The statistic is equal to the ratio of (34) and (35) multiplied by D/q, and its distribution is F(q,D) if'y = RH is true. We replaced the unknown 2 by S (and hence D by D) in both (34) and (35). For (34) we thus use: -1 (y - Rfi)‘ {RIX' (5'1 9 I) X]’1 R'} (y - Rfi). (36) Since the quadratic form (34) is continuous in z, and S is a con- sistent estimator, the substitute form (36) converges in distri- bution to (34), so that its limiting distribution is X2(q) under the null hypothesis and the normality condition. By dividing the quadratic form (35) by the number of degrees of freedom (0), we obtain a ratio which converges in probability to 33 Since the form 1 as T (and hence also 0) increases indefinitely. (35) is a continuous function of z, the corresponding fraction %-of the substitute form. 335ee Theil (1971), pp. 143-144, pp. 313-314, pp. 402 -403. 152 (v - xfir (5'1 9 I) (v - xfi) (37) also converges in probability to l. The ratio of (36) and (37) multiplied by D/q has its limiting distribution, (l/q) X2(q)’ under the null hypothesis and the normality condition. Instead of this limiting distribution we use F(q,D). This makes no difference asymptotically, since F(q,D) converges in distribution to (l/q) X2(q) as D +w. For finite D the F approxi- mation is more cautious than the x2 approximation because it gives a negative verdict on the null hypothesis in a smaller number of cases. This cautious attitude is preferable since the procedure implies that the value of the quadratic form (37) nay just as well be replaced by its expectation (0).34 4.4.3. Other Statistics One other statistic, R2, was investigated in the restricted model but in vain. Under the OLS, R2 can be defined: , ..,.. -2 x2 e e _ y y ' Ty =1- 1-1 '7 yy yy-Ty 34Alternative procedures for testing linear restrictions on the coefficients of a linear regression model may produce conflicting decisions. In the cases where conflict among tests arises, the con- flict may be resolved if one test can be shown to be more powerful than the others. But, in practice, economic theory almost invariably suggests a range of alternative hypotheses, i.e., tests of composite rather than simple hypothesis. It is well known that when testing a composite hypothesis there may be no uniformly most powerful test. At present, relatively little is known about the comparative power of various alternative criteria. Some recent researches on this fog;;)have been done. See N. E. Savin (1966) and T. S. Breusch 163 as far as there exists orthogonality between e'e and y'y. But in the restricted model, we can no longer expect this nice property nor use the above definitional relation in order to get the multiple correlation coefficients.35 But in the present work, in spite of all these deficiencies we try to measure the portion of the error sum of squares in a total sum of squares by calculating our substi- 2 tute for R in the restricted model as in (38). Another statistic for testing the significance of Lagrangian multipliers, 1i in (23), is computed for the standard error opri as follows: 52(X,) = [RCR'1;}. (39) where V(X) = [RCR']'1 . Hence the Student-t values for the null hypotheses of X's are calculated, assuming that the Lagrangians are asymptotically 35Although it was not attempted, there is an interesting definition for the case with no constant term in the regression equation, which also does not have the orthogonality property, such that: ~2 = r|h - TE? y'y - 192 ’ where I the number of observations y = the sample mean of the estimated values of y. But this concept, unfortunately, dggs not have any comprghensive correspondence, in the sense that R does not generate R for the case with the constant term. See The Wharton School of Finance and Commerce, University of Pennsylvania (1973). 164 normal, where the degrees of freedom on the Lagrangians is the large sample number, N = 30.36 8.3.5. Monte Carlo Experiments In the connection with the estimating methods it is worth- while to note the two following Monte Carlo experiments,the first done recently by Byron (1977) and the second more specifically designed and simulated in this study. 5.1. The Truncation Error in the TRANSLOG Approximation The TRANSLOG approximation is based on a truncated Taylor series expansion with an excluded and unknown remainder term. That is, using production theory the unknown production function is approximated by a second order Taylor series expansion. Since the production function is monotonic with nonnegative inputs it may be expressed logarithmically, and after appropriate scaling the Taylor series expansion may be taken around zero. This step is intended to minimize higher order terms and the convergencies of the series. The log quadratic Taylor series expansion is linear in the unknown parameters and these parameters, which are simply the derivatives evaluated at zero, enable inferences to be made about the character- istics of the underlying production function. To illustrate, consider any general logarithmic production function with 2 inputs, 36The reason is that the small sample theory has not been worked out and we are assuming the Lagrangians are asymptotically normal, consequently N1; 30. 165 1n y = f[ln X1, ln X2] (40) The quadratic Taylor series approximation has the general form 31], .. _ _ a_z__2 Z Z + 3101 40) + 5(4 901—3¢3¢1(¢’¢o)+ HOT. where (41) Z=1ny,Z o=f[1nXo¢], =lnX,¢,o=lnXo HOT is called "the remainder term" consisting of the third and higher order terms. The derivatives are evaluated at 80 corresponding to X0 = [1]. Thus 2 2 2 =ao + .2 a. 1n X. +'3 z z y. jln X1 1n Xj + HOT, (42) 1=l i= -1 j= —1 In the above, 3 ln y 32 1n y a = Z , a = = o o i ’ Yij a 1n X1.o a 1n X1.o a 1n xjo 37 and HOT can be expressed, due to Cauchy, as follows: 37Due to Lagrange, the another form of the third order term can be written 332 Z Z 2 lnX1 lnX. lnXk, 1 J k31nX1. :11nx‘1.31nxk x=x* J 1 HOT =— 3 6 where lnX* is some number between zero and lnX. 166 332 lnX1 lan lnXk X=X ° (43) Z Z k j k BlnX1 alnXj BlnXk ( where HOT consists of the fourth and the higher order terms. Here it is not possible to make analytic statements about the TRANSLOG approximation because the remainder term (HOT) cannot be expressed with mathematical exactness even with some additional constraints on the production function concerned, unless we can find some relationship between the first, the second, and the higher order terms. The statistical bias in the estimation from this specifi- cation error can be shown as follows by comparing the estimates of the true equation with the present estimating one: The true least squares estimates of B in the equation, y=XB+R+u (44) where R is HOT may be written é= (x'x1'1 x'(.v- R). (45) and the bias is obviously E18) - s = (.X'x1'1 M (46) what does emerge is that the higher the correlation between X and R the larger will be the bias in the least squares estimates of B. It is true, however, that this will be moderated by the magnitude of R and, in particular, by the moment of X'R. Since it is not sufficient 167 to argue that if R is of a small order of magnitude its effect on the bias in the parameter estimates is negligible, the results of the Monte Carlo experiments reported by Byron (1977, p. 18) should be noted, "Direct estimation of the production function was found to be inferior to indirect estimation based on the first order conditions. What emerged in both cases is the indisputable results that the parameter estimates are biased, but not seriously; . . . To the author the translog procedure emerged from these experiments better than anticipated, especially in relation to the estimates of the elasticity of substitution." 5.2. Experiments on the TRANSLOG Approach with the Existence of the Zero-Valued Variables As noticed in the previous section, the closeness of the TRANSLOG approximation depends on the proximity of the quantities of the inputs (X1's) to unity. There is no general limit to the approximation error incurred in such power series expansion. How- ever, as one is free to choose the scaling of the measurement of the inputs, one can minimize the approximation error by setting the sample means of the Xi's at unity.38 38It is worthwhile to note here that the choice of the sample means in the normalization of the variables, such as between the arithmetic and the geometric means, does not affect the coefficients of the first order terms (i.e., constant terms in the share equa- tions). But does it not affect the second order terms of the approximation from the different dispersion (or deviations) under different sample means? 168 Note also that the use of the equations (15) and (16) depends on the quantities of the inputs being strictly positive, otherwise the expression is not well defined. Aside from the case of a uni- product production technology, the micro-reality of a multiproduct technology does, almost everywhere, allow a situation where some sample establishments do produce some of the products produced in the industry but not all of them. Thus the reality of the system construction necessarily bring us the model(s) of (15) and (16) with the zero-valued variables in the cases of more than one product. There have been suggested two general methods available to circumvent this problem: (1) One can insert nonnegative nonhomogeneity parameters and redefine the product such that 21. = x1. + X? (47) (2) One can construct a new variable as a sub-aggregate of several original variables having the form of N éij 21 = jg] Bij ij (48) where xij is the j-th kind of product in the i-th category of 39 products and N is the number of products classifiable into the i-th group of commodity. 39Jorgenson, Christensen, and Lau (1970). 169 In the first method, the xg's, of course, must be either known a pgip£i_or estimated by nonlinear methods and the concept of nonhomogeneity parameters may arise in the theory of consumption as minimum quantities at a subsistence level, measured in the unit (such as calory) of characteristics of commodities. However, in the theory of multiproduct production the corresponding concept seems to be hard to define, except for by-products common to all establishments in the industry. The second method seems to be very useful particularly in the present study, i.e., in constructing a quantity index for the nonmajor commodities.40 In general, we do need _a prio_r_1_ knowledge of 811 and 61j in this aggregation, which will in turn be studied through investigation of a technology. Hence, in the present study we assume 611 = l for the nonmajor commodities.41 Nevertheless, in the reality of a multiproduct technology, this problem is encountered in the variables of the major products for some sample establishments. Allowing the zero-valued variables in the estimating TRANSLOG share equations, by replacing certain negligible small figures, the Monte Carlo experiments here were set up under the statistical 40 41If one tries not to distinguish the major and the nonmajor commodity groups and tries only to avoid zero-valued products for all establishments then almost every industry turns out to be the case of uniproduct technology. See Chapter II on this sub- aggregation of the structure of industry's multiproduct production. Chemical fertilizer is a kind of fertilizer, for instance. 170 assumptions made by the proponents of the translog procedure to examine its performance under their conditions. The TRANSLOG production equation below, 2 2 2 1nV=a+ Za.lnX.+ 23 Zy..lnX.lnX.+e (49) O 1:] 1 1 1=j j=1 13 1 J was based on an underlying CES production function, 1 V — a[51 x] + 62 x2 1 w1th a] + 52 1. (50) . _ a 1n V = a 1n V . . S1nce a1 —-§eifi—if~and 311 81nX1 STan the der1vat1ves for the CES function at 1n X = [l] are a1 u 51, v11 = ppa1(51 - l) and 1U D 51 5° . <51) 1’11 1 Based on the first order conditions, we have the following share equations to be estimated, V a 1n X. 1 2 0 $1 = a1 + .2 Yij 1n Xj + e, 1=1,2 - (54) 3:1 171 To generate data for this model X1 and V are assumed exogeneous and the shares, 51's, are endogeneous. In other words, the systematic part of the shares, based on a CES production function, is a('p/1J) vp/1J x'_|'p , (55) (I) I 1'“51 2 1.152 a('p/U) VD/U X513 . (56) M II Given S1 + $2 = l, the disturbance has to be introduced with the property e1 = -e2. X1 and X2 were first generated as uniformly distributed variables, secondly the zero-valued observations with various frequencies were inserted in X1 and X2 separately and finally they were transformed into random normal variates. Also the generated X1 and X2 were held as fixed regressors in repeated samples and experiments. V was generated using (49). The systematic part of the share equations was then generated using (55) and (56) and the disturbance introduced additively with a pre-specified signal-noise ratio. The TRANSLOG estimates were obtained by applying generalized least squares (GLS) to (54) with the usual singular covariance adjustment as described in the previous section. The characteristics of the CES parameters, the translog parameters at X1 = 1, the exogenous variables and the disturbances were as follows: CES Parameters; a = l, 61 = 0.4, 62 = O.6,c>= -O.5, p = 1. 172 Translog Parameters; 01 = 0.4, 02 = 0.6, Yll = 0.12, Y12 = -O.12, Y22 = 0.12 . Exogenous Variables; Means: X1 = 20000, X2 = 32000. Covariance structure (x 105): 0.9 0.5 T“-9.0 5.0 160. 70. 2Lv = ’ Z14v = ’ zHv = 0.5 1.6 5.0 16.0 70. 360. Disturbances: REN = 0.99, RfiN = 0.95, RfiN = 0.80 . The exogenous variables before scaling were generated with the above means and covariance matrices: LV - low variability, MV = medium variability and HV = high variability. The variance of the disturbances was related to the variance of the systematic part of the dependent variables in order to correspond to the R2 indicated above: again LN = low noise, MN = medium noise, HN = high noise. All the results were based on 50 replications with a sample size of 50. 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A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0- 00000.0- 00000.0- 00000.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 0.0 00000.0- 00000.0- 0.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 0.0 00000.0 00000.0- 0.0 00000.0- 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0- 00000.0- 00000.0- 00000.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0 00000.0 00000.0 00000.0 00000.0 00000.0 0 00000000000m 0000550w 00000000000: 00000000000m 0000550w 00000000000: 000050000 00000000 00000000 00000000 0000 00000000 00000000 0000000000 00000000 0000000000-.0-0-0> 00000 203 00000000 0000 00000000 00000000 0000000000 00000000 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0 00000.0 00000.0 00000.0- 00000.0- 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0 00000.0 00000.0- 00000.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 0.0 00000.0- 00000.0- 0.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 0.0 00000.0 00000.0 0.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0- 00000.0- 00000.0- 00000.0- 00000.0- 00000.0- 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0- 00000.0- 00000.0- 00000.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0 00000.0 00000.0 00000.0- 00000.0- 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0- 00000.0- 00000.0- 00000.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 00000.0- 00000.0- 00000.0- 00000.0 00000.0 00000.0- 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 0.0 00000.0- 00000.0- 0.0 00000.0 00000.0 0 000000000000 00005500 000000000002 000000000000 00005500 00000000000: .000050000 00000000 000000x0 .000000000-.0umu0> m0m 00000 205 'y 00000000 0000 00000000 00000000 0000000000 00000000 A 0.00 A 0.00 A00000.00 A 0.00 A 0.00 A00000.00 00 0.0 00000.0- 00000.0 0.