Econometrics, Vol. 1: Econometric Modeling of Producer Behavior 0262100827, 9780262100823

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Econometrics. Vol. 1, Economic Modeling of Producer Behavior Jorgenson, Dale Weldeau. MI T Press 0262100827 9780262100823 9780585362793 English Production (Economic theory) --Econometric models. 2000 HB241.J67 2000eb 330/ .01/ 5195 Production (Economic theory) --Econometric models.

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Econometrics: Volume 1 Econometric Modeling of Producer Behavior Dale W. Jorgenson

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© 2000 Dale W. Jorgenson All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Jorgenson, Dale Weldeau, 1933 Econometrics / Dale W. Jorgenson. p. cm. Includes bibliographical references and index. ISBN 0-262-10082-7 (v. 1: hc: alk. paper) 1. Production (Economic theory)Econometric models. I. Title. HB241.J67 2000 330'.01'5195dc21 99 046138

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Contents List of Tables Preface List of Sources 1 Econometric Methods for Modeling Producer Behavior Dale W. Jorgenson

xi xiii xxxi

1 1

1.1 Introduction 8 1.2 Price Functions 20 1.3 Statistical Methods 31 1.4 Applications of Price Functions 45 1.5 Cost Functions 55 1.6 Applications of Cost Functions 63 1.7 Conclusion 2 Empirical Studies of Depreciation Dale W. Jorgenson

73 75

2.1 Econometric Models 81 2.2 Studies of Depreciation 87 2.3 Applications 93

2.4 Conclusion 3 An Economic Theory of Agricultural Household Behavior Dale W. Jorgenson and Lawrence J. Lau

97 97

3.1 Introduction 98 3.2 The Welfare Function 108 3.3 The Utility Functions 109 3.4 The Technology 112 3.5 The Constraints 115 3.6 The Complete Model 122 3.7 An Example of Functional Form Econometric Implementation: Bernoulli Utility Function and Cobb-Douglas Production Function

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4 Transcendental Logarithmic Production Frontiers Laurits R. Christensen, Dale W. Jorgenson, and Lawrence J. Lau

125 125

4.1 Introduction 127 4.2 Additivity and Homogeneity 134 4.3 Transcendental Logarithmic Frontiers 138 4.4 Testing the Theory of Production 146 4.5 Empirical Results 153 4.6 Summary and Conclusion 5 The Duality of Technology and Economic Behavior Dale W. Jorgenson and Lawrence J. Lau

159 159

5.1 Introduction 160 5.2 Technology 170 5.3 Economic Behavior 177 5.4 Duality 182 5.5 Conclusion 185 5.6 Historical Note 6 Duality and Differentiability in Production Dale W. Jorgenson and Lawrence J. Lau

189

189 6.1 Introduction 190 6.2 Differentiability 194 6.3 Duality 198 6.4 Production Possibilities 202 6.5 Extensions 7 Efficient Estimation of Nonlinear Simultaneous Equations with Additive Disturbances Dale W. Jorgenson and Jean-Jacques Laffont

209 209

7.1 Introduction 211 7.2 The Model 212 7.3 Cramer Rao Bound 222 7.4 Minimum Distance 229 7.5 Instrumental Variables 232 7.6 Efficiency 8 Tests of a Model of Production for the Federal Republic of Germany, 1950 1973 Klaus Conrad and Dale W. Jorgenson

241 241

8.1 Introduction 242 8.2 Translog Production and Price Functions

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8.3 Integrability 249 8.4 Monotonicity and Convexity of Translog Functions 252 8.5 Tests of the Theory of Production 255 8.6 Estimation and Test Statistics 260 8.7 Economic Interpretation 9 The Structure of Technology: Nonjointness and Commodity Augmentation, Federal Republic of Germany, 1950 1973 267 Klaus Conrad and Dale W. Jorgenson 267 9.1 Introduction 269 9.2 Translog Production and Price Functions 273 9.3 Nonjointness 277 9.4 Technical Change 281 9.5 Tests 283 9.6 Estimation and Test Statistics 286 9.7 Conclusion 10 The Structure of Technology and Changes of Technology over Time, Federal Republic of Germany, 1950 1973 Klaus Conrad and Dale W. Jorgenson

291 291

10.1 Introduction 295 10.2 Separability 298 10.3 Technical Change 301 10.4 Tests 303 10.5 Estimation and Test Statistics 307 10.6 Conclusion 310 10.7 Summary 11 Statistical Inference for a System of Simultaneous, Nonlinear, Implicit Equations in the Context of Instrumental Variable Estimation 311 A. Roland Gallant and Dale W. Jorgenson 311 11.1 Introduction 313 11.I Three-Stage Least-Squares Estimation 313 11.2 The Statistical Model Subject to a Maintained Hypothesis 315 11.3 The Test Statistics T0: An Analog of the Likelihood Ratio Test 316 11.4 Regularity Conditions 317 11.5 The Asymptotic Distribution of T0

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11.6 The Test Statistic TA: The Wald Test 324 11.7 The Asymptotic Distribution of TA 326 11.8 An Example: The Hypothesis of Symmetry 329 11.II Two-Stage Least-Squares Estimation 329 11.9 The Statistical Model 330 11.10 The Test Statistic

: An Analog of the Likelihood Ratio Test 331

11.11 Regularity Conditions 332 11.12 The Asymptotic Distribution of 333 11.13 The Test Statistics

: The Wald Test 333

11.14 The Asymptotic Distribution of 334 11.15 An Example: The Hypothesis of Homogeneity Appendix

335

12 Relative Prices and Technical Change Dale W. Jorgenson and Barbara M. Fraumeni

341 341

12.1 Introduction 344 12.2 Econometric Models

354 12.3 Empirical Results 369 12.4 Conclusion 13 Relative Price Changes and Biases of Technical Change in Japan Masahiro Kuroda, Kanji Yoshioka, and Dale W. Jorgenson

373 373

13.1 Introduction 379 13.2 Theoretical Framework 388 13.3 Design of Experiment and Data Sources 392 13.4 Estimated Results 14 The Role of Energy in Productivity Growth Dale W. Jorgenson

403 403

14.1 Introduction 408 14.2 Econometric Models 437 14.3 Summary and Conclusion 442 14.4 Appendix 15 Bilateral Models of Production for Japanese and U.S. Industries Dale W. Jorgenson, Masahiro Kuroda, Hikaru Sakuramoto, and Kanji Yoshioka

457 457

15.1 Introduction

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15.2 Theoretical Framework 462 15.3 Empirical Results 472 15.4 Conclusion Appendix A

476

Appendix B

480

References

489

Index

527

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List of Tables 1.1 Classification of Industries by Biases of Technical Change

40

1.2 Cost Function for U.S. Electric Power Industry (Parameter Estimates, 1955 and 1970; t-ratios in Parentheses)

57

2.1 Rates of Economic Depreciation

82

2.2 Rates of Economic Depreciation: Business Assets

83

4.1 Estimates of the Parameters of the Translog Production Possibility Frontier

148

4.2 Estimates of the Parameters of the Translog Price Possibility Frontier

149

4.3 Critical Values of F(v1, v2) and c2/v1

151

4.4 F Ratios for Direct and Indirect Tests of the Theory of Production and of Restrictions on the Form of the Production and Price Possibility Frontiers

152

4.5 F Ratios for Direct and Indirect Tests of Factor Augmentation and Group-wise Additivity

153

8.1 Parameter Estimates, Translog Production Function

256

8.2 Parameter Estimates, Translog Price Function

257

8.3 Critical Values of c2/Degrees of Freedom and N(0, 1)

259

8.4 Test Statistics for Translog Production and Price Functions

259

8.5 Value Shares and Rate of Technical Change, Translog Production Function

260

9.1 Parameter Estimates, Translog Production Function

284

9.2 Parameter Estimates, Translog Price Function

285

9.3

Critical Values of c2/Degree of Freedom

286

9.4 Test Statistics for Translog Production and Price Functions

287

10.1 Parameter Estimates, Translog Production Function

305

10.2 Parameter Estimates, Translog Price Function

306

10.3 Critical Values of c2

307

10.4 Test Statistics for Translog Production and Price Functions

308

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11.1 Three-Stage Least-Squares Estimates of the Parameters of the Translog Demand System

329

11.2 Two-Stage Least-Squares Estimates of the Parameters of the Translog Demand System

335

12.1 Parameter Estimates for Sectoral Models of Production and Technical Change

356

12.2 Classification of Industries by Biases of Technical Change

365

13.1 Source of Economic Growth in Japan

374

13.2 List of Industries

378

13.3 Own Price Elasticity of Input: International Comparison between Japan and the United States

395

13.4 Allen Partial Substitutability among Inputs

396

13.5 Cross Share Elasticity and Biases of Technical Change between U.S. and Japanese Sectors

397

14.1 Industrial Sectors

412

14.2 Parameter Estimates for Sectoral Models of Production and Technical Change

414

14.3 Classification of Industries by Biases of Technical Change

435

15.1 Rates of Technical Change and Differences in Technology

464

15.2 Share Elasticities

466

15.3 Biases of Technical Change

470

15.4 Biases of the Differences in Technology

471

15.5 Accelerations of Technical Change and Differences in the Difference in Technology 474 15.B

Parameter Estimates for Bilateral Models of Production in Japan and the United States

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Preface Dale W. Jorgenson This volume contains my econometric studies of producer behavior. The volume includes a self-contained presentation of duality in the theory of production, statistical methods for estimation and inference in systems of nonlinear simultaneous equations, and econometric models based on flexible functional forms. The innovations embodied in these modelsduality, simultaneity, and flexibilityhave become standard in modeling producer behavior. The centerpiece of the volume is a suite of econometric models generated from the dual formulation of the theory of producer behavior. The companion volume, Aggregate Consumer Behavior, provides a parallel treatment of my econometric studies of consumer behavior. These flexible representations of technology and preferences serve as building blocks for the general equilibrium models presented in my earlier volumes, Econometric General Equilibrium Modeling and Energy, the Environment, and Economic Growth. In chapter 1 I survey econometric methods for modeling producer behavior. The goal of empirical research is to determine the nature of substitution among inputs, the character of differences in technology, and the role of economies of scale. Econometric methodology based on duality in the theory of production has generated an extensive body of empirical work. I summarize studies of substitution, technical change, and economies of scale that draw on this methodology. The traditional approach to econometric modeling begins with additive and homogeneous production functions. Demand and supply functions are derived from the conditions for producer equilibrium. However, the constraints imposed by additivity and homogeneity frustrate the objective of characterizing technology empirically. For example, the production function originated by Charles Cobb and Paul Douglas (1928) requires that elasticities of substitution among all inputs must be equal to unity.

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The constant elasticity of substitution (CES) production function introduced by Kenneth Arrow, Hollis Chenery, Bagicha Minhas, and Robert Solow (1961) achieves flexibility by treating the elasticity of substitution as an unknown parameter. However, the CES production function retains additivity and homogeneity and imposes stringent limitations on patterns of substitution. Daniel McFadden (1963) and Hirofumi Uzawa (1962) have shown, essentially, that elasticities of substitution among all inputs must be the same. The innovations in econometric methodology for modeling producer behavior, summarized in this volume, stem from the dual formulation of production theory originated by Harold Hotelling (1932). Lawrence Lau and I give a self-contained presentation of duality in production theory in chapters 5 and 6. Technology is characterized by a price or cost function that is dual to the production function. Demand and supply functions are generated without imposing arbitrary restrictions on the underlying technology. The responses of demands and supplies to changes in prices, technology, and economies of scale characterize the behavior of producers. For example, measures of substitution are specified in terms of the impacts of price changes on demands and supplies. Similarly, measures of technical change are specified in terms of the impacts of changes in technology. A judicious choice of these measures results in a flexible approach to econometric modeling. In chapter 1 I outline the generation of transcendental logarithmic or translog price and cost functions. I define the share elasticity as the impact on the share of an input in the value of output of a proportional change in the price of an input. If a share elasticity is positive, the corresponding value share increases with the input price. If a share elasticity is negative, the value share decreases with the price. Finally, if a share elasticity is zero, the value share is independent of the price, as in the Cobb-Douglas production function. Similarly, the bias of technical change is the impact of a change in technology on the input value share. If the bias of some technical change is positive, the corresponding value share increases with a change in the level of technology and we say that the technical change is input-using. If the bias is negative, the corresponding value share decreases with a change in the level of technology and the technical change is input-saving. Finally, if the bias is zero, the value share is independent of technology; in this case we say that the technical change is neutral.

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An important feature of models of production based on the translog price function is that the rate of technical change is endogenous, but does not affect future production possibilities. The biases of technical change can be used to derive the implications of changes in input prices for the rate of technical change. If the bias is positive, the rate of technical change decreases with the input price. If the bias is negative, the rate of technical change increases with the input price. Finally, if the bias is zero, so that technical change is neutral, the rate of technical change is independent of the price. The description of technology is completed by the deceleration of technical change. This is defined as the negative of the rate of change of the rate of technical change. If the deceleration is positive, negative, or zero, the rate of technical change is decreasing, increasing, or independent of the level of technology. In the system of demand and supply functions generated from the translog price function the share elasticities and biases of technical change are unknown parameters. The dependent variables are the value shares of all inputs and the rate of technical change. All the dependent variables are functions of the same independent variables, namely, prices and the level of technology. These functions are nonlinear in the variables; the functions may also be nonlinear in the parameters. Finally, the parameters may be subject to nonlinear constraints arising from the theory of production. Additional constraints arise from restrictions on technology such as additivity and homogeneity. Myopic decision rules for econometric models of producer behavior can be derived by treating the price of capital input as the rental price of capital services. Production decisions depend only on current prices, including the price of investment goods. More details about myopic decision rules are given in my paper, ''Technology and Decision Rules in the Theory of Investment Behavior," in the companion volume, Tax Policy and the Cost of Capital. These decision rules greatly facilitate the implementation of the econometric models. The constraints on the system of demand and supply functions implied by the theory of production are: 1. Homogeneity. The value shares and the rate of technical change are homogeneous of degree zero in the input prices. 2. Product exhaustion. The sum of the value shares is equal to unity. 3. Symmetry. The matrix of share elasticities, biases of technical change, and the deceleration of technical change must be symmetric.

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4. Nonnegativity. The value shares must be nonnegative. 5. Monotonicity. The matrix of share elasticities must be nonpositive definite. Statistical methods for estimating the unknown parameters of systems of demand and supply functions depend on the character of the data set. For cross-section observations on individual producing units, prices can be treated as exogenous variables. Unknown parameters can be estimated by the nonlinear multivariate regression techniques introduced by Robert Jennrich (1969) and Edmond Malinvaud (1970, 1980). These techniques deal with nonlinearities in the parameters, nonlinearities in the variables, or both. For time series observations on industry groups, the prices that determine demands and supplies must be treated as endogenous variables. Unknown parameters can be estimated by techniques for nonlinear simultaneous equations. In chapter 7 Jean-Jacques Laffont and I present the method of nonlinear three-stage least squares for estimation of parameters in systems of demand and supply functions. These methods deal with simultaneity, as well as nonlinearities in the parameters and the variables. The theory of production can be tested statistically by deriving constraints on the parameters of a system of demand and supply functions implied by the theory. Additional constraints, for example, the constraints implied by additivity and homogeneity, can also be tested. Jennrich and Malinvaud have introduced methods for statistical inference in nonlinear multivariate regression models. Ronald Gallant and I present methods for statistical inference in systems of nonlinear simultaneous equations in chapter 11. I conclude chapter 1 by considering flexible representations of technology for econometric general equilibrium modeling. I also describe the use of panel data techniques for modeling technical change and economies of scale simultaneously. Finally, I outline methods for constructing dynamic models of production that incorporate internal costs of adjustment. The optimal production plan at each point of time depends on the initial level of "quasi-fixed" inputs, such as capital inputs, as well as expectations about future prices of outputs and inputs. In chapter 2 I survey empirical studies of depreciation, an important special topic in econometric modeling of producer behavior. The measurement of depreciation requires modeling substitution among

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different vintages of capital inputs, corresponding to units of capital accumulated at different points of time. In principle each vintage could be treated as a separate input in an econometric model of production. However, the number of parameters would increase with the square of the number of inputs in a flexible functional form like the translog, rendering this approach infeasible. The key simplifying assumption in the vintage model of capital is that different vintages are perfect substitutes in production. The services of these different vintages are proportional to initial investments with constants of proportionality given by relative efficiencies in production. The prices of different vintages of capital inputs are proportional to the relative efficiencies. Empirical research on the vintage model of capital reduces to modeling the relative efficiencies. Although the assumption of perfect substitutes is restrictive, the vintage approach has become the method of choice for modeling substitution among capital inputs. I have presented a vintage model of capital in "The Economic Theory of Replacement and Depreciation," chapter 5 of the companion volume, Tax Policy and the Cost of Capital. This model, originally formulated by Hotelling (1925), is characterized by price-quantity duality. Capital goods decline in efficiency with age, requiring replacement investments to maintain productive capacity. The price of a capital good also falls with age, reflecting both the current decline and the present value of future declines in efficiency. Depreciation is the decrease in the value of a capital good with age. Laurits Christensen and I have presented a vintage accounting system for prices and quantities of capital goods in our paper, "Measuring the Performance of the Private Sector of the U.S. Economy, 1929 1969," chapter 5 in the companion volume, Postwar U.S. Economic Growth. This accounting system provides an internally consistent framework for measuring depreciation and capital stock. We have extended this framework to encompass income, product, and wealth data for the econometric general equilibrium models described below. In the model of capital goods prices introduced by Robert Hall (1971) the relative efficiencies of a capital good are expressed as functions of age and calendar time. The unknown parameters of the model can be estimated from observations on the prices of capital goods of different vintages. This model can be generalized to capital goods with different varieties that are perfect substitutes in production.

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Relative efficiencies are represented as functions of the technical characteristics of each variety. Providing an illustration of modeling the relative efficiencies of different vintages of capital, Charles Hulten and Frank Wykoff (1981b) have constructed econometric models for the prices of eight categories of capital goods. Making use of the asset classification scheme of the Bureau of Economic Analysis (1987) capital stock study, KunYoung Yun and I (1991b) have derived economic depreciation rates for thirty-five asset categories. These estimates of depreciation have been incorporated into price and quantity indices of capital services for thirty-five industries in "Productivity and Economic Growth," chapter 1 of the companion volume, International Comparisons of Economic Growth. The research of Hulten and Wykoff has been successfully exploited by the Bureau of Economic Analysis in measuring depreciation in the U.S. national accounts, as described by Barbara Fraumeni (1997). Ellen Dulberger (1989) has employed speed of processing and main memory as technical characteristics of different varieties of computer processors. The Bureau of Economic Analysis (1986) has introduced price indices for computers based on this model of relative efficiencies into the U.S. National Income and Product Accounts. Kevin Stiroh and I (1999) have derived price and quantity indices for the capital services of computers in our paper, "Information Technology and Economic Growth." In chapter 3 Lawrence Lau and I present an economic theory of agricultural household behavior. This theory relates household consumption and production decisions to the prices of outputs, variable inputs, and consumption goods. Additional determinants of these decisions include stocks of quasi-fixed inputs, household wealth, and the composition of the household. Extensive empirical studies of agricultural households based on this approach were published in Resource Use in Agriculture, Applications of the Profit Function to Selected Countries, a special issue of Food Research Institute Studies, edited by Pan Yotopoulos and Lau (1978). Our theory of agricultural household behavior expresses household welfare as a function of the utility functions of individual household members. An important simplifying assumption is that the utility functions for all individuals are identical, except for proportional transformations of units of measurement. These transformations are equivalence scales that depend on the characteristics of the individual

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such as age and sex. Daniel Slesnick and I have used this approach to modeling household behavior in our paper, "Aggregate Consumer Behavior and Household Equivalence Scales," chapter 5 in the companion volume, Measuring Social Welfare. The objective of the agricultural household is to maximize household welfare, subject to the technology of the enterprise, the constraints on the time available to household members, and the total expenditure of the household. The household takes the profits of the agricultural enterprise and non-agricultural income as given in making consumption decisions. It maximizes welfare with respect to leisure, consumption of goods produced within the agricultural enterprise, and purchased consumption goods. Given competitive markets for agricultural inputs, including hired labor, production decisions depend only on technology and are independent of preferences. Lau and I represent the technology of the agricultural enterprise in terms of outputs, variable inputs such as labor, materials, energy, and quasi-fixed inputs such as land and reproducible capital. We derive a profit function that is dual to the agricultural production function. This gives the maximized value of profit of the enterprise as a function of the prices of the outputs and the variable inputs and the quantities of the quasi-fixed inputs. In chapter 4 Christensen, Lau and I, present an econometric model of production that embodies three innovations. The model utilizes translog functional forms, statistical methods for nonlinear systems of simultaneous equations, and duality in production theory. In this model the economy supplies outputs of consumption and investment goods and demands inputs of capital and labor services. Price and quantity data for the inputs and outputs of the U.S. private domestic economy are taken from the system of U.S. national accounts that Christensen and I have constructed for the period 1929 1969. An increase in the output of investment goods requires foregoing a part of the output of consumption goods, so that adjusting the rate of investment is costly. However, costs of adjustment are fully reflected in the market price of investment goods. The cost of capital input is a function of this price, so that costs of adjustment are external to the production process. In models of production with internal costs of adjustment, like those presented in section 1.7 of chapter 1, the cost of capital input must be inferred from the shadow value of the adjustment costs. Further details are given in my paper, "Technology and

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Decision Rules in the Theory of Investment Behavior," in the companion volume, Tax Policy and the Cost of Capital. Our first objective is to develop tests of the theory of production that do not employ additivity and homogeneity as part of the maintained hypothesis. For this purpose we generate an econometric model of aggregate producer behavior from the translog production possibility frontier. The dependent variables are ratios of the values of investment goods and labor services to the value of capital services. The independent variables are logarithms of the quantities of outputs of investment and consumption goods, quantities of inputs of capital and labor services, and the level of technology. Under constant returns to scale our model of aggregate producer behavior implies the existence of a price possibility frontier, defined by the set of prices consistent with zero profits. The price possibility frontier and the system of demand and supply functions are dual to the production possibility frontier and the necessary conditions for producer equilibrium. An econometric model generated from the translog price possibility frontier has the same dependent variables. The independent variables are logarithms of the prices of investment and consumption goods, logarithms of the prices of capital and labor services, and the level of technology. The translog production and price possibility frontiers correspond to two distinct representations of technology. We have estimated the unknown parameters of both models by the method of nonlinear three-stage least squares presented in chapter 7. We have tested hypotheses implied by the theory of production for both models, using the test statistics presented in chapter 10. Results for both models are consistent with the validity of an extensive set of restrictions implied by the theory. Our second objective is to test the additivity and homogeneity restrictions that underly the constant elasticity of the substitution production function. We employ the same data and econometric methodology as in our tests of the theory of production. The constraints implied by additivity and homogeneity conflict sharply with the empirical evidence. We further simplify the technology by requiring that the elasticity of substitution between capital and labor inputs is equal to unity, as in the Cobb-Douglas production function. Conditional on additivity and homogeneity, this is also strongly rejected by our tests.