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.0v A00000.00 A00000.00 A00000.00 00 00000.0- 00000.0- 00000.0- 00000.0- 00000.0- 00000.0- 0 A00000.00 A00000.00 A00000.0v A00000.00 A00000.00 A00000.0v 00 00000.0 00000.0 00000.0 00000.0 00000.0 00000.0 0 A00000.0v A00000.00 A00000.0V A00000.0v A00000.00 A00000.00 00 0.0 00000.0- 00000.0 0.0 00000.0- 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.0v 00 0.0 00000.0- 00000.0- 0.0 00000.0 00000.0 0 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 A00000.00 00 0.0 00000.0 00000.0 0.0 00000.0- 00000.0- 0 A00000.0v A00000.0v A00000.0v A00000.0v A00000.00 A00000.0V 00 0.0 00000.0 00000.0- 0.0 00000.0- 00000.0 0 A00000.00 A00000.00 A00000.0V A00000.00 A00000.0v A00000.0v 00 0.0 00000.0 00000.0 0.0 00000.0- 00000.0- 0 A 0.00 A 0.00 A00000.0V A 0.00 A 0.00 A00000.0V 00 0.0 00000.0 00000.0 0.0 00000.0 00000.0- 0 A 0.00 A 0.00 A00000.0V A 0.00 A 0.00 A00000.0V 00 0.0 00000.0- 00000.0 0.0 00000.0- 00000.0- 0 000000000000 00005500 00000000000: 000000000000 00005500 00000000000: 000050000 00000000 00000000 0000000000-.0-0-0> 00000 206 .0-00 00000 00 A00 00000000 000 0000 A 0.00 A 0.00 A00000.00 A 0.00 A 0.00 A00000.00 00 00000.0 00000.0 00000.0 00000.0 00000.0 00000.0 0 A 0.00 A 0.00 A00000.00 A 0.00 A 0.00 A00000.00 00 00000.0- 00000.0- 00000.0- 00000.0- 00000.0- 00000.0- 0 A 0.00 A 0.00 A00000.00 A 0.00 A 0.00 A00000.00 00 0.0 00000.0 00000.0 0.0 00000.0 00000.0- 0 A 0.00 A 0.00 A00000.00 A 0.00 A 0.00 A00000.00 00 0.0 00000.0 00000.0 0.0 00000.0- 00000.0- 0 A 0.00 A 0.00 A00000.00 A 0.00 A 0.00 A00000.00 0.0 00000.0- 00000.0- 0.0 00000.0 00000.0 000 A 0.00 A 0.00 A00000.00 A 0.00 A 0.00 A00000.00 00 0.0 00000.0- 00000.0 0.0 00000.0 00000.0- 0 000000000000 00005500 000000000002 000000000000 00005500 0000000000:: 000050000 00000000 00000000 00000000 0000 00000000 00000000 0000000000 00000000 .000000000-.0-0-0> 00000 207 TABLE VI-4-a.--Parameter Estimates for the Transiog Functionsa-- Manufacture of Knitted Underwear. Trans1og Production Function Trans1og Cost Function Parameter Unrestricted Symmetry Unrestricted Symmetry up 0.12064 0.11602 0.12446 0.11980 (0.01235) (0.00811) (0.00942) (0.00809) “A 0.01617 0.01624 0.02431 0.02273 (0.00262) (0.00207) (0.00236) (0.00206) “K 0.21565 0.22754 0.26835 0.25476 (0.02028) (0.01253) (0.01308) (0.01052) “F 0.02064 0.02157 0.01267 0.01471 (0.00136) (0.00127) (0.00119) (0.00105) “R 0.62691 0.61862 0.57020 0.58801 (0.02219) (0.0 ) (0.01778) (0.0 ) yPP 0.07416 0.06069 0.08612 0.05682 (0.01430) (0.00928) (0.02205) (0.00962) yPA -0.03284 -0.01617 0.00413 -0.01275 (0.01423) (0.00248) (0.01896) (0.00417) YPK -0.00066 -0.00398 -0.02264 -0.03137 (0.00996) (0.00834) (0.00697) (0.00583) YPF -0.00254 -0.00319 -0.01869 -0.00196 (0.00602) (0.00127) (0.00814) (0.00180) YPR -0.03744 -0.03735 -0.00529 -0.01074 (0.00636) (0.00621) (0.00858) (0.00770) YAP -0.01310 -0.01617 -0.00860 -0.01275 (0.00303) (0.00248) (0.00553) (0.00417) yAA 0.02628 0.02532 0.02071 0.01762 (0.00302) (0.00203) (0.00476) (0.00370) YAK —0.00168 -0.00290 -0.00721 -0.00830 (0.00211) (0.00200) (0.00175) (0.00164) YAF -0.00221 -0.00070 0.00254 0.00302 (0.00128) (0.00077) (0.00204) (0.00127) yAR -0.00593 -0.00556 0.00187 0.00042 (0.00135) (0.00132) (0.00215) (0.00193) YKP 0.00199 -0.00398 -0.01246 -0.03137 (0.02349) (0.00834) (0.03061) (0.00583) TABLE VI-4-b.--Continued. 208 Trans1og Production Function Trans1og Cost Function Unrestricted Parameter Unrestricted Symmetry Symmetry YKA 0.02540 -0.00290 0.00493 -0.00830 (0.02337) (0.00200) (0.02633) (0.00164) YKK 0.02984 0.02617 0.06515 0.05605 (0.01636) (0.01168) (0.00968) (0.00766) YKF -0.01280 0.00106 -0.00298 -0.00233 (0.00989) (0.00104) (0.01130) (0.00085) YKR -0.02355 -0.02036 0.00235 -0.01405 (0.01045) (0.00898) (0.01191) (0.00848) YFP -0.00458 -0.00319 -0.00360 -0.00196 (0.00158) (0.00127) (0.00279) (0.00180) YFA -0.00091 -0.00070 -0.00235 0.00302 (0.00157) (0.00077) (0.00240) (0.00127) YFK 0.00033 0.00106 -0.00209 -0.00233 (0.00110) (0.00104) (0.00088) (0.00085) YFF 0.00521 0.00490 -0.00336 -0.00108 (0.00066) (0.00061) (0.00103) (0.00089) YFR -0.00201 -0.00207 0.00057 0.00235 (0.00070) (0.00070) (0.00109) (0.00097) YRP -0.05847 -0.03735 -0.06146 -0.01074 (0.02570) (0.0 ) (0.04161) (0.0 ) YRA -0.01793 -0.00556 -0.02781 0.00042 (0.02557) (0.0 ) (0.03579) (0.0 ) YRK -0.02783 -0.02036 -0.03321 -0.01405 (0.01791) (0.0 ) (0.01316) (0.0 ) YRF 0.01234 -0.00207 0.02249 0.00235 (0.01082) (0.0 ) (0.01537) (0.0 ) YRR 0.06892 0.06534 0.00050 0.02202 (0.01144) (0.0 ) (0.01618) (0.0 ) aSee the footnote (a) in Table VI-1. 209 TABLE VI-5-a.--Parameter Estimates for the Trans1og Functionsa-- Manufacture of Briquettes. Trans1og Production Function Trans1og Cost Function Parameter Unrestricted Symmetry Unrestricted Symmetry 0P 0.03747 0.04134 0.03385 0.03956 (0.00440) (0.00357) (0.00532) (0.00438) “A 0.03096 0.03307 0.02281 0.02271 (0.00216) (0.00175) (0.00284) (0.00241) aK 0.21490 0.17562 0.25783 0.24605 (0.01174) (0.00810) (0.00976) (0.00701) “F 0.01344 0.01322 0.00842 0.01015 (0.00092) (0.00076) (0.00111) (0.00094) aR 0.70323 0.73673 0.67709 0.68154 (0.01160) (0.0 ) (0.01379) (0.0 ) YPP 0.03109 0.04208 0.02652 0.02470 (0.00580) (0.00348) (0.01090) (0.00492) YPA -0.00856 -0.00870 -0.01411 -0.00334 (0.00457) (0.00175) (0.00885) (0.00336) YPK 0.00837 0.00948 -0.02563 -0.02700 (0.00384) (0.00368) (0.00227) (0.00209) YPF 0.00173 0.00039 -0.01130 0.00137 (0.00245) (0.00078) (0.00378) (0.00118) YPR -0.04108 -0.04325 0.00053 0.00426 (0.00318) (0.00291) (0.00584) (0.00491) YAP -0.01233 -0.00890 0.00506 -0.00334 (0.00285) (0.00175) (0.00581) (0.00336) YAA 0.02488 0.02522 0.01021 0.01360 (0.00225) (0.00164) (0.00472) (0.00347) YAK -0.00000 0.00050 -0.00835 -0.00885 (0.00189) (0.00185) (0.00121) (0.00117) YAF 0.00019 0.00034 -0.00191 0.00104 (0.00121) (0.00059) (0.00201) (0.00104) YAR -0.01622 -0.01736 -0.00160 -0.00246 (0.00156) (0.00146) (0.00311) (0.00266) YKP 0.02304 0.00948 -0.01121 -0.02700 (0.01549) (0.00368) (0.02000) (0.00209) TABLE VI-S-b.--Continued. 210 Trans1og Production Function Translog Cost Function Parameter Unrestricted Symmetry Unrestricted Symmetry YKA 0.02847 0.00050 -0.00521 -0.00885 (0.01221) (0.00185) (0.01625) (0.00117) YKK -0.01150 -0.01697 0.05950 0.05689 (0.01026) (0.00794) (0.00416) (0.00380) YKF 0.01043 0.00135 —0.00780 -0.00337 (0.00655) (0.00079) (0.00693) (0.00046) YKR -0.00963 0.00564 0.00158 -0.01767 (0.00849) (0.00652) (0.01072) (0.00509) YFP 0.00006 0.00039 0.00065 0.00137 (0.00122) (0.00078) (0.00228) (0.00118) YFA 0.00071 0.00034 -0.00103 0.00104 (0.00096) (0.00059) (0.00185) (0.00104) YFK 0.00134 0.00135 -0.00329 -0.00337 (0.00081) (0.00079) (0.00047) (0.00046) YFF 0.00403 0.00397 -0.00263 -0.00065 (0.00051) (0.00042) (0.00079) (0.00060) YFR -0.00608 -0.00604 -0.00023 0.00160 (0.00067) (0.00061) (0.00122) (0.00104) YRP -0.04186 -0.04325 -0.02102 0.00426 (0.01530) (0.0 ) (0.02827) (0.0 ) YRA -0.04549 -0.01736 0.01014 -0.00246 (0.01206) (0.0 ) (0.02297) (0.0 ) YRK 0.00179 0.00564 -0.02224 -0.01767 (0.01013) (0.0 ) (0.00588) (0.0 ) YRF -0.01638 -0.00604 0.02363 0.00160 (0.00647) (0.0 ) (0.00980) (0.0 ) YRR 0.07301 0.06100 -0.00028 0.01426 (0.00839) (0.0 ) (0.01515) (0.0 ) aSee the footnote (a) in Tab1e VI-1. 211 TABLE VI-6-a.--Parameter Estimates for the Translog Functionsa-- Mo1ding Industry. Trans1og Production Function Transiog Cost Function Parameter Unrestricted Symmetry Unrestricted Symmetry up 0.06467 0.08149 0.10325 0.10587 (0.00983) (0.00888) (0.1188) (0.01075) “A 0.01937 0.02178 0.02470 0.02416 (0.00323) (0.00305) (0.00417) (0.00382) “K 0.24182 0.24213 0.23530 0.19254 (0.02103) (0.01452) (0.01915) (0.01258) “F 0.06395 0.08153 0.08283 0.07010 (0.00881) (0.00698) (0.00940) (0.00704) “R 0.61019 0.57307 0.55392 0.60733 (0.02294) (0.0 ) (0.02434) (0.0 ) ypp 0.02433 0.03598 0.02430 0.02190 (0.00890) (0.00418) (0.01582) (0.00526) yPA -0.01438 -0.01499 -0.04535 -0.01240 (0.00953) (0.00206) (0.01279) (0.00327) YPK -0.00894 -0.00944 -0.00873 -0.01149 (0.00561) (0.00484) (0.00385) (0.00337) YPF -0.00885 -0.00523 0.00729 0.00563 (0.00336) (0.00236) (0.00335) (0.00230) yPR -0.00316 -0.00633 -0.00524 -0.00364 (0.00345) (0.00323) (0.00352) (0.00348) YAP -0.01668 -0.01499 -0.00490 -0.01240 (0.00293) (0.00206) (0.00556) (0.00327) YAA 0.02221 0.02035 0.01189 0.01769 (0.00313) (0.00208) (0.00449) (0.00299) YAK -0.00203 -0.00246 -0.00384 -0.00447 (0.00184) (0.00171) (0.00135) (0.00129) YAF -0.00231 -0.00094 0.00191 0.00109 (0.00111) (0.00092) (0.00118) (0.00097) VAR -0.00156 -0.00197 -0.00250 -0.00191 (0.00113) (0.00109) (0.00124) (0.00122) yKP 0.05326 -0.00944 -0.02373 -0.01149 (0.01905) (0.00484) (0.02552) (0.00337) TABLE VI-6-b.--Continued. 212 Trans1og Production Function Trans1og Cost Function Parameter Unrestricted Symmetry Unrestricted Symmetry YKA -0.03197 -0.00246 0.01368 -0.00447 (0.02040) (0.00171) (0.02062) (0.00129) YKK -0.00411 -0.00385 0.05884 0.05154 (0.01200) (0.00958) (0.00621) (0.00522) yKF -0.00223 0.00846 -0.00547 -0.01403 (0.00720) (0.00398) (0.00540) (0.00240) yKR 0.00603 0.00728 -0.01465 -0.02155 (0.00738) (0.00663) (0.00567) (0.00475) yFP -0.00426 -0.00523 -0.00924 0.00563 (0.00798) (0.00236) (0.01252) (0.00230) YFA -0.00431 -0.00094 0.00054 0.00109 (0.00855) (0.00092) (0.01012) (0.00097) YFK 0.00532 0.00846 -0.01145 -0.01403 (0.00503) (0.00398) (0.00305) (0.00240) YFF 0.00247 0.00241 0.01085 0.00899 (0.00312) (0.00286) (0.00265) (0.00231) YFR -0.00665 -0.00472 0.00146 -0.00169 (0.00309) (0.00280) (0.00278) (0.00251) YRP -0.05665 -0.00633 0.01357 -0.00364 (0.02078) (0.0 ) (0.03244) (0.0 ) YRA 0.02845 -0.00197 0.01925 -0.00191 (0.02225) (0.0 ) (0.02621) (0.0 ) YRK 0.00976 0.00728 -0.03481 -0.02155 (0.01309) (0.0 ) (0.00789) (0.0 ) YRF 0.01091 -0.00472 -0.01458 -0.00169 (0.00785) (0.0 ) (0.00686) (0.0 ) YRR 0.00534 0.00574 0.02094 0.02879 (0.00806) (0.0 ) (0.00721) (0.0 ) aSee the footnote (a) in Tab1e VI-1. 213 constant terms, a11 the own variab1e's coefficients and most of the cross variabTes' coefficients in each share equation of the restricted estimation of the trans1og production function are significant, except few of the cross variables' coefficients between inputs and outputs (such as 6K3. 61p. €1A’ ETF’ 52p: €2A’ 52K,and 52F etc.). These insignificant coefficients may be attributed to the existence of the weak separabi1ity between inputs and outputs in the mu1ti-input, mu1ti-output production technoiogy. The va1idity of this weak separabiTity is again discussed in the next section 2.2.2. in the significance of alternative estimations with respect to different hypothesis. The significance of the parameters estimated in the trans1og cost function is a1so simi1ar to that in the trans]og production function over a11 six industries, in genera1. 2.2. Significance of the Estimation In this subsection to investigate statistica] significance of the current empirica1 estimation, two different measures are considered, i.e., the first for the goodness of fit and the second for testing the va1idity of the restrictions imposed. 2.2.1. The Goodness of Fit As a measure for the goodness of fit in each share equation, the R2 is ca1cu1ated as one minus the ratio of the error sum of 'squares (about its mean) to the tota1 sum of squares (about its mean). A1though on1y four of the input shares are inc1uded in the 214 estimation procedure, the R2 for the de1eted input share can be inferred using the 1inear homogeneity constraints. The R2 for the de1eted output share a1so can be inferred in a simi1ar way. Since we have more than one equation to be estimated, no definite infer- ences can be made on the goodness of fit for the equations system as a whoTe. Hence, two arbitrary measures to the goodness of fit for the who1e system are considered, i.e., the simp1e average of the R2' 5 of each share equations and their weighted average by each factor share. And as a1ready noted in the previous section, the interpretation of the Rz's in the case of estimation with restrictions imposed on the parameters shou1d be different, in the sense that the sum of the unexp1ained variation in the exp1ained variab1e (i.e. the error sum of aquares) and the exp1ained variance is no more equa1 to the tota1 variance of the exp1ained variab1e (i.e., the tota1 sum of squares).6 Tab1es VII-1 through VII-3 contain, for six industries, the R2's of each share equations and their averages by row, and by co1umn those in the unrestricted and the restricted estimations, separated by the translog production and cost functions. Measured by the mu1tip1e corre1ation coefficient, R2, the fit is very poor in the input share equations, averaging under 0.5, and is much better in the output equations, averaging above 0.8, for a11 of six industries. 6See the section 4.4.3., Chapter III, Part B. 215 .ueeuee Peace ea ueeemee saw: xuwewame—m m.geaee$ some x: eeugmwez emece>e ecu mm emege>e eep;m_e3 we» use museuee ecu mange? ee meeeue» _pe Lew m. m »e cues e_eesguwce es» mm eme5e>e e~e5_m .xgumeecm mcpcceu ecu :_ xuweesEee genes anew; es» ee a 5» ea weaves pm ecu page? 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_.> mcowpuwgpmmm emeu cepueeeege e a zgpmeeca we m we Eeeeeem _m>e4 Peewuwgu we eegmea muwumwpmpm ummh a .mcewuuwgpmem m>mpmcgeu—< Le» ummhuman.NuHHH> m4meg e3 gewgz Eeeeege me meegmee mg» gem eegeeeem peg wee meepe> maveceemeggee egg neg; .cewu:g_gumwe -m ee apnea mg» cw eewmpeeam eeeeegm me meegmee pgegemmwe gem meepe> e3» geezpeg gemuepeegeucm he eeuepeeyee we .o.o we Pe>ep eegeewmwcmwm we gewuegwgumweum me espe> _eewuwge eghg ._-HH> mpnmh cw gev meoceooe age meme _ee.N mop.m mem.~ com op geamseaw gee ggemsece a=_epoz .H> cam.~ m-.¢ mem.~ mom o_ seemssxw gee meuumecwgm .> mam.~ eom.F mem.m mme op seemsexw gov amazemeca emeegex .>H cerueceu :ewpegem Po.o u e N> _> mgeppe_gpmmm emeu :eweueeege g a zgumeegH we m ee Eeeeegm mewumpaeum amok _m>eg _eewuwgu me megmeo [ [I'll ".I|ll'll .Umacwpcouuu.~uHHH> m4m N m cevueceg :ewuegau gewpegeu geweugsm gewueczm gewuegeu cewuewgpmem umeu zeppezeege umeu gewuezeega umeu gewpeeeege g agemeecm xgumzeca zgpmeegH ucwgceu muueeege :mgem Leezpeeu gegpeeg c.5gewpevppzz cewmcegmeg age we eepm>-ev cepuewgumem euegeeem gem moppm_ueem ewe»--.F-m-~HH> mgm

.+ aa>v eege meeeueeep .a _n. up mgmgz .o n he» a .esmN on Peeem me Ame» + me» + gm» w .m amen .HHH gmaeegu ..m.e cewuemm cw Fweume cememgwepexm we mgmg emgmewmcee mcemuewgummmg me cememegee xewmgmmeeeg gemcep mg» .mFeEexm gem .xuwpwgegege _o.o e meg m:_e> mpepemge cw omn.~ me Fm>m_ Feewewge mgp :egu empemgm seepewgum 1mg geem Lem me—e>uu memegeemmggee mgp peg» mmueemecw mpgeu mgu gm Rev gmwgmume gee: mgmgsec mg» .m “gee .HHH empeego ..m.e.e ece .N.p.e cewuemm cw emcwepexm mm gmwpewepez gewmcegmeg mghe -- -- -- -- eke., emm.P Pee Nee -- -- -- -- emN._ _ee._- emu New NN_.o emm.o- ANN.F- _Nm.~- Nop.o m-.F- g_m Pew -- -- -- -- _em.o mmm.o- xee New ee_.P- Ppe.o- ome.o em~.o NeP.F- mom.o- ¥_u _gu -- -- -- -- oee.o Pop._- emu New _~_.o Pep.o- Neo.o moo.o _gp.o N_m._- <_w _ew -- -- -- -- eme.o mmo._- awe New -o.e Ame.o- ee_.o mme.o- mee.o Nee.F- ape Pee em~._- eeg.o- emem.e- «m_e.m- mmo.o eme.o- xe» ax» g_e.o- Pmm.o o_~.F mme._ mmm.P 5mm.o em» me» eeN.F omm.o eem.o- *Nmp.g- mue._ meo.~ ex» xe» compose; cowauceu coweegzm :ewpegam cewuecem :ewuecem :ewuewgpmmm pmeu :ewueeeege emeu seepeaeege pmeo cewueaeege g mueeeege szem gunmeegH gemzpeem gmguemg xgemeecH newcceu .eeseweeoe--.-_-m-eee> “geek 230 .anuHHH> mpgeh cw any m»o=»oom mg» mmmn ..-m-HHH> epge» cw gem m»o=»ooe mg» meme __~.o- Nee.m *m_m.~ mmm.~- «_mm.m *Nmo.m gm» u ex» oee.» mg_.~ ee~.N mem.~- *oem.m ¥FN_.m em» u e<> gee.» Nee.» 5mm.o new.»- emo._ 94¢.N ex» u x<> eme.o- «em.o- er.» 4eee.~- ego~.m 3e»¢.n me» n ma» ee_.N- «Nee.m- wee..- emee.m- no... ome.o- ax» u gm» em~.~- ea».~- 4-~.~ eme.o- Pmm.o- ege.~- me» u ~m~.m «No~.m- *mmo.m- em_e.m *_mp.e- «Neo.m- an» ~mm.~ e~_.o eee.~ *_Ne.e geo.~ o_o.» en» »m_._ ewe.» Nee.» e¢N.o- m_e._ eem.m en» a _ue Ppe.~- 3ego.m- eoo.P- emm.m- me~.~ tomg.m- enema"..> N m =e»»e=eu ce»»eg=m ge»»ec:m :e»»ec=m ge»»ec:u ce»»eczm ce»»e»g»mmm »meu ge»»e:eege »meu ce»»e:eege »meu :e»»e=eege xu»m:e:H mgwepez mm»»m=e»gm gengmega em»»»=x me mg=»ee»egez Lr me mg:»eew==ez e.»gm»»e»»_=z sewagegmeg mg» me m:»e>-»v =e»»e»g»mmm m»egeemm gem mem»mw»e»m »mmp--.~nm--~> mgmg»m:egm xg»m=egH mcwgceu muuauosm 398m meznwoo... Lwnummg .Ammmmg»=mgee :» mme»:mugmev meow»wegeu »»»e»ge»ecez mngump»em »ez .m»gmsgmw_ge»mm we gmgszz--.»-»-xH mgmg»mse>w ce»»ec:m ge»»e::m :ew»eg:m gew»ec=m ce»»e==m =e_»e==u :e»»esw»mm emeu ce»»u=eege »meu ce»»e=eeee »meu :e»»eeeege em»e»g»mmm >g»m:eg~ ag»m:egm >g»m:e:H mcwcceu mugs—yoga ZmLUW megoou— meumwA (F(LL (1} Lr')iyf (77(7) 7 r .xwg»ez cewmmm: mg» yo mmepe>cmmpmuu.Fi~uxH ugm=eOem--.N-N-xH OOOe» 239 concave) regions and have increasing returns to sca1e in certain ranges of the inputs. In particu1ar, this concavity condition seems to be satisfied better in the separate estimations by the size of samp1e estab1ishments, even in the cost function of the 1eather footwear industry and in the production functions of the manufacture of knitted underwear, of briquettes, and of the mo1ding industry.22 On the other hand, none of the output-transformation curves in a11 3 industries satisfy the convexity conditions. That is, the eigenva1ues of the Hessians for the mu1tioutputs are -0.0000, -0.2596 and -0.3731 in the canning industry, -0.0000 and -0.4586 in the leather footwear, and -0.0000 and -0.4078 in the screw products industry. These resu1ts seem to be very confusing at first where the conventiona1 textbooks show a transformation curve expressing a technica1 re1ation in transforming one output into another given fixed input bundIes, which is convex to the origin, not concave to the origin as in this resu1t. Two possib1e exp1anations on these non-convex output— transformation curves may be investigated, i.e., the heterogeneity of the output with different qua1ity and the existence of technica1 changes, between two groups of samp1e estab1ishments, one of which 22This resu1ts are shown in the Tater section of the supp1ementary resu1ts on the production and cost function, separate1y estimated by the size of samp1e estab1ishments. The existence of increasing returns to sca1e may exp1ain the non-concavity of the estimated isoquants in these industries. 240 produce on1y a uniproduct (group A) and the other of which produce mu1tiproducts (group B). Corporating with the distribution of samp1e estab1ishments producing a uniproduct and mu1tiproducts by industry, shown in Tab1e 11-4 of the section 5, Chapter II, the estimated technoIogy imp1ies such a shape of the transformation curve from its concavity as shown in the Figure II, in the case of two outputs. Product 1 ‘33.). Group A 31 Product 2 Figure II.--Estimated Transformation Curve The first testab1e argument can be stated as fo11ows: The qua1ity of the output produced by the uniproduct estab1ishments may be different from that of the output produced by the mu1ti- product estab1ishments, even if these products are c1assified into the same commodity category in the census data base (i.e. KSIC 7-digit commodity code). When it can be assumed safe1y that the degree of the heterogeneity is we11 ref1ected in their individua1 output prices, it is possib1e that the actua1 transformation curve is convex to the origin in terms of va1ues if the output price of 241 the uniproduct estab1ishments is significant1y different from that of the mu1tiproduct estab1ishments. That is, the possibi1ity of the actua1 transformation curves, against the estimated curve, from the existence of a heterogeneous output can be shown as in the Figure III. Product 1 : estimated curve : actua1 curve Product 2 Figure III.--Possib1e Transformation Curve With Heterogeneous Output In genera1, two different measures for the average price of each c1assified output can be defined separate1y for the group of samp1e estab1ishments producing one output on1y and the other producing more than one output. The first measure is a quantity- weighted average price, assuming that each c1assified output is homogeneous whether they are produced by uniproduct estab1ishments or not. The second is a va1ue-weighted average price, assuming that the c1assified outputs are of different qua1ity from whether they are produced by uniproduct firms or not. 242 The Tab1e IX-2-3 contains these two different measures of the average prices for three mu1tioutput industries, representing the differentia1s in average prices by the average price ratio between the uniproduct and the mu1tiproduct estab1ishments. In the canning industry the quantity-weighted average price differentia1 is 28.90% in the output 1, 46.54% in the output 2 and 1.90% in the output 3, whi1e the va1ue-weighted average price differentia1 is 21.20% in the output 1, 21.01% in the output 2 and 4.24% in the output 3. Hence it may we11 be said that the outputs 1 and 2 are significant1y heterogeneous in the sense that the output price differentia1s between two types of estab1ishments are above 30% (see the co1umn C) and a1so the differentia1s in the va1ue-weighted average price are above 20% in the outputs 1 and 2 (see the co1umn F), showing the 6% changes in average price differentia1s between two a1ternative measures for the output 1 and the 17.42% for the output 2 (see the cqumn G). This resu1t may imp1y that the heterogeneity of these c1assified outputs between two groups of estab1ishments not on1y resu1ts from a significant degree of price differentia1s but a11 resu1ts from some other factors such as manageria1 efficiency in marketing, or possib1y technica1 change, etc. The 1eather footwear industry shows a1so 90.86% average price differentia1s in the output 1. And the screw products industry shows 68.86% in the output 1 and 27.88% in the output 2. The comon phenomena in a11 three industries are that the output 243 .m»=msgm»»ge»mm »eeeege»:= mg» gem emme m» epesgem msem mgh .»:e»=e » P mce :eg» mgee memeeeege m»:msgm»_ge»mm mg» gem apge mce m.» mgmgz .pm.rwmmw w wepesgom mngeppe» mg» Ag em»e»=e»ee m» m»:mEgm»»ge»mm »eeeegew»»:s mg» n» mewge mmegm>e em»gm»mzim:»e> mgh g m»:msgm»»ge»mm »eeeege»»—ee mg» gem emm: m» e—32Lem » msem mg» .»e=ee.a.E= e mgweeeema m»:msgm.£ge»mm mg» gen. 3.8 mge m.» . .éww n J lawn” 3.26:?» we em»e»=upee m» m»=mEgm»Pge»mm »eseege»== mg» a» mewge mmegm>e em»gm»mzux»»»=e=e mghe mOmN._ OO_O.» NNOO.O PNOO.O OONN.» Om_e.O OONm.O N-»OQ»OO NOeO.N OOON.¢ ON»_.O Opem.O Oeee.» eNe0.0 Oe__.O »-»OO»OO m»e=eegm 3mgum OOe0.0 eNe0.0 OFON.» _ON»._ OO_0.0 Ome_._ eOmO._ N-»OO»OO NOOO.O OmOO.» _Oee._ _PFO.N eOOO._ OON»._ »NeN.N F-»OO»OO gem3»eem gmg»emg ONNO._ eNeO._ NOO0.0 Nee0.0 OOPO.» OONN.O NOON.O m-»OO»OO OON0.0 FOFN._ eOFO.O ONO~.O Omee.» O»O_.O eeON.O N-»OO»OO mon.O ONPN.» eONN.O ONON.O OOON.» OOO_.O OOO».O P-»OO»OO au»meecm.mgmcceu gO\gvugOO gO\Ovugmv ng seem AOO seem AON< em»gm»m31m:»e> ememgm mmegm>< em»gm»m3-z»»»ce=o .mELPm »ezeege»»_ez mg» ece »e:eege»== mg» g» m»=e»:o we mmewgm mmegm><--.m-~-x~ mgme mg» warm: »g em»e»:u»em mge Omme »e»e»e N0.0» OO.O OO.N OO.O »O.O OO.O ».OO>OOOOO .O>< O.OO» O.NOO 0.00 O.N» O.NON O.N»N OOOO »OOOO O» OO O O» NN OO OOL»O OO emOeOz O O» O O O » ONO».O » O» O O » O NNO»gO O N » N O N »NO»g O ON O N O» N» ONO»-OOO» »O O O» O O » N OOO»-»OO» »O O O O O O » OOO»-OOO»»O » N » » O O OOO»-»OO» »O O O O O » » OOO»-OOO» »N O O O O » » OOO» OOOOOO»» OELOO OSLOO OELOO mag»; OELOO OSLOO meeu mm< »eeeege »e:eege»g= »ezeege »eeeege»g= »eeeege »ezeega»g= .»»»32 .»»»O2 .»»»O2 m»e:eegm 3mgem gem3»eeg gmg»emg xg»m:eg» mgmgceo O.OEO»O »OOOOOO»»»Oz OOO »OOeocO»OO »O OOOOOOcOO OOO--.O-N-x» OOOOO 247 the parameter estimation of the first-order terms (i.e., the constant terms in the share equations such as 01 and Bi) and of the second order terms (Yij’ eij’ gij and aij) in the specification of the transcendenta1 1ogarithmic functions. 23 2.4.1. Output E1asticity of Factor and Factor Share 24 where In the imp1icit production function of t(x,Y) = 1, X is a vector of inputs and Y is a vector of output, the output e1asticity of the i-th factor, denoted by Qi’ is defined as the proportionate rate of change of t with respect to xi: 3 1n t(X,Y) x, a t(X,Y)25 Q. 1 (13) a 1n xi t(X,Y) 3 X, This e1asticity again becomes the i-th factor share in tota1 output, when the margina1 productivity theory of distribution is emp1oyed 23This is often referred to as "the e1asticity of a function." See A11en (1938), pp. 251-2. 24Compare the specification of the functiona1 form, t(Y) = 1, in the section 1, Chapter II, Part 8. Here the input and output are separated. 25This can be shown, in particu1ar, in the production function of exp1icit (or mu1tip1icative) separabi1ity between input and output. When we have the function form of t(X,Y) = F(X)G(Y) = 1, then a 1n t(X,Y) x, a t(X,Y) xi 0 a F(x) x, a F(X) _ -—————- Y - a 1n xi t(x,Y) a x, F(x)G(v) a x, F(x) 3 xi This is the conventiona1 definition of the output e1asticity of the i-th factor, shown in most economic textbooks. For examp1e, see Henderson and Quandt (1971), p. 57. 248 in factor pricing.26 In re1ation to the specification of trans1og production function, the output e1asticity of the i-th factor can be identified as the constant term of each share equation a1so, eva1uated at X = Y = [1].27 In the same way as in the production function, the cost e1asticity of the i-th factor price can be defined, denoted by O: as the proportionate rate of change of tota1 cost with respect to the i-th factor price: ,, a 1n C(W,Y) w,- a C(W,Y) Q. = = 1 a 1n w, C(W,Y) a w, (14) The cost e1asticity of the i-th factor price a1so becomes the i-th factor's cost share and hence is identified as the constant term of the i-th cost share equation, eva1uated at W = Y [1].28 26Again this can be shown for a homogeneous production function with the mu1tip1icative separabi1ity as before: 3 1n t(X,Y) X1 3 F(X) X1 = —_ ”i = $1, a 1n x,- F(X) 3 x1. F(X) where W. is the i-th factor price and S. is the i—th factor share in tota1 output. 1 27See the equations (1) through (5) in the section 2,1, of this chapter. 28Since the cost function for a cost-minimizing firms must be homogeneous of degree one in the input prices, the cost e1asticity of the i-th factor price becomes: * Ni 3 C(W,Y) W-i Q'i = = X. = S" C(W,Y) a w, C(W,Y) ‘ 1 M from the cost equation of C(W,Y) = .2 W.X., where M = tota1 number 1_] 1 1 249 As shown in the Tables VI-l-a through VI-6-a, the estimates of these output e1asticities are drastically different in the unrestricted estimation from those in the restricted estimations of both functions. But there is no significant differences found in the estimations of different restrictions. For example, in the canning industry, the output elasticity of the operative worker in the production function is estimated as 0.053 in the unrestricted case, while it is 0.110 in the restricted case of symmetry and is 0.113 in that of explicit separability.29 Also its share in the cost function is 0.047, while it is 0.124 in the restricted case of symmetry and is 0.127 in that of explicit separability. But it is verified that the estimates of its output e1asticity in the restricted estimation are very close to the average factor shares which are directly calculated from the observations on sample establishments, over all six industries, relative to its estimates in the unrestricted estimations. For example in the canning industry, the average factor share of fuel input in the observations 30 is calculated as 0.047, while its output elasticity is estimated as 0.022 in the unrestricted estimation of the production function of input, and X, = quantity of the i-th factor input. Also see the equation (9) in the section 2.1 of this chapter. 29The parameter estimates of 0.053, 0.110, and 0.113 can be found in the first, second, and third column of the first row of the Table VI-l-a, the section 2.1. of this chapter. 30Data synthesis on factor shares is explained in the section 6.3., Chapter II, Part B. 250 and as 0.009 in that of the cost function. 0n the other hand, this elasticity is estimated as 0.048 (or 0.049) in the restricted estimations of the production function, and estimated as 0.031 (or 0.033) in that of the cost function. These verifications are found as a consistent phenomena in both the production function and the cost function of all six industries selected in this study. Hence the estimates of the output elasticities of factor inputs are found to be good and stable both in the restricted estimations of the production function and the cost function over all six industries. The importance of each factor input in the production activity is also found significantly different from industry to industry. The output elasticity of the operative worker ranges from 0.10 to 0.15 in average among the industries, except it is 31 And the output only about 0.04 in the manufacture of briquettes. elasticity of the administrative worker is less than 0.05 in average. Hence for labor inputs as a whole they range from 0.15 to 0.20 in all five industries, except about 0.07 in the manufacture of briquettes again.32 31This lowest elasticity of operative workers is quite under- standable, where the industry was found to have quite a high capital- 1abor ratio, high average products of that input and hence high wage level, relative to the other industries. See the section 6, Chapter II, Part B, on "Some characteristics of the Industries selected." 32Compared with the average labor shares calculated directly ‘from the observations, their drastic drops are found in some industries. For'example, the output elasticity of labor inputs in gross outputs (are reduced from about 0.28 to 0.18 in the leather footwear industry, and from 0.24 to 0.19 in the screw products industry. 251 The output elasticity of the capital input has a large variation in the different industries and also between the estimates of the production function and the cost function. For example, in the manufacture of briquettes they are estimated as 0.18 in the production function and as 0.25 in the cost function. And in the molding industry they are 0.24 in the production function and 0.19 in the cost function. Again we find that the output elasticity of capital input estimated in the production function is more reliable and close to the factor share directly calculated from the original data base, relative to its estimates 33 Based on the estimates of the production in the cost function. function, we also find the relatively high output elasticities of capital input in the industries which have relatively high level of capital-labor ratio. For example, the canning industry, the screw products industry, and the molding industry are shown to have significantly high capital-labor ratio in the Table III-2, where the Tables of VI-l-a, VI-3-a, and VI-6-a show quite a high output elasticity of capital input in the estimations of their production functions. The output elasticity of fuel input is estimated around 0.05 in average of all six industries and its estimates are rather stable both in the production function and in the cost function, 33One exceptional case is found in the canning industry. The estimated capital share in the production function is 0.348 and that in the cost function is 0.291, while the calculated share in the data is 0.254. See the Tables III-4 of the section 6.4., Chapter II, Part B and the Table VI-l-a in this chapter. 252 relative to the average share in gross output, calculated directly from the observations. But this dependency of energy input in gross outputs varies very much among different industries. In general, the industries of higher capital-labor ratio show relatively high level of energy consumptions. For example in the canning industry, the screw products industry, and in the molding industry, the factor share of fuel consumption ranges from 0.05 to 0.08, while in the other three industries it ranges from 0.01 to 0.02. The output elasticity of raw material input ranges between 0.5 and 0.6 in the estimations of the production and the cost function of the five industries selected, except the highest elasticity of about 0.7 is found in the manufacture of briquettes.34 Here also some significant differences in the size of the elasticity estimated are noticed between the production function and the cost function of the leather footwear industry and the molding industry. But still the estimated elasticities in the production function are found close to the calculated average factor shares previously in the four industries out of the six selected ones, except the canning industry and the leather footwear industry. Based on these empirical results it becomes noteworthy that the output elasticity of raw material input usually varies from industry to industry so widely that the conventional hypothesis on the role of raw materials as an fixed proportion to gross outputs may well be rejected. 34In the manufacture of briquettes, the lowest elasticities of labor and capital inputs seem also to be attributed to this extremely high elasticity of the raw materials. 253 2.4.2. Elasticity of Factor Substitution and ElastiCity of Pr660ct Transformation The most common quantitative indices of production factor substitutability are forms of the elasticity of substitution (E.S.).35 The defining formulae for these indices have the dis- advantage of not allowing direct empirical evaluation. But the translog function in general exhibits the property of variable elasticities of substution (V.E.S.) in each observed establishment.36 However, the assumption of constant E.S. leads to simple estimation 37 On the other hand, in the methods, and has been widely used. current empirical study on production technology with a larger number of factors (and products), there is no traditional defini- tion of the E.S., but three forms have been suggested in the literature:38 35For two factors of production, the E.S. is defined along an equal-product curve as the elasticity of the factor input ratio with respect to the marginal rate of substitution. See Allen (1938), pp. 340-3. 