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Our overall conclusion is that flexible representations of technology are appropriate for dynamic general equilibrium modeling at the aggregate level. A representation incorporating additivity and homogeneity is much less satisfactory. Yun and I have employed the translog price function in our dynamic general equilibrium model of the impact of U.S. tax policy. We have presented the model in our paper, "The Efficiency of Capital Allocation," chapter 10 in the companion volume, Tax Policy and the Cost of Capital. We have used this model in analyzing the impact of U.S. tax reforms in our paper, "Tax Policy and Capital Allocation," chapter 11 in the same volume. Our dynamic general equilibrium model of tax policy also incorporates a flexible representation of preferences. This is based on the translog indirect utility function presented in my paper with Christensen and Lau in the companion volume, Aggregate Consumer Behavior. Equilibrium in the tax model is characterized by an intertemporal price system that clears markets for consumption and investment goods and for capital and labor services. The price of investment goods reflects the present value of capital services and links the present to the future. Capital has been accumulated through previous investments, linking the present to the past. In chapters 8, 9, and 10 Klaus Conrad and I have applied the econometric methodology presented in chapter 1 to aggregate data for the Federal Republic of Germany. These data are presented in our 1975 book, Measuring Performance in the Private Economy of the Federal Republic of Germany, 1950 1973. The economy supplies investment and consumption goods and demands capital and labor services. An additional feature of this model is that the rate of technical change is endogenous and depends on the same independent variables as the demands and supplies. We utilize translog price and production functions to generate systems of nonlinear simultaneous equations that describe aggregate producer behavior. In chapter 8 we derive constraints on the parameters of econometric models of producer behavior implied by the theory of production. Since the price and production functions provide two distinct representations of technology, we present tests of these constraints for both. The theory of production is consistent with the results of both sets of tests, corroborating and extending the findings of chapter 4. We test inequality restrictions implied by monotonicity and convexity of the price and production functions, as well as the equality restrictions tested in chapter 4.

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In chapter 10 Conrad and I have tested and rejected restrictions on technology associated with the additivity and homogeneity implied by the constant elasticity of substitution production function. These findings also corroborate and extend those of chapter 4. In chapter 9 we represent technical change by commodity augmentation factors that are analogous to the equivalence scales of chapter 3. We find that technical change is factor-augmenting, so that inputs of capital and labor services can be transformed into efficiency units, while investment and consumption goods outputs can be represented in natural units. In chapter 12 Fraumeni and I present econometric models for each of thirty-five industrial sectors of the U.S. economy. These models are based on a translog price function for each sector. The price of output is a function of the prices of the primary factors of productioncapital and labor servicesprices of inputs of energy and materials, and time as an index of technology. An important feature of these models is that the rate of technical change is endogenous, but does not affect future production possibilities. The econometric model of producer behavior for each of the thirty-five industries consists of a system of nonlinear simultaneous equations. The equations give the value shares of capital, labor, energy, and materials (KLEM) inputs and the rate of technical change as functions of relative prices and time. Price and quantity data for the inputs and outputs of each industry are taken from the system of national accounts presented in my 1980 paper, "Accounting for Capital," while the rate of technical change is an index number constructed from these data. This paper extends the vintage accounting system I had developed with Christensen to include both sectoral and aggregate production accounts. The parameters of the system of input demand equations for each industry are estimated by the method of nonlinear three-stage least squares presented in chapter 7. These parameters included the share elasticities that describe substitution and the biases that describe technical change. We have estimated these parameters from timeseries data for each industry. The industry-level data for the U.S. are described in my 1980 paper with Fraumeni, "The Role of Capital in U.S. Economic Growth, 1948 1976." In 1987 we published updated sectoral and aggregate production accounts in our book with Frank Gollop, Productivity and U.S. Economic Growth. The results are sum-

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marized in chapter 1 of the companion volume, Postwar U.S. Economic Growth. As before, we describe substitution patterns by share elasticities, giving the impact of a proportional price change on the share of an input in the value of an input. As an illustration, the share elasticity of capital with respect to the price of labor is zero if the elasticity of substitution between the two inputs is equal to unity, since the share of capital is constant. We describe patterns of technical change by biases, giving the impact of a change in technology on the input value share. For example, we say that technical change is capital-using if the capital share increases with time, holding input prices constant. The empirical findings on patterns of substitution and technical change reveal striking similarities among industries. In general, share elasticities are nonnegative, so that shares increase with proportional input price changes and elasticities of substitution are greater than unity. The elasticities of the shares of capital with respect to the price of labor are nonnegative for thirty-three of the thirty-five industries. Elasticities of substitution between capital and labor are greater than unity for these industries. Similarly, elasticities of the shares of capital with respect to the price of energy are nonnegative for thirty-four industries and elasticities with respect to the price of materials are nonnegative for all thirty-five industries. The share elasticities of labor with respect to the price of materials are nonnegative for all thirty-five industries. However, the share elasticities of labor with respect to the price of energy are nonnegative for only nineteen of the thirty-five industries. Finally, the share elasticities of energy with respect to the price of materials are nonnegative for thirty of the thirty-five industries. A classification of industries by patterns of the biases of technical change is given in table 1.1 of chapter 1. The most common pattern is capital-using, labor-using, energy-using, and materials-saving technical change. The economic interpretation is that changes in technology conserve material inputs or increase value added through inputs of capital, labor, and energy. This occurs for nineteen of the thirty-five industries. Technical change is capital-using for twenty-five of the thirty-five industries, labor-using for thirty-one, energy-using for twenty-nine, and materials-using for only two. We have emphasized that rates of technical change are endogenous in our econometric models of producer behavior. These rates depend on prices of inputs and the level of technology. If the bias of technical

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change is capital-using, then an increase in the price of a capital input reduces the rate of technical change. Since this is typical of the patterns we have described, an increase in the price of capital inputs reduces the rate of technical change. Similarly, increases in the prices of labor and energy inputs typically depress the rate of technical change, while an increase in the price of materials inputs raises the rate of technical change. Over extended periods of time, energy prices have fallen relative to the prices of other inputs, elevating rates of technical change at the industry level. However, prices of labor inputs have risen relative to other input prices, depressing these rates of technical change. The substantial increases in energy prices after 1973 have had the effect of reducing sectoral rates of technical change, decreasing the aggregate rate of technical change, and diminishing the rate of growth of the U.S. economy. In chapter 1 of the companion volume, Energy, the Environment, and Economic Growth, Peter Wilcoxen and I present a dynamic general equilibrium model of the U.S. economy with a flexible representation of technology for each of thirty-five industries. For this purpose we have employed econometric models of producer behavior for these industries based on translog price functions. We have used our dynamic general equilibrium model to analyze the economic impact of alternative energy, environmental, and tax policies. Mun Ho and I use this model to analyze the impact of trade policies in chapters 8, 9 and 10 of the same volume. Our dynamic general equilibrium model of the U.S. economy also incorporates a flexible representation of preferences. This is presented in my paper with Lau and Thomas Stoker, "Transcendental Logarithmic Model of Aggregate Consumer Behavior," chapter 8 of the companion volume, Aggregate Consumer Behavior. The model is based on exact aggregation over systems of household demand functions derived from the translog indirect utility function. Equilibrium is characterized by an intertemporal price system that clears markets for the outputs of all thirty-five industries as well as for capital and labor services. The price of investment goods is forward-looking and depends on future capital service prices, while the stock of capital is backward-looking and depends on past investments. In chapter 14 I consider the relationship between energy prices and rates of technical change in greater detail. I present an econometric model of producer behavior for thirty-five U.S. industries, based on

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data for the period 1958 1979. I divide energy inputs between electricity and nonelectrical energy, so that the model for each sector includes the shares of five inputs capital and labor services, electricity and nonelectrical energy, and materials. The shares are functions of the prices of these inputs, as well as time is an index of technology. Finally, the model includes an endogenous rate of technical change, also a function of the five input prices and time. The patterns of substitution for the models presented in chapter 14 are similar to those for the models of chapter 12. Technical change is electricity-using for twenty-three of the thirty-five industries and nonelectrical energyusing for twenty-eight of the thirty-five industries. An increase in the price of electricity reduces the rate of technical change for twenty-three industries and reduces this rate for the remaining twelve industries. An increase in the price of non-electrical energy reduces the rate of technical change for twenty-eight industries and reduces the rate for the remaining seven. Historically, the price of electricity has fallen relative to the price of nonelectrical energy over extended periods of time. Prices of both types of energy have fallen relative to prices of capital and labor services and materials inputs. Electrification associated with the positive bias of technical change for electricity has raised rates of technical change in a wide range of industries. However, greater use of nonelectrical energy has increased rates of technical change in an even broader range. Jumps in the prices of both forms of energy after 1973 have had a depressing effect on rates of technical change at the industry level, slowing the aggregate rate of technical change and the growth rate of the U.S. economy. In chapter 13 Masahiro Kuroda, Kanji Yoshioka, and I present econometric models of producer behavior for thirty industries of the Japanese economy. These models are based on the translog price function introduced in chapter 12. We implement this model for price and quantity data for inputs and outputs of Japanese industries, as well as rates of technical change for these industries. The data are also employed in my paper with Kuroda and Mieko Nishimizu, ''Japan-U.S. Industry-Level Productivity Comparisons, 1960 1979," chapter 7 in the companion volume, International Comparisons of Economic Growth. In table 13.3 we compare patterns of substitution between U.S. and Japanese industries. These are broadly similar. In figure 13.2 we compare patterns of technical change for the two countries. The bias of technical change is laborusing for all thirty industries in Japan, while

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the bias is material-saving for twenty-eight of the thirty industries. The bias is energy-using for twenty-six of these industries and capital-saving for twenty-two. Capital-saving bias predominates in Japan, while capital-using bias predominates in the U.S. For both countries an increase in the price of energy results in a reduction in the rate of technical change and a slowdown in economic growth. In chapter 15 Kuroda, Hikaru Sakuramoto, Yoshioka, and I present bilateral econometric models of producer behavior for twenty-eight Japanese and U.S. industries. We treat data on production patterns for Japan and the U.S. as separate sets of observations. However, we assume that econometric models of producer behavior for the two countries have common parameters. The point of departure for the econometric model is a bilateral translog production function. Output depends on a dummy variableone for the U.S. and zero for Japanthat allows for differences in technology between the two countries, as well as inputs and time as an index of the level of technology. We combine data for the U.S. and Japan to estimate the parameters that describe substitution and technical change. These data are employed for bilateral productivity comparisons in my paper with Kuroda and Nishimizu. For the same input prices and level of technology production is more capital-intensive and intermediate input-intensive in Japanese industries and more labor-intensive in U.S. industries. Rates of technical change are higher for Japanese industries and lower for U.S. industries. Not surprisingly, the technology gap between Japan and the U.S. is gradually closing, as Kuroda, Nishimizu, and I have shown. One of the key innovations in the econometric models of production presented in this volume is the application of duality in production theory. The starting point of the theory of production is the set of production possibilities, containing all the production plans available to the producing unit. The production function gives the maximum net output of any commodity as a function of the net outputs of all other commodities. The theory of marginal productivity completes the theory of production; this may be identified with the gradient of the production function. In chapter 5 Lau and I present equivalent specifications of the set of production possibilities, the production function, and the marginal productivities. We can take a characterization of any one of the three as a starting point of the theory of production and derive the properties of the other two. Marginal productivities provide the vehicle for

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generating econometric models of production in chapters 4, 8, 9, and 10 at the aggregate level and in chapter 15 at the sectoral level. The theory of production provides the links between these econometric models and representations of technology in terms of the production function and the set of production possibilities. The second objective of chapter 5 is to characterize the set of feasible production plans from the point of view of economic behavior. For this purpose Lau and I present a theory of supply that parallels the theory of marginal productivity. We develop equivalent specifications of the profit function, the net supplies, and the set of price and profit possibilities. Any one of the three can be taken as the starting point for the theory of production. The net supplies provide the vehicle for generating econometric models of production in chapters 4, 8, 9, and 10 and chapters 12, 13, and 14. The theory of production provides links between these econometric models and the alternative representations of technology we consider in chapter 5. The final objective of chapter 5 is to link the theory of supply with the theory of marginal productivity. Lau and I demonstrate the equivalence of technological ands behavioral viewpoints of the theory of production. For this purpose we employ equivalent specifications of the production and profit functions. The theory also implies equivalent specifications of marginal productivities and net supplies and of the sets of production and price possibilities. The econometric models of production considered in this volume can be linked to any of these six alternative specifications of the theory of production. Sets of production and price possibilities are not employed in econometric modeling. However, Kenneth Hoffman and I have utilized these specifications of technology in our paper, "Economic and Technological Models for Evaluation of Energy Policy" included in the companion volume, Econometric General Equilibrium Modeling. We present a linear activity analysis model of the U.S. energy sector. This model is based on information from detailed engineering studies of technologies that are not currently available, but could be implemented under alternative technology policies. The activity analysis model of the energy sector is linked to econometric models for the nonenergy sectors of the U.S. economy to provide a complete representation of technology. The linear activity analysis model does not require that the marginal products corresponding to a given production plan and the net supplies corresponding to a given price system are unique. How-

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ever, uniqueness of the marginal products and the net supplies is essential for econometric modeling. In chapter 6 Lau and I consider the implications of differentiability of the production and profit functions or, equivalently, uniqueness of the marginal products and net supplies. By strengthening convexity assumptions for the production and profit functions we are able to develop the theory of production in terms of properties of differentiable convex functions and their gradients. A second key innovation in the econometric models of production we present in this volume is the application of methods for systems of nonlinear simultaneous equations. In chapter 7 Laffont and I present the method of nonlinear three-stage least squares for estimation of the parameters of these models. Gallant and I provide the corresponding methods for statistical inference in chapter 11. These methods were greatly extended by Lars Hansen (1982) and became the basis for the Generalized Method of Moments that is now the standard approach to estimation and inference in macroeconometric modeling. In chapter 7 Laffont and I consider two lines of attack on efficient estimation of systems of nonlinear simultaneous equations. The nonlinear three-stage least squares estimator is obtained by minimizing a weighted sum of squared residuals, where the weights depend on a set of instrumental variables. The first step in constructing this estimator is to linearize the system of nonlinear simultaneous equations. Arnold Zellner and Henri Theil's (1962) method of three-stage least squares is applied to the linearized model. This process is reiterated until the weighted sum of squared residuals is a minimum. An alternative approach to efficient estimation of systems of nonlinear simultaneous equations is an extension of the method of efficient instrumental variables for linear systems developed in my papers with James Brundy (1971, 1973). Laffont and I show that efficient instrumental variables and minimum distance estimators achieve the same efficiency. Both are less efficient than the full information maximum likelihood estimator for systems of nonlinear simultaneous equations; however, the maximum likelihood estimator is computationally burdensome, and requires estimating all the equations of the model at the same time. In chapter 11 Gallant and I present statistics for testing hypotheses about the parameters of systems of nonlinear simultaneous equations. We consider statistics based on likelihood ratio and Wald approaches, applied to the nonlinear three-stage least-squares estimator. The likeli-

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hood ratio approach involves a comparison of values of the minimized criterion function with and without the constraints implied by the hypothesis to be tested. Both approaches can be used to generate confidence intervals and regions for the unknown parameters. These methods are available in many econometric software packages. The econometric models of producer behavior presented in this volume have been incorporated into the dynamic general equilibrium models. These general equilibrium models also include econometric models of consumer behavior presented in the companion volume, Aggregate Consumer Behavior. The advantage of the econometric approach to general equilibrium modeling is that responses of production and consumption decisions to changes in energy prices, environmental controls, trade restrictions, and tax policies can be derived from historical experience. This experience is an indispensable guide to economic policy making. Implementation of the econometric approach to general equilibrium modeling has necessitated innovations in economic theory and econometric method. Duality has been especially critical in generating econometric models that provide flexible representations of technology and preferences. These models have required the development of new statistical methods for systems of nonlinear simultaneous equations. Duality, simultaneity, and flexibility in econometric modeling have led to a burgeoning empirical literature, characterizing technology and preferences in a wide range of empirical settings. I would like to thank June Wynn of the Department of Economics at Harvard University for her excellent work in assembling the manuscripts for this volume in machine-readable form. Renate D'Arcangelo of the Editorial Office of the Division of Engineering and Applied Science at Harvard edited the manuscripts, proofread the machinereadable versions and prepared them for typesetting. Warren Hrung, then a senior at Harvard College, checked the references and proof-read successive versions of the typescript. William Richardson and his associates provided the index. Gary Bisbee of Chiron Incorporated typeset the manuscript and provided camera-ready copy for publication. The staff of the MIT Press, especially Terry Vaughn, Victoria Richardson, Lindsey Kistler, and Michael Sims, has been very helpful at every stage of the project. Financial support was provided by the Program on Technology and Economic Policy of the Kennedy School of Government at Harvard. As always, the author retains sole responsibility for any remaining deficiencies in the volume.

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List of Sources 1. Dale W. Jorgenson. 1986. Econometric Methods for Modeling Producer Behavior. In Handbook of Econometrics, eds. Z. Griliches and M. D. Intriligator, vol. 3, pp. 1841 1915. Reprinted with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands. 2. Dale W. Jorgenson. 1996. Empirical Studies of Depreciation. Economic Inquiry 34, no. 1 (January): 24 42. Reprinted by permission. 3. Dale W. Jorgenson and Lawrence J. Lau. An Economic Theory of Agricultural Household Behavior, not previously published. 4. Laurits R. Christensen, Dale W. Jorgenson, and Lawrence J. Lau. 1973. Transcendental Logarithmic Production Frontiers. Review of Economics and Statistics 55, no. 1 (February): 28 45. Reprinted by permission. 5. Dale W. Jorgenson and Lawrence J. Lau. 1974. The Duality of Technology and Economic Behavior. Review of Economic Studies 41(2), no. 126 (April): 181 200. Reprinted by permission. 6. Dale W. Jorgenson and Lawrence J. Lau. 1974. Duality and Differentiability in Production. Journal of Economic Theory 9, no. 1 (September): 23 42. Reprinted by permission from Academic Press. 7. Dale W. Jorgenson and Jean-Jacques Laffont. 1974. Efficient Estimation of Nonlinear Simultaneous Equations with Additive Disturbances. Annals of Social and Economic Measurement 3, no. 4 (October): 615 640. Reprinted by permission. 8. Klaus Conrad and Dale W. Jorgenson. 1977. Tests of a Model of Production for the Federal Republic of Germany, 1950 1973. European Economic Review 10, no. 1 (October): 51 75. Reprinted with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands. 9. Klaus Conrad and Dale W. Jorgenson, 1978. The Structure of Technology: Nonjointness and Commodity Augmentation, Federal Republic of Germany, 1950 1973. Empirical Economics 3, Issue 2: 91 113. Reprinted by permission.

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10. Klaus Conrad and Dale W. Jorgenson. 1978. The Structure of Technology and Changes of Technology over Time, Federal Republic of Germany, 1950 1973. Zeitschrift für Wirtschafts- und Sozialwissenschaften 3: 259 279. Reprinted by permission. 11. A. Ronald Gallant and Dale W. Jorgenson. 1979. Statistical Inference for a System of Simultaneous, Nonlinear, Implicit Equations in the Context of Instrumental Variables Estimation. Journal of Econometrics 11, nos. 2/3 (October/December): 275 302. Reprinted with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands. 12. Dale W. Jorgenson and Barbara M. Fraumeni. 1983. Relative Prices and Technical Change. In Quantitative Studies on Production and Prices, eds. W. Eichhorn, R. Henn, K. Neumann, and R. W. Shephard, pp. 241 269. Würzburg: Physica-Verlag. Reprinted by permission. 13. Masahiro Kuroda, Kanji Yoshioka, and Dale W. Jorgenson. 1984. Relative Price Changes and Biases of Technical Change in Japan. Economic Studies Quarterly 35, no. 2 (August): 116 138. Reprinted by permission. 14. Dale W. Jorgenson. 1984. The Role of Energy in Productivity Growth. In International Comparisons of Productivity and Causes of the Slowdown, ed. J. W. Kendrick, pp. 279 323. Cambridge: Ballinger. Reprinted by permission. 15. Dale W. Jorgenson, Masahiro Kuroda, Kikaru Sakuramoto, and Kanji Yoshioka. 1990. Bilateral Models of Productions for Japanese and U.S. Industries. In C. Hulten (ed.), Productivity Growth in Japan and the United States, Studies in Income and Wealth, ed. C. R. Hulten, vol. 51, pp. 59 83. Chicago, IL: University of Chicago Press. Reprinted by permission of the University of Chicago Press.

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1 Econometric Methods for Modeling Producer Behavior Dale W. Jorgenson 1.1 Introduction The purpose of this chapter is to provide an exposition of econometric methods for modeling producer behavior. The objective of econometric modeling is to determine the nature of substitution among inputs, the character of differences in technology, and the role of economies of scale. The principal contribution of recent advances in methodology has been to exploit the potential of economic theory in achieving this objective. Important innovations in specifying econometric models have arisen from the dual formulation of the theory of production. The chief advantage of this formulation is in generating demands and supplies as explicit functions of relative prices. By using duality in production theory, these functions can be specified without imposing arbitrary restrictions on patterns of production. The econometric modeling of producer behavior requires parametric forms for demand and supply functions. Patterns of production can be represented in terms of unknown parameters that specify the responses of demands and supplies to changes in prices, technology, and scale. New measures of substitution, technical change, and economies of scale have provided greater flexibility in the empirical determination of production patterns. Econometric models of producer behavior take the form of systems of demand and supply functions. All the dependent variables in these functions depend on the same set of independent variables. However, the variables and the parameters may enter the functions in a nonlinear manner. Efficient estimation of these parameters has necessitated the development of statistical methods for systems of nonlinear simultaneous equations.

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The new methodology for modeling producer behavior has generated a rapidly expanding body of empirical work. We illustrate the application of this methodology by summarizing empirical studies of substitution, technical change, and economies of scale. In this introductory section we first review recent methodological developments and then provide a brief overview of the chapter. 1.1.1 Production Theory The economic theory of productionas presented in such classic treatises as Hicks's Value and Capital (1946) and Samuelson's Foundations of Economic Analysis (1983)is based on the maximization of profit, subject to a production function. The objective of this theory is to characterize demand and supply functions, using only the restrictions on producer behavior that arise from optimization. The principal analytical tool employed for this purpose is the implicit function theorem. 1 Unfortunately, the characterization of demands and supplies as implicit functions of relative prices is inconvenient for econometric applications. In specifying an econometric model of producer behavior the demands and supplies must be expressed as explicit functions. These functions can be parameterized by treating measures of substitution, technical change, and economies of scale as unknown parameters to be estimated on the basis of empirical data. The traditional approach to modeling producer behavior begins with the assumption that the production function is additive and homogeneous. Under these restrictions demand and supply functions can be derived explicitly from the production function and the necessary conditions for producer equilibrium. However, this approach has the disadvantage of imposing constraints on patterns of productionthereby frustrating the objective of determining these patterns empirically. The traditional approach was originated by Cobb and Douglas (1928) and was employed in empirical research by Douglas and his associates for almost two decades.2 The limitations of this approach were made strikingly apparent by Arrow, Chenery, Minhas, and Solow (1961, henceforward ACMS), who pointed out that the CobbDouglas production function imposes a priori restrictions on patterns of substitution among inputs. In particular, elasticities of substitution among all inputs must be equal to unity.

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The constant elasticity of substitution (CES) production function introduced by ACMS adds flexibility to the traditional approach by treating the elasticity of substitution as an unknown parameter. 3 However, the CES production function retains the assumptions of additivity and homogeneity and imposes very stringent limitations on patterns of substitution. McFadden (1963) and Uzawa (1962) have shown, essentially, that elasticities of substitution among all inputs must be the same. The dual formulation of production theory has made it possible to overcome the limitations of the traditional approach to econometric modeling. This formulation was introduced by Hotelling (1932) and later revived and extended by Samuelson (1953, 1960)4 and Shephard (1953, 1970).5 The key features of the dual formulation are, first, to characterize the production function by means of a dual representation such as a price or cost function and, second, to generate explicit demand and supply functions as derivatives of the price or cost function.6 The dual formulation of production theory embodies the same implications of optimizing behavior as the theory presented by Hicks (1946) and Samuelson (1983). However, the dual formulation has a crucial advantage in the development of econometric methodology: Demands and supplies can be generated as explicit functions of relative prices without imposing the arbitrary constraints on production patterns required in the traditional methodology. In addition, the implications of production theory can be incorporated more readily into an econometric model. 1.1.2 Parametric Form Patterns of producer behavior can be described most usefully in terms of the behavior of the derivatives of demand and supply functions.7 For example, measures of substitution can be specified in terms of the response of demand patterns to changes in input prices. Similarly, measures of technical change can be specified in terms of the response of these patterns to changes in technology. The classic formulation of production theory at this level of specificity can be found in Hicks's Theory of Wages (1932). Hicks (1932) introduced the elasticity of substitution as a measure of substitutability. The elasticity of substitution is the proportional

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change in the ratio of two inputs with respect to a proportional change in their relative price. Two inputs have a high degree of substitutability if this measure exceeds unity and a low degree of substitutability if the measure is less than unity. The unitary elasticity of substitution employed in the Cobb-Douglas production function is a borderline case between high and low degrees of substitutability. Similarly, Hicks introduced the bias of technical change as a measure of the impact of changes in technology on patterns of demand for inputs. The bias of technical change is the response of the share of an input in the value of output to a change in the level of technology. If the bias is positive, changes in technology increase demand for the input and are said to use the input; if the bias is negative, changes in technology decrease demand for the input and are said to save the input. If technical change neither uses nor saves an input, the change is neutral in the sense of Hicks. By treating measures of substitution and technical change as fixed parameters the system of demand and supply functions can be generated by integration. Provided that the resulting functions are themselves integrable, the underlying price or cost function can be obtained by a second integration. As we have already pointed out, Hicks's elasticity of substitution is unsatisfactory for this purpose, since it leads to arbitrary restrictions on patterns of producer behavior. The introduction of a new measure of substitution, the share elasticity, by Christensen, Jorgenson, and Lau (1971, 1973) and Samuelson (1973) has made it possible to overcome the limitations of parametric forms based on constant elasticities of substitution. 8 Share elasticities, like biases of technical change, can be defined in terms of shares of inputs in the value of output. The share elasticity of a given input is the response of the share of that input to a proportional change in the price of an input. By taking share elasticities and biases of technical change as fixed parameters, demand functions for inputs with constant share elasticities and constant biases of technical change can be obtained by integration. The shares of each input in the value of output can be taken to be linear functions of the logarithms of input prices and of the level of technology. The share elasticities and biases of technical change can be estimated as unknown parameters of these functions. The constant share elasticity (CSE) form of input demand functions can be integrated a second time to obtain the underlying price or cost function. For example, the logarithm of the price of output can be

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expressed as a quadratic function of the logarithms of the input prices and the level of technology. The price of output can be expressed as a transcendental or, more specifically, an exponential function of the logarithms of the input prices. 9 Accordingly, Christensen, Jorgenson, and Lau refer to this parametric form as the translog price function.10 1.1.3 Statistical Method Econometric models of producer behavior take the form of systems of demand and supply functions. All the dependent variables in these functions depend on the same set of independent variablesfor example, relative prices and the level of technology. The variables may enter these functions in a nonlinear manner, as in the translog demand functions proposed by Christensen, Jorgenson, and Lau. The functions may also be nonlinear in the parameters. Finally, the parameters may be subject to nonlinear constraints arising from the theory of production. The selection of a statistical method for estimation of systems of demand and supply functions depends on the character of the data set. For cross-section data on individual producing units, the prices that determine demands and supplies can be treated as exogenous variables. The unknown parameters can be estimated by means of nonlinear multivariate regression techniques. Methods of estimation appropriate for this purpose were introduced by Jennrich (1969) and Malinvaud (1970, 1980).11 For time-series data on aggregates such as industry groups, the prices that determine demands and supplies can be treated as endogenous variables. The unknown parameters of an econometric model of producer behavior can be estimated by techniques appropriate for systems of nonlinear simultaneous equations. One possible approach is to apply the method of full information maximum likelihood. However, this approach has proved to be impractical, since it requires the likelihood function for the full econometric model, not only for the model of producer behavior. Jorgenson and Laffont (1974) have developed limited information methods for estimating the systems of nonlinear simultaneous equations that arise in modeling producer behavior. Amemiya (1974) proposed to estimate a single nonlinear structural equation by the method of nonlinear two-stage least squares. The first step in this procedure is to linearize the equation and to apply the method of

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two-stage least squares to the linearized equation. Using the resulting estimates of the coefficients of the structural equation, a second linearization can be obtained and the process can be repeated. Jorgenson and Laffont extended Amemiya's approach to a system of nonlinear simultaneous equations by introducing the method of nonlinear three-stage least squares. This method requires an estimate of the covariance matrix of the disturbances of the system of equations as well as an estimate of the coefficients of the equations. The procedure is initiated by linearizing the system and applying the method of three-stage least squares to the linearized system. This process can be repeated, using a second linearization. 12 It is essential to emphasize the role of constraints on the parameters of econometric models implied by the theory of production. These constraints may take the form of linear or nonlinear restrictions on the parameters of a single equation or may involve restrictions on parameters that occur in several equations. An added complexity arises from the fact that the restrictions may take the form of equalities or inequalities. Estimation under inequality restrictions requires nonlinear programming techniques.13 The constraints that arise from the theory of production can be used to provide tests of the validity of the theory. Similarly, constraints that arise from simplification of the patterns of production can be tested statistically. Methods for statistical inference in multivariate nonlinear regression models were introduced by Jennrich (1969) and Malinvaud (1970, 1980). Methods for inference in systems of nonlinear simultaneous equations were developed by Gallant and Jorgenson (1979) and Gallant and Holly (1980).14 1.1.4 Overview This chapter begins with the simplest form of the econometric methodology for modeling producer behavior. This methodology is based on production under constant returns to scale. The dual representation of the production function is a price function, giving the price of output as a function of the prices of inputs and the level of technology. An econometric model of producer behavior is generated by differentiating the price function with respect to the prices and the level of technology. We present the dual formulation of the theory of producer behavior under constant returns to scale in section 1.2. We parameterize this

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model by taking measures of substitution and technical change to be constant parameters. We then derive the constraints on these parameters implied by the theory of production. In section 1.3 we present statistical methods for estimating this model of producer behavior under linear and nonlinear restrictions. Finally, we illustrate the application of this model by studies of data on individual industries in section 1.4. In section 1.5 we consider the extension of econometric modeling of producer behavior to nonconstant returns to scale. In regulated industries the price of output is set by regulatory authority. Given the demand for output as a function of the regulated price, the level of output can be taken as exogenous to the producing unit. Necessary conditions for producer equilibrium can be derived from cost minimization. The minimum value of total cost can be expressed as a function of the level of output and the prices of all inputs. This cost function provides a dual representation of the production function. The dual formulation of the theory of producer behavior under nonconstant returns to scale parallels the theory under constant returns. However, the level of output replaces the level of technology as an exogenous determinant of production patterns. An econometric model can be parameterized by taking measures of substitution and economies of scale to be constant parameters. In section 1.6 we illustrate this approach by means of studies of data on individual firms in regulated industries. In section 1.7 we conclude the chapter by outlining frontiers for future research. Current empirical research has focused on the development of more elaborate and more detailed data sets. We consider, in particular, the modeling of consistent time series of interindustry transactions tables and the application of the results to general equilibrium analysis of the impact of economic policy. We also discuss the analysis of panel data sets, that is, time series of cross sections of observations on individual producing units. Current methodological research has focused on dynamic modeling of production. At least two promising approaches to this problem have been proposed. Both employ optimal control models of producer behavior. The first is based on static expectations with all future prices taken to be equal to current prices. The second approach is based on stochastic optimization under rational expectations, utilizing information about expectations of future prices contained in current production patterns.

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1.2 Price Functions The purpose of this section is to present the simplest form of the econometric methodology for modeling producer behavior. We base this methodology on a production function with constant returns to scale. Producer equilibrium implies the existence of a price function, giving the price of output as a function of the prices of inputs and the level of technology. The price function is dual to the production function and provides an alternative and equivalent description of technology. An econometric model of producer behavior takes the form of a system of simultaneous equations, determining the distributive shares of the inputs and the rate of technical change. Measures of substitution and technical change give the responses of the distributive shares and the rate of technical change to changes in prices and the level of technology. To generate an econometric model of producer behavior we treat these measures as unknown parameters to be estimated. The economic theory of production implies restrictions on the parameters of an econometric model of producer behavior. These restrictions take the form of linear and nonlinear constraints on the parameters. Statistical methods employed in modeling producer behavior involve the estimation of systems of nonlinear simultaneous equations with parameters subject to constraints. These constraints give rise to tests of the theory of production and tests of restrictions on patterns of substitution and technical change. 1.2.1 Duality In order to present the theory of production we first require some notation. We denote the quantity of output by y and the quantities of J inputs by xj (j = 1, 2, . . . , J). Similarly, we denote the price of output by q and the prices of the J inputs by pj (j = 1, 2, . . . , J). We find it convenient to employ vector notation for the input quantities and prices: x = (x1, x2, . . . , xJ)vector of input quantities, p = (p1, p2, . . . , pJ)vector of input prices. We assume that the technology can be represented by a production function, say F, where

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and t is an index of the level of technology. In the analysis of time series data for a single producing unit the level of technology can be represented by time. In the analysis of cross-section data for different producing units the level of technology can be represented by one-zero dummy variables corresponding to the different units. 15 We can define the shares of inputs in the value of output by

Under competitive markets for output and all inputs the necessary conditions for producer equilibrium are given by equalities between the share of each input in the value of output and the elasticity of output with respect to that input

where v = (v1, v2, . . . , vJ)vector of value shares. In x = (ln x1, ln x2, . . . , ln xJ)vector of logarithms of input quantities. Under constant returns to scale the elasticities and the value shares for all inputs sum to unity

where i is a vector of ones. The value of output is equal to the sum of the values of the inputs. Finally, we can define the rate of technical change, say vt, as the rate of growth of the quantity of output holding all inputs constant

It is important to note that this definition does not impose any restriction on patterns of substitution among inputs. Given the identity between the value of output and the value of all inputs and given equalities between the value share of each input and the elasticity of output with respect to that input, we can express the

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price of output as a function, say Q, of the prices of all inputs and the level of technology

We refer to this as the price function for the producing unit. The price function Q is dual to the production function F and provides an alternative and equivalent description of the technology of the producing unit. 16 We can formalize this description in terms of the following properties of the price function: 1. Positivity. The price function is positive for positive input prices. 2. Homogeneity. The price function is homogeneous of degree one in the input prices. 3. Monotonicity. The price function is increasing in the input prices. 4. Concavity. The price function is concave in the input prices. Given differentiability of the price function, we can express the value shares of all inputs as elasticities of the price function with respect to the input prices

where ln p = (ln p1, ln p2, . . . , ln pJ)vector of logarithms of input prices. Further, we can express the negative of the rate of technical change as the rate of growth of the price of output, holding the prices of all inputs constant

Since the price function Q is homogeneous of degree one in the input prices, the value shares and the rate of technical change are homogeneous of degree zero and the value shares sum to unity

Since the price function is increasing in the input prices the value shares must be nonnegative,

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Since the value shares sum to unity, we can write

where v ³ 0 implies

and v ¹ 0.

1.2.2 Substitution and Technical Change We have represented the value shares of all inputs and the rate of technical change as functions of the input prices and the level of technology. We can introduce measures of substitution and technical change to characterize these functions in detail. For this purpose we differentiate the logarithm of the price function twice with respect to the logarithms of input prices to obtain measures of substitution

We refer to the measures of substitution (1.2.7) as share elasticities, since they give the response of the value shares of all inputs to proportional changes in the input prices. If a share elasticity is positive, the corresponding value share increases with the input price. If a share elasticity is negative, the value share decreases with the input price. Finally, if a share elasticity is zero, the value share is independent of the price. 17 Second, we can differentiate the logarithm of the price function twice with respect to the logarithms of input prices and the level of technology to obtain measures of technical change

We refer to these measures as biases of technical change. If a bias of technical change is positive, the corresponding value share increases with a change in the level of technology and we say that technical change is input-using. If a bias of technical change is negative, the value share decreases with a change in technology and technical change is input-saving. Finally, if a bias is zero, the value share is independent of technology; in this case we say that technical change is neutral.18 Alternatively, the vector of biases of technical change upt can be employed to derive the implications of changes in input prices for the

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rate of technical change. If a bias of technical change is positive, the rate of technical change decreases with the input price. If a bias is negative, the rate of technical change increases with the input price. Finally, if a bias is zero so that technical change is neutral, the rate of technical change is independent of the price. To complete the description of technical change we can differentiate the logarithm of the price function twice with respect to the level of technology

We refer to this measure as the deceleration of technical change, since it is the negative of rate of change of the rate of technical change. If the deceleration is positive, negative, or zero, the rate of technical change is decreasing, increasing, or independent of the level of technology. The matrix of second-order logarithmic derivatives of the logarithm of the price function Q must be symmetric. This matrix includes the matrix of share elasticities Upp,the vector of biases of technical change upt, and the deceleration of technical change utt. Concavity of the price function in the input prices implies that matrix of second-order derivatives, say H, is nonpositive definite, so that the matrix Upp + vv' V is nonpositive definite, where

the price of output q is positive and the matrices N and V are diagonal

We can define substitution and complementarity of inputs in terms of the matrix of share elasticities Upp and the vector of value shares v. We say that two inputs are substitutes if the corresponding element of the matrix Upp + vv' V is negative. Similarly, we say that two inputs are complements if the corresponding element of this matrix is positive. If the element of this matrix corresponding to the two inputs is zero, we say that the inputs are independent. The definition of substitution and complementarity is symmetric in the two inputs, reflecting

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the symmetry of the matrix Upp + vv' V. If there are only two inputs, nonpositive definiteness of this matrix implies that the inputs cannot be complements. 19 We next consider restrictions on patterns of substitution and technical change implied by separability of the price function Q. The most important applications of separability are associated with aggregation over inputs. Under separability, the price of output can be represented as a function of the prices of a smaller number of inputs by introducing price indexes for input aggregates. By treating the price of each aggregate as a function of the prices of the inputs making up the aggregate, we can generate a second stage of the model. We say that the price function Q is separable in the K input prices {p1, p2, . . . , pK} if and only if the price function can be represented in the form

where the function P is independent of the J K input prices {pK+1, pK+2, . . . , pJ} and the level of technology t.20 We say that the price function is homothetically separable if the function P in (2.10) is homogeneous of degree one.21 Separability of the price function implies homothetic separability.22 The price function Q is homothetically separable in the K input prices {p1, p2, . . . , pK} if and only if the production function F is homothetically separable in the K input quantities {x1,x2, . . . , xK}

where the function G is homogeneous of degree one and independent of J K quantities {xK+1, xK+2, . . . , xJ} and the level of technology t.23 We can interpret the function P in the definition of a separability of the price function as a price index; similarly, we can interpret the function G as a quantity index. The price index is dual to the quantity index and has properties analogous to those of the price function: 1. Positivity. The price index is positive for positive input prices. 2. Homogeneity. The price index is homogeneous of degree one in the input prices. 3. Monotonicity. The price index is increasing in the input prices. 4. Concavity. The price index is concave in the input prices.

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The total cost of the K inputs included in the price index P, say c, is the sum of expenditures on all K inputs

We can define the quantity index G for this aggregate as the ratio of total cost to the price index P

The product of the price and quantity indexes for the aggregate is equal to the cost of the K inputs. 24 We can analyze the implications of homothetic separability by introducing price and quantity indexes of aggregate input and defining the value share of aggregate input in terms of these indexes. An aggregate input can be treated in precisely the same way as any other input, so that price and quantity indexes can be used to reduce the dimensionality of the space of input prices and quantities. The price index generates a second stage of the model, by treating the price of each aggregate as a function of the prices of the inputs making up the aggregate.25 1.2.3 Parameterization In the theory of producer behavior the dependent variables are value shares of all inputs and the rate of technical change. The independent variables are prices of inputs and the level of technology. The purpose of an econometric model of producer behavior is to characterize the value shares and the rate of technical change as functions of the input prices and the level of technology. To generate an econometric model of producer behavior a natural approach is to treat the measures of substitution and technical change as unknown parameters to be estimated. For this purpose we introduce the parameters

where Bpp is a matrix of constant share elasticities, bpt is a vector of constant biases of technical change, and btt is a constant deceleration of technical change.26

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We can regard the matrix of share elasticities, the vector of biases of technical change, and the deceleration of technical change as a system of second-order partial differential equations. We can integrate this system to obtain a system of first-order partial differential equations

where the parametersap, atare constants of integration. To provide an interpretation of the parametersap, atwe first normalize the input prices. We can set the prices equal to unity where the level of technology t is equal to zero. This represents a choice of origin for measuring the level of technology and a choice of scale for measuring the quantities and prices of inputs. The vector of parameters ap is the vector of value shares and the parameter at is the negative of the rate of technical change where the level of technology t is zero. Similarly, we can integrate the system of first-order partial differential equations (1.2.14) to obtain the price function

where the parameter a0 is a constant of integration. Normalizing the price of output so that it is equal to unity where t is zero, we can set this parameter equal to zero. This represents a choice of scale for measuring the quantity and price of output. For the price function (1.2.15) the price of output is a transcendental or, more specifically, an exponential function of the logarithms of the input prices. We refer to this form as the transcendental logarithmic price function or, more simply, the translog price function, indicating the role of the variables. We can also characterize this price function as the constant share elasticity or CSE price function, indicating the role of the fixed parameters. In this representation the scalarsat, btthe vectorsap, bptand the matrix Bpp are constant parameters that reflect the underlying technology. Differences in levels of technology among time periods for a given producing unit or among producing units at a given point of time are represented by differences in the level of technology t. For the translog price function the negative of the average rates of technical change at any two levels of technology, say t and t 1, can be

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expressed as the difference between successive logarithms of the price of output, less a weighted average of the differences between successive logarithms of the input prices with weights given by the average value shares

In the expression (1.2.16)

is the average rate of technical change,

and the vector of average value shares

is given by

We refer to the expression (1.2.16), introduced by Christensen and Jorgenson (1970), as the translog rate of technical change. We have derived the translog price function as an exact representation of a model of producer behavior with constant share elasticities and constant biases and deceleration of technical change. 27 An alternative approach to the translog price function, based on a Taylor's series approximation to an arbitrary price function, was originated by Christensen, Jorgenson, and Lau (1971, 1973). Diewert (1976, 1980) has shown that the translog rate of technical change (2.16) is exact for the translog price function and the converse. Diewert (1971, 1973, 1974b) introduced the Taylor's series approach for parameterizing models of producer behavior based on the dual formulation of the theory of production. He utilized this approach to generate the ''generalized Leontief" parametric form, based on square root rather than logarithmic transformations of prices. Earlier, Heady and Dillon (1961) had employed Taylor's series approximations to generate parametric forms for the production function, using both square root and logarithmic transformations of the quantities of inputs. The limitation of Taylor's series approximations has been emphasized by Gallant (1981) and Elbadawi, Gallant, and Souza (1983). Taylor's series provide only a local approximation to an arbitrary price or production function. The behavior of the error of approximation must be specified in formulating an econometric model of producer behavior. To remedy these deficiencies Gallant (1981) has introduced global approximations based on Fourier series.28

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1.2.4 Integrability The next step in generating our econometric model of producer behavior is to incorporate the implications of the econometric theory of production. These implications take the form of restrictions on the system of equations (1.2.14), consisting of value shares of all inputs v and the rate of technical change vt. These restrictions are required to obtain a price function Q with the properties we have listed above. Under these restrictions we say that the system of equations is integrable. A complete set of conditions for integrability is the following: 1.2.4.1 Homogeneity The value shares and the rate of technical change are homogeneous of degree zero in the input prices. We first represent the value shares and the rate of technical change as a system of equations (1.2.14). Homogeneity of the price function implies that the parametersBpp, bptin this system must satisfy the restrictions

where i is a vector of ones. For J inputs there are J + 1 restrictions implied by homogeneity. 1.2.4.2 Product Exhaustion The sum of the value shares is equal to unity. Product exhaustion implies that the value of the J inputs is equal to the value of the product. Product exhaustion implies that the parametersap, Bpp, bptmust satisfy the restrictions

For J inputs there are J + 2 restrictions implied by product exhaustion. 1.2.4.3 Symmetry The matrix of share elasticities, biases of technical change, and the deceleration of technical change must be symmetric.