36But the estimation of these V.E.S. for each establishment contains really heavy computational burdens. Hence it is left for the later study subject in this study. 37The references in Arrow, Chenery, Minhas, and Solow (1961), and Morrissett (1953) include many of the empirical studies of the E.S. which make this assumption. 38The definitions of the A.E.S. and the D.E.S. appear in the section 4.2.2., Chapter I, Part A. And the definition of the S.E.S. also appear in the section 3.2.2., Chapter II, Part A. The Allen E.S. and the D.E.S. were introduced in Allen and Hicks (1934), pp. 202-6, 211-l4, in the terminology "elasticity of complimentary" and "elasticity of substitution between Y and Z in the YZ indifference direction," respectively. The Allen E.S. is developed further in Arrow, Chenery, Minhas, and Solow (1961), p. 503. Also Uzawa (1962) has reformulated the definition of the A.E.S., and has characterized 254 (a) the Allen partial elasticity of substitution (A.E.S.), (b) the Direct partial elasticity of substitution (D.E.S.), (c) the Shadow partial elasticity of substitution (S.E.S.). Owing to the two theorems,39 presented in Jorgenson, Christensen, and Lau (1970) we measure the corresponding constant E.S.‘s in this empirica1 works, mainly because of heavy computa- tional burdens involved in measuring various E.S. for sample establishments.4O Again, the defining formulae for these indices of the D.E.S. and the A.E.S. have been worked out in Mundlak and Razin (1973) in terms of parameters of the translog production function, and in Brown, Caves, and Christensen (1975) in terms of the translog cost function. Also the defining formulae for the S.E.S. in terms of parameters of the translog cost function is worked out at the section 3.2.2., Chapter II, Part A in this study. Tab1es X-l-l through X-l-6 contain the estimates of these various elasticities of substitution for the translog production its class of constant E.S. production functions. The D.E.S. has been used in Morrissett (1953), pp. 42, 49-52, and Meade (1961), pp. 77-82. Also McFadden (1963) has characterized its class of constant E.S. production functions. The S.E.S. is originally reformulated by McFadden (1963) and was characterized its class :of constant E.S. cost functions. But no estimations of the S.E.S. seem to be tried in any empirical works, as far as we have surveyed. 39See the section 4.2.2., Chapter I, Part A, about the :existence of a transcental logarithmic production function which attains a given arbitrary set of either the D.E.S. or the A.E.S. at given quantities of out uts and inputs. See also Jorgenson, Christensen, and Lau (1970), pp. 24-27. 40See the footnote 36 in this section. 255 TABLE X-l-l.--Estimated Elasticities of Substitutiona--Canning Industry. Translog Prod. Funct. Translog Cost Funct. Demand Factors D.E.S. A.E.S. S.E.S. A.E.S. Elasticity (P : P) -1.0000 -5.2265 -l.0000 -6.2075 -0.8299 (P : A 2.5982 -O.1482 0.3563 -2.1768 -0.0770 (P : K 1.2382 0.9485 0.8542 0.7974 0.2414 P : F) 1.2498 0.7516 1.0050 1.7939 0.0824 P : R) 1.3669 0.7669 0.9185 1.0552 0.5090 A : A -l.0000 -17.4930 -1.0000 -28.1500 -0.9959 A : K 2.1636 0.7548 0.4704 0.7253 0.2195 (A : F 1.8980 -0.1879 0.6441 0.3184 0.0146 A : R 2.1854 0.8533 0.4734 0.9568 0.4615 K : K -1.0000 -2.1164 -1.0000 -2.3464 -0.7102 K : F 1.1627 1.1031 0.9386 0.7187 0.0330 (K : R) 1.1252 0.9061 0.8844 0.8900 0.4293 (F : F) -1.0000 -18.0010 -1.0000 -20.1600 -0.9259 (F : R) 1.2170 0.8053 0.9673 0.9481 0.4573 (R : R) -1.0000 -0.9111 -1.0000 -l.0783 -0.5201 (1 : 1) 1.0000 0.6021 1.0000 0.5997 0.3748 (1 : 2) -1.0246 -1.0195 -0.9753 -0.9802 -0.l397 (1 : 3) -1.0161 -1.0117 -0.9845 -0.9887 -0.2298 (2 : 2; 1.0000 5.9396 1.0000 6.0215 0.8582 2 : 3 -1.0288 -1.0497 0.9717 -0.9523 -0.2214 (3 : 3) 1.0000 3.5114 1.0000 3.3019 0.7675 a(i) Various elasticities substitution (E.S.) are referred by their abbreviations respectively. Hence the D.E.S. indicates the Direct partial elasticity of substitution. The A.E.S. indicates that the Allen partial elasticity of substitution, the S.E.S. indicates the shadow partial elasticity of substitution. The demand elasticity indicates the elasticity of factor demand with respect to their price changes. (ii) The column of "factors" indicates the factors inter- acted directly in the elasticity concerned, where p indicates the. operative worker, A indicates the administrative worker, K does the capital input, F does the fuel input, and R indicates the raw material input. Also for the output commodities, the numeric number is used for the mu1tiproduct industries, such that 1 indicates the first major-commodity produced in each industry and 2 does the second major-commodity, etc. (iii) For example, the estimated E.S. of -0.1482 in the second row of the second column indicates that the Allen par- tial elasticity of substitution between the inputs of operative worker and of administrative worker is measured as -0.l482. (iv) Since the elasticity of substitution between the i-th factor and the j-th factor is, by definition, equal to that between the j-th factor and the i-th factor (that is, 015 = Oji for example), the table only contains one of these two same measures. 256 TABLE X-l-2.--Estimated Elasticities of Substitutiona--Leather Footwear Industry. See the footnote (a) in the Table X-l-l. Translog Prod. Funct. Translog Cost Funct. Demand Factors D.E.S. A.E.S. S.E.S. A.E.S. Elasticity (P : P) -1.0000 -4.2228 -l.0000 -4.9295 -0.8361 (P : A) 2.0094 -0.8166 -0.6245 -6.5607 -0.2024 (P : K) 1.1199 1.1646 0.5577 0.3209 0.0870 (P : F) 2.7025 0.3455 0.7953 0.6032 0.0088 (P : R) 1.2983 0.8096 0.7052 1.0306 0.5295 (A : A) -1.0000 -20.4400 -1.0000 -28.6790 -0.8849 (A : K) 1.4163 1.5272 -0.2940 0.2347 0.0636 (A : F) 2.1781 1.1163 0.5535 6.8011 0.0988 (A : R) 1.5825 0.8316 0.2973 1.0515 0.5402 (K : K) -1.0000 -2.4786 -1.0000 -2.7278 -0.7399 (K : F) 2.6968 0.4467 0.8055 0.2365 0.0034 (K : R) 1.2116 0.8082 0.9348 1.0398 0.5342 (F : F) -1.0000 -40.3870 -1.0000 -55.7320 -0.8097 (F : R) 2.6820 0.7480 0.8250 0.8435 0.4334 (R : R) -1.0000 -0.7685 -1.0000 -0.9508 -0.4885 (1 : 1) 1.0000 0.5884 1.0000 0.4899 0.3288 (1 : 2) -l.0135 -l.0135 -0.9860 -0.9860 -0.3242 (2 : 2) 1.0000 1.7457 1.0000 2.0414 0.6712 A 257 TABLE X-1-3.--Estimated Elasticities of Substitutiona--Screw Products Industry. Translog Prod. Funct. Translog Cost Funct. Demand Factors D.E.S. A.E.S. S.E.S. A.E.S. Elasticity (P : P) -1.0000 -3.4723 -1.0000 -4.3295 -0.7756 (P : A) 7.2532 -0.3284 0.2414 -l.5003 -0.07l6 (P : K) 1.3039 0.7691 0.4220 -0.0310 -0.0080 (P : F) 1.5947 0.2412 1.0891 2.0695 0.0877 (P : R) 1.2926 0.9338 0.7036 0.9365 0.4412 (A : A) -1.0000 -11.2220 -1.0000 -16.7410 -O.7990 (A : K) 3.5491 0.3546 0.3257 -0.1779 -0.0462 (A : F) 1.8731 1.0988 0.9263 4.6157 0.1956 (A : R) 3.0625 0.9955 0.4266 0.9521 0.4486 (K : K) -1.0000 -2.7922 -1.0000 -3.0391 -0.7890 (K : F) 1.3586 0.8632 0.9610 -0.1456 -0.0062 (K : R) 1.0048 1.0826 0.7288 0.8891 0.4189 (F : F) -1.0000 -17.0810 -1.0000 -25.1320 -1.0651 (F : R) 1.3725 0.8974 1.1021 1.0863 0.5118 (R : R) -l.0000 -1.1229 -1.0000 -l.1147 -0.5252 (1 : 1) 1.0000 0.4199 1.0000 0.4204 0.2960 (1 : 2) -1.0151 -1.0151 -0.9852 -0.9852 -0.2915 (2 : 2) 1.0000 2.4543 1.0000 2.3789 0.7039 aSee the footnote (a) in the Table X-l-l. 258 TABLE X-1-4.--Estimates Elasticities of Substitutiona--Manufacture of Knitted Underwear. Translog Prod. Funct. Translog Cost Funct. Demand Factors D.E.S. A.E.S. S.E.S. A.E.S. Elasticity (P : P) -1.0000 -0.6540 -1.0000 -6.3139 -0.8755 (P : A) -8.1768 -55.4660 0.1847 -2.6777 -0.0670 (P : K) 1.5101 1.7747 0.4852 0.0333 0.0078 (P : F) _ 1.5973 5.2598 0.9978 0.0762 0.0012 (P : R) 1.9167 1.6732 0.6318 0.8681 0.5096 (A : A) -1.0000 271.3300 -1.0000 -34.1600 -0.8543 (A : K) 19.2350 -1.8209 0.2764 -0.4174 0.0977 (A : F) 2.6161 -0.3817 0.9269 8.9067 0.1360 (A : R) 164.6800 2.2774 0.3242 1.0282 0.6036 (K : K) -1.0000 -3.8042 -1.0000 -3.3146 -O.7759 (K : F) 1.4277 0.4265 1.0332 0.3473 0.0053 (K : R) 1.1922 1.1643 0.7839 0.8978 0.5270 (F : F) -l.0000 -96.7760 -1.0000 -69.1560 -1.0556 (F : R) 1.4760 1.1203 1.0758 1.2623 0.7410 (R : R) -1.0000 -0.9857 -1.0000 -0.6968 -0.4090 aSee the footnote (a) in the Table X-1-1. 259 TABLE X-l-5.--Estimated Elasticities of Substitutiona--Manufacture of Briquettes. Translog Prod. Funct. Translog Cost Funct. Demand Factors D.E.S. A.E.S. S.E.S. A.E.S. Elasticity (P : P) -1.0000 -37.2550 -1.0000 -11.7880 -0.9053 (P : A) 5.6468 32.5300 0.5749 -0.2317 -0.0082 (P : K) 1.3752 -0.7432 0.4584 -1.0452 -0.l796 (P : F) 1.4362 -0.5766 1.0163 2.2256 0.0324 (P : R) 2.5454 2.7462 0.7191 1.0791 0.7570 (A : A) -l.0000 -135.4900 -1.0000 -26.5230 -0.9351 (A : K) 2.4175 2.2060 0.5381 -0.4609 -0.0792 (A : F) 1.6429 -2.8721 0.9607 3.0308 0.0442 (A : R) 3.8845 2.6336 0.6251 0.9008 0.6319 (K : K) -1.0000 -4.2213 -1.0000 -4.9326 -0.8477 (K : F) 1.2929 0.2974 0.8791 -0.3448 -0.0050 (K : R) 0.9336 1.0461 0.6897 0.8534 0.5987 (F : F) -1.0000 -92.5390 -1.0000 -70.6660 -1.0301 (F : R) 1.3924 2.0970 1.0478 1.1569 0.8116 (R : R) -1.0000 -0.7515 -1.0000 -0.4223 -0.2962 aSee the footnote (a) in the Table X-1-1. 260 TABLE X-1-6.--Estimated Elasticities of Substitutiona--Molding Industry. Translog Prod. Funct. Translog Cost Funct. Demand Factors D.E.S. A.E.S. S.E.S. A.E.S. Elasticity (P : P) -1.0000 -15.6390 -1.0000 -7.0575 -0.8342 (P : A) 4.7683 24.8740 0.3379 -2.4180 -0.0742 (P : K) 1.3267 1.4944 0.7250 0.5434 0.1157 (P : F) 1.2371 1.9551 0.9125 1.6438 0.1216 (P : R) 1.3735 1.1026 0.8273 0.9455 0.5334 (A : A) -1.0000 -130.3600 -1.0000 -30.4150 -0.9337 (A : K) 2.4829 1.4562 0.4293 0.3168 0.0675 (A : F) 1.9839 1.0130 0.5781 1.4818 0.1096 (A : R) 2.7461 1.1993 0.4446 0.8896 0.5019 (K : K) -1.0000 -3.6647 -1.0000 -4.0058 -0.8530 (K : F) 0.9620 0.4104 0.7496 0.1094 0.0081 (K : R) 0.9718 0.9370 0.7548 0.8206 0.4630 (F : F) -1.0000 -13.1180 -1.0000 -10.8730 -0.8045 (F : R) 1.0469 1.1007 0.8814 0.9596 0.5414 (R : R) -1.0000 -0.7942 -1.0000 -0.7778 -0.4388 aSee the footnote (a) in the Table X-1-1. 261 and cost functions by industry. Each table consists of three parts. The first part in the production function contains the D.E.S. in their first column and the A.E.S. in their second column. The second part of the table contains the S.E.S. in their first column and the A.E.S. in their second column estimated from the parameters of the cost function. In the last column of the Table X-l, the elasticity of demand with respect to price changes is presented, which relates, denoted by "ij’ the proportionate change in the i-th factor quantity to the proportionate change in the j-th factor pricez4] n-. = -—-————-= -—--——- (15) The index of short-run responsiveness in factor substitution between the operative and the administrative workers, measured by the D.E.S.(P:A), is a positive value of 2.5982 in the production function of the canning industry, drastically different from 0.3563 of the S.E.S.(P:A) in its cost function. The differences between the D.E.S.(P:A) and the S.E.S.(P:A) may be interpreted as the different reaction of the producers in the canning industry upon the different situations, where the D.E.S. measures the producers' responsiveness in their optimizing behavior given the fixed levels of all the other inputs in given production technology and the 41For example, see Brown, Caves, and Christensen (1975), p. 26. 262 S.E.S. is measured given the fixed prices of all the other inputs on the same given production technology. 0n the other hand, the index of long-run responsiveness in factor substitution between these two factor inputs, measured by the A.E.S. (P:A) is -0.1482 in its production function, also significantly different from -2.1768 in its cost function. However, one thing in common in the production technology of the canning industry is that a proportionate increase of the operative workers reduces the input level of the administrative worker in the short- run, but it increases also the input level of the administrative worker eventually. This complementary relationship between two different labor inputs in the long-run responsiveness of factor substitution42 is identified by the negative sign of the A.E.S. (P:A) in the production function and in the cost function, not only of the canning industry but also all the other five industries 43 Hence the complementarity in these two selected in this study. labor inputs is well reflected in the measure of the demand elasticity of the operative worker with respect to the salary change of the administrative worker. That is, the demand elasticity of the operatives with respect to the administratives' salary 42The meaningfu1 interpretations on the sign of the A.E.S. are well explained in Ferguson (1971), pp. 107-100. 43Two exceptions are found in the production functions of the manufacture of briquettes and the molding industry, but still the cost function of these industry show the complementary relation- ship between the two 1abor inputs. 263 increase is measured as -0.0770 in the canning industry, as -0.2024 in the leather footwear industry, and -0.0716 in the screw products industry, etc. The short-run responsiveness in factor substitution between the labor inputs and the capital input, denoted by the D.E.S. (P:K) and the D.E.S. (A:K) in the production function of the Table X-l-l for the canning industry, are shown also quite different from those of the S.E.S. in the cost function. For example, the D.E.S. (P:K) is shown as of 1.2382 in Table X-1-1, while the S.E.S. (P:K) is of 0.8542. On the other hand, the long-run measures of the A.E.S. (P:K) and the A.E.S. (A:K) are relatively stable in the production and the cost functions, where the A.E.S.(P:K) is 0.9485 in the production function and the A.E.S. (P:K) is 0.7974 in the cost 44 At the same time, we find that the short-run responsive- function. ness in factor substitution between labor and capital inputs is more elastic than its long-run responsiveness, where the D.E.S. (P:K) is 1.2382, the D.E.S. (A:K) is 2.1636, the A.E.S. (P:K) is 0.9485, and the A.E.S. (A:K) is 0.7548, in the canning industry. The D.E.S. between labor inputs and capital input, greater than 1.0 and the A.E.S. between labor inputs and capital input, less than 1.0, are found not only in the canning industry but also in the 44But the stable measures of the A.E.S. (P:K) and the A.E.S. (A:K) between the production function and the cost function do not seem to hold in the other industries any more. For example, see the A.E.S. (P:K) of 1.1646 in the third row of the second column in the Table X-1-2 for the leather footwear industry and compare the A.E.S. (P:K) of 0.3209 in the third row of the fourth column on- the same table. 264 other industries.45 Also the A.E.S. (P:K) is greater than the A.E.S. (A:K), while the D.E.S. (P:K) is smaller than the D.E.S. (A:K) in the canning industry. This implies that the input of administrative workers is more elastic in factor substitution to the capital input than the operative worker in the short-run, but in the long-run the input of the operative worker becomes more elastic to the capital input in factor substitution. Again it is found not only in the canning industry but also in the most other industries.46 The demand elasticities of two labor inputs with respect to the price changes in capital input is estimated as 0.2414 and 0.2195 respectively in the canning industry. These labor demand e1asticities with respect to capital price changes are found to be the second largest, next to that with respect to the price change in the raw materials, not only in the canning industry but also all the other five industries. The D.E.S.'s between the labor inputs and the fuel input are found to be greater than 1.0 and their A.E.S.'s are less than 1.0 in the canning industry. But the A.E.S.'s between these labor and fuel inputs seem to vary very wildly from the function estimated to the industry se1ected. Also the elasticity of substitution 45Two exceptions are found in the leather footwear industry and the manufacture of knitted underwear. Also the A.E.S. (P:K) is smaller than the D.E.S. (P:K) in the molding industry. 46Only exception in this phenomena is found in the leather footwear industry with the A.E.S. (P:K) of 1.1646 and the A.E.S. (A:K) of 1.5272, and in the manufacture of briquettes with the A.E.S. (P:K) of -0.7432 and the A.E.S. (A:K) of 2.2060. 