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A necessary and sufficient condition for symmetry is that the matrix of parameters must satisfy the restrictions

For J inputs the total number of symmetry restrictions is 1/2J(J + 1). 1.2.4.4 Nonnegativity The value shares must be nonnegative. Nonnegativity is implied by monotonicity of the price function

For the translog price function the conditions for monotonicity take the form

Since the translog price function is quadratic in the logarithms of the input prices, we can always choose prices so that the monotonicity of the price function is violated. Accordingly, we cannot impose restrictions on the parameters that would imply nonnegativity of the value shares for all prices and levels of technology. Instead, we consider restrictions that imply monotonicity of the value shares wherever they are nonnegative. 1.2.4.5 Monotonicity The matrix of share elasticities must be nonpositive definite. Concavity of the price function implies that the matrix Bpp + vv' V is nonpositive definite. Without violating the product exhaustion and nonnegativity restrictions we can set the matrix vv' V equal to zero. For example, we can choose one of the value shares equal to unity and all the others equal to zero. A necessary condition for the matrix Bpp + vv' V to be nonpositive definite is that the matrix of constant share elasticities Bpp must be nonpositive definite. This condition is also sufficient, since the matrix vv' V is nonpositive definite and the sum of two nonpositive definite matrices is nonpositive definite. 29

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We can impose concavity on the translog price functions by representing the matrix of constant share elasticities Bpp in terms of its Cholesky factorization

where T is a unit lower triangular matrix and D is a diagonal matrix. For J inputs we can write the matrix Bpp in terms of its Cholesky factorization as follows:

where

The matrix of constant share elasticities Bpp must satisfy restrictions implied by symmetry and product exhaustion. These restrictions imply that the parameters of the Cholesky factorization must satisfy the following conditions

Under these conditions there is a one-to-one transformation between the elements of the matrix of share elasticities Bpp and the parameters of the Cholesky factorizationT, D. The matrix of share elasticities is nonpositive definite if and only if the diagonal elements {d1, d2, . . . , dJ 1} of the matrix D are nonpositive. 30

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1.3 Statistical Methods Our model of producer behavior is generated from a translog price function for each producing unit. To formulate an econometric model of production and technical change we add a stochastic component to the equations for the value shares and the rate of technical change. We associate this component with unobservable random disturbances at the level of the producing unit. The producer maximizes profits for given input prices, but the value shares of inputs are subject to a random disturbance. The random disturbances in an econometric model of producer behavior may result from errors in implementation of production plans, random elements in the technology not reflected in the model of producer behavior, or errors of measurement in the value shares. We assume that each of the equations for the value shares and the rate of technical change has two additive components. The first is a non-random function of the input prices and the level of technology; the second is an unobservable random disturbance that is functionally independent of these variables. 31 1.3.1 Stochastic Specification To represent an econometric model of production and technical change we require some additional notation. We consider observations on the relative distribution of the value of output among all inputs and the rate of technical change. We index the observations by levels of technology (t = 1,2, . . . , T). We employ a level of technology indexed by time t as an illustration throughout the following discussion. The vector of value shares in the t-th time period is denoted vt (t = 1,2, . . . , T). Similarly, the rate of technical change in the t-th time period is denoted The vector of input prices in the t-th time period is denoted pt (t = 1,2, . . . , T). Similarly, the vector of logarithms of input prices is denoted ln pt (t = 1,2, . . . , T). We obtain an econometric model of production and technical change corresponding to the translog price function by adding random disturbances to the equations for the value shares and the rate of technical change

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where et is the vector of unobservable random disturbances for the value shares of the t-th time period and is the corresponding disturbance for the rate of technical change. Since the value shares for all inputs sum to unity in each time period, the random disturbances corresponding to the J value shares sum to zero in each time period

so that these disturbances are not distributed independently. We assume that the unobservable random disturbances for all J + 1 equations have expected value equal to zero for all observations

We also assume that the disturbances have a covariance matrix that is the same for all observations; since the random disturbances corresponding to the J value shares sum to zero, this matrix is nonnegative definite with rank at most equal to J. We assume that the covariance matrix of the random disturbances corresponding to the value shares and the rate of technical change, say S, has rank J, where

Finally, we assume that the random disturbances corresponding to distinct observations in the same or distinct equations are uncorrelated. Under this assumption the covariance matrix of random disturbances for all observations has the Kronecker product form

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1.3.2 Autocorrelation The rate of technical change is not directly observable; we assume that the equation for the translog price index of the rate of technical change can be written

where

is the average disturbance in the two periods

Similarly, is a vector of averages of the logarithms of the input prices and index of technology in the two periods.

is the average of time as an

Using our new notation, the equations for the value shares of all inputs can be written

where is a vector of averages of the disturbances in the two periods. As before, the average value shares sum to unity, so that the average disturbances for the equations corresponding to value shares sum to zero

The covariance matrix of the average disturbances corresponding to the equation for the rate of technical change for all observations is proportional to a Laurent matrix

where

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The covariance matrix of the average disturbance corresponding to the equation for each value share is proportional to the same Laurent matrix. The covariance matrix of the average disturbances for all observations has the Kronecker product form

Since the matrix W in (1.3.9) is known, the equations for the average rate of technical change and the average value shares can be transformed to eliminate autocorrelation. The matrix W is positive definite, so that there is a matrix P such that

To construct the matrix P we first invert the matrix W to obtain the inverse matrix W 1, a positive definite matrix. We then calculate the Cholesky factorization of the inverse matrix W 1,

where T is a unit lower triangular matrix and D is a diagonal matrix with positive elements along the main diagonal. Finally, we can write the matrix P in the form

where D1/2 is a diagonal matrix with elements along the main diagonal equal to the square roots of the corresponding elements of D.

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We can transform equations for the average rates of technical change by the matrix P = D1/2T' to obtain equations with uncorrelated random disturbances

since

The transformation P = D1/2T' is applied to data on the average rates of technical change and data on the average values of the variables that appear on the right-hand side of the corresponding equation. We can apply the transformation P = D1/2T' to the equations for average value shares to obtain equations with uncorrelated disturbances. As before, the transformation is also applied to data on the average values of variables that appear on the right-hand side of the corresponding equations. The covariance matrix of the transformed disturbances from the equations for the average value shares and the equation for the average rates of technical change has the Kronecker product form

To estimate the unknown parameters of the translog price function we combine the first J 1 equations for the average value shares with the equation for the average rate of technical change to obtain a complete econometric model of production and technical change. We can estimate the parameters of the equation for the remaining average value share, using the product exhaustion restrictions on these parameters. The complete model involves 1/2 J(J + 3) unknown parameters. A total of 1/2(J2 + 4J + 5) additional parameters can be estimated as functions of these parameters, using the homogeneity, product exhaustion, and symmetry restrictions. 32 1.3.3 Identification and Estimation We next discuss the estimation of the econometric model of production and technical change given in (1.3.5) and (1.3.6). The assump-

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tion that the input prices and the level of technology are exogenous variables implies that the model becomes a nonlinear multivariate regression model with additive errors, so that nonlinear regression techniques can be employed. This specification is appropriate for cross-section data on individual producing units. For aggregate time series data the existence of supply functions for all inputs makes it essential to treat the prices as endogenous. Under this assumption the model becomes a system of nonlinear simultaneous equations. To estimate the complete model of production and technical change by the method of full information maximum likelihood it would be necessary to specify the full econometric model, not merely the model of producer behavior. Accordingly, to estimate the model of production in (1.3.5) and (1.3.6) we consider limited information techniques. For nonlinear multivariate regression models we can employ the method of maximum likelihood proposed by Malinvaud (1980). 33 For systems of nonlinear simultaneous equations we outline the estimation of the model by the nonlinear three-stage least squares (NL3SLS) method originated by Jorgenson and Laffont (1974). Wherever the right-hand side variables can be treated as exogenous, this method reduces to limited information maximum likelihood for nonlinear multivariate regression models. Application of NL3SLS to our model of production and technical change would be straightforward, except for the fact that the covariance matrix of the disturbances is singular. We obtain NL3SLS estimators of the complete system by dropping one equation and estimating the resulting system of J equations by NL3SLS. The parameter estimates are invariant to the choice of the equation omitted in the model. The NL3SLS estimator can be employed to estimate all parameters of the model of production and technical change, provided that these parameters are identified. The necessary order condition for identification is that

where V is the number of instruments. A necessary and sufficient rank condition is given below; this amounts to the nonlinear analogue of the absence of multicollinearity. Our objective is to estimate the unknown parametersap, Bpp, bptsubject to the restrictions implied by homogeneity, product exhaustion, symmetry, and monotonicity. By dropping the equation

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for one of the value shares, we can eliminate the restrictions implied by summability. These restrictions can be used in estimating the parameters that occur in the equation that has been dropped. We impose the restrictions implied by homogeneity and symmetry as equalities. The restrictions implied by monotonicity take the form of inequalities. We can write the model of production and technical change in (1.3.5) and (1.3.6) in the form

where vj (j = 1,2, . . . , J 1) is the vector of observations on the distributive share of the j-th input for all time periods, transformed to eliminate autocorrelation, vJ is the corresponding vector of observations on the rates of technical change; the vector g includes the parametersap, at, Bpp, bpt, btt; fj (j = 1,2, . . . , J) is a vector of nonlinear functions of these parameters; finally, ej (j = 1,2, . . . , J) is the vector of disturbances in the j-th equation, transformed to eliminate autocorrelation. We can stack the equations in (1.3.13), obtaining

By the assumptions in section 1.3.1 above the random vector e has mean zero and covariance matrix Se Ä I where Se is obtained from the covariance matrix S in (1.3.11) by striking the row and column corresponding to the omitted equation. The nonlinear three-stage least squares (NL3SLS) estimator for the model of production and technical change is obtained by minimizing the weighted sum of squared residuals

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with respect to the vector of unknown parameters g, where Z is the matrix of T 1 observations on the V instrumental variables. Provided that the parameters are identified, we can apply the Gauss-Newton method to minimize (1.3.15). First, we linearize the model (1.3.14), obtaining

where g0 is the initial value of the vector of unknown parameters g and

where g1 is the revised value of this vector. The fitted residuals u depend on the initial and revised values. To revise the initial values we apply Zellner and Theil's (1962) three-stage least-squares method to the linearized model, obtaining

If S(g0) > S(g1), a further iteration is performed by replacing g0 by g1 in (1.3.16) and (1.3.17), resulting in a further revised value, say g2, and so on. If this condition is not satisfied, we divide the revision Dg by two and evaluate the criteria S(g) again; we continue reducing the revision Dg until the criterion improves or the convergence criterion maxj Dgj/gj is less than some prespecified limit. If the criterion improves, we continue with further iterations. If not, we stop the iterative process and employ the current value of the vector of unknown parameters as our NL3SLS estimator. 34 The final step in estimation of the model of production and technical change is to minimize the criterion function (1.3.15) subject to the restrictions implied by monotonicity of the distributive shares. We have eliminated the restrictions that take the form of equalities. Monotonicity of the distributive shares implies inequality restrictions on the parameters of the Cholesky factorization of the matrix of constant share elasticities Bpp. The diagonal elements of the matrix D in this factorization must be nonpositive.

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We can represent the inequality constraints on the matrix of share elasticities Bpp in the form

where J 1 is the number of restrictions. We obtain the inequality constrained nonlinear three-stage least-squares estimator for the model by minimizing the criterion function subject to the constraints (1.3.18). This estimator corresponds to the saddlepoint of the Lagrangian function

where l is a vector of J 1 Lagrange multipliers and f is a vector of J 1 constraints. The Kuhn-Tucker (1951) conditions for a saddlepoint of the Lagrangian (1.3.19) are the first-order conditions

and the complementary slackness condition

To find a saddlepoint of the Lagrangian (1.3.19) we begin by linearizing the model of production and technical change (1.3.14) as in (1.3.16). Second, we linearize the constraints as

where g0 is a vector of initial values of the unknown parameters. We apply Liew's (1976) inequality constrained three-stage least-squares method to the linearized model, obtaining

where Dd is the change in the values of the parameters (1.3.17) and l * is the solution of the linear complementarity problem

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Given an initial value of the unknown parameters g0 that satisfies the J 1 constraints (1.3.18), if S(g1) < S(g0) and d1 satisfies the constraints, the iterative process continues by linearizing the model (1.3.14) as in (1.3.16) and the constraints (1.3.18) as in (1.3.22) at the revised value of the vector of unknown parameters g1 = g0 + Dg. If not, we shrink Dg as before, continuing until an improvement is found subject to the constraints or maxj Dgj/gj is less than a convergence criterion. The nonlinear three-stage least-squares estimator obtained by minimizing the criterion function (1.3.15) is a consistent estimator of the vector of unknown parameters g. A consistent estimator of the covariance matrix Se with typical element sjk is given by

Under suitable regularity conditions the estimator

is asymptotically normal with covariance matrix

We obtain a consistent estimator of this matrix by inserting the consistent estimators and in place of the parameters g and Se. The nonlinear three-stage least-squares estimator is efficient in the class of instrumental variables estimators using Z as the matrix of instrumental variables. 35 The rank condition necessary and sufficient for identifiability of the vector of unknown parameters g is the nonsingularity of the following matrix in the neighborhood of the true parameter vector

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The order condition (1.3.12) given above is necessary for the nonsingularity of this matrix. Finally, we can consider the problem of testing equality restrictions on the vector of unknown parameters g. For example, suppose that the maintained hypothesis is that there are r = 1/2 J(J + 3) elements in this vector after solving out the homogeneity, product exhaustion, and symmetry restrictions. Additional equality restrictions can be expressed in the form

where d is a vector of unknown parameters with S elements, s < r. We can test the hypothesis

against the alternative

Test statistics appropriate for this purpose have been analyzed by Gallant and Jorgenson (1979) and Gallant and Holly (1980). 36 A statistic for testing equality restrictions in the form (1.3.27) can be constructed by analogy with the likelihood ratio principle. First, we can evaluate the criterion function (1.3.15) at the minimizing value , obtaining

Second, we can replace the vector of unknown parameters g by the function g(d) in (1.3.27)

minimizing the criterion function with respect to d, we obtain the minimizing value , and the constrained value of the criterion itself . The appropriate test statistic, say values of the criterion function

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, is equal to the difference between the constrained and unconstrained

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Gallant and Jorgenson (1979) show that this statistic is distributed asymptotically as chi-squared with r s degrees of freedom. Wherever the right-hand side variables can be treated as exogenous, this statistic reduces to the likelihood ratio statistic for nonlinear multivariate regression models proposed by Malinvaud (1980). The resulting statistic is distributed asymptotically as chi-squared. 37 1.4 Applications of Price Functions We first illustrate the econometric modeling of substitution among inputs in section 1.4.1 by presenting an econometric model for nine industrial sectors of the U.S. economy implemented by Berndt and Jorgenson (1973). The Berndt Jorgenson model is based on a price function for each sector, giving the price of output as a function of the prices of capital and labor inputs and the prices of inputs of energy and materials. Technical change is assumed to be neutral, so that all biases of technical change are set equal to zero. In section 1.4.2 we illustrate the economectric modeling of both substitution and technical change. We present an econometric model of producer behavior that has been implemented for thirty-five industrial sectors of the U.S. economy by Jorgenson and Fraumeni (1983). In this model the rate of technical change and the distributive shares of productive inputs are determined simultaneously as functions of relative prices. Although the rate of technical change is endogenous, this model must be carefully distinguished from models of induced technical change. Aggregation over inputs has proved to be an extremely important technique for simplifying the description of technology for empirical implementation. The corresponding restrictions can be used to generate a two-stage model of producer behavior. Each stage can be parameterized separately; alternatively, the validity of alternative simplifications can be assessed by testing the restrictions. In section 1.4.3 we conclude with illustrations of aggregation over inputs in studies by Berndt and Jorgenson (1973) and Berndt and Wood (1975). 1.4.1 Substitution In the Berndt Jorgenson (1973) model, production is divided among nine sectors of the U.S. economy:

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1. Agriculture, nonfuel mining, and construction 2. Manufacturing, excluding petroleum refining 3. Transportation 4. Communications, trade, and services 5. Coal mining 6. Crude petroleum and natural gas 7. Petroleum refining 8. Electric utilities 9. Gas utilities. The nine producing sectors of the U.S. economy included in the Berndt Jorgenson model can be divided among five sectors that produce energy commoditiescoal, crude petroleum and natural gas, refined petroleum, electricity, and natural gas as a product of gas utilitiesand four sectors that produce nonenergy commoditiesagriculture, manufacturing, transportation, and communications. For each sector output is defined as the total domestic supply of the corresponding commodity group, so that the input into the sector includes competitive imports of the commodity, inputs of energy, and inputs of nonenergy commodities. The Berndt Jorgenson model of producer behavior includes a system of equations for each of the nine producing sectors giving the shares of capital, labor, energy and materials inputs in the value of output as functions of the prices of the four inputs. To formulate an econometric model stochastic components are added to this system of equations. The rate of technical change is taken to be exogeneous, so that the adjustment for autocorrelation described in section 1.3.2 is not required. However, all prices are treated as endogenous variables; estimates of the unknown parameters of the econometric model are based on the nonlinear three-stage least-squares estimator presented in section 1.3.3. The endogenous variables in the Berndt Jorgenson model of producer behavior include value shares of capital, labor, energy, and materials input for each sector. Three equations can be estimated for each sector, corresponding to three of the value shares, as in (1.2.14). The unknown parameters include three elements of the vector {ap} and six share elasticities in the matrix {Bpp}, which is constrained to be symmetric, so that there is a total of nine unkown parameters. Berndt and Jorgenson estimate these parameters from time series data for the

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period 1947 1971 for each industry; the estimates are presented by Hudson and Jorgenson (1974). As a further illustration of modeling of substitution among inputs, we consider an econometric model of the total manufacturing sector of the U.S. economy implemented by Berndt and Wood (1975). This sector combines the manufacturing and petroleum refining sectors of the Berndt-Jorgenson model. Berndt and Wood generate this model by expressing the price of aggregate input as a function of the prices of capital, labor, energy, and materials inputs into total manufacturing. They find that capital and energy inputs are complements, while all other pairs of inputs are substitutes. By comparison with the results of Berndt and Wood, Hudson and Jorgenson (1978) have classified patterns of substitution and complementarity among inputs for the four nonenergy sectors of the Berndt-Jorgenson model. For agriculture, nonfuel mining and construction, capital and energy are complements and all other pairs of inputs are substitutes. For manufacturing, excluding petroleum refining, energy is complementary with capital and materials, while other pairs of inputs are substitutes. For transportation, energy is complementary with capital and labor while other pairs of inputs are substitutes. Finally, for communications, trade and services, energy and materials are complements and all other pairs of inputs are substitutes. Berndt and Wood have considered further simplification of the Berndt-Jorgenson model of producer behavior by imposing separability restrictions on patterns of substitution among capital, labor, energy, and materials inputs. 38 This would reduce the number of input prices at the first stage of the model through the introduction of additional input aggregates. For this purpose additional stages in the allocation of the value of sectoral output among inputs would be required. Berndt and Wood consider all possible pairs of capital. Labor, energy, and materials inputs, but find that only the input aggregate consisting of capital and energy is consistent with the empirical evidence.39 Berndt and Morrison (1979) have disaggregated the Berndt-Wood data on labor input between blue collar and white collar labor and have studied the substitution among the two types of labor and capital, energy, and materials inputs for U.S. total manufacturing, using a translog price function. Anderson (1981) has reanalyzed the BerndtWood data set, testing alternative specifications of the model of substitution among inputs. Gallant (1981) has fitted an alternative model of substitution among inputs to these data, based on the Fourier func-

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tional form for the price function. Elbadawi, Gallant, and Souza (1983) have employed this approach in estimating price elasticities of demand for inputs, using the Berndt-Wood data as a basis for Monte Carlo simulations of the performance of alternative functional forms. Cameron and Schwartz (1979), Denny, May, and Pinto (1978), Fuss (1977a), and McRae (1981) have constructed econometric models of substitution among capital, labor, energy, and materials inputs based on translog functional forms for total manufacturing in Canada. Technical change is assumed to be neutral, as in the study of U.S. total manufacturing by Berndt and Wood (1975), but nonconstant returns to scale are permitted. McRae and Webster (1982) have compared models of substitution among inputs in Canadian manufacturing, estimated from data for different time periods. Friede (1979) has analyzed substitution among capital, labor, energy, and materials inputs for total manufacturing in the Federal Republic of Germany. He assumes that technical change is neutral and utilizes a translog price function. He has disaggregated the results to the level of fourteen industrial groups, covering the whole of the West German economy. He has separated materials inputs into two groupsmanufacturing and transportation services as one group and other nonenergy inputs as a second group. Ozatalay, Grumbaugh, and Long (1979) have modeled substitution among capital, labor, energy and materials inputs, on the basis of a translog price function. They use time-series data for total manufacturing for the period 1963 1974 in seven countriesCanada, Japan, the Netherlands, Norway, Sweden, the U.S., and West Germany. Longva and Olsen (1983) have analyzed substitution among capital, labor, energy, and materials inputs for total manufacturing in Norway. They assume that technical change is neutral and utilize a generalized Leontief price function. They have disaggregated the results to the level of nineteen industry groups. These groups do not include the whole of the Norwegian economy; eight additional industries are included in a complete multisectoral model of production for Norway. Dargay (1983) has constructed econometric models of substitution among capital, labor, energy, and materials inputs based on translog functional forms for total manufacturing in Sweden. She assumes that technical change is neutral, but permits nonconstant returns to scale. She has disaggregated the results to the level of twelve industry groups within Swedish manufacturing.