265 between capital and fuel inputs varies from industry to industry, reflecting its different production technology. Also the elasticity of substitution between fuel and raw material inputs is estimated as 0.8053 and 0.9481 in the production and the cost functions respectively, in the canning industry. The substitution elasticities of fuel input with respect to the other inputs seem to vary differ- ently from industry to industry. For example, the elasticity of fuel substitution with respect to the capital and the raw material inputs are greater than the substitution elasticity of fuel with respect to the labor inputs in the screw products industry, while those with respect to labor inputs are greater than those with respect to capital and raw material inputs in the molding industry. The direct elasticity of substitution of the raw materials for the operative worker is 1.3369, for the administrative is 2.1854, for the capital input is 1.1252, and for the fuel input is 1.2170 in the canning industry, which are all higher than their A11en partial e1asticities respectively. The distribution of these A.E.S. between the raw materials and the other factors ranges from 0.7669 for the operatives to 0.9061 for the capital input, showing rather small variations among them seemingly. These ranges are slightly different from industry to industry47 but they seem to be quite stable relative to the other substitution elasticities of 47For example, the leather footwear industry shows their ranges from 0.7480 for the fuel to 0.8316 for the administrative worker. And also the molding industry shows their ranges from 0.9370 for the capital input to 1.1993 for the administrative worker. 266 factors other than raw material input. This seemingly constant, around 1.0,if we can say, substitution elasticities of the raw material input may suggest the separabi1ity between the raw material 48 input and the other inputs, imp1ying again the usual separability between the raw material input and the value added to justify the 49 in the study of a production conventiona1 value added approach function. Even if the value added procedure can be justified under the assumption of the separabi1ity between the value added and the raw material input, still it is worthwhile to note that the A.E.S. of raw materials with respect to the other factor inputs are neither zero nor infinity, but around unity, as found in this study on the 48Berndt and Christensen (1973b) have established that 'separability restrictions are equivalent to certain equality 'restrictions on the Allen partial elasticities of substitution (A.E.S.). To illustrate, we note that the following are equivalent :restrictions for a production function of three inputs, i.e., y = F(X], X2, X3): (i) inputs x] and X2 are functionally separable from X3, i.e., f(x,. x2. x3) = H(JIX]. x2]. x3); (ii) equality of the A.E.S., i.e., the A.E.S. (X1, X3) = the A.E.S. (X2, X3) (iii) the existence of a cogsistent aggregate price index P* and a consistent aggregate index X with components P1 and P2, X1 and X2, ;respectively. See also Berndt and Christensen (1973a) on the application in their empirica1 works. 49The conventional value added approach in the production study has been discussed in detail, in the section 3.5, Chapter II, Part B. 267 six industries se1ected. Another interesting finding in this empirica1 estimation is that the demand elasticities of all the factor inputs with respect to the price change in the raw material input are also constant seemingly, around 0.5, not only in the canning industry but also in all other five industries. For example, the canning industry shows its range from 0.4293 for the capital input to 0.5201 for the raw material input itself. The estimated own-substitution elasticity of each factor is more or less stable between the production function and the cost function and it is drastically different from factor to factor. But in average the highest own-e1asticity of substitution is shown for the fuel input, the second for the input of adminis- trative worker, the third for the operative worker, the next for the capital input and the smallest for the raw material input, in all six industries. 0n the other hand, the own elasticity of factor demand is shown as the highest for the administratives, as the second for the fuel input, the third for the operative worker, the next for the capital input, and finally the lowest for the raw material input in the canning industry. And this seems to be more or less in common in the other five industries, except that the fuel and the labor inputs are reversed in their order in some industries.50 a 50The leather footwear industry shows the reversed order between the fuel input and the operative workers. But the screw products industry, the manufacture of briquettes, and the manufacture of knitted underwear show the highest own elasticity of demand for the fuel input, less than -l.0. 268 Finally, the cross-e1asticities of products transformation in the canning industry are measured around unity, i.e., slightly greater than 1.0 in the production function, and slightly less than 1.0 in the cost function. The bigger figures in the production function than in the cost function are in common in the three multi- products industries selected in this study. Also the cross-price elasticities of demand for the major commodities range from 0.1387 to 0.2298 in the canning industry and around 0.3 in the leather footwear and the screw products industry, indicating quite a stable responsiveness. 0n the other hand, the own-transformation elasticities of each product in the long-run show that the nonmajor commodities are more elastic than the first major commodities not only in the canning industry but also in the other two industries. Hence the own-price elasticities of commodity demand also show the lower (around 0.4) for the first major-commodity produced and the higher (around 0.8) for the less-major-commodities in the canning industry. This holds also in the other two industries. Hence the first major-commodity in these industries is less price- elastic then the less-major-commodities are. In summary, what we find from the evaluation of various E.S. estimated over the six industries selected in this study are the followings: (1) The D.E.S. between factor inputs are, in general, greater than the A.E.S. where the former reflects the short-run 269 responsiveness in factor substitution and the latter shows its long- run responsiveness. (2) The D.E.S. measured under the fixed levels of all the other factors, other than two factors concerned are found much bigger than the S.E.S. measured under the fixed prices of all the other factors, other than two factors concerned. This implies that the elasticity of factor substitution in general becomes much smaller when there exist a strong stability (or rigidity) in all factor prices. (3) The A.E.S. between the operative worker and the adminis- trative worker is of a negative value, implying that the complementary relationship between these two labor inputs hold in the production activities of all the six industries. (4) The A.E.S. between the operative worker and the capital input is quite different from the A.E.S. between the administrative worker and the capital input, implying that the hypothesis on the existence of a proper aggregate quantity (or price) index for the labor input as a whole should be rejected, based on the strong agreement on the separability between the operative worker and the administrative worker.5] (5) The conventional agreement on the unitary elasticity of capital-labor substitution may or may not be accepted, i.e., inconclusive in this study, since they vary from industry to industry but by and large they seem to range around unity. 515ee the footnote (48) on the separability restrictions. 270 (6) The D.E.S. between the operative worker and the capital input is smaller than that between the administrative worker and the capital input, while the A.E.S. between the operative worker and the capital input is greater than that between the administrative worker and the capital input. This implies that the factor substi- tution between the administrative worker and the capital input may happen strongly in the short run, but eventually after producers's full adjustment in the production process is done the factor substitution between the operative worker and the capital input become significant. (7) The factor substitution of the fuel input with respect to other input factors happen differently from industry to industry. For example, in some industries the substitutions become more elastic with respect to the labor inputs, and in some other industries with respect to the capital input and the raw material input. (8) The seemingly constant elasticities of substitution of raw material input with respect to all the other factors are found around 1.0 in all the six industries, implying that there may exist the separabi1ity between the value added and the raw material input to justify the conventional value added procedure in the empirical study on a production function, but still nothing is found to support the elasticity between the value added and the raw'material input be either zero or infinity. Our findings on this e1asticity seem to be more or less unity, instead of either zero or infinity. 271 (9) The own-factor substitution e1asticities are found the highest for the fuel, the next for the administrative worker, the operative worker, the capital input, and the lowest for the raw material input. (10) The own-price elasticities of factor demand are shown the highest for either the administrative worker or the fuel input, the next for the operative, the capital input, and the lowest for the raw material input. (11) The cross-price elasticities of factor demand are found the highest (around 0.5) from the price changes in the raw material input, the next from the price changes in the capital input, those in the labor inputs, and the lowest from the price changes in the fuel input. (12) The own-product transformation e1asticities are found to be higher for the nonmajor commodities produced in an industry than for the first major-commodity. (13) The own-price elasticities of each product demand are also found to be higher for the nonmajor commodities produced in an industry than for the first major-commodity. (14) The cross-product transformation e1asticities are found quite stable, ranging around unity, in the three mu1tiproducts industries. (15) The cross-price elasticities of product demand are found stable, ranging around 0.3, in the three multiproducts industries. 272 2.4.3. Share Elasticities with Respect to Quantity Change§,and to Price Changes 7 .i’ 6ii translog production function can also be interpreted as estimated The estimates of the Yi and 6ij parameters in the share e1asticities with respect to quantity changes. The cost a 1n F share of input i is equal to The cross partial derivative 3 1n Xi 32F = y.. can be defined as a constant share e1asticity 13 3 1n xi 3 1n Xj summarizing the response of cost share Si to a change in 1n Xj. Alternatively the share elasticity can be defined as = -—— (l6) 3 1n Si Yij . S. 3 1n XJ 1 In the latter case, the estimated share elasticities at the means of the data will be equal to the estimates of Yij/ai' In the same way, the estimates of the same parameters in the translog cost function can be interpreted as estimated share elasticities with respect to price changes and the alternative definition of the share elasticity with respect to price changes can also be defined simi1ar1y. Tab1e X-2 contains only the alternative measures of the estimated share elasticities with respect to the own quantity changes and with respect to the own price changes respectively. .zg»m=e:O geem gO mmmgegm »»O»ge:e e» »ememmg g»O3 O»OeO»meOm mgegm m .»e=eege »OLOO mg» mgeme Om ege »Om>O»emeOmg Ommgege muOge O»O 0» age Ommgege z»O»ge=e O»O e» »ememmg g»O3 O»OeO»me»m mgegm m .gegeO m>O»enge mg» Ogems Om .mOeEexm Lem .O»=e»=e ege O»eecO ge»eeO geem e» OmO»OeO»OeOm mgegm mgOeceOmmggee mg» m»egme mOge» .mg» gO OOegEOO mgOe ONOO. OONO. ONOO. OOOO. OO»O. »NNO. Om OON». »OOO. OONO. OOO». NOO». O»OO. Om NNON. N»ON. OONN. OOOO. OON». NOO». Om ONON. OOOO. »ONN. OO»N. O»NO.» ONOO. Om OOON. OONO. OONO. OOOO. OOOO. OO»». Om - Ommgegu meOgO gzo mg» e» »emmwmm g»O3 he -- -- -- -- -- OO»O. OO 9. . . . N -- -- -- OOOO OOOO O»»O m -- -- -- OOOO. OOOO. NOOO. »O OO»O. ONOO. OOO». ONOO. OOOO. NNOO. Om OONO. OOOO. ONNN. O»O». OOOO. OOO». Om OO»O. OOOO. OO»». OOOO. OOO». OOOO. Om NOOO. NNON. »OOO. OONO. OOOO. ONOO.» O. O»OO. OO»O.» »ONO. OOOO. O»ON. OOON. OO Ommgegulx»O»geac gzo mg» e» »ememwm g»O2 »g»m=egH Om»»mecOgm gengmeg: O»»Oeegg »emz»eem O»»maeg» mgOeOez em»»ng segum gmg»emg mgcheu 1V| .Ommgegu meOgO gzo mg» ege Ommgegu »»O»geeo gzo mg» e» »emgmmm g»O3 OmO»OeO»meOm mgegm--.N-x mgmO»eOmg mg» me emgOOme mO egev O.ge»eeO geem an em»ngm3 OgeO»eOcm mgegm O»ea»=e egev »OO mg» mge OgeO»e:em mgegm A»:O»=e mg» egev »eegO mg» geO m muOOeea mg» Oe mmegm> .»»Og: EegO em»eeg» miOOeee mg» mgmgz .mgegm »O.»OO»ee O mg» OO O. miOmeOe mg» Oe mmegm>e em»ngmz mgOe .N mNmO.o OOON.o meO.o mONo.o one».o OONo.o »OOO» »O V OO»OOOOOTOOOO»O2 O»»O.o mNmO.o NOO».O moom.o OONO.o ONON.o »Oeg» »O V mm»»m:uOgm NOO».O NNON.O ONN».O NNO».O OONO.O OOO».O »OOO» »O O gengmecg em»»ng Nmmm.o m»om.o mOOO.o Ommm.o mOom.o momm.o »OO»=o »OOV OOON.o omON.o oomN.o OOOo.o OmmO.o mono.o »Oeg» »O V O»e=eegg 3mgem OOON.o NOON.o OOOm.o OOON.o OmmO.o NONm.o »OO»Oo »OOV OmNO.o OOON.o NomO.o mmmm.o NOOO.o OOOO.o »Oeg» »O v .xu»m:egwlmgOggeu mmOOEem enema Ozego OmOOsem geese Oeegw Og»m:egH »e»eO emNOm- emNOm- Oe»eO emNOm- emNOm- mmgeg OOeEm mmgeg OOesm geO»eg=g »Oeu geO»eg:g geO»e:eegO (1|) Ir. e.m.~m-OOe=o mg» Oe mmegm>< em»ngm2 mgO--.O-Ox mgmgp 279 small-sized and 0.7354 for the 1arge-sized groups in the production function of the canning industry, while it is 0.8191 and 0.7357 respectively in the cost function. As shown earlier, it was 0.7975 in the production function and 0.7969 in the cost function for the total-estab1ishments of the canning industry. As a result, the poor fit shown in the estimations of the input share equations for the total establishments in each industry seems to come mostly from the poorer fit for the small-sized estab- 1ishment group, where the differences of the weighted average of the quasi-Rz's in the output share equations between the small- and the large-sized groups of the mu1ti-output industries are found to be very negligible. This may imply that the very concept of either a production function or a cost function become poorer when it is applied for a moderately small size of establishments.55 3.1.2. Properties of the Functions Estimated The hypotheses of linear homogeneity and symmetry are more specifically accepted in the different size groups of establishments within each industry, compared with those in its total establishments. For example, judged by the likelihood ratio test, those hypotheses are accepted at the significance level of 1% in the estimations of both the production and the cost functions for the 1arge-sized group of the canning industry. The details of these test statistics are shown in the Table XI-2. In addition, these hypotheses are accepted 55This may become more convincing when the significance test of the function estimated is discussed. 280 .xg»m=egO guem Oe O»nggOOOge»mm Oe»e» mg» ege emNOmimmgeO mg» .emNOmiOOeEO mg» ..m.O .OmOeEem »gmngOOe mmgg» gm>e OgeO»eEO»Om m»egenmm mg» go emOeOEO Og»meeam ege O»Omgmmeseg gemgO» Oe OgeO»eOg»mmg mg» geO mge OOO»OO»e»O »Om» mgOe Omo.mm mOm.mO mO0.0» mom.m~ oO geO»ug=O »Oeu Ouw.~m mmo.mO oO~.OO mo~.m~ o» geO»egem geO»e=ee»O xy»meeg»imgOeOez oom.m~ NO0.0~ Nmm.Om mom.m~ oO geO»egeg »Oeu Omm.~O Omw.ON mmO.mO mom.m~ oO ceO»eg:g geO»eOeegO Om»»m:aOgm ONO.»m Omo.O~ mON.OO mo~.m~ oO geO»eg:u »Oeu Oom.mO OO0.0N NmO.N mom.mm o» geO»egeg geO»e=eegO gengmegg em»»ng omo.~O mmm.OO mmm.oe mom.Om om geO»eg=g »meu ooe.OO moe.~m www.me mem.Om om geO»eg=g geO»u=eegO m»e=eegO 3mgem ome.~m omm.wm mOO.mm NOO.mO NN geO»eg:g »meu NN0.00 mem.mm mOm.oO NO0.00 ON geO»egeg geO»e=eegO xu»meeg» mgOggeu OmOOEem Oeegu Oeego Oo.o n O »OV Oe»eO emNOm- emNOm- O Eeemmgg mmgeg OOeEm »e O va Oe Oe meOe> mmgmmo meO»OO»e»m »OmO OeeO»Ogo O.OOOOOOOOOO »OOO OO»OO OOOO»»OOOO--.N-OO OOOOO 281 in the estimations of the production function for the large-sized estab1ishment group of the screw product industry and the molding industry, and also are accepted in those of the cost function for the 1arge-sized group of the briquettes and the molding industries.56 0n the other hand, the convexity conditions of the isoquants are also well satisfied in the production and the cost functions estimated for two groups of different estab1ishment size in the most industries, particularly for the 1arge-sized group in all 57 while the monotonicity conditions seem not to five industries, be improved in terms of the absolute number of sample units in any industry. As a result, a production technology in the industries selected in this study may be significantly different from the different size of establishments included, when the results are combined from the goodness of fit, significance test of the restrictions imposed and from the properties of the technology estimated by two different establishment groups.58 But not 56One exception is found in the manufacture of knitted underwear. Both in the production and the cost functions for the small-sized groups of the industry are these restrictions accepted significantly at a = 0.01, while in both functions for the large- sized group are these rejected. 57The non-convex isoquants are found in both functions esti- mated for the small-sized group of the three uniproduct industries, i.e., the knitted underwear, briquettes and molding industries. 58This statement can be verified when an additional test is employed which is not covered here. That is the so-called Chou- test, defined as Fh+m_2k = [(SSEt-SSEn-SSEm)/k]/[(SSEn+SSE )/ (n+m-2k)], where SSE is the square sum of errers, n, m, an t are the'sample size of the small, the large-sized and the total estab- 1ishment groups respectively. And k is the number of explanatory variables. Also see Kmenta (1971), p. 373. 282 necessarily all these differentials are attributable to the effects of returns to scale naively, since these results in this study are very restricted by the constant returns to scale in the technology estimated. 3.1.3. Parameter Estimates The scale effect on output elasticities of factor inputs is shown in the Table XI-3, in terms of the sign in direction which changes as the firm size gets bigger. Output elasticities of two labor inputs as a whole decrease as the size of establishments becomes larger in most industries, but the manufacture of knitted underwear, the characteristics of which was revealed by a low capital-labor use ratio and relatively lowest wage rates for 59 On the other hand, the scale effect on the capital workers. input share is found to be positive in all industries. Together with the negative sca1e effect in the labor shares, the share ratio between capital labor inputs are increasing in all industries, 60 This may imply that most industries selected in this in general. study reveal themselves to be capital-using and labor share- decreasing as the firm size becomes bigger, The scale effect on the share of raw materials in total input are all negative, and that of the fuel consumption is also 59About the detail characteristics of this industry, see the subsection 2.6., Chapter II, Part B. 60One exception is found in the cost function of the knitted underwear. 283 .mmgm>mg mg» Ocems ngm Riv m>O»emm: mg» ege .gmmgeO »mm mNOO »nggOOOge»mm mg» me mgOOemgegO mge emggmegee mgegm me»eeO mg» »eg» Ogeme ngm O+v OOOO mg» eOO< .OOm>O»umOOmg OgeO»eg:O »mee mg» ege geO»e=eege mg» m»egme =.O.e= ege =.O.e= .mOge» mg» gHe + + + + u + + + + + gone» e» »OOOOOO OO eO»ea mgegm . + u u u i i n u . OOeOgm»ez 3em . i u i u u + + + + mOmzm + + + + + + + + + + mOe»OOeu - - - - + + - i - - Ogegeg Oe»eO . i i i + + + + + . mm>O»eg»mOgOEe< . u n i + + u u - . Om>O»egmeo oF-U o$oQ .L..U o$ofl oFoU onvon— .%.U onvoa .%.U tuna LOHUCK Og»mzegH mm»»m:eOgm gengmegg O»e:eegm Og»meecO mgOeOez em»»Og¥ 3mgem . mgcheu 1’ LI . [ L’l O.OOOOOOOOOO»O OOOOOO OO OOOOOOO O»OOO--.O-»x OOOOO 284 all negative, except in the canning and the screw product industries which have relatively low capital-fuel use ratio and cheaper fuel prices in common.61 0n the other hand, there does not seem to be any uniform tendency in the scale effects on the partial elasticities of factor substitutions (or output transformations), as shown in the Tables III-l through III-5 of the Appendix B. In particular, the comple- mentary relationship between two heterogeneous labor inputs does not hold for some of the 1arge-sized estab1ishments either in the production or the cost functions, i.e., of the canning, knitted underwear and the briquettes industries. Factor substitution between operatives and capital input becomes slightly more elastic for the large size of establishments than for the small size, either in the production function of most industries (except the molding industry) or in the cost function of most industries (except the manufacture of knitted underwear). The fact that factor substitution elasticity between labor and capital inputs becomes higher as the firm size gets bigger, implies that the shape of isoquants reflected in the labor-capital subspace becomes steeper as the firm size (or alternatively speaking, the level of output) becomes bigger, when it can be said safe1y that the level of output in establishment becomes higher as the firm size measured by the number of workers gets bigger. 6]See the footnote (59). 285 Further analyses on the relationships between the elasticities of factor substitution and the output level (or the factor use ratio) could be done more deliberately in the sense that the substitution elasticity may be significantly influenced and directly determined 62 But either by the output level or by the capital-labor ratio. these experiments are postponed to later researches and not covered in this study. 3.2. Alternative Exclusion Rules in Sampling The current empirical study on the micro-rea1ity of produc- tion technology in the Korean manufacturing industries is very much conditioned by the exclusion rules in the sense that the exclusion criteria from data availability of capital input reduced sharply the number of sample establishments in each industry.63 The absence of informations on capital input (such as the horse- power equipment or the net capital stock) for certain establishments in the manufacturing census can be interpreted in three different ways. First, the establishments have significant level of capital input in their production process but not reported in the census 62The hypotheses that the common, constant elasticity of factor substitution is a function of either the level of output or the capital-labor ratio, have been suggested by Revankar (1971), Lu and Fletcher (1968), Sato and Beckmann (1968) and Lovell (1973). The detailed discussion on this topic was already discussed in the section 3.1., Chapter I, Part A. 63Refer the section 2.4., Chapter II, Part B. 286 questionnaires, i.e., the case of the missing data. Second, the establishments have some but very neligible amounts of capital input hence recorded as if they have zero capital input. Third, the establishments have no capital input in their production process. The first two possibilities bring us to the problem of errors in measurement, i.e., "underestimation" of capital input. The inclusion of such an establishment into the empirical 64 estimation also implies that the results would be consistent with those of aggregate production (or cost) function at a certain macro (or sector) level based on highly aggregative data.65 The number of total sample units covered in this inclusive case is 79 (i.e., 11 more than before) establishments in the canning industry, 206 (i.e., 186 more) establishments in the leather footwear industry, 104 (i.e., 20 more) in the screw product industry, 139 (i.e., 25 more) in the manufacture of knitted underwear, 262 (i.e., 64The inclusion of such an establishment in the estimation of the translog function again involves the problem of the log- transformation of the zero-valued variable, discussed already in details in the section 3.5., Chapter III, Part 8. Hence the actual estimation in this section was done after replacing this zero- valued varaable by certain neglibibly small figure, i.e., .0 X 10’ . 65This can be valid in the sense that many empirical studies on aggregate production function are based on the capital data, which again come from the same census data file. Hence the aggre- gative capital data based on the census file are usually measured by the simple summation of capital input of all estab1ishments recorded in the file. 287 29 more) in the manufacture of briquettes and 152 (i.e., 21 more) establishments in the molding industry. Hence the increment of the sample size due to the different exclusion rules is rather minor in most industries, except the leather footwear industry. 3.2.1. Goodness of Fit On average, the quasi-R2 of each share equation, their simple average and their weighted average in the inclusive cases are worse than those in the previous exclusive cases for most industries. In particular, the quasi-R2 in the capital share equation changes as drastically as expected in each industry. 2 of the capital share equation Tab1e XII-1 includes only the quasi-R and the weighted average of the quasi-Rz's of input- and output- share equations. 3.2.2. Properties of the Functions Estimated The imposed restrictions of linear homogeneity and symmetry on the functions estimated are more strongly rejected by the likeli- hood ratio test, for the inclusive cases than for the exclusive cases in most industries. 0n the other hand, the number of sample estab1ishments, not satisfying the monotonicity conditions in any share equations, is increasing slightly but decreasing as apercentage for the inclusive cases, relative to the exclusive cases in most industries. And the convexity conditions are satisfied in the inclusive cases similarly as in the exclusive cases in most indus- tries. Hence the properties of the functions estimated do not seem 288 .»-»x O»OeO OO» OO »eg OOOOOOOO oz» meme i- mNmO.o oemm.o .. ammo.o ONOo.ou mmeu m>OmeOexm -u Omoo.o OOOo.o .. NmNo.o ammo.o mmeu m>OmeOegO Og»meegH mgOeOe: .. NOO».O mme.o .. ONmO.o mNoo.o mmeu m>OmeOuxu .. mmmo.o mono.o .. »OOO.o oOOo.o mmeu m>Om=OugO mm»»mmmOgm u- OO»O.o mwm~.o .. moom.o mmOo.o mmeu m>Om=Oexu in OO»0.0 mmoo.o n- OOOO.o omoo.o mmeo m>OOOOeOH gengmegg em»»ng mmmm.o OOm~.o Ommm.o »mmm.o memo.o mOoo.o mmeu m>OmeOexm ommm.o Ommo.o mooo.o ~mmm.o mmeo.o mOOo.o mmeo m>OOOOUgH m»u:eegg 3mgem NOOm.o com».o Omo~.o OOOO.o oONm.o NomO.o mmeo m>Om=»exm mmOO.o Nmmo.o Owoo.o NOON.o mmmo.o Omoo.o mmeo m>OmeOegH »em3»eeg gmg»emg momm.o OwOO.o mOmO.o ONON.o mmm~.o OOoO.o mmeu m>OmeOexm omom.o OooO.o OOmc.o Oeow.o o~O~.o mmmo.o mmeu m>Om=Ouc» Nu»meeg» mgOggeu »ee»eo »Oeg» geO»e:cm »Oe»eo »egg» geO»eeem mgegm mgegm mmegm>< em»ngm3 OO»OOeu mmegm>< em»ngm3 Oe»Oeeu geO»ugeg »meu geO»ec=O geO»e=eegO e.ag»meegH Ag O.N O-OOOOO OOO--.»-O»x OOOOO 289 to have much difference between the inclusive and the exclusive cases, except that the goodness of fit and the significance of the restrictions imposed are influenced significantly from the inclusion of sample establishments with zero capital input. 3.2.3. Parameter Estimates The effects of the inclusion of establishments with zero capital input on the parameter estimates of a production technology seem to be so variant over the six industries, as shown in the Table XII-2. Particularly in the leather footwear industry where the most drastic change occurs in the sample size with alternative exclusion rules, the increased share of capital input in the esti- mated production function is accompanied, together with the increased labor share, with the drastica11y decreased share of raw materials, compared to those in the exclusive case. On the other hand, the decreased capital shares in the cost function is accompanied with the increased share of raw materials, leaving labor shares as same as in the exclusive case. Apparently, the inclusion of those estab- 1ishments with zero capital input into the estimation samples results in the lower average level of capital input in the production function and higher average price of capital input in the cost function than the previous exclusion rules result in. But the effects of these possible errors in measurements of capital input on the capital share seem to be inconclusive across the industries selected in this study. 290 oOmm.o Ommo.o NOO~.o mm~o.o mmoO.o geO»eg=g »Oeu memm.o mmmo.o OmO~.o Ommo.c Oomo.o :eO»egeg geO»eeeegO zg»maeg» mgOeOez NooO.o mOOo.o OwOO.o memo.o mmOo.o :eO»egeg »meu mNOO.o ONOo.o omOO.o oOmo.o ammo.o geO»eg:g :eO»e=eegO mm»»m:aOgm momm.o meo.o mOm~.o Ommo.o mmmO.o geO»eg=g »Oeu Ommo.o oomo.o oemm.o omOo.o mOmo.o :eO»eceg geO»e=ee»O gengmega em»»Og¥ mmom.o omeo.o mmm~.o ammo.o ONOO.o geO»eg:g »Oeo mNNO.o oemo.o Namm.o meo.o mOmO.o geO»eg:g ceO»e=eegO O»eeeege 3mgem NONm.o wOOo.o moO~.o OO»0.0 OoOO.o geO»eg=u »Oeu mmom.o mOOo.o OONN.o mmOo.o NNmO.o geO»eg:m geO»e:eegO gem3»eem gmg»em» momm.o memo.o mem~.o mmmo.o OONO.o geO»egeg »meu mmOm.o OOOo.o mmmm.o memo.o OO»O.o geO»eg=g geO»eeeegO Nu»m=egO mgOggeu OOeOgm»ez Omen »egg» mm>O»eg» mm>O»engo sea »OOOOOO -OOOOEO< “trl "1‘1. ltfif r .OOOOEOOOO OOOOOO OOOOOO--.N-HOO OOOOO 291 More interesting results are found in the estimated elastici- ties of factor substitutions. The complementary relationship between operatives and administratives, as shown in the Tables IV-l and IV-2 in the Appendix B, are also identified by the negative sign of the A.E.S. between two factors in each function of most industries. Secondly, the elasticity of substitution between either of two labors and capital input approaches to unity more closely in the inclusive cases than in the exclusive cases for most industries, leaving its own substitution elasticity almost the same as before.66 This finding may have some implication for the popular hypothesis of unitary elasticity of substitution between labor and capital inputs in most empirical studies on aggregate production functions of manufacturing industry. Since the inclusive cases here have the same data base as have most studies done at the aggregate level from the census file, the above result may not be ignored as trivial. Hence the conventionally accepted, unitary elasticity of substitution between labor and capital may be partly attributed to the use of the aggregate data base where problems of errors in records, particularly the undermeasurement of the capital variab1e, prevail. The elasticities of substitution of raw material with respect to other inputs in the inclusive cases are again found to be within some stable regions around unity. In summary, the * 66Two exceptions are found in the cost functions of the manufacture of briquettes and the molding industry. See the Table IV-2 of the Appendix B. 292 empirical estimations in the inclusive cases are not particularly preferable unless there exist significant improvements either in the goodness of fit or in the significance of the restrictions acceptable conventionally, or in eliminating possible errors in measurements, in addition to a simple merit of using a large sample size. 3.3. Alternative Variables of Different Quality The estimations with alternative variab1e sets of different quality are investigated in this section, which contains 4 sets: the set A of the man-day workers for labor inputs and of the power equipment for capital input, the set 8 of the man-day workers and of the net capital stock, the set C of the number of workers and of the power equipment, and the set D of the number of workers and of the net capital stock. The resulting sample size is 66 establish- ments in the sets A and C while 45 establishments in the sets B and D of the canning industry, 20 in the sets A and C while 13 in the sets 8 and D of the leather footwear industry, 94 in the sets A and C while 15 in the sets 8 and D of the screw product industry, 144 in the sets A and C while 10 in the sets B and D of the manu- facture of knitted underwear, 233 in the sets A and C while 32 in B and D of the manufacture of briquettes, and 131 establishments in the sets A and C while 55 establishments in the sets B and D of the molding industry. Hence the reduction of the sample size occurs 293 very drastica11y, depending on the choice of capital variab1e between the power equipment and the net capital stock. 3.3.1. Goodness of Fit The estimated quasi-Rz's vary from the share equations to the function considered in the industries. Measured by the simple (and the weighted) average(s) of the quasi-Rz's of each share equation, the estimations of both the production and the cost functions with the net capital stock (i.e., the sets 8 and 0) are found to fit better than those with the power equipment (i.e., the sets A and C) in most industries. 0n the other hand, no discrimi- nations between the man-day workers and the number of workers (i.e., the sets A and C v.s. B and D) are found at all. 3.3.2. Properties of the Functions Estimated The significance test of the hypotheses of linear homogeneity and symmetry are found to be different from the industry concerned to the function estimated. For example in the canning industry, these hypotheses in the sets 8 and D are more strongly rejected in the production function, but well accepted in the cost function where the likelihood ratio statistics are 33.492 and 36.615 re- spectively, compared to the critical value of 45.642 at the signifi- cance level of 1%. 0n the other hand, in the leather footwear industry the hypotheses in all four sets are accepted in the 294 production function but rejected in the sets 8 and D of the cost function.67 The monotonicity conditions in both the production and the cost functions are satisfied better in the sets A and C in terms of the percentages of such estab1ishments among total sample units. 3.3.3. Parameter Estimates The output elasticities of two labor inputs are estimated relatively bigger in the sets A and C than in the sets 8 and D, in general. Hence the capital shares are found to be smaller in the sets A and C than in the sets 8 and D of the screw product, 68 Also this knitted underwear, briquettes and molding industries. alternative choice for capital data have shown more significant changes in the estimated cost function than in the estimated production function, implying that the choice of capital variab1e here results in more drastic changes in factor price than in the level of capital input. These results may indicate the direction of a bias in the parameter estimates, if the level of net capital stock recorded in the data file is a true measure of capital input used in the production process, instead of power equipment. 67Here the hypotheses are accepted in the sets A and C of the cost function where the test statistics are 31.756 and 31.083 respectively, compared to the critical value of 37.566 at the significance level of 1%. 68One exception is the leather footwear industry where the higher shares of labor and capital inputs in the set A are shifted into that of raw materials in the set C. 295 On the other hand, the discrimination in the choise of two labor inputs (i.e., utilization rate of labors) does not result in any significant differences in the estimated A.E.S.'s in general, as shown in the Tables V-l through V-6 of the Appendix B. This implicates that the utilization of labor forces, measured by working hours, may be quite similar across most estab1ishments within each industry. But a1ternative choice for capital data results in some differences in the estimated A.E.S.'s. First, the complementary relationship between two heterogeneous labor inputs does not hold for the sets B and D in both the production and the cost functions of the manufacture of knitted underwear. Second, the A.E.S.'s between operatives and capital input are higher, above unity, in the sets B and 0 than in the sets A and C for the production function of most industries. Finally, the A.E.S.'s of raw materials with respect to other inputs are also stable in all four alternative sets. In summary, no meaningful gains are found from the selection of net capital stock for capital variab1e, where no decisive improvements are noticed either in the goodness of fit, or in the validity of the popularly acceptable hypotheses, or in the satis- faction of certain properties of a production technology, which may well compensate its clear disadvantage of drastic reductions in sample estab1ishments to be used in empirica1 estimations. CHAPTER V. CONCLUSIONS AND RECOMMENDATIONS The subject of a production technology is one of the areas of economics where the gap between theoretical formulations and empirical knowledges is still quite wide. This is why the nature and magnitude of scale, share and substitution parameters continue to attract research interest. Furthermore, there seem to have been few theoretical and empirical studies on the mu1ti-input mu1ti-output production technology until recently. Hence the very concept of the mu1ti-input mu1ti-output production function, together with the multi-input multi-output cost function via their duality relationship, have become popular in most of recent theoretical studies in the theory of production and a few empirica1 knowledges are being accumulated in its beginning stage, mostly limited to a very aggregate (or sector) level. The present study has purported (i) to understand the theory of a multi-input mu1ti-output production technology as an extension of a theory of two-input uni-output production technology, (ii) to investigate the workability of the mu1ti-input multi-output production theory, using a cross-section data system of the Korean manufacturing census, (iii) to find some knowledges on the first and second order properties of a production technology, using the translog approach, at an establishment level which is close to the 296 297 reality of most manufacturing activities, and finally (iv) to collect informations on the usefulness of the Korean manufacturing census data system which is quite a common type of data system in most of the other countries. It is hard to summarize and evaluate such a study as ours succinctly, particularly when it occurs for only a few, specific industries, which have seemingly few substantive characteristics in their production technologies in common. If one judges by one of the major purposes of this study, the first and second order properties of the multi-input mu1ti-output (and also uni-output) production technology, the returns have been moderately high from using the translog approach. Also another of our purposes was to enter into and analyze a body of data at a level of disaggregation rarely encountered before. Micro-data at the establishment level are largely terra incognita for economists. While the promise of great discoveries was not met, we did learn something about the structure of production in a few of Korean manufacturing industries, and much more about the structure of such data and problems that they pose for the analyst. This knowledge should be helpful to other research workers who will undoubtedly want to continue exploring such data. We shall divide our concluding comments into two parts: a review of the empirical findings and lessons for further research. The validity of the empirical findings should also be limited to the six industries selected in this study in principle. 298 In the estimation of the share equations system in each industry, the degree of factor market imperfection is found to be so high that the producers' decisions on factor demand deviate very widely from the first order conditions for either profit maximization or cost minimization. This implies that there exists quite a high degree of input hoardings in reality in the sense that considerably high portions of factor demands are not price- responsive and can not be adjusted appropriate1y during given production period. This phenomena seem to be more serious in the small-sized estab1ishments, as recognized from their poorer quasi-Rz's in the input share equations appeared in our supplementary investigations. 0n the other hand, most properties considered here seem to be acceptable in the estimated production technologies of most industries, even if the exact acceptances of the hypotheses of linear homogeneity, symmetry and separability between inputs and outputs are rather inconclusive such that it partly depends on the choice of test statistics between the likelihood ratio test and the F-test discussed earlier and it also varies from the industry concerned to the function estimated. Further, the monotonicity and convexity conditions are found to be well satisfied in general, particularly better in the estimations of the translog cost function than in those of the production functions. But the shape of the output transformation curves in three mu1ti-output industries is not found as usual in the sense that they are all convex to the 299 origin. Two possible explanations are also investigated in terms of the existence of output heterogeneity and of technical changes between the uni- and mu1ti-output estab1ishment groups in each industry. The first order properties of the mu1ti-input multi-output (and uni-output) production technologies are again synthesized in terms of the output elasticity of factor input and output share. And the second order properties of the technologies are investigated in terms of various elasticities of factor substitution, of e1astici- ties of output transformation and of share elasticities. First, the wide differences in the estimates of factor shares between the production function and the cost function in some industries made us to reject the self duality between two functions. This also implies that the validity of the naive Cobb- Douglas form should be rejected for the specification of a tech- nology proper. This is again supported by the null hypothesis test on the second order terms of the translog functions where most of them are significantly rejected from being zero. Second, the labor share ranges from 0.15 to 0.20 in most industries, and the capital share ranges from 0.18 to 0.25. Hence the labor share in value added ranges from 0.38 to 0.53 and the capital share from 0.47 to 0.62. Third, the share of raw materials varies wildly from industry to industry, ranging from 0.4 to 0.7 at most. This implies that the conventional hypotheses on the role of raw materials should be 300 rejected and hence the so-called va1ue-added approach should also be tested, not g_prjgrj_given assumption that the value added can be dealt properly as a good proxy for output measure in the empirical study of a production technology. This suspecion is well supported again from the findings such that the substitution elasticities of raw materials with respect to other factor inputs are close to neither zero nor infinity but unity. Fourth, many interesting results are found in the close looking at the estimated, various elasticities of substitutions, the least of which are as follows: (1) All the D.E.S.'s are greater than the A.E.S.'s. (2) There exists a supplementary relationship between operatives and administrative workers, holding a strong separability condition between them. (3) The elasticity of substitution between labor and capital inputs are around unity, but not very conclusive. (4) The substitution elasticities of raw materials with respect to other inputs are around unity. (5) The own substitution elasticities of factors are the greatest in fuel, the second in administratives, operatives, capital and the least in raw materials. And this ordering holds also for their demand elasticities with respect to their own price changes. (6) The cross-demand elasticities with respect to other factors's price changes are the highest for raw materials, the second for capital input, labors, and the lowest for fuel input. 301 (7) The higher own transformation e1asticities are found for the non-major output(s) of the industries than for the major output. This ordering also holds for the own-supply elasticities of outputs with respect to their own price changes. (8) The cross-output transformation elasticities are around unity, while the cross-output supply elasticities with respect to other outputs's price changes are around 0.3 in three mu1ti-output industries. Fifth, several results from the analyses on the share elasticities of factor inputs are also shown in this study. The least of them is that the share elasticity of capital input is greater with respect to the level of capital input changes than to its price changes. Further results are included in this volume from the supplementary estimations, supporting the main findings above mentioned. First, much poorer fits are found in the estimation of two translog functions for the small-sized estab1ishment group within each industry, even bringing us a suspecion on the very concept of the production or the cost functions for this group of establishments in manufacturing activities. On average, the share ratio between capital and labor inputs are increasing from the increasing capital share and the decreasing labor share, as the firm size gets bigger. Also the substitution e1asticities between two factors are getting bigger in most industries. Like the labor share, the shares of 302 fuel and raw materials are decreasing as the firm size gets bigger in most industries. Second, no significant gains are found from the close investigations on the estimation results inclusive of such estab- lishments with zero capital input, but much poorer fits in general in all six industries. Again one interesting result is noticed such that the estimated substitution e1asticity between labor and capital is very close to unity in both the production and the cost functions of all six industries. This may implicate that the most popular finding of unitary substitution elasticity between labor and capital in an aggregate production function may be partly attributed to the use of such an aggregate data that includes establishments of zero capital input in samples. Third, the alternative estimations covering several variables of different quality do not seem to result in any significant differences in the properties of the functions estimated in general and suggest no specific preference in choosing a1ternative labor and capital variables in this empirical estimation. Specifi- cally, the choice of the number of workers for labor input does not seem to be preferable not only from any improvements found in the actual estimations but also from the very concept of labor input in a production process. The choice between the horsepower equip- ment and the net capital stock also does not show any significant, discriminating results, except such a drastic reduction in the estimation sample size in this study. 303 The suggestive directions for further researches are not only restricted to the improvement toward more profound economic analyses and to the specification of production technology for empirical works, but also to the refinements in data problems. First of all, several interesting analyses can be designed, based on the results from this study. First, the relationships between the factor substitution e1asticity (in particular, with respect to labor and capital inputs) and either the level of output or the factor use ratio (i.e., capital-labor ratio) have been questioned in many production studies. This can be focused rather easily here but with heavy computationa1 burdens involved, since the translog functional forms assume the variable elasticities of factor substitutions over different sample establishments in the industry concerned. One immediate suggestive work can be formulated for testing a function of capital-labor substitution e1asticity with explanatory variable of either the level of output1 or the capital-labor ratio. Second, the complete system of a production technology which consists of the production function and certain conditional equations derived from the first order conditions for profit maximization, can be estimated for further knowledges on returns to scale, based on the same data system. 1The level of output can indicate either the quantity data or the amounts in the unioutput case, but only the amount data can be used for the level of output in the mu1tioutput case. 304 Third, certain investigations on technological changes can also be possible by introducing directly the business-beginning year for time variable into the translog functions. As already shown in the previous section, for example, a significant evidence was noticed in our three mu1tioutput industries that the average age of single output-producing estab1ishments is moderately smaller than that of multiproducts-producing establishments. Fourth, the validity of the functional forms such as CD, CES and many variants of VES, can be identifiable by either checking the relationships among the parameter estimates of the translog functions, or by testing the significances of alternative estimations under various restrictions implied by those specific functional forms. Fifth, in this study the translog production function and the translog cost function are estimated and compared each other due to their dual relationship. But also the translog profit function and the translog revenue function can be estimated and verified together, since the duality among these four functions are well specified already in some recent theoretical works and there exist many data available for quantity and price of various inputs and outputs in the Korean manufacturing census system. Sixth, the specification of disturbance terms are defined for a deviations in producers' decision from the optimal factor share decision, since we adopt the estimation method of linear regression for the share equations system. More exact specification 305 of errors in the first order conditions of profit maximization, will certainly introduce some nonlinearity in the function to be estimated, as already clearified. Hence an introduction of a proper, nonlinear estimation method can be helpful for estimating further the new translog share equations system which is consistent with the specification of errors in the first order conditions. Seventh, since one of the most weak data point in this study is capital input, it may be also worthwhile to formulate and estimate a production technology which does not have capital input specifically, definable for the short-run technology. That is, capital input can be viewed as the nonproducible input L in the earlier specification of a mu1ti-input mu1ti-output technology, and hence the following relation can be shown; Xk = F(Xp,XA,XF,XR, Y],Y2,Y3). Eighth, all five production inputs are defined, in this study, in a horizontal way such that they are all variable with respect either to establishment or to product. But in the reality of a production technology, this may not be valid a1ways. In general we can classify three categories for factor inputs, i.e., the factors variable for firms such as labors and fuels, the factors fixed for estab1ishment such as capital stock and the factors fixed for each output such as raw materials different from one output to another. Different characteristics of each factor input may be specified differently in the specification of a production technology. Some further efforts on the specification 306 of a technology should be worthwhile along this line. Earlier studies on this issue can be traced back to Bradford and Johnson (1953), Beringer (1955) and Baquet (1976), mostly in the farm management analyses.2 Hence the following specification for the input bundle, in the case of weak separability between input and output bundles, can be shown; 1 . . . X9 5 X94:I . . . .< ll f(X and Y2 g(X1 . . . X X 9 9+] 0 O o where (Y1, Y2) are outputs. (X1 . . . Xg) are factors variable for firm, (Xg+1 . . . Xk) are factors variable for establishments, and (X . Xz) and (X . X“) are factors fixed for k+1'° (ii-1" each output. Because the data we used are a bit unusual, there are also a few lessons to be learned from our experience with them. The Korean census includes estab1ishments with more than 5 workers. This is a very low limit indeed. While for many purposes dispersion 21 am very grateful to professor Glenn L. Johnson, Michigan State University, on his remarks. In the agricultural production studies, this topic seems to have been very popular so long time and discussed by the problem of "horizontally versus vertically combined production technology." 307 along the scale dimension is desirable, the data for very small units seem to be less complete and accurate than those of larger units. Also in some industries, the production structure of such very small units is very different from the larger units, even though both types belong to the same industry group. In general, there is much more "noise" in the smaller units. Hence in studies of this type, the very small units might well be either excluded or subjected to some other special treatment. The other important missing ingredient in our data is information on variation in labor qua1ity across establishments. "Quality" is a many dimensional concept, the most important being education, occupation and other indices of skill levels. No data are available in Korea on the education and skill level of the labor force at the establishment level but only for relatively crude industrial groupings. It should not be too difficult to expand the present operative/administrative workers questions and inquire about the education, sex and skill composition of the establishments workforce. Data that are collected could be also significantly improved. For example, the only wage rate derivable from the figures is an annual average which may be quite a far removed from any relevant concept of the marginal cost of labor input. Significant improve- ment could be achieved if overtime hours and payrolls were segregated from the total. Similarly, the capital data should come closer to the con- cept of capital used rather than capital owned. Either one should 308 ask about the value of capital used irrespective of ownership or one should inquire about the rental costs of leased equipment as well as about rental receipts from rented out capital. The final important missing ingredient in the census data is information on variations in output quality across estab1ishments. Even a naive way of specifying output quality by several grades which an establishment sells at different prices should help a lot for this type of study. The heterogeneity of output was briefly evidenced such that the average price of a certain output, .produced by one product-producing establishments,is significantly different from the average price of that product, produced by more than one product-producing establishment in each industry, where the product concerned is classified into the same 7-digit KSIC commodity code in the census. We have learned something from our investigations, not the least of which is that just "more data" will not do. If we persist in asking rather complicated questions, we shall need much better and more relevant figures before we can hope to answer them precisely. APPENDICES 309 APPENDIX A 310 APPENDIX A-I DERIVATION OF THE SHARE EQUATIONS SYSTEM IN THE MULTIPRODUCT TRANSFORMATION FUNCTION The Lagrangian function for the profit maximization is; m .2 w.xj f AF(§, x). (1) n L = z POYO - = J 1 1 1 J_] And the first-order conditions are; 3L 8F 3F P1 —=Pi+}\—=0=>—_.—=-—’i=1’...n, (2) 3)". 3y,- ay, 4 8 L BF 3F Wj ___=-wj+)\——=0=>—=——,j=l,...m, (3) awj axj axj A 3.L —=F(§_a l)=0- 8 A In the Translog function, the first-derivatives become 81nF 3F yi P1 yi -1 alnF =-———= --——= Piyi — O Piyi = (-),F) , (4) alnyi 3y, F X F 1F alnyi alnF 3F Xj Wj xi 1 alnF = —__.——-= ---—-= w.x. —- #>W.X. = (AF) (5) alnxj axj F X F J 3 AF 3 J alnxj 311 312 From (4) and (5), we can have; n n alnF alnF .2: Piyi = - AF )3 ; O-AF = Epiyi/z , (6) 1=1 i=1 alnyi i i alnyi and m m alnF alnF z w.x. = AF )3 ;=> AF =zw.x./z , (7) j=1 J J j=1 alnx. ,- 3 ~13- 31m. J J Hence from (4) and (5); BlnF -l Piyi alnF alnyi AF EPiyi i alnyi and 1 BlnF 1 WJXJ' alnF . A . . ' . alnxJ F waJ j alnxJ The derived equations (8) and (9) can be written as; alnF ( s. = = e.+-z e..ln x.i-z a. 1n y , 10) 1 alnyi 1 j 13 j k 1k k and alnF - , = = . + . 1 + .. . , SJ alnx. 01.J E 732 n x2 g 831 1n y1 (ll) .1 where alnF 2 = X B. + Z Z 3.. 1n x. + Z Z 6. 1n y = 1.0, i,k=l, . . .n, and 313 BlnF Z ' alnx. J .1 Y0 1" X + 2 z €°° 1" yo = -100, j’]=], o o o 32 2 j 1 31 1 (13) Here the relations, (12) and (13), hold under the sufficient condi- tions for linear homogeneity, that is, Z a. -1.0, 2 Bi = 1.0, j i E 8.. = g Eij = 0.0, E Sik = 0.0, and g th = 0.0. APPENDIX A-II INTRINSIC PROPERTY OF THE SHARE EQUATIONS SYSTEM y1=f(2(-) + e1 * Here the GLS estimator (Bi) is identical to the OLS estimator (Bi)' Therefore, 8: = Bi = (x'x)'1 x'yi , (l) where r . - n *‘ - r ‘— 81 Boi ’ Yi 5’11 811 .Ygi B - y m1 t1 )— L— .._11 --- J— —-i‘ Since 2 yji = 1.0 for j = 1, . . . T, the sum of the coefficients; i becomes: 2 Bi - 2(x'x)'] x'yi = (x'x)'] x' Z yi = (x'x)' x'I, (2) i i i where I = ‘_—-_ d. . .d 0n the other hand, the vector I can be expressed in our estimating system as 314 315 _.ooo—.I oonoo Hence the relation, (2), can be represented as _ _ F- : I ‘1 I =‘ g B. - 2 801 (x x) x x l 1 Z .Bli 2 8mi 0 b L. 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Production Function A B C D (P : P) - 5.2265 - 5.9661 - 5.1913 - 5.9653 (P : A) - 0.1482 0.2393 - 0.1493 0.1396 (P : K) 0.9485 0.9402 0.9467 0.9678 (P : F) 0.7516 0.1148 0.7026 0.0474 (P : R) 0.7669 0.8793 0.7665 0.8781 (A : A) -17.4930 -20.0070 -17.4500 -19 7030 (A : K) 0.7548 1.0259 0.7258 1 0506 (A : F) - 0.1879 - 0.9732 - 0.0157 - 0.9248 (A : R) 0.8533 0.8512 0.8607 0.8615 K : K) - 2.1164 - 1.8139 - 2.1246 - 1.8133 K : F) 1.1031 1.2799 1.0620 1.2962 (K : R) 0.9061 0.8040 0.9194 0.8011 (F : F) -18.0010 -17.0420 -17.8890 -16.7680 (F : R) 0.8053 0.8379 0.8247 0.8515 (R : R) - 0.9111 - 0.9330 - 0.9247 - 0 9458 Cost Function P : P; - 6.2075 - 6.3487 - 6.1989 - 6.3667 P : A - 2.1768 - 1.1662 - 2.0014 - 1.1506 (P : K 0.7974 0.5422 0.8097 0.5230 (P : F 1.7939 3.1030 1.7842 3.0736 (P : R) 1.0552 1.0186 1.0178 1.0118 (A z A) -28.1500 -22.6630 -27.5350 -22.5740 (A : K) 0.7253 0.3404 0.7740 0.3345 A : F) 0.3184 2.7251 0.8774 2-8923 (A : R) 0.9568 0.9654 0.9449 0.9707 (K : K) - 2.3464 - 2.1006 - 2.3381 - 2.0933 K : F) 0.7187 0.3852 0.7697 0.4038 (K : R) 0.8900 0.9754 0.8695 0.9771 (F : F) -20.1600 -22.8070 -20.4790 -22.8720 (F : R) 0.9481 0.9936 0.9065 0.9729 (R : R) - 1.0783 - 1.1758 - 1.0824 - 1.1782 aSet A (man-day workers : horsepower equipment); Set 8 (man- day workers : net capita1 stock); Set C (No. of workers : horsepower equipment); Set 0 (No. of workers : net capita1 stock); See the footnote (a) in the Tab1e X-1-1, Chapter IV, Part B. 342 TABLE V-2.--E1asticity of Substitution Estimated: Leather Footwear Industry. Production Function A B C D (P : P) - 4.2228 - 5.8606 - 4.1898 - 5.9699 (P : A; - 0.8166 - 2.2873 - 0.9180 - 2.4500 P : K 1.1646 1.9197 1.1917 1.9897 P : F) 0.3455 1.0485 0.2974 0.9745 P : R) 0.8096 0.6865 0.8009 0.6937 (A : A) -20.4400 -15.9750 -22.3200 -16.3200 (A : K) 1.5272 3.6744 1.5375 3.7796 (A : F) 1.1163 — 0.1459 1.0727 - 0.1022 A : R) 0.8316 0.3770 0.8014 0.3918 (K : K) - 2.4786 - 5.7787 - 2.4813 - 5.7769 K : F) 0.4467 - 0.5880 0.5427 - 0.5701 (K : R) 0.8082 1.2477 0.8088 1.2262 F : F) -40.3870 -60.7050 -42.4200 -60.3890 F : R) 0.7480 0.9413 0.7504 0.9415 (R : R) - 0.7685 - 0.5828 - 0.7607 - 0.5772 Cost Function (P : P) - 4.9295 - 4.3092 - 4.9429 - 4.2616 P : A) - 6.5607 -12.9270 - 6.6702 -13.1930 (P : K) 0.3209 - 1.4373 0.3128 - 1.4337 (P : F) 0.6032 14.4470 0.4465 14.5510 (P : R) 1.0306 0.9274 1.0392 0.9256 (A : A) -28.6790 -41.2250 -28.2560 -42.3390 (A : K) 0.2347 1.9161 0.1377 2.0893 (A : F) 6.8011 -30.2620 7.7002 -31.4920 (A : R) 1.0515 1.6330 1.0846 1.6290 (K : K) - 2.7278 - 5.2466 - 2.7251 - 5.2692 K : F) 0.2365 - 1.5883 0.2872 - 1.7114 (K : R) 1.0398 0.8766 1.0327 0.8644 (F : F) -55.7320 -88.7540 -57.5150 -86.2190 (F : R) 0.8435 0.7047 0.8649 0.7294 (R : R) - 0.9508 - 0.5855 - 0.9502 - 0.5855 aSee the footnote in Tab1e V-1, Appendix B. 343 TABLE V-3.--E1asticity of SubstitutionaEstimated: Screw Product Industry. Production Function A B C D (P : P) - 3.4723 - 4.7685 - 3.4174 - 4.7602 (P : A) - 0.3284 - 0.6814 0.3491 - 0.7757 P : K) 0.7691 1.2943 0.7074 1.2770 (P : F) 0.2412 - 0.9874 0.3284 - 1.1575 (P : R) 0.9338 1.1136 0.9399 1.1073 (A : A) -11.2220 -16.5530 -11.3220 -17.6660 (A : K) 0.3546 1.6234 0.4135 1.6861 (A : F) 1.0988 - 1.2560 1.0192 — 1.5598 (A : R) 0.9955 1.3899 0.9886 1.4379 (K : K) - 2.7922 - 2.1921 - 2.7115 - 2.1548 (K : F) 0.8632 1.4086 0.7870 1.4613 (K : R) 1.0826 0.8164 1.0641 0.8185 F : F) -17.0810 -13.6730 -16.9720 -14.7270 F : R) 0.8974 1.3614 0.9040 1.3999 (R : R) - 1.1229 - 1.4809 - 1.1151 - 1.4635 Cost Function (P : P) - 4.3295 - 4.5618 - 4.3066 - 4.5387 (P : A) - 1.5003 1.3427 - 1.4549 1.4984 P : K - 0.0310 0.7130 - 0.0566 0.7206 P : F 2.0695 2.9893 2.0044 3.0594 (P : R) 0.9365 0.9377 0.9403 0.9469 (A : A) -16.7410 -16.4090 -16.7280 -16.9830 (A : K) - 0.1779 0.3784 - 0.1932 0.3140 (A : F) 4.6157 2.6018 4.4706 2.0312 (A : R) 0.9521 0.7374 0.9533 0.7109 K : K) - 3.0391 - 2.1274 - 3.0744 - 2.1244 K : F) - 0.1456 0.5790 - 0.0776 0.5978 (K : R) 0.8891 0.9683 0.8929 0.9688 (F : F) -25.1320 -22.3000 -25.0550 -22.0170 (F : R) 1.0863 0.9155 1.0881 0.9097 (R : R) - 1.1147 - 1.4693 - 1.1100 - 1.4700 aSee the footnote in Tab1e V-1, Appendix B. TABLE V-4.--E1asticity of Substitution Estimated: Knitted Underwear.a 344 Manufacture of Production Function A B C D (P P) - 0.6540 -15.4760 1.8864 -14.9310 (P A) -55.4660 2.1888 -69.3430 6.8133 (P K) 1.7747 3.3659 1.6541 3.7110 P F) 5.2598 7.5327 6.4901 11.0890 (P R) 1.6732 3.6025 1.6804 2.8064 (A : A) 271.3300 -68.2990 337.0000 -77.3110 (A : K) - 1.8209 1.7248 - 0.5644 1.3651 (A : F) - 0.3817 - 5.4368 - 6.2888 -26.4190 (A : R) 2.2774 3.1576 2.4098 2.8584 (K : K) - 3.8042 - 3.1778 - 3.8651 - 3.4564 (K : F) 0.4265 - 0.1246 0.3664 - 0.4765 (K : R) 1.1643 0.4403 1.1651 0.4768 (F : F) -96.7760 -116.3200 -96.2110 —128.3600 (F : R) 1.1203 0.6417 1.0906 1.2856 (R : R) - 0.9857 - 1.8726 - 0.9926 - 1.5869 Cost Function (P : P) - 6.3139 - 4.7331 - 6.2990 - 4.6459 (P : A) - 2.6777 0.7884 - 2.5255 1.1806 (P : K) 0.0333 0.4845 0.0418 0.4176 (P : F) 0.0762 5.8002 0.2123 6.9653 P : R) 0.8681 0.2090 0.8375 0.3938 (A : A) -34.1600 -33.3320 -34.4600 —36.7950 (A : K) - 0.4174 0.2236 - 0.4330 0.1785 (A : F) 8.9067 - 9.1790 8.4156 -18.4200 (A : R) 1.0282 0.9452 1.0276 1.2975 K : K; - 3.3146 - 2.4775 - 3.3067 - 2.4783 K : F 0.3473 0.5195 0.4672 0.4996 (K : R) 0.8978 1.1632 0.9351 1.1572 (F : F) -69.1560 -122.1900 -70.7260 -122.4000 (F : R) 1.2623 1.3931 1.2441 1.6321 (R : R) - 0.6968 - 0.9842 - 0.6973 - 0.9784 aSee the footnote in Tab1e V-1, Appendix B. 345 TABLE V-5.--E1asticity of Substitution Estimated: Manufacture of Briquettes.a Production Function A B C D (P : P) -37.2550 -85.1360 -36.2660 -79.4500 P : A) 32.5300 - 8.9121 32.7820 -11.4560 (P : K) - 0.7423 1.3426 - 0.8289 1.4961 P : F) - 0.5766 -10.4710 - 0.7154 -10.5580 (P : R) 2.7462 4.5347 2.6494 4.4236 (A : A) -135.4900 -72.6910 -140.7500 -74.5320 (A : K) 2.2060 1.8974 2.6325 2.0971 (A : F) - 2.8721 -17.6200 - 3.5722 -17.4980 (A : R) 2.6336 2.8157 2.7770 3.0118 (K : K; - 4.2213 - 2.9283 - 4.2105 - 2.9379 K : F 0.2974 0.9239 0.4831 0.9893 (K : R) 1.0461 1.1500 1.0268 1.1379 (F : F) -92.3590 ~139.8700 -93.1240 -136.2000 (F : R) 2.0970 3.1485 2.1170 3.1440 (R : R) - 0.7515 - 0.9254 - 0.7434 - 0.9355 Cost Function (P : P) -11.7880 -26.8720 -11.8470 -26.8830 (P : A; - 0.2317 5.0023 - 0.5029 4.2466 (P : K - 1.0452 - 0.2867 - 0.9862 - 0.2634 (P : F) 2.2256 4.1361 2.2371 4.1811 P : R) 1.0791 1.5135 1.1102 1.5755 (A : A) -26.5230 -38.5340 -26.5730 -38.4210 (A : K) - 0.4609 0.0372 - 0.3995 0.0447 {A : F) 3.0308 0.1037 3.1769 0.3962 A : R) 0.9008 1.0134 0.9212 1.0674 (K : K) - 4.9326 - 2.6136 - 4.8978 - 2.6142 (K : F) - 0.3448 0.1565 - 0.2587 0.1532 (K : R) 0.8534 0.7321 0.8560 0.7292 (F : F; -70.6660 -122.8900 -71.6140 -123.6200 F : R 1.1569 1.6846 1.1429 1.6847 (R : R) - 0.4223 - 0.5271 - 0.4230 - 0.5270 aSee the footnote in Tab1e V-1, Appendix B. 346 TABLE V-6.--E1asticity of Substitution Estimated: Mo1ding Industry.a Production Function A B C D (P : P) -15.6390 -24.3890 -15.7890 -24.3400 (P : A) 24.8740 19.5130 25.1900 20.8510 (P : K) 1.4944 1.1750 1.5464 1.1148 (P : F) 1.9551 0.6025 1.9786 0.5240 (P : R) 1.1026 1.7042 1.0940 1.6570 (A : A) -130.3600 ~109.9300 -132.3100 -110.0000 (A : K; 1.4562 1.1189 1.5721 0.9488 (A : F 1.0130 4.7376 0.9429 4.6474 (A 3 R) 1.1993 2.4523 1.2048 2.3529 (K : K) - 3.6647 - 3.6754 - 3.6408 — 3.6650 (K : F) 0.4104 0.9045 0.3184 0.9169 (K : R 0.9370 1.1464 0.9228 1.1588 (F : F) -13.1180 -15.3890 -12.9580 -15.4010 (F : R) 1.1007 1.0614 1.1133 1.0736 (R : R) - 0.7942 - 0.9503 - 0.7890 - 0.9449 Cost Function (P : P) - 7.0575 -11.3050 - 7.0997 -11.3770 P : A) - 2.4180 - 0.6748 - 2.5011 - 0.6597 (P : K) 0.5434 0.9782 0.5726 0.9943 (P : F) 1.6438 1.5224 1.5764 1.4385 (P : R) 0.9455 0.9576 0.9339 0.9635 (A : A -30.4150 -26.8180 -30.3500 -27.0470 (A : K 0.3168 0.8998 0.3498 0.8987 (A : F) 1.4818 2.6690 1.5088 2.5595 (A : R) 0.8896 0.9179 0.8894 0.9047 K : K) - 4.0058 - 3.3610 - 4.0088 - 3.3576 K : F) 0.1094 0.7355 0.1006 0.7475 (K : R) 0.8206 0.7444 0.8280 0.7401 F : F) -10.8730 -10.7060 -10.7940 -10.7430 F : R) 0.9596 0.5789 0.9653 0.5955 (R : R) - 0.7778 - 0.7449 - 0.7770 - 0.7430 aSee the footnote in Tab1e V-1, Appendix B. 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