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Although the breakdown of inputs among capital, labor, energy, and materials has come to predominate in econometric models of production at the industry level Humphrey and Wolkowitz (1976) have grouped energy and materials inputs into a single aggregate input in a study of substitution among inputs in several U.S. manufacturing industries that utilizes translog price functions. Friedlaender and Spady (1980) have disaggregated transportation services between trucking and rail service and have grouped other inputs into capital, labor and materials inputs. Their study is based on cross-section data for ninety-six three-digit industries in the United States for 1972 and employs a translog functional form with fixed inputs. Parks (1971) has employed a breakdown of intermediate inputs among agricultural materials, imported materials and commercial services, and transportation services in a study of Swedish manufacturing based on the generalized Leontief functional form. Denny and May (1978) have disaggregated labor input between white collar and blue collar labor, capital input between equipment and structures, and have grouped all other inputs into a single aggregate input for Canadian total manufacturing, using a translog functional form. Frenger (1978) has analyzed substitution among capital, labor, and materials inputs for three industries in Norway, breaking down intermediate inputs in a different way for each industry, and utilizing a generalized Leontief functional form. Griffin (1977a,b,c, 1978) has estimated econometric models of substitution among inputs for individual industries based on translog functional forms. For this purpose he has employed data generated by process models of the U.S. electric power generation, petroleum refining, and petrochemical industries constructed by Thompson et al. (1977). Griffin (1979) and Kopp and Smith (1980a,b, 1981a,b) have analyzed the effects of alternative aggregations of intermediate inputs on measures of substitution among inputs in the steel industry. For this purpose they have utilized data generated from a process analysis model of the U.S. steel industry constructed by Russell and Vaughan (1976). 40 Although we have concentrated attention on substitution among capital, labor, energy, and materials inputs, there exists a sizable literature on substitution among capital, labor, and energy inputs alone. In this literature the price function is assumed to be homothetically separable in the prices of these inputs. This requires that all possible pairs of the inputscapital and labor, capital and energy, and labor and

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energyare separable from materials inputs. As we have observed above, only capital-energy separability is consistent with the results of Berndt and Wood (1975) for U.S. total manufacturing. Appelbaum (1979b) has analyzed substitution among capital, labor, and energy inputs in the petroleum and natural gas industry of the United States, based on the data of Berndt and Jorgenson. Field and Grebenstein (1980) have analyzed substitution among physical capital, working capital, labor, and energy for ten two-digit U.S. manufacturing industries on the basis of translog price functions, using cross-section data for individual states for 1971. Griffin and Gregory (1976) have modeled substitution among capital, labor, and energy inputs for total manufacturing in nine major industrialized countriesBelgium, Denmark, France, Italy, the Netherlands, Norway, the U.K., the U.S., and West Germanyusing a translog price function. They pool four cross sections for these countries for the years 1955, 1960, 1965, and 1969, allowing for differences in technology among countries by means of one-zero dummy variables. Their results differ substantially from those of Berndt and Jorgenson and Berndt and Wood. These differences have led to an extensive discussion among Berndt and Wood (1979, 1981), Griffin (1981a,b), and Kang and Brown (1981), attempting to reconcile the alternative approaches. Substitution among capital, labor, and energy inputs requires a price function that is homothetically separable in the prices of these inputs. An alternative specification is that the price function is homothetically separable in the prices of capital, labor, and natural resource inputs. This specification has been utilized by Humphrey, Burras, and Moroney (1975), Moroney and Toevs (1977, 1979) and Moroney and Trapani (1981a,b) in studies of substitution among these inputs for individual manufacturing industries in the United States based on translog price functions. A third alternative specification is that the price function is separable in the prices of capital and labor inputs. Berndt and Christensen (1973b, 1974) have used translog price functions employing this specification in studies of substitution among individual types of capital and labor inputs for U.S. total manufacturing. Berndt and Christensen (1973b) have divided capital input between structures and equipment inputs and have tested the separability of the two types of capital input from labor input. Berndt and Christensen (1974) have divided labor input between blue collar and white collar inputs and have tested the separability of the two types of labor input from capital

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input. Hamermesh and Grant (1979) have surveyed the literature on econometric modeling of substitution among different types of labor input. Woodland (1975) has analyzed substitution among structures, equipment and labor inputs for Canadian manufacturing, using generalized Leontief price functions. Woodland (1978) has presented an alternative approach to testing separability and has applied it in modeling substitution among two types of capital input and two types of labor input for U.S. total manufacturing, using the translog parametric form. Field and Berndt (1981) and Berndt and Wood (1979, 1981) have surveyed econometric models of substitution among inputs. They focus on substitution among capital, labor, energy and materials inputs at the level of individual industries. 1.4.2 Technical Change The Jorgenson-Fraumeni (1983) model is based on a production function characterized by constant returns to scale for each of thirty-five industrial sectors of the U.S. economy. Output is a function of inputs of primary factors of productioncapital and labor servicesinputs of energy and materials, and time as an index of the level of technology. While the rate of technical change is endogenous in this econometric model, the model must be carefully distinguished from models of induced technical change, such as those analyzed by Hicks (1932), Kennedy (1964), Samuelson (1965b), von Weizsäcker (1962), and many others. In those models the biases of technical change are endogenous and depend on relative prices. As Samuelson (1965b) has pointed out, models of induced technical change require intertemporal optimization since technical change at any point of time affects future production possibilities. 41 In the Jorgenson-Fraumeni model of producer behavior myopic decision rules can be derived by treating the price of capital input as a rental price of capital service.42 The rate of technical change at any point of time is a function of relative prices, but does not affect future production possibilities. This greatly simplifies the modeling of producer behavior and facilitates the implementation of the econometric model. Given myopic decision rules for producers in each industrial sector, all of the implications of the economic theory of production can be described in terms of the properties of the sectoral price functions given in section 1.2.1.43

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The Jorgenson-Fraumeni model of producer behavior consists of a system of equations giving the shares of capital, labor, energy, and materials inputs in the value of output and the rate of technical change as functions of relative prices and time. To formulate an econometric model a stochastic component is added to these equations. Since the rate of technical change is not directly observable, we consider a form of the model with autocorrelated disturbances; the data are transformed to eliminate the autocorrelation. The prices are treated as endogenous variables and the unknown parameters are estimated by the method of nonlinear three-stage least squares presented in section 1.3.3. The endogenous variables in the Jorgenson-Fraumeni model include value shares of sectoral inputs for four commodity groups and the sectoral rate of technical change. Four equations can be estimated for each industry, corresponding to three of the value shares and the rate of technical change. As unknown parameters there are three elements of the vector {ap}, the scalar {at}, six share elasticities in the matrix {Bpp}, which is constrained to be symmetric, three biases of technical change in the vector {bpt}, and the scalar {btt}, so that there is a total of fourteen unknown parameters for each industry. Jorgenson and Fraumeni estimate these parameters from timeseries data for the period 1958 1974 for each industry, subject to the inequality restrictions implied by monotonicity of the sectoral input value shares. 44 The estimated share elasticities with respect to price {Bpp} describe the implications of patterns of substitution for the distribution of the value of output among capital, labor, energy, and materials inputs. Positive share elasticities imply that the corresponding value shares increase with an increase in price; negative share elasticities imply that the value shares decrease with price; zero share elasticities correspond to value shares that are independent of price. The concavity constraints on the sectoral price functions contribute substantially to the precision of the estimates, but require that the share of each input be nonincreasing in the price of the input itself. The empirical findings on patterns of substitution reveal some striking similarities among industries.45 The elasticities of the shares of capital with respect to the price of labor are nonnegative for thirty-three of the thirtyfive industries, so that the shares of capital are non-decreasing in the price of labor for these thirty-three sectors. Similarly, elasticities of the share of capital with respect to the price of energy are nonnegative for thirty-four industries and elasticities with respect to

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the price of materials are nonnegative for all thirty-five industries. The share elasticities of labor with respect to the prices of energy and materials are nonnegative for nineteen and for all thirty-five industries, respectively. Finally, the share elasticities of energy with respect to the price of materials are nonnegative for thirty of the thirty-five industries. We continue the interpretation of the empirical results with estimated biases of technical change with respect to price {bpt}. These parameters can be interpreted as changes in the sectoral value shares (1.2.14) with respect to time, holding prices constant. This component of change in the value shares can be attributed to changes in technology rather than to substitution among inputs. For example, if the bias of technical change with respect to the price of capital input is positive, we say that technical change is capital-using; if the bias is negative, we say that technical change is capital-saving. Considering the rate of technical change (1.2.14). The biases of technical change {bpt} can be interpreted in an alternative and equivalent way. These parameters are changes in the negative of the rate of technical change with respect to changes in prices. As substitution among inputs takes place in response to price changes, the rate of technical change is altered. For example, if the bias of technical change with respect to capital input is positive, an increase in the price of capital input decreases the rate of technical change; if the bias is negative, an increase in the price of capital input increases the rate of technical change. A classification of industries by patterns of the biases of technical change is given in table 1.1. The pattern that occurs with greatest frequency is capital-using, labor-using, energy-using, and materials-saving technical change. This pattern occurs for nineteen of the thirty-five industries for which biases are fitted. Technical change is capitalusing for twenty-five of the thirty-five industries, labor-using for thirty-one industries, energy-using for twentynine industries, and materials-using for only two industries. The patterns of biases of technical change given in table 1.1 have important implications for the relationship between relative prices and the rate of economic growth. An increase in the price of materials increases the rate of technical change in thirty-three of the thirty-five industries. By contrast, increases in the prices of capital, labor, and energy reduced the rates of technical change in twenty-five, thirty-one, and twenty-nine industries, respectively. The substantial in-

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Table 1.1 Classification of industries by biases of technical change Pattern of Industries biases Agriculture, metal mining, crude petroleum and natural gas, Capital using Labor nonmetallic mining, textiles, apparel, lumber, furniture, printing, leather, fabricated metals, electrical machinery, using motor vehicles, instruments, miscellaneous manufacturing, Energy transportation, trade, finance, insurance and real estate, using Material services saving Coal mining, tobacco manufacturers, communications, Capital using Labor government enterprises using Energy saving Material saving Petroleum refining Capital using Labor saving Energy using Material saving Construction Capital using Labor saving Energy saving Material using Electric utilities Capital saving Labor saving Energy using Material saving Primary metals Capital saving Labor using Energy saving Material saving Paper, chemicals, rubber, stone, clay and glass, machinery Capital except electrical, transportation equipment and ordnance, gas saving Labor usingutilities Energy using Material

saving Food Capital saving Labor saving Energy using Material using Source: Jorgenson and Fraumeni (1983b), p. 264.

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creases in energy prices since 1973 have had the effect of reducing sectoral rates of technical change, slowing the aggregate rate of technical change, and diminishing the rate of growth for the U.S. economy as a whole. 46 While the empirical results suggest a considerable degree of similarity across industries, it is necessary to emphasize that the Jorgenson-Fraumeni model of producer behavior requires important simplifying assumptions. First, conditions for producer equilibrium under perfect competition are employed for all industries. Second, constant returns to scale at the industry level are assumed. Finally, a description of technology that leads to myopic decision rules is employed. These assumptions must be justified primarily by their usefulness in implementing production models that are uniform for all thirty-five industrial sectors of the U.S. economy. Binswanger (1974a,c, 1978c) has analyzed substitution and technical change for U.S. agriculture, using cross sections of data for individual states for 1949, 1954, 1959, and 1964. Binswanger was the first to estimate biases of technical change based on the translog price function. He permits technology to differ among time periods and among groups of states within the United States. He divides capital inputs between land and machinery and divides intermediate inputs between fertilizer and other purchased inputs. He considers substitution among these four inputs and labor input. Binswanger employs time-series data on U.S. agriculture as a whole for the period 1912 1964 to estimate biases of technical change on an annual basis. Brown and Christensen (1981) have analyzed time-series data on U.S. agriculture for the period 1947 1974. They divide labor services between hired labor and self-employed labor and capital input between land and all othermachinery, structures, and inventories. Other purchased inputs are treated as a single aggregate. They model substitution and technical change with fixed inputs, using a translog functional form. Berndt and Khaled (1979) have augmented the Berndt-Wood data set for U.S. manufacturing to include data on output. They estimate biases of technical change and permit nonconstant returns to scale. They employ a Box-Cox transformation of data on input prices, generating a functional form that includes the translog, generalized Leontief, and quadratic as special cases. The Box-Cox transformation is also employed by Appelbaum (1979a) and by Caves, Christensen, and

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Trethaway (1980). Denny (1974) has proposed a closely related approach to parameterization based on mean value functions. Kopp and Diewert (1982) have employed a translog parametric form to study technical and allocative efficiency. For this purpose they have analyzed data on U.S. total manufacturing for the period 1947 1971 compiled by Berndt and Wood (1975) and augmented by Berndt and Khaled (1979). Technical change is not required to be neutral and nonconstant returns to scale are permitted. They have interpreted the resulting model of producer behavior as a representation of average practice. They have then rescaled the parameters to obtain a ''frontier" representing best practice and have employed the results to obtain measures of technical and allocative efficiency for each year in the sample. 47 Wills (1979) has modeled substitution and technical change for the U.S. steel industry, using a translog price function. Norsworthy and Harper (1981) have extended and augmented the Berndt-Wood data set for total manufacturing and have modeled substitution and technical change, using a translog price function. Woodward (1983) has reanalyzed these data and has derived estimates of rates of factor augmentation for capital, labor, energy, and materials inputs, using a translog price function. Jorgenson (1984b) has modeled substitution and technical change for thirty-five industries of the United States for the period 1958 1979, dividing energy inputs between electricity and nonelectrical energy inputs. He employs translog price functions with capital, labor, two kinds of energy, and materials inputs and finds that technical change is electricity-using and nonelectrical energy-using for most U.S. industries. Nakamura (1984) has developed a similar model for twelve sectors covering the whole of the economy for the Federal Republic of Germany for the period 1960 1974. He has disaggregated intermediate inputs among energy, materials, and services. We have already discussed the work of Kopp and Smith on substitution among inputs, based on data generated by process models of the U.S. steel industry. Kopp and Smith (1981c, 1982) have also analyzed the performance of different measures of technical change, also using data generated by these models. They show that measures of biased technical change based on the methodology developed by Binswanger can be explained by the proportion of investment in specific technologies.

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Econometric models of substitution among inputs at the level of individual industries have incorporated intermediate inputsbroken down between energy and materials inputsalong with capital and labor inputs. However, models of substitution and technical change have also been constructed at the level of the economy as a whole. Output can be divided between consumption and investment goods, as in the original study of the translog price function by Christensen, Jorgenson, and Lau (1971, 1973), and input can be divided between capital and labor services. Hall (1973) has considered nonjointness of production of investment and consumption goods outputs for the United States. Kohli (1981, 1983) has also studied nonjointness in production for the United States. Burgess (1974) has added imports as an input to inputs of capital and labor services. Denny and Pinto (1978) developed a model with this same breakdown of inputs for Canada. Conrad and Jorgenson (1977, 1978a) have considered nonjointness of production and alternative models of technical change for the Federal Republic of Germany. 1.4.3 Two-Stage Allocation Aggregation over inputs has proved to be a very important means for simplifying the description of technology in modeling producer behavior. The price of output can be represented as a function of a smaller number of input prices by introducing price indexes for input aggregates. These price indexes can be used to generate a second stage of the model by treating the price of each aggregate as a function of the prices of the inputs making up the aggregate. We can parameterize each stage of the model separately. The Berndt-Jorgenson (1973) model of producer behavior is based on two-stage allocation of the value of output of each sector. In the first stage the value of sectoral output is allocated among capital, labor, energy, and materials inputs, where materials include inputs of nonenergy commodities and competitive imports. In the second stage the value of energy expenditure is allocated among expenditures on individual types of energy and the value of materials expenditure is allocated among expenditures on competitive imports and nonenergy commodities. The first stage of the econometric model is generated from a price function for each sector. The price of sectoral output is a function of

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the prices of capital and labor inputs and the prices of inputs of energy and materials. The second stage of the model is generated from price indexes for energy and materials inputs. The price of energy is a function of the prices of five types of energy inputs, while the price of materials is a function of the prices of four types of nonenergy inputs and the price of competitive imports. The Berndt-Jorgenson model of producer behavior consists of three systems of equations. The first system gives the shares of capital, labor, energy and materials inputs in the value of output, the second system gives the shares of energy inputs in the value of energy input, and the third system gives the shares of nonenergy inputs and competitive imports in the value of materials inputs. To formulate an econometric model stochastic components are added to these systems of equations. The rate of technical change is taken to be exogenous; all pricesincluding the prices of energy and materials inputs for each sectorare treated as endogenous variables. Estimates of the unknown parameters of all three systems of equations are based on the nonlinear three-stage least-squares estimator. The Berndt-Jorgenson model illustrates the use of two stage allocation to simplify the description of producer behavior. By imposing the assumption that the price of aggregate input is separable in the prices of individual energy and materials inputs, the price function that generates the first stage of the model can be expressed in terms of four input prices rather than twelve. However, simplifications of the first stage of the model requires the introduction of a second stage, consisting of price functions for energy and materials inputs. Each of these price functions can be expressed in terms of five prices of individual inputs. Fuss (1977a) has constructed a two-stage model of Canadian total manufacturing using translog functional forms. He treats substitution among coal, liquid petroleum gas, fuel oil, natural gas, electricity, and gasoline as a second stage of the model. Friede (1979) has developed two-stage models based on translog price functions for fourteen industries of the Federal Republic of Germany. In these models the second stage consists of three separate modelsone for substitution among individual types of energy and two for substitution among individual types of nonenergy inputs. Dargay (1983) has constructed a two-stage model of twelve Swedish manufacturing industries utilizing a translog functional form. She has analyzed substitution among electricity, oil, and solid fuels inputs at the second stage of the model.

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Nakamura (1984) has constructed three-stage models for twelve industries of the Federal Republic of Germany, using translog price functions. The first stage encompasses substitution and technical change among capital, labor, energy, materials, and services inputs. The second stage consists of three modelsa model for substitution among individual types of energy, a model for substitution among individual types of materials, and a model for substitution among individual types of services. The third stage consists of models for substitution between domestically produced input and the corresponding imported input of each type. Pindyck (1979a,b) has constructed a two-stage model of total manufacturing for ten industrialized countriesCanada, France, Italy, Japan, the Netherlands, Norway, Sweden, the U.K., the U.S., and West Germanyusing a translog price function. He employs annual data for the period 1959 1973 in estimating a model for substitution among four energy inputscoal, oil, natural gas, and electricity. He uses annual data for the period 1963 1973 in estimating a model for substitution among capital, labor, and energy inputs. Magnus (1979) and Magnus and Woodland (1980) have constructed a two-stage model for total manufacturing in the Netherlands along the same lines. Similarly, Ehud and Melnik (1981) have developed a two-stage model for the Israeli economy. Halvorsen (1977) and Halvorsen and Ford (1979) have constructed a two-stage model for substitution among capital, labor, and energy inputs for nineteen two-digit U.S. manufacturing industries on the basis of translog price functions. For this purpose they employ cross-section data for individual states in 1971. The second stage of the model provides a disaggregation of energy input among inputs of coal, oil, natural gas, and electricity. Halvorsen (1978) has analyzed substitution among different types of energy on the basis of cross-section data for 1958, 1962, and 1971. 1.5 Cost Functions In section 1.2, we have considered producer behavior under constant returns to scale. The production function (1.2.1) is homogeneous of degree one, so that a proportional change in all inputs results in a change in output in the same proportion. Necessary conditions for producer equilibrium (1.2.2) are that the value share of each input is equal to the elasticity of output with respect to that input. Under constant returns to scale the value shares and the elasticities sum to unity.

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In this section, we consider producer behavior under increasing returns to scale. Under increasing returns and competitive markets for output and all inputs, producer equilibrium is not defined by profit maximization, since no maximum of profit exists. However, in regulated industries the price of output is set by regulatory authority. Given demand for output as a function of the regulated price, the level of output is exogenous to the producing unit. With output fixed from the point of view of the producer, necessary conditions for equilibrium can be derived from cost minimization. Where total cost is defined as the sum of expenditures on all inputs, the minimum value of cost can be expressed as a function of the level of output and the prices of all inputs. We refer to this function as the cost function. We have described the theory of production under constant returns to scale in terms of properties of the price function (1.2.4); similarly, we can describe the theory under increasing returns in terms of properties of the cost function. 1.5.1 Duality Utilizing the notation of section 1.2, we can define total cost, say c, as the sum of expenditures on all inputs

We next define the shares of inputs in total cost by

With output fixed from the point of view of the producing unit and competitive markets for all inputs, the necessary conditions for producer equilibrium are given by equalities between the shares of each input in total cost and the ratio of the elasticity of output with respect to that input and the sum of all such elasticities

where i is a vector of ones and v = (v1, v2, . . . , vJ)vector of cost shares.

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Given the definition of total cost and the necessary conditions for producer equilibrium, we can express total cost, say c, as a function of the prices of all inputs and the level of output

We refer to this as the cost function. The cost function C is dual to the production function F and provides an alternative and equivalent description of the technology of the producing unit. 48 We can formalize the theory of production in terms of the following properties of the cost function: 1. Positivity. The cost function is positive for positive input prices and a positive level of output. 2. Homogeneity. The cost function is homogeneous of degree one in the input prices. 3. Monotonicity. The cost function is increasing in the input prices and in the level of output. 4. Concavity. The cost function is concave in the input prices. Given differentiability of the cost function, we can express the cost shares of all inputs as elasticities of the cost function with respect to the input prices

Further, we can define an index of returns to scale as the elasticity of the cost function with respect to the level of output

Following Frisch (1965), we can refer to this elasticity as the cost flexibility. The cost flexibility vy is the reciprocal of the degree of returns to scale, defined as the elasticity of output with respect to a proportional increase in all inputs

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If output increases more than in proportion to the increase in inputs, cost increases less than in proportion to the increase in output. Since the cost function C is homogeneous of degree one in the input prices, the cost shares and the cost flexibility are homogeneous of degree zero and the cost shares sum to unity

Since the cost function is increasing in the input prices, the cost shares must be nonnegative and not all zero

The cost function is also increasing in the level of output, so that the cost flexibility is positive

1.5.2 Substitution and Economies of Scale We have represented the cost shares of all inputs and the cost flexibility as functions of the input prices and the level of output. We can characterize these functions in terms of measures of substitution and economies of scale. We obtain share elasticities by differentiating the logarithm of the cost function twice with respect to the logarithms of input prices

These measures of substitution give the response of the cost shares of all inputs to proportional changes in the input prices. Second, we can differentiate the logarithm of the cost function twice with respect to the logarithms of the input prices and the level of output to obtain measures of economies of scale

We refer to these measures as biases of scale. The vector of biases of scale upy can be employed to derive the implications of economies of

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scale for the relative distribution of total cost among inputs. If a scale bias is positive, the cost share of the corresponding input increases with a change in the level of output. If a scale bias is negative, the cost share decreases with a change in output. Finally, if a scale bias is zero, the cost share is independent of output. Alternatively, the vector of biases of scale upy can be employed to derive the implications of changes in input prices for the cost flexibility. If the scale bias is positive, the cost flexibility increases with the input price. If the scale bias is negative, the cost flexibility decreases with the input price. Finally, if the bias is zero, the cost flexibility is independent of the input price. To complete the description of economies of scale we can differentiate the logarithm of the cost function twice with respect to the level of output

If this measure is positive, zero, or negative, the cost flexibility is increasing, decreasing, or independent of the level of output. The matrix of second-order logarithmic derivatives of the logarithms of the cost function C must be symmetric. This matrix includes the matrix of share elasticities Upp, the vector of biases of scale upy, and the derivative of the cost flexibility with respect to the logarithm of output uyy. Concavity of the cost function in the input prices implies that the matrix of second-order derivatives, say H, is nonpositive definite, so that the matrix Upp + vv' V is nonpositive definite, where

Total cost c is positive and the diagonal matrices N and V are defined in terms of the input prices p and the cost shares v, as in section 1.2. Two inputs are substitutes if the corresponding element of the matrix Upp + vv' V is negative, complements if the element is negative, and independent if the element is zero. In section 1.2.2 we have introduced price and quantity indexes of aggregate input implied by homothetic separability of the price function. We can analyze the implications of homothetic separability of the cost function by introducing price and quantity indexes of aggregate input and defining the cost share of aggregate input in terms

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of these indexes. An aggregate input can be treated in precisely the same way as any other input, so that price and quantity indexes can be used to reduce the dimensionality of the space of input prices and quantities. We say that the cost function C is homothetic if and only if the cost function is separable in the prices of all J inputs {p1, p2, . . . , pJ}, so that

where the function P is homogeneous of degree one and independent of the level of output y. The cost function is homothetic if and only if the production function is homothetic, where

where the function G is homogeneous of degree one. 49 Since the cost function is homogeneous of degree one in the input prices, it is homogeneous of degree one in the function P, which can be interpreted as the price index for a single aggregate input; the function G is the corresponding quantity index. Furthermore, the cost function can be represented as the product of the price index of aggregate input P and a function, say H, of the level of output

Under homotheticity, the cost flexibility vy is independent of the input prices

If the cost flexibility is also independent of the level of output, the cost function is homogeneous in the level of output and the production function is homogeneous in the quantity index of aggregate input G. The degree of homogeneity of the production function is the degree of returns to scale and is equal to the reciprocal of the cost flexibility. Under constant returns to scale the degree of returns to scale and the cost flexibility are equal to unity.

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1.5.3 Parameterization and Integrability In section 1.2.3 we have generated an econometric model of producer behavior by treating the measures of substitution and technical change as unknown parameters to be estimated. In this section we generate an econometric model of cost and production by introducing the parameters

where Bpp is a matrix of constant share elasticities, bpy is a vector of constant biases of scale, and byy is a constant derivative of the cost flexibility with respect to the logarithm of output. We can regard the matrix of share elasticities, the vector of biases of scale, and the derivative of the cost flexibility with respect to the logarithm of output as a system of second-order partial differential equations. We can integrate this system to obtain a system of first-order partial differential equations

where the parametersap, ayare constants of integration. Choosing scales for measuring the quantities and prices of output and the inputs, we can consider values of input prices and level of output equal to unity. At these values the vector of parameters ap is equal to the vector of cost shares and the parameters ay is equal to the cost flexibility. We can integrate the system of first-order partial differential equations (1.5.14) to obtain the cost function

where the parameter a0 is a constant of integration. This parameter is equal to the logarithm of total cost where the input prices and level of output are equal to unity. We can refer to this form as the translog cost function, indicating the role of the variables, or the constant share elasticity (CSE) cost function, indicating the role of the parameters. To incorporate the implications of the economic theory of production we consider restrictions on the system of equations (1.5.14)

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required to obtain a cost function with properties listed above. A complete set of conditions for integrability is the following: 1.5.3.1 Homogeneity The cost shares and the cost flexibility are homogeneous of degree zero in the input prices. Homogeneity of degree zero of the cost shares and the cost flexibility implies that the parametersBpp and bpymust satisfy the restrictions

where i is a vector of ones. For J inputs there are J + 1 restrictions implied by homogeneity. 1.5.3.2 Cost Exhaustion The sum of the cost shares is equal to unity. Cost exhaustion implies that the value of the J inputs is equal to total cost. Cost exhaustion implies that the parametersap, Bpp, bpymust satisfy the restrictions

For J inputs there are J + 2 restrictions implied by cost exhaustion. 1.5.3.3 Symmetry The matrix of share elasticities, biases of scale, and the derivative of the cost flexibility with respect to the logarithm output must be symmetric. A necessary and sufficient condition for symmetry is that the matrix of parameters must satisfy the restrictions

For J inputs the total number of symmetry restrictions is 1/2 J(J + 1).

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1.5.3.4 Nonnegativity The cost shares and the cost flexibility must be nonnegative. Since the translog cost function is quadratic in the logarithms of the input prices and the level of output, we cannot impose restrictions on the parameters that imply nonnegativity of the cost shares and the cost flexibility. Instead, we consider restrictions on the parameters that imply monotonicity of the cost shares wherever they are nonnegative. 1.5.3.5 Monotonicity The matrix of share elasticities Bpp + vv' V is nonpositive definite. The conditions on the parameters assuring concavity of the cost function wherever the cost shares are nonnegative are precisely analogous to the conditions given in section 1.2.4 for concavity of the price function wherever the value shares are nonnegative. These conditions can be expressed in terms of the Cholesky factorization of the matrix of constant share elasticities Bpp. 1.5.4 Stochastic Specification To formulate an econometric model of cost and production we add a stochastic component to the equations for the cost shares and the cost function itself. To represent the econometric model we require some additional notation. Where there are K producing units we index the observations by producing unit (k = 1,2, . . . , K). The vector of cost shares for the k-th unit is denoted vk and total cost of the unit is ck (k = 1,2, . . . , K). The vector of input prices faced by the k-th unit is denoted pk and the vector of logarithms of input prices is ln pk (k = 1,2, . . . , K). Finally, the level of output of the i-th unit is denoted yk (k = 1,2, . . . , K). We obtain an econometric model of cost and production corresponding to the translog cost function by adding random disturbances to the equations for the cost shares and the cost function

where ek is the vector of unobservable random disturbances for the

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is the corresponding disturbance for the cost function (k = 1,2, . . . , cost shares of the k-th producing unity and K). Since the cost shares for all inputs sum to unity for each producing unit, the random disturbances corresponding to the J cost shares sum to zero for each unit

so that these disturbances are not distributed independently. We assume that the unobservable random disturbances for all J + 1 equations have expected value equal to zero for all observations

We also assume that the disturbances have a covariance matrix that is the same for all producing units and has rank J, where

Finally, we assume that random disturbances corresponding to distinct observations are uncorrelated, so that the covariance matrix of random disturbances for all observations has the Kronecker product form

We can test the validity of restrictions on economies of scale by expressing them in terms of the parameters of an econometric model of cost and production. Under homotheticity the cost flexibility is independent of the input prices. A necessary and sufficient condition for homotheticity is given by

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the vector of biases of scale is equal to zero. Under homogeneity the cost flexibility is also independent of output, so that

the derivative of the flexibility with respect to the logarithm of output is zero. Finally, under constant returns to scale, the cost flexibility is equal to unity; given the restrictions implied by homogeneity, constant returns requires

1.6 Applications of Cost Functions To illustrate the econometric modeling of economies of scale in section 1.6.1, we present an econometric model that has been implemented for the electric power industry in the United States by Christensen and Greene (1976). This model is based on cost functions for cross sections of individual electric utilities in 1955 and 1970. Total cost of steam generation is a function of the level of output and the prices of capital, labor, and fuel inputs. Steam generation accounts for more than ninety percent of total power generation for each of the firms in the ChristensenGreene sample. A key feature of the electric power industry in the United States is that individual firms are subject to price regulation. The regulatory authority sets the price for electric power. Electric utilities are required to supply the electric power that is demanded at the regulated price. This model must be carefully distinguished from the model of a regulated firm proposed by Averch and Johnson (1962). 50 In the Averch-Johnson model firms are subject to an upper limit on the rate of return rather than price regulation. Firms minimize costs under rate of return regulation only if the regulatory constraint is not binding. The literature on econometric modeling of scale economies in U.S. transportation and communications industries parallels the literature on the U.S. electric power industry. Transportation and communications firms, like electric utilities, are subject to price regulation and are required to supply all the services that are demanded at the regulated price. However, the modeling of transportation and communications services is complicated by joint production of several outputs. We review econometric models with multiple outputs in section 1.6.2.

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1.6.1 Economies of Scale The Christensen-Greene model of the electric power industry consists of a system of equations giving the shares of all inputs in total cost and total cost itself as functions of relative prices and the level of output. To formulate an econometric model Christensen and Greene add a stochastic component to these equations. They treat the prices and levels of output as exogenous variables and estimate the unknown parameters by the method of maximum likelihood for nonlinear multivariate regression models. The endogenous variables in the Christensen-Greene model are the cost shares of capital, labor, and fuel inputs and total cost. Christensen and Greene estimate three equations for each cross section, corresponding to two of the cost shares and the cost function. As unknown parameters they estimate two elements of the vector ap, the two scalarsa0 and aythree elements of the matrix of share elasticities Bpp, two biases of scale in the vector bpy, and the scalar byy. They estimate a total of ten unknown parameters for each of two cross sections of electric utilities for the years 1955 and 1970. 51 Estimates of the remaining parameters of the model are calculated by using the cost exhaustion, homogeneity, and symmetry restrictions. They report that the monotonicity and concavity restrictions are met at every observation in both cross-section data sets. The hypothesis of constant returns to scale can be tested by first considering the hypothesis that the cost function is homothetic; under this hypothesis the cost flexibility is independent of the input prices. Given homotheticity the additional hypothesis that the cost function is homogeneous can be tested; under this hypothesis the cost flexibility is independent of output as well as prices. These hypotheses can be nested, so that the test of homogeneity is conditional on the test of homotheticity. Likelihood ratio statistics for these hypotheses are distributed, asymptotically, as chi-squared. We present the results of Christensen and Greene for 1955 and 1970 in table 1.2. Test statistics for the hypotheses of homotheticity and homogeneity for both cross-section data sets and critical values for chi-squared are presented in table 1.2. Homotheticity can be rejected, so that both homotheticity and homogeneity are inconsistent with the evidence; homogeneity, given homotheticity, is also rejected. If all other parameters involving the level of output were set equal to zero, the parameter ay would be the reciprocal of the degree of returns to

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Table 1.2 Cost function for U.S. electric power industry (parameter estimates, 1955 and 1970; t-ratios in parentheses).a Parameter 1955 1970 8.412 7.14 a0 (31.52) (32.45) 0.386 0.587 ay (6.22) (20.87) 0.094 0.208 aK (0.94) (2.95) 0.348 0.151 aL (4.21) ( 1.85) 0.558 0.943 aE (8.57) (14.64) 0.059 0.049 byy (5.76) (12.94) 0.008 0.003 bKy ( 1.79) ( 1.23) 0.016 0.018 bLy ( 10.10) ( 8.25) 0.024 0.021 bEy (5.14) (6.64) 0.175 0.118 bKK (5.51) (6.17) 0.038 0.081 bLL (2.03) (5.00) 0.176 0.178 bEE (6.83) (10.79) 0.018 0.011 bKL ( 1.01) ( 0.749) 0.159 0.107 bKE ( 6.05) ( 7.48) 0.020 0.070 bLE ( 2.08) ( 6.30) Test statistics for restrictions on economies of scaleb Statistic Homotheticity Homogeneity 78.22 102.27 1955 57.91 157.46 1970 9.21 11.35 Critical value (1%) aSource: Christensen and Greene (1976, table 4, p. 666). bSource: Christensen and Greene (1976, table 5, p. 666).

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scale. For both 1955 and 1970 data sets this parameter is significantly different from unity. Christensen and Greene employ the fitted cost functions presented in table 1.2 to characterize scale economies for individual firms in each of the two cross sections. For both years the cost functions are U-shaped with a minimum point occurring at very large levels of output. In 1955, 118 of the 124 firms have significant economies of scale; only six firms have no significant economies or diseconomies of scale, but these firms produce 25.9 percent of the output of the sample. In 1970, ninety-seven of the 114 firms have significant economies of scale, sixteen have none, and one has significant scale diseconomies. Econometric modeling of economies of scale in the U.S. electric power industry has generated a very extensive literature. The results through 1978 have been surveyed by Cowing and Smith (1978). More recently, the Christensen-Greene data base has been extended by Greene (1983) to incorporate cross sections of individual electric utilities for 1955, 1960, 1965, 1970, and 1975. By including both the logarithm of output and time as an index of technology in the translog total cost function (1.5.15), Greene is able to characterize economies of scale and technical change simultaneously. Stevenson (1980) has employed a translog total cost function incorporating output and time to analyze cross sections of electric utilities for 1964 and 1972. Gollop and Roberts (1981) have used a similar approach to study annual data on eleven electric utilities in the United States for the period 1958 1975. They use the results to decompose the growth of total cost among economies of scale, technical change, and growth in input prices. Griffin (1977b) has modeled substitution among different types of fuel in steam electricity generation using four cross sections of twenty OECD countries. Halvorsen (1978) has analyzed substitution among different fuel types, using cross-section data for the United States in 1972. Cowing, Reifschneider, and Stevenson (1983) have employed a translog total cost function similar to that of Christensen and Greene to analyze data for eighty-one electric utilities for the period 1964 1975. For this purpose they have grouped the data into four cross sections, each consisting of three-year totals for all firms. If disturbances in the equations for the cost shares (1.5.19) are associated with errors in optimization, costs must increase relative to the minimum level given by the cost function (1.5.15). Accordingly, Cowing,

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Reifschneider and Stevenson employ a disturbance for the cost function that is constrained to be positive. 52 An alternative to the Christensen-Greene model for electric utilities has been developed by Fuss (1977b, 1978). In Fuss's model, the cost function is permitted to differ ex ante, before a plant is constructed, and ex post, after the plant is in place.53 Fuss employs a generalized Leontief cost function with four input pricesstructures, equipment, fuel, and labor. He models substitution among inputs and economies of scale for seventy-nine steam generation plants for the period 1948 1961. We have observed that a model of the behavior of a regulated firm based on cost minimization must be carefully distinguished from the model originated by Averch and Johnson (1962). In addition to allowing a given rate of return, regulatory authorities may permit electric utilities to adjust the regulated price of output for changes in the cost of specific inputs. In the electric power industry a common form of adjustment is to permit utilities to change prices with changes in fuel costs. Peterson (1975) has employed a translog cost function for the electric utility industry to test the Averch-Johnson hypothesis. For this purpose he introduces three measures of the effectiveness of regulation into the cost function: a one-zero dummy variable distinguishing between states with and without a regulatory commission, a similar variable differentiating between alternative methods for evaluation of public utility property for rate making purposes, and a variable representing differences between the rate of return allowed by the regulatory authority and the cost of capital. He analyzes annual observations on fifty-six steam generating plans for the period 1966 to 1968. Cowing (1978) has employed a quadratic parametric form to test the Averch-Johnson hypothesis for regulated firms. He introduces both the cost of capital and the rate of return allowed by the regulatory authority as determinants of input demands. Cowing analyzes data on 114 steam generation plants constructed during each of three time periods1947 1950, 1955 1959, and 1960 1965. Gollop and karlson (1978) have employed a translog cost function that incorporates a measure of the effectiveness of regulatory adjustments for changes in fuel costs. This measure is the ratio of costs that may be recovered under the fuel cost adjustment mechanism to all fuel costs. Gollop and karlson analyze data for cross sections of individual electric utilities for the years 1970, 1971, and 1972.

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Atkinson and Halvorsen (1980) have employed a translog parametric form to test the effects of both rate of return regulation and fuel cost adjustment mechanisms. For this purpose they have analyzed cross-section data for electric utilities in 1973. Gollop and Roberts (1983) have studied the effectiveness of regulations on sulfur dioxide emissions in the electric utility industry. They employ a translog cost function that depends on a measure of regulatory effectiveness. This measure is based on the legally mandated reduction in emissions and on the enforcement of emission standards. Gollop and Roberts analyze cross sections of fifty-six electric utilities for each of the years 1973 1979 and employ the results to study the impact of environmental regulation on productivity growth. 1.6.2 Multiple Outputs Brown, Caves, and Christensen (1979) have introduced a model for joint production of freight and transportation services in the railroad industry based on the translog cost function (1.5.15). 54 A cost flexibility (5.4) can be defined for each output. Scale biases and derivatives of the cost flexibilities with respect to each output can be taken to be constant parameters. The resulting cost function depends on logarithms of input prices and logarithms of the quantities of each output. Caves, Christensen, and Trethaway (1980) have extended this approach by introducing Box-Cox transformations of the quantities of the outputs in place of logarithmic transformations. This generalized translog cost function permits complete specialization in the production of a single output. The generalized translog cost function has been applied to cross sections of Class I railroads in the United States for 1955, 1963, and 1974 by Caves, Christensen, and Swanson (1981). They consider five categories of inputs: labor, way and structures, equipment, fuel, and materials. For freight transportation services they take ton-miles and average length of freight haul as measures of output. Passenger services are measured by passenger-miles and average length of passenger trip. They employ the results to measure productivity growth in the U.S. railroad industry for the period 1951 1974. Caves, Christensen, and Swanson (1981) have employed data for cross sections of Class I railroads in the United States to fit a variable cost function, treating way and structures as a fixed input and combining equipment and materials into a single variable input. They have

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employed the results in measuring productivity growth for the period 1951 1974. Friedlaender and Spady (1981) and Harmatuck (1979) have utilized a translog total cost function to analyze crosssection data on Class I railroads in the United States. Jara-Diaz and Winston (1981) have employed a quadratic cost function to analyze data on Class III railroads, with measures of output disaggregated to the level of individual point-to-point shipments. Bräutigan, Daugherty, and Turnquist (1982) have used a translog variable cost function to analyze monthly data for nine years for a single railroad. Speed of shipment and quality of service are included in the cost function as measures of the characteristics of output. The U.S. trucking industry, like the U.S. railroad industry, is subject to price regulation. Spady and Friedlaender (1978) have employed a translog cost function to analyze data on a cross section of 168 trucking firms in 1972. They have disaggregated inputs into four categorieslabor, fuel, capital, and purchased transportation. Freight transportation services are measured in ton-miles. To take into account the heterogeneity of freight transportation services, five additional characteristics of output are included in the cost functionaverage shipment size, average length of haul, percentage of less than truckload traffic, insurance costs, and average load per truck. Friedlaender, Spady, and Chiang (1981) have employed the approach of Spady and Friedlaender (1978) to analyze cross sections of 154, 161, and 47 trucking firms in 1972. Inputs are disaggregated in the same four categories, while an additional characteristics of output is included, namely, terminal density, defined as ton-miles per terminal. Separate models are estimated for each of the three samples. Friedlaender and Spady (1981) have employed the results in analyzing the impact of changes in regulatory policy. Harmatuck (1981) has employed a translog cost function to analyze a cross secton of 100 trucking firms in 1977. He has included data on the number and size of truck load and less-than-truckload shipments and average length of haul as measures of output. He disaggregates input among five activitiesline haul, pickup and delivery, billing and collecting, platform handling, and all other. Finally, Chiang and Friedlaender (1984) have disaggregated the output of trucking firms into four categoriesless than truckload hauls of under 250 miles, between 250 500 miles, and over 500 miles, and truck load trafficall measured in ton miles. Inputs are disaggre-

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gated among five categorieslabor, fuel, revenue equipment, ''other" capital, and purchased transportation. Characteristics of output similar to those included in earlier studies by Chiang, Friedlaender, and Spady are incorporated into the cost function, together with measures of the network configuration of each firm. They have employed this model to analyze a cross section of 105 trucking firms for 1976. The U.S. air transportation industry, like the U.S. railroad and trucking industries, is subject to price regulation. Caves, Christensen, and Trethaway (1983) have employed a translog cost function to analyze a panel data set for all U.S. trunk and local service airlines for the period 1970 1981. Winston (1985) has provided a survey of econometric models of producer behavior in the transportation industries, including railroads, trucking, and airlines. In the United States the communications industries, like the transportation industries, are largely privately owned but subject to price regulation. Nadiri and Schankerman (1981) have employed a translog cost function to analyze time-series data for 1947 1976 on the U.S. Bell System. They include the operating telephone companies and Long Lines, but exclude the manufacturing activities of Western Electric and the research and development activities of Bell Laboratories. Output is an aggregate of four service categories; inputs of capital, labor, and materials are distinguished. A time trend is included in the cost function as an index of technology; the stock of research and development is included as a separate measure of the level of technology. Christensen, Cummings, and Schoech (1983) have employed alternative specifications of the translog cost function to analyze time-series data for the U.S. Bell System for 1947 1977. They employ a distributed lag of research and development expenditures by the Bell System to represent the level of technology. As alternative representations they consider the proportion of telephones with access to direct distance dialing, the percentage of telephones connected to central offices with modern switching facilities, and a more comprehensive measure of research and development. They also consider specifications with capital input held fixed and with experienced labor and management held fixed. Evans and Heckman (1983, 1984) have provided an alternative analysis of the same data set. They have studied economies of scope in the joint production of telecommunications services. Bell Canada is the largest telecommunications firm in Canada. Fuss and Waverman (1981) have employed a translog cost function to

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analyze time series data on Bell Canada for the period 1952 1975. Three outputs are distinguished: message toll service, other total service, and local and miscellaneous service. Capital, labor, and materials are treated as separate categories of input. The level of technology is represented by a time trend. Denny, Fuss, Everson, and Waverman (1981) have analyzed time series data for the period 1952 1976. The percentage of telephones with access to direct dialing and the percentage of telephones connected to central offices with modern switching facilities are incorporated into the cost function as measures of the level of technology. Kiss, Karabadjian, and Lefebvre (1983) have compared alternative specifications of output and the level of technology. Fuss (1983) has provided a survey of econometric modeling of telecommunications services. 1.7 Conclusion The purpose of this concluding section is to suggest possible directions for future research on econometric modeling of producer behavior. We first discuss the application of econometric models of production in general equilibrium analysis. The primary focus of empirical research has been on the characterization of technology for individual producing units. Application of the results typically involves models for both demand and supply for each commodity. The ultimate objective of econometric modeling of production is to construct general equilibrium models encompassing demands and supplies for a wide range of products and factors of production. A second direction for future research on producer behavior is to exploit statistical techniques appropriate for panel data. Panel data sets consist of observations on several producing units at many points of time. Empirical research on patterns of substitution and technical change has been based on time series observations on a single producing unit or on cross-section observations on different units at a given point of time. Research on economies of scale has been based primarily on cross-section observations. Our exposition of econometric methods has emphasized areas of research where the methodology has crystallized. An important area for future research is the implementation of dynamic models of technology. These models are based on substitution possibilities among outputs and inputs at different points of time. A number of promising avenues for investigation have been suggested in the literature on the

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theory of production. We conclude the chapter with a brief review of possible approaches to the dynamic modeling of producer behavior. 1.7.1 General Equilibrium Modeling At the outset of our discussion it is essential to recognize that the predominant tradition in general equilibrium modeling does not employ econometric methods. This tradition originated with the seminal work of Leontief (1951), beginning with the implementation of the static input-output model. Leontief (1953) gave a further impetus to the development of general equilibrium modeling by introducing a dynamic input-output model. Empirical work associated with input-output analysis is based on estimating the unknown parameters of a general equilibrium model from a single interindustry transactions table. The usefulness of the "fixed coefficients" assumption that underlies input-output analysis is hardly subject to dispute. By linearizing technology it is possible to solve at one stroke the two fundamental problems that arise in the practical implementation of general equilibrium models. First, the resulting general equilibrium model can be solved as a system of linear equations with constant coefficients. Second, the unknown parameters describing technology can be estimated from a single data point. The first successful implementation of a general equilibrium model without the fixed coefficients assumption of input-output analysis is due to Johansen (1974). Johansen retained the fixed coefficients assumption in modeling demands for intermediate goods, but employed linear logarithmic or Cobb-Douglas production functions in modeling the substitution between capital and labor services and technical change. Linear logarithmic production functions imply that relative shares of inputs in the value of output are fixed, so that the unknown parameters characterizing substitution between capital and labor inputs can be estimated from a single data point. In modeling producer behavior Johansen employed econometric methods only in estimating constant rates of technical change. The essential features of Johansen's approach have been preserved in the general equilibrium models surveyed by Fullerton, Henderson, and Shoven (1984). The unknown parameters describing technology in these models are determined by "calibration" to a single data point. Data from a single interindustry transactions table are supplemented

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by a small number of parameters estimated econometrically. The obvious disadvantage of this approach is that arbitrary constraints on patterns of production are required in order to make calibration possible. An alternative approach to modeling producer behavior for general equilibrium models is through complete systems of demand functions for inputs in each industrial sector. Each system gives quantities demanded as functions of prices and output. This approach to general equilibrium of modeling producer behavior was originated by Berndt and Jorgenson (1973). As in the descriptions of technology by Leontief and Johansen, production is characterized by constant returns to scale in each sector. As a consequence, commodity prices can be expressed as functions of factor prices, using the nonsubstitution theorem of Samuelson (1951). This greatly facilitates the solution of the econometric general equilibrium model constructed by Hudson and Jorgenson (1974) by permitting a substantial reduction in dimensionality of the space of prices to be determined by the model. The implementation of econometric models of producer behavior for general equilibrium analysis is very demanding in terms of data requirements. These models require the construction of a consistent time series of interindustry transactions tables. By comparison, the noneconometric aproaches of Leontief and Johansen require only a single interindustry transactions table. Second, the implementation of systems of input demand functions requires methods for the estimation of parameters in systems of nonlinear simultaneous equations. Finally, the restrictions implied by the economic theory of producer behavior require estimation under both equality and inequality constraints. Jorgenson and Fraumeni (1983) have constructed an econometric model of producer behavior for thirty-five industrial sectors of the U.S. economy. The next research objective is to disaggregate the demands for energy and materials by constructing a hierarchy of models for allocation within the energy and materials aggregates. A second research objective is to incorporate the production models for all thirty-five industrial sectors into an econometric general equilibrium model of production for the U.S. economy along the lines suggested by Jorgenson (1983b, 1984a). A general equilibrium model will make it possible to analyze the implications of sectoral patterns of substitution and technical change for the behavior of the U.S. economy as a whole.

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1.7.2 Panel Data The approach to modeling economies of scale originated by Christensen and Greene (1976) is based on the underlying assumption that individual producing units at the same point of time have the same technology. Separate models of production are fitted for each time period, implying that the same producing unit has a different technology at different points of time. A more symmetrical treatment of observations at different points of time is suggested by the model of substitution and technical change in U.S. agriculture developed by Binswanger (1974c, 1978c). In this model technology is permitted to differ among time periods and among producing units. Caves, Christensen, and Trethaway (1983) have employed a translog cost function to analyze a panel data set for all U.S. trunk and local service airlines for the period 1970 1981. Individual airlines are observed in some or all years during the period. Differences in technology among years and among producing units are incorporated through one-zero dummy variables that enter the cost function. One set of dummy variables corresponds to the individual producing units. A second set of dummy variables corresponds to the time periods. Although airlines provide both freight and passenger service, the revenues for passenger service greatly predominate in the total, so that output is defined as an aggregate of five categories of transportation services. Inputs are broken down into three categorieslabor, fuel, and capital and materials. The number of points served by an airline is included in the cost functions as a measure of the size of the network. Average stage length and average load factor are included as additional characteristics of output specific to the airline. Caves, Christensen, and Trethaway introduce a distinction between economies of scale and economies of density. Economies of scale are defined in terms of the sum of the elasticities of total cost with respect to output and points served, holding input prices and other characteristics of output constant. Economies of density are defined in terms of the elasticity of total cost with respect to output, holding points served, input prices, and other characteristics of output constant. Caves, Christenen, and Trethaway find constant returns to scale and increasing returns to density in airline service. The model of panel data employed by Caves, Christensen, and Trethaway in analyzing air transportation service is based on "fixed

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effects." The characteristics of output specific to a producing unit can be estimated by employing one-zero dummy variables for each producing unit. An alternative approach based on "random effects" of output characteristics is utilized by Caves, Christensen, Trethaway, and Windle (1984) in modeling rail transportation service. They consider a panel data set for forty-three Class I railroads in the United States for the period 1951 1975. Caves, Christensen, Trethaway, and Windle employ a generalized translog cost functon in modeling the joint production of freight and passenger transportation services by rail. They treat the effects of characteristics of output specific to each railroad as a random variable. They estimate the resulting model by panel data techniques originated by Mundlak (1963, 1978). The number of route miles served by a railroad is included in the cost function as a measure of the size of the network. Length of haul for freight and length of trip for passengers are included as additional characteristics of output. Economies of density in the production of rail transportation services are defined in terms of the elasticity of total cost with respect to output, holding route miles, input prices, firm-specific effects, and other characteristics of output fixed. Economies of scale are defined holding only input prices and other characteristics of output fixed. The impact of changes in outputs, route miles, and firm specific effects can be estimated by panel data techniques. Economies of density and scale can be estimated from a single cross section by omitting firm-specific dummy variables. Panel data techniques require the construction of a consistent time series of observation on individual producing units. By comparison, the cross-section methods developed by Christensen and Greene require only a cross section of observations for a single time period. The next research objective in characterizing economies of scale and economies of density is to develop panel data sets for regulated industrieselectricity generation, transportation, and communicationsand to apply panel data techniques in the analysis of economies of scale and economies of density. 1.7.3 Dynamic Models of Production The simplest intertemporal model of production is based on capital as a factor of production. A less restrictive model generates costs of

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adjustment from changes in the level of capital input through investment. As the level of investment increases, the amount of marketable output that can be produced from given levels of all inputs is reduced. Marketable output and investment can be treated as outputs that are jointly produced from capital and other inputs. Models of production based on costs of adjustment have been analyzed, for example, by Lucas (1967b) and Uzawa (1969). Optimal production planning with costs of adjustment requires the use of optimal control techniques. The optimal production plan at each point of time depends on the initial level of capital input, so that capital is a "quasi-fixed" input. Obviously, labor and other inputs can also be treated as quasi-fixed in models of production based on costs of adjustment. The optimal production plan at each point of time depends on the initial levels of all quasi-fixed inputs. The optimal production plan with costs of adjustment depends on all future prices of outputs and inputs of the production process. Unlike the prices of outputs and inputs at each point of time employed in the production studies we have reviewed, future prices cannot be observed on the basis of market transactions. To simplify the incorporation of future prices into econometric models of production, a possible aproach is to treat these prices as if they were known with certainty. A further simplification is to take all future prices to be equal to current prices, so that expectations are "static." Dynamic models of production based on static expectations have been employed by Denny, Fuss, and Waverman (1981), Epstein and Denny (1983), and Morrison and Berndt (1981). Denny, Fuss, and Waverman have constructed models of substitution among capital, labor, energy, and materials inputs for two-digit industries in Canada and the United States. Epstein and Denny have analyzed substitution among these same inputs for total manufacturing in the United States. Morrison and Berndt have utilized a similar data set with labor input divided between blue collar and white collar labor. Berndt, Morrison, and Watkins (1981) have surveyed dynamic models of production. The obvious objection to dynamic models of production based on static expectations is that current prices change from period to period, but expectations are based on unchanging future prices. An alternative approach is to base the dynamic optimization on forecasts of future prices. Since these forecasts are subject to random errors, it is natural to require that the optimization process take into account the uncertainty that accompanies forecasts of future prices. Two alterna-

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tive approaches to optimization under uncertainty have been proposed. We first consider the approach to optimization under uncertainty based on certainty equivalence. Provided that the objective function for producers is quadratic and constraints are linear, optimization under uncertainty can be replaced by a corresponding optimization problem under certainty. This gives rise to linear demand functions for inputs with prices replaced by their certainty equivalents. This approach has been developed in considerable detail by Hansen and Sargent (1980, 1981) and has been employed in modeling producer behavior by Epstein and Yatchew (1985), Meese (1980), and Sargent (1978). An alternative approach to optimization under uncertainty is to employ the information about expectations of future prices contained in current input levels. This approach has the advantage that it is not limited to quadratic objective functions and linear constraints. Pindyck and Rotemberg (1983a) have utilized this approach in analyzing the Berndt-Wood (1975) data set for U.S. manufacturing, treating capital and labor input as quasi-fixed. They employ a translog variable cost function to represent technology, adding costs of adjustment that are quadratic in the current and lagged values of the quasi-fixed inputs. Pindyck and Rotemberg (1983b) have employed a similar approach to the analysis of production with two kinds of capital input and two types of labor input. Notes 1. This approach to production theory is employed by Carlson (1939), Frisch (1965), and Schneider (1934). The English edition of Frisch's book is a translation from the ninth edition of his lectures, published in Norwegian in 1962; the first edition of these lectures dates back to 1926. 2. These studies are summarized by Douglas (1948). See also: Douglas (1967, 1976). Early econometric studies of producer behavior, including those based on the Cobb-Douglas production function, have been surveyed by Heady and Dillon (1961) and Walters (1963). Samuelson (1979) discusses the impact of Douglas's research. 3. Econometric studies based on the CES production function have been surveyed by Griliches (1967), Jorgenson (1974), Kennedy and Thirlwall (1972), Nadiri (1970), and Nerlove (1967). 4. Hotelling (1932) and Samuelson (1954) develop the dual formulation of production theory on the basis of the Legendre transformation. This approach is employed by Jorgenson and Lau (1974a,b) and Lau (1976, 1978b). 5. Shephard utilizes distance functions to characterize the duality between cost and

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production functions. This approach is employed by Diewert (1974a, 1982), Hanoch (1978), McFadden (1973), and Uzawa (1964). 6. Surveys of duality in the theory of production are presented by Diewert (1982) and Samuelson (1983). 7. This approach to the selection of parametric forms is discussed by Diewert (1974a), Fuss, McFadden, and Mundlak (1978), and Lau (1974). 8. A more detailed discussion of this measure is presented in section 1.2.2. 9. An alternative approach, originated by Diewert (1971, 1973, 1974b), employs the square roots of the input prices rather than the logarithms and results in the "generalized Leontief" parametric form. 10. Surveys of parametric forms employed in econometric modeling of producer behavior are presented by Fuss, McFadden, and Mundlak (1978) and Lau (1986). 11. Methods for estimation of nonlinear multivariate regression models are summarized by Malinvaud (1980). 12. Nonlinear two and three stage least squares methods are also discussed by Amemiya (1977), Gallant (1977), and Gallant and Jorgenson (1979). 13. Constrained estimation is discussed in more detail in section 1.3.3. 14. Surveys of methods for estimation of nonlinear multivariate regressions and systems of nonlinear simultaneous equations are given by Amemiya (1983) and Malinvaud (1980), esp. Chs. 9 and 20. Computational techniques are surveyed by Quandt (1983). 15. Time-series and cross-section differences in technology have been incorporated into a model of substitution and technical change in U.S. agriculture by Binswanger (1974a,c, 1978c). Binswanger's study is summarized in section 1.4.2. 16. The dual formulation of production theory under constant returns to scale is due to Samuelson (1954). 17. The share elasticity was introduced by Christensen, Jorgenson, and Lau (1971, 1973) and Samuelson (1973). 18. This definition of the bias of technical change is due to Hicks (1932). Alternative definitions of biases of technical change are compared by Binswanger (1978b). 19. Alternative definitions of substitution and complementarity are discussed by Samuelson (1974). 20. The concept of separability is due to Leontief (1947a,b) and Sono (1961). 21. The concept of homothetic separability was introduced by Shephard (1953, 1970). 22. A proof of this proposition is given by Lau (1969, 1978b). 23. A proof of this proposition is given by Lau (1978b). 24. This characterization of price and quantity indexes was originated by Shephard (1953, 1970). 25. Gorman (1959) has analyzed the relationship between aggregation over commodities and two-stage allocation. A presentation of the theory of two-stage allocation and references to the literature are given by Blackorby, Primont, and Russell (1978). 26. Share elasticities were introduced as constant parameters of an econometric model of producer behavior by Christensen, Jorgenson, and Lau (1971, 1973). Constant share elasticities, biases, and deceleration of technical change are employed by Jorgenson and Fraumeni (1983) and Jorgenson (1983b, 1984b). Binswanger (1974a,c, 1978c) uses a different definition of biases of technical change in parameterizing an econometric model with

constant share elasticities. 27. Arrow, Chenery, Minhas, and Solow (1961) have derived the CES production function as an exact representation of a model of producer behavior with a constant elasticity of substitution. 28. An alternative approach to the generation of the translog parametric form for the production function by means of Taylor's series was originated by Kmenta (1967). Kmenta employs a Taylor's series expansion in terms of the parameters of the CES

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production function. This approach imposes the same restrictions on patterns of production as those implied by the constancy of the elasticity of substitution. The Kmenta approximation is employed by Griliches and Ringstad (1971) and Sargan (1971), among others, in estimating the elasticity of substitution. 29. This approach to global concavity was originated by Jorgenson and Fraumeni (1983). Caves and Christensen (1980) have compared regions where concavity obtains for alternative parametric forms. 30. The Cholesky factorization was first employed in imposing local concavity restrictions by Lau (1978b). 31. Different stochastic specifications are compared by Appelbaum (1978), Burgess (1975), and Geary and McDonnell (1980). The implications of alternative stochastic specifications are discussed in detail by Fuss, McFadden, and Mundlak (1978). 32. This approach to estimation is presented by Jorgenson and Fraumeni (1983). 33. Maximum likelihood estimation by means of the "seemingly unrelated regressions" model analyzed by Zellner (1962) would not be appropriate here, since the symmetry constraints we have described in section 1.2.4 cannot be written in the bilinear form considered by Zellner. 34. Computational techniques for constrained and unconstrained estimation of nonlinear multivariate regression models are discussed by Malinvaud (1980). Techniques for computation of unconstrained estimators for systems of nonlinear simultaneous equations are discussed by Berndt, Hall, Hall, and Hausman (1974) and Belsley (1974, 1979). 35. The method of nonlinear three-stage least squares introduced by Jorgenson and Laffont (1974) was extended to nonlinear inequality constrained estimation by Jorgenson, Lau, and Stoker (1982), esp. pp. 196 204. 36. A nonstatistical approach to testing the theory of production has been presented by Afriat (1972), Diewert and Parkan (1983), Hanoch and Rothschild (1972), and Varian (1984). 37. Statistics for testing linear inequality restrictions in linear multivariate regression models have been developed by Gourieroux, Holly, and Montfort (1982); statistics for testing nonlinear inequality restrictions in nonlinear multivariate regression models are given by Gourieroux, Holly, and Monfort (1980). 38. Restrictions on patterns of substitution implied by homothetic separability have been discussed by Berndt and Christensen (1973a), Jorgenson and Lau (1975), Russell (1975), and Russell and Blackorby (1976). 39. The methodology for testing separability restrictions was originated by Jorgenson and Lau (1975). This methodology has been discussed by Blackorby, Primont and Russell (1977) and by Denny and Fuss (1977). An alternative approach has been developed by Woodland (1978). 40. The advantages and disadvantages of summarizing data from process analysis models by means of econometric models have been discussed by Maddala and Roberts (1980, 1981) and Griffin (1980, 1981c). 41. A review of the literature on induced technical change is given by Binswanger (1978a). 42. The model of capital as a factor of production was originated by Walras (1954). This model has been discussed by Diewert (1980) and by Jorgenson (1973a, 1980). 43. Myopic decision rules are derived by Jorgenson (1973b). 44. Data on energy and materials are based on annual interindustry transactions tables for the United States compiled by Jack Faucett Associates (1977). Data on labor and capital are based on estimates by Fraumeni and Jorgenson (1980). 45. Parameter estimates are given by Jorgenson and Fraumeni (1981b), pp. 255 264.

46. The implications of patterns of biases of technical change are discussed in more detail by Jorgenson (1981).

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47. A survey of the literature on frontier representations of technology is given by Forsund, Lovell, and Schmidt (1980). 48. Duality between cost and production functions is due to Shephard (1953, 1970). 49. The concept of homotheticity was introduced by Shephard (1953). Shephard shows that homotheticity of the cost function is equivalent to homotheticity of the production function. 50. A model of a regulated firm based on cost minimization was introduced by Nerlove (1963). Surveys of the literature on the Averch-Johnson model have been given by Bailey (1973) and Baumol and Klevorick (1970). 51. Christensen and Greene have assembled data on cross sections of individual firms for 1955 and 1970. The quantity of output is measured in billions of kilowatt hours (kwh). The quantity of fuel input is measured by British thermal units (Btu). Fuel prices per million Btu are averaged by weighting the price of each fuel by the corresponding share in total consumption. The price of labor input is measured as the ratio of total salaries and wages and employee pensions and benefits to the number of full-time employees plus half the number of part-time employees. The price of capital input is estimated as the sum of interest and depreciation. 52. Statistical methods for models of production with disturbances constrained to be positive or negative are discussed by Aigner, Amemiya and Poirier (1976) and Greene (1980). 53. A model of production with differences between ex ante and ex post substitution possibilities was introduced by Houthakker (1955). This model has been further developed by Johansen (1972) and Sato (1975) and has been discussed by Hildenbrand (1981) and Koopmans (1977). Recent applications are given by Forsund and Hjalmarsson (1979, 1983), and Forsund and Jansen (1983). 54. A review of the literature on regulation with joint production is given by Bailey and Friedlaender (1982).

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2 Empirical Studies of Depreciation Dale W. Jorgenson This chapter surveys empirical research on depreciation based on an econometric model of asset prices introduced by Hall in 1971 and shows how Hall's model provides a vehicle for unifying research on constant quality price indices and depreciation. An additional objective of the chapter is to illustrate the use of this research in constructing an integrated system of income, product, and wealth accounts. A system of vintage accounts, like that originated by Christensen and Jorgenson in 1973, is the key to successful integration. The purpose of this chapter is to survey empirical research on depreciation and its applications. As a framework for the survey I employ the econometric model of asset prices introduced by Hall (1971) that has been the primary vehicle for this research for the past two decades. Since 1985 research on asset prices has been utilized successfully to generate constant quality price indices for investment in computers; these indices are included in the U.S. National Income and Product Accounts. The challenge facing economic statisticians now is to employ asset price information effectively while making badly needed revisions in the treatment of depreciation in the national accounts. Empirical research on constant quality price indices focuses on the relationship between the prices of assets and their characteristicsprocessing speed and memory capacity in the case of computers. By contrast, empirical research on depreciation centers on the relationship between the price of an asset and its age, so that prices of assets of different ages or vintages must be analyzed. Since the connection between research on constant quality price indices and depreciation is not obvious, I describe Hall's ''hedonic" model of asset prices, which comprehends both, in the next section.

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My second objective is to summarize empirical research on depreciation, beginning with the landmark studies of Hulten and Wykoff (1981a,b,c) for the Office of Tax Analysis of the Department of the Treasury. As a consequence of the rapid assimilation of the results of Hulten and Wykoff, depreciation has been transformed from one of the most contentious and problematic areas in economic measurement to one of the best understood and most useful. Empirical research has generated the information needed for revising the treatment of depreciation in the U.S. national accounts. My third objective is to illustrate the use of information on asset prices in constructing an integrated system of income, product, and wealth accounts. For this purpose I describe a system of vintage accounts introduced by Christensen and Jorgenson (1973). This system includes accumulation equations that generate the perpetual inventory of assets required for the wealth accounts and asset pricing equations that produce the information on depreciation needed for income and product accounts. A system of vintage accounts is the key to the successful integration of measures of income and product with a measure of wealth. My overall conclusion is that the U.S. National Income and Product Accounts have failed to provide internally consistent measures of capital stocks and depreciation. Top priority should be assigned to improving the national accounts by incorporating information now available from empirical research on depreciation. Fortunately, this task can be facilitated by utilizing concepts and methods already familiar to economic statisticians from their work on constant quality price indices. The following section outlines Hall's econometric model of asset prices, focusing on the version of this model employed by Hulten and Wykoff. I also consider an alternative version recently introduced by Oliner (1993). In section 2.2, I summarize empirical research on depreciation, including studies of asset prices and retirements. Studies of both types are required for constructing a system of vintage asset prices. In section 2.3 I outline the role of depreciation in an integrated system of income, product and wealth accounts. The system of vintage accounts originated by Christensen and Jorgenson provides a natural framework for incorporating the results of empirical research on depreciation. Section 2.4 concludes the chapter.

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2.1 Econometric Models To measure depreciation one needs a set of prices and quantities of investment goods. Investment represents the acquisition of capital goods, for example, a certain number of computers with a given performance. The price of acquisition of a durable good is the unit cost of acquiring it. For instance, the price of acquisition of a computer is the unit cost of purchasing the machine. Implicit in these definitions is the assumption that prices and quantities are measured in units of constant quality. Capital services are defined in terms of the use of an investment good for a specified period of time. For example, computers can be leased for days, months, or years. The rental price of capital services is the unit cost of renting an investment good rather than purchasing it, so that the rental price of a computer is the unit cost of using a machine for a specified period of time. Depreciation is the component of unit cost associated with the aging of assets. This component can be isolated by comparing prices of assets of different ages. The objective of empirical research on depreciation is to construct a system of asset prices for this purpose. I begin the description of a system of vintage accounts with quantities of investment goods. I refer to investment goods acquired at different points of time as different vintages of capital. A system of vintage accounts can be generated by the perpetual inventory method. This method is based on the assumption that the quantity of capital is proportional to the initial level of investment. The constants of proportionality are given by the relative efficiencies of different vintages of capital. An important simplifying assumption is that relative efficiencies of different vintages of an investment good depend only on the age of the investment good and not on the time at which it was acquired. The quantity of capital input is the flow of capital services into production. Since the capital services provided by a given investment good are proportional to the initial investment, the services provided by different vintages at the same point of time are perfect substitutes. Under perfect substitutability the flow of capital services is a weighted sum of past investments; the weights correspond to the relative efficiencies of the different vintages of capital. A system of vintage

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accounts containing data on investments of every age in every period is essential for the measurement of depreciation. We turn next to the price data required for a vintage accounting system. The rental price of capital services is the price of capital input. Under perfect substitutability the rental prices for all vintages of capital are proportional to a single rental price with constants of proportionality given by the relative efficiencies of the different vintages. The price of acquisition of a capital good is the sum of present values of future rental prices of capital services, weighted by relative efficiencies of the capital good in future time periods. To measure rental prices, a system of vintage accounts containing data on prices of capital goods of every age in every period is required. Durable goods decline in efficiency with age, and thus require replacement in order to maintain productive capacity. This is the quantity interpretation of the intuitive notion of "maintaining capital intact." Similarly, the price of a durable good declines with age, resulting in depreciation that reflects both the current decline in efficiency and the present value of future declines in efficiency. Depreciation provides the price interpretation of "maintaining capital intact." Both concepts are used in a system of vintage accounts. 1 Price and quantity indices of capital services in each period can be constructed for each durable good given a full set of vintage price and quantity data. With less complete information a simplified set of price and quantity indices can be constructed from empirical estimates of the relative efficiencies of investment goods of different ages. This is the point at which empirical research on depreciation can be particularly useful. As an illustration, consider a system of vintage accounts introduced by Christensen and Jorgenson (1973). This system is based on the assumption that the decline in efficiency of assets with age is geometric. To construct this simplified accounting system, estimate capital stock at the end of each period Kt as a weighted sum of investments of age v at time t {At v}:

This is the capital accumulation equation, relating the stock of assets to past investments, required for a perpetual inventory of assets. With a constant rate of decline in efficiency d, replacement Rt is proportional to capital stock:

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Similarly, the price of acquisition of new investment goods pA,t is a weighted sum of of the present values of future rentals {pK,t}. This is the capital asset pricing equation, relating the asset price to the values of future capital services, needed for a vintage price system. With a constant rate of decline in efficiency the rental price becomes

where rt is the rate of discount. Depreciation pD,t is proportional to the acquisition price:

Finally, the acquisition price of investment goods of age v at time t, say pA,t,v is

The vintage price system consists of the acquisition prices {pA,t,v} while the vintage quantity system consists of the quantities of past investments {At v}. The price and quantity of capital services can be derived from these two systems of vintage accounts, together with depreciation and replacement of capital goods. 2 Under the assumption that the decline in efficiency of a durable good is geometric, the vintage price system required for construction of price and quantity indices for capital input depends on the price for acquisition of new capital goods pA,t. At each point of time the prices for acquisition of capital goods of age v, say {pA,t,v}, are proportional to the price for new capital goods. The constants of proportionality decline geometrically at the rate d. The rate of decline can be treated as an unknown parameter in an econometric model and estimated from a sample of prices for acquisition of capital goods of different vintages. I obtain an econometric model for vintage price functions by taking logarithms of the acquisition prices {pA,t,v} and adding a random disturbance term:

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where the rate of decline in efficiency d and the rate of inflation in the price of the asset g are unknown parameters and et,v is an unobservable random disturbance. Assume that the disturbance term has an expected value equal to zero and constant variance, say s2, so that

and that disturbances corresponding to distinct observations that are uncorrelated:

Under these assumptions the unknown parameters of the econometric model can be estimated by linear regression methods. The econometric model for vintage price functions can be generalized in several directions. First, the age of the durable good v and the time period t can enter nonlinearly into the vintage price function. Hall (1971) has proposed an analysis of variance model for the vintage price function. In this model each age is represented by a dummy variable equal to one for that age and zero otherwise. Similarly, each time period can be represented by a dummy variable equal to one for that time period and zero otherwise. Hall's analysis of variance model for vintage price functions can be written

where Dv is a vector of dummy variables for age v and Dt is a vector of dummy variables for time t; bv and bt are the corresponding vectors of parameters. In the estimation of this model dummy variables for one vintage and one time period can be dropped in order to obtain a matrix of observations on the independent variables of full rank. An alternative approach to nonlinearity has been proposed by Hulten and Wykoff (1981b). They transform the prices of acquisition {pA,t,v}, age v, and time t by means of the Box-Cox transformation, obtaining

where the parameters qp, qv, and qt can be estimated by nonlinear regression methods from the model for vintage price functions:

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The model giving the logarithm of an asset price as a linear function of age v and time period t is a limiting case of the Hulten-Wykoff model with parameter values

A third approach to nonlinearity has been introduced by Oliner (1993). He proposes to augment the linear model by introducing polynomials in age and time. For example, a quadratic model can be represented by

This has the advantage of flexibility in the representation of time and age effects, but economizes on the number of unknown parameters. A further generalization of the econometric model of vintage price functions has been proposed by Hall (1971). This is appropriate for durable goods with a number of different models that are perfect substitutes in production. Each model is characterized by a number of technical characteristics that affect relative efficiency. We can express the price for acquisition of new capital goods at time zero as a function of the characteristics

where C is a vector of characteristics and bc is the corresponding vector of parameters. An econometric model of prices of new capital goods makes it possible to correct these prices for quality change. Changes in quality can be incorporated into price indices for capital goods by means of the "hedonic technique" originated by Waugh (1929) in dealing with the heterogeneity of agricultural commodities. This approach was first applied to capital goods in an important study of automobile prices by Court (1939). A seminal article by Griliches (1961) revived the hedonic methodology and applied it to postwar automobile prices. Chow (1967) first applied this methodology to computer prices in research conducted at IBM. 3 Cole, Chen, Barquin-Stolleman, Dulberger, Helvacian, and Hodge (1986) reported the results of a joint project conducted by the Bureau

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of Economic Analysis (BEA) of the Department of Commerce and IBM to construct a constant quality price index for computers. Triplett (1986) discussed the economic interpretation of constant quality price indices in an accompanying article. Subsequently, the Bureau of Economic Analysis (1986) described the introduction of constant quality price indices for computers into the U.S. National Income and Product Accounts. A more detailed report on the the Bureau of Economic Analysis IBM research on computer processors is presented by Dulberger (1989), who employs speed of processing and main memory as technical characteristics in modeling the prices of processors. An extensive survey of research on hedonic price indices for computers is given by Triplett (1989). 4 Hall's (1971) methodology provides a means for determining both a quality-corrected price index for new capital goods and relative efficiencies for different vintages of capital goods. Substituting the "hedonic" model of prices of net assets into the econometric model of vintage price functions

The unknown parameters of this model can be estimated by linear regression methods. Hall's methodology has been applied to prices of mainframe computers and computer peripherals by Oliner (1993, 1994a). In Oliner's applications the chronological age of assets is replaced by the "model" age, that is, the time that a model has been available from the manufacturer. An alternative approach is to substitute observations on the price of new capital goods pA,0 into the econometric model, so that the dependent variable is the difference between the logarithms of new and used assets

This makes it possible to separate the modeling of the vintage price function and the quality-corrected price index of new capital goods. Hulten, Robertson, and Wykoff (1989), Wykoff (1989), and OTA (1990, 1991a, 1991b) have employed this approach. A further decomposition of the econometric model of asset prices has been suggested by Biorn (1992b). In order to isolate the vintage

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effect from the other determinants of asset prices, Biorn proposes to substitute observations on the prices of new capital goods in each time period pA,t into the econometric model

This decomposition is implicit in the vintage price model originated by Terborgh (1954). The decomposition of vintage price functions is discussed in more detail by Biorn (1992a, 1992b). The concepts and methods employed in developing constant quality price indices for the U.S. National Income and Product Accounts can also be used in measuring depreciation. A constant quality price index can be constructed by focusing on a cross section of prices on assets with different characteristics. This approach is used in constructing price indices for computers in the U.S. National Income and Product Accounts. Alternatively, prices of assets of different vintages can be analyzed to obtain estimates of depreciation suitable for introduction into the U.S. national accounts. Finally, these two objectives can be combined in Hall's "hedonic" model of asset prices. This model provides a unified framework for modeling asset prices for these different purposes. 2.2 Studies of Depreciation To illustrate the econometric modeling of vintage price functions I present a model implemented by Hulten and Wykoff (1981b) for eight categories of assets in the United States. Their study includes tractors, construction machinery, metalworking machinery, general industrial equipment, trucks, autos, industrial buildings, and commercial buildings. In 1977 investment expenditures on these categories amounted to fifty-five percent of spending on producers' durable equipment and forty-two percent of spending on nonresidential structures. 5 In the estimation of econometric models of vintage price functions, the sample of used asset prices is "censored" by the retirement of assets. The price of acquisition for assets that have been retired is equal to zero. If only observations on surviving assets are included in a sample of used asset prices, estimates of depreciation are biased by excluding observations on assets that have been retired. In order to correct this bias Hulten and Wykoff (1981b) multiply the prices of surviving assets of each vintage by the probability of survival, expressed as a function of age.

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Table 2.1 Rates of economic depreciation Without censored sample With censored sample correction correction Age Commercial Industrial Commercial Industrial 5 2.85 2.99 2.66 2.02 10 2.64 3.01 1.84 1.68 15 2.43 3.04 1.48 1.50 20 2.30 3.07 1.27 1.39 30 2.15 3.15 1.02 1.25 40 2.08 3.24 0.88 1.17 50 2.04 3.34 0.79 1.11 60 2.02 3.45 0.72 1.06 70 2.02 3.57 0.66 1.03 BGA*2.47 3.61 1.05 1.28 R2 0.985 0.997 0.971 0.995 Source: Hulten and Wykoff (1981a, table 5, p. 387); commercial corresponds to office and industrial corresponds to factory. * BGA = Best Geometric Rate Vintage price functions for commercial and industrial buildings are summarized in table 2.1. For each class of assets the rate of economic depreciation is tabulated as a function of the age of the asset. The natural logarithm of the price is regressed on age and time to obtain an average rate of depreciation, which Hulten and Wykoff refer to as the best geometric rate (BGA). The square of the multiple correlation coefficient (R2) is given as a measure of the goodness of fit of the geometric approximation to the fitted vintage price function for each asset. Vintage price functions are estimated with and without the correction for censored sample bias. The first conclusion that emerges from table 2.1 is that a correction for censored sample bias is extremely important in the estimation of vintage price functions. The Hulten-Wykoff study is the first to employ such a correction. The second conclusion reached by Hulten and Wykoff (1981b, p. 387) is that . . . a constant rate of depreciation can serve as a reasonable statistical approximation to the underlying Box-Cox rates even though the latter are not geometric [their italics]. This result, in turn, supports those who use the single parameter depreciation approach in calculating capital stocks using the perpetual inventory method.

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Table 2.2 Rates of economic depreciation: Business assets Producers durable equipment 1. Furniture and fixtures 2. Fabricated metal products 3. Engines and turbines 4. Tractors 5. Agricultural machinery 6. Construction machinery 7. Mining and oil field machinery 8. Metalworking machinery 9. Special industry machinery 10. General industrial equipment 11. Office, computing, and accounting machinery 12. Service industry machinery 13. Electrical machinery 14. Trucks, buses, and truck trailers 15. Autos 16. Aircraft 17. Ships and boats 18. Railroad equipment 19. Instruments 20. Other equipment

0.1100 0.0917 0.0786 0.1633 0.0971 0.1722 0.1650 0.1225 0.1031 0.1225 0.2729 0.1650 0.1179 0.2537 0.3333 0.1833 0.0750 0.0660 0.1500 0.1500

Nonresidential structures 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

Industrial buildings Commercial buildings Religious buildings Educational buildings Hospital and institutional buildings Other nonfarm buildings Railroad structures Telephone and telegraph structures Electric light and power structures Gas structures Other public utility structures Farm structures Mining exploration, shaft and wells Other nonbuilding structures

0.0361 0.0247 0.0188 0.0188 0.0233 0.0454 0.0176 0.0333 0.0300 0.0300 0.0450 0.0237 0.0563 0.0290

Residential structures 35. Residential structures Source: Jorgenson and Sullivan (1981b, table 1).

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After 1973 energy prices increased sharply and productivity growth rates declined dramatically at both aggregate and sectoral levels. Baily (1981) attributed part of the slowdown in economic growth to a decline in relative efficiencies of older capital goods resulting from higher energy prices. Hulten, Robertson, and Wykoff (1989) tested the stability of vintage price functions during the 1970s. Wykoff (1989) has analyzed price data for four models of business-use automobiles collected from a large leasing company, applying the Hulten-Wykoff methodology. Hulten, Robertson, and Wykoff (1989, p. 255) have carefully documented the fact that the relative efficiency functions for nine types of producers' durable equipment were unaffected by higher energy prices: ''While depreciation almost certainly varies from year to year in response to a variety of factors, we have found that a major event like the energy crises, which had the potential of significantly increasing the rate of obsolescence, did not in fact result in a systematic change in age-price profiles." They also conclude that "the use of a single number to characterize the process of economic depreciation [of a given type of asset] seems justified in light of the results." Table 2.2 presents rates of economic depreciation derived from the best geometric approximations of Hulten and Wykoff (1981b) for thirty-four categories of nonresidential business assets and one category of residential assets. These rates were extended to include public utility and residential property by Jorgenson and Sullivan (1981). Hulten and Wykoff compare the best geometric depreciation rates presented in table 2.1 with depreciation rates employed by the Bureau of Economic Analysis (1977) in estimating capital stock. The Hulten-Wykoff rate for equipment averages 0.133, while the Bureau's rate averages 0.141, so that the two rates are very similar. The Hulten-Wykoff rate for structures is 0.037, while the Bureau's rate is 0.060, so these rates are substantially different. The distribution of retirements used by Hulten and Wykoff (1981b) to correct for censored sample bias are based on the Winfrey (1935) S-3 curve with the Bureau of Economic Analysis (1977) lifetimes. These lifetimes are taken, in turn, from Bulletin F, compiled by the Internal Revenue Service and published in 1942. Between 1971 and 1981, the Office of Industrial Economics conducted forty-six studies of survival probabilities, based on vintage accounts for assets reported under the Asset Depreciation Range System introduced in 1962. The results of twenty-seven of these studies have been summarized by Brazell,

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Dworin, and Walsh (1989). These results provide estimates of the distribution of useful lives based on the actual retention periods for the assets examined. A very important objective for future research on vintage price functions is to incorporate information from the Office of Industrial Studies research into corrections for sample selection bias. Before 1981, the tax law linked tax depreciation to retirement of assets. Between 1981 and 1986 the Accelerated Cost Recovery System severed the link between tax depreciation and economic depreciation altogether. The Tax Reform Act of 1986 re-instituted tax depreciation based on economic depreciation. 6 Under the 1986 Act, the Office of Tax Analysis was mandated by Congress to undertake empirical studies of economic depreciation, including "the anticipated decline in value over time,"7 and report the results in the form of a useful life. This is the lifetime for straight-line depreciation that yields the same present value as economic depreciation. For this purpose Office of Tax Analysis (1990, 1991a, 1991b) conducted major surveys of used asset prices and retirements for scientific instruments, business-use passenger cars, and business-use light trucks and analyzed the results. Oliner (1996b) conducts an extensive survey of used asset prices and retirement patterns for machine tools with the assistance of the Machinery Dealers National Association. He compares his results with those from previous empirical studies of economic depreciation for this industry, including those of Beidleman (1976) and Hulten and Wykoff (1981b). Finally, he compares estimates of depreciation and capital stock with those of the Bureau of Economic Analysis's (1987) Capital Stock Study. Oliner's study, like those of the Office of Tax Analysis, combines information on used asset prices and retirements for the same or similar populations of assets. This is an important advance over previous studies based on the vintage price approach. Oliner (1993) collects and analyzes used asset prices for IBM mainframe computers and estimates constant-quality price change and economic depreciation simultaneously. Previous studies of computer prices, such as the studies surveyed by Triplett (1989), were limited to constant-quality price change. The primary data source for computer prices used by Oliner is the Computer Price Guide, published by Computer Merchants, Inc. The data on retirement patterns were obtained from data on the installed stock of IBM computers tabulated by the International Data Corporation. Oliner (1996a) conducts a similar study of computer peripheralslarge and intermediate disk drives, printers, displays, and card readers and punches. The prices of used

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assets are based on Computer Price Guide and estimates of retirement patterns are inferred from the duration of price listings. An alternative to the vintage price approach is to employ rental prices rather than prices of acquisition to estimate the pattern of decline in efficiency. 8 This approach is employed by Malpezzi, Ozanne, and Thibodeau (1987) to analyze rental price data on residential structures and by Taubman and Rasche (1969) to study rental price data on commercial structures. While leases on residential property are very frequently one year or less in duration, leases on commercial property are typically for much longer periods of time. Since the rental prices are constant over the period of the lease, estimates based on annual rental prices for commercial property are biased toward the onehoss-shay pattern found by Taubman and Rasche; Malpezzi, Ozanne, and Thibodeau find rental price profiles for residential property that decline with age. A second alternative to the vintage price approach originated by Meyer and Kuh (1957) is to analyze investment for replacement purposes. Coen (1980) compares the explanatory power of alternative patterns of decline in efficiency in a model of investment behavior that also includes the price of capital services. For equipment he finds that eleven of twenty-one two-digit manufacturing industries are characterized by geometric decline in efficiency, three by sum of the years' digits and seven by straight-line. For structures he finds that fourteen industries are characterized by geometric decline, five by straight-line and two by one-hoss-shay patterns. Hulten and Wykoff (1981c, p. 110) conclude that: "The weight of Coen's study is evidently on the side of the geometric and neargeometric forms of depreciation." Alternative approaches for analyzing investment for replacement purposes are employed by Pakes and Griliches (1984) and Doms (1994). Pakes and Griliches relate profits for U.S. manufacturing firms to past investment expenditures. The weights on investments of different ages can be interpreted as relative efficiencies of these assets. Doms includes a weighted average of past investment expenditures in a production function. Treating the weights as unknown parameters to be estimated, he obtains estimates of relative efficiences of assets. While Pakes and Griliches find patterns of relative efficiencies that rise and then decline, Doms obtains relative efficiencies that decline geometrically. Empirical research on depreciation is now available for the principal categories of assets included in the U.S. National Income and

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Product Accounts. The most extensive body of research is that of Hulten and Wykoff (1981a, 1981b, 1981c), Hulten, Robertson, and Wykoff (1989), and Wykoff (1989). However, important additional studies have been completed by the Office of Tax Analysis (1990, 1991a, 1991b) and Oliner (1993, 1995a, 1995b). Finally, estimates of retirement distributions required for correcting sample selection bias have been completed by the Office of Industrial Economics and summarized by Brazell, Dworin, and Walsh (1989). Taken together, these empirical studies of depreciation provide the information needed to revise the treatment of depreciation in the national accounts. 2.3 Applications The measurement of depreciation has been an important objective of research at the Bureau of Economic Analysis for several decades. National income, net of depreciation, is included in the U.S. National Income and Product Accounts. Stocks of depreciable assets are estimated as a component of national wealth. The culmination of this research was the magisterial Bureau of Economic Analysis's (1987) study, Fixed Reproducible Tangible Wealth in the United States, 1925 1989, updated in 1993 to cover the period 1925-89. Perhaps surprisingly, the Bureaus's estimates of capital stocks and depreciation do not incorporate the results of the extensive empirical literature on vintage price functions that has been summarized in section 2.2. 9 The Bureau employs measures of capital stocks for equipment and structures with relative efficiencies that are constant over the lifetime of each capital good. This produces a measure of gross capital stock. Depreciation based on the straight-line method is used to produce a measure of net capital stock. The first issue is whether the Bureau's (1987, 1993) studies are internally consistent. This issue arises because patterns of decline in efficiency of assets are used in estimating both depreciation and capital stocks, but these patterns can be the same or different. Obviously, internal consistency requires that the same relative efficiencies be used in both sets of estimates.10 If we consider economic depreciation under the assumption that assets do not decline in efficiency until the end of their useful lives, we can simplify the analysis by assuming that rates of return {rt+s} and prices of capital services {pK,t+t} are constant to obtain depreciation on an asset of age v

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Under the Bureau's assumption that relative efficiencies of assets of different ages are constant over the lifetimes of assets, economic depreciation declines geometrically. This reflects the time discounting of the retirement of an asset. The Bureau's straight-line estimates of depreciation obviously reflect a different pattern of relative efficiencies than that employed in estimating capital stock, so that its measurements of depreciation and capital stock are internally inconsistent. This issue was first analyzed in detail by Jorgenson and Griliches (1972a, esp. pp. 81 87). Denison (1972, esp. pp. 101 109) defends the Bureau's use of straight-line depreciation. However, he fails to respond to the criticism that straight-line depreciation is inconsistent with the relative efficiencies of capital goods of different ages that the Bureau employs in measuring capital stock. The logical inconsistencies in the Bureau's methodology arise from the definition of depreciation. Denison's (1972, pp. 104 105) defense of the straight-line formula is based on the "capital input definition" of depreciation that he originally introduced in 1957. This definition was subsequently adopted in the U.S. National Income and Product Accounts. The capital input definition is described by Young and Musgrave (1980, p. 32), as follows: "Depreciation is the cost of the asset allocated over its service life in proportion to its estimated service at each date." Within the vintage accounting framework presented in section 2.1, Denison's capital input definition of depreciation allocates the cost of an asset over its lifetime in proportion to the relative efficiencies of capital goods of different ages. 11 Young and Musgrave (1980, pp. 33 37) contrast the Denison definition with the "discounted value definition" employed in the vintage accounting system outlined in section 2.1. Among the advantages for the capital input definition claimed by Denison (1957, p. 240) and by Young and Musgrave (1980, p. 33) is that this definition avoids discounting of future capital services. In fact, discounting can be avoided in the measurement of depreciation only if the decline in the efficiency of capital goods is geometric. The Bureau of Economic Analysis's assumption that efficiency is constant over the lifetime of an asset requires discounting, as I have already demonstrated. Only if relative efficiencies decline geometrically does economic depreciation coincide

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with Denison's capital input definition. But this definition then implies declining balance rather than straight-line depreciation. The second issue that arises in the measurement of depreciation is the incorporation of up-to-date information from empirical studies like those summarized in section 2.2. The available empirical evidence on efficiency functions for different classes of assets, growing out of the work of Hulten and Wykoff (1981a, 1981b, 1981c), is very extensive. This evidence includes the results of econometric modeling of vintage price functions, like those of Hulten, Wykoff, and Robertson (1989), Wykoff (1989), the Office of Tax Analysis (1990, 1991a, 1991b), and Oliner (1993, 1996a, 1996b). It also includes studies of retirement patterns like those developed by the Office of Industrial Economics and summarized by Brazell, Dworin, and Walsh (1989). Both types of information are needed for the measurement of depreciation. The Bureau of Economic Analysis (1987, 1993) employs only information about retirements in measuring depreciation. This is a consequence of applying the "capital input definition" of depreciation under the assumption that relative efficiencies of assets are constant over the lifetime of each capital good. This assumption is contradicted by the empirical evidence on depreciation I have reviewed in section 2.2. The incorporation of this evidence into the U.S. National Income and Product Accounts requires that Bureau's definition of depreciation be replaced by the "discounted value definition" presented in section 2.1. A vintage accounting system for prices and quantities of investment goods such as that originated by Christensen and Jorgenson (1973), provides an internally consistent framework for measuring capital stocks and depreciation. Christensen and Jorgenson use this framework in constructing an integrated system of income, product and wealth accounts. They distinguish between two alternative measures of economic performance, identifying the production approach with the production possibility frontier employed by Jorgenson and Griliches (1967). This approach is implemented by means of a production account with an accounting identity between the values of outputs and inputs. These data are used to allocate the growth of output between the growth of capital and labor inputs and productivity growth. Christensen and Jorgenson have also described a welfare approach to the measurement of economic performance, based on a social welfare function like that employed by Jorgenson and Yun (1991a). This approach can be implemented by means of an income and expen-

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diture account with an accounting identity between income net of depreciation and the sum of saving and consumption. Data from this account can be used to allocate growth of income between the growth of current consumption and the growth of future consumption through saving. Both production and income and expenditure accounts are essential components of the U.S. National Income and Product Accounts and accounting systems that employ the United Nations (1968) System of National Accounts. Saving is linked to the asset side of the wealth account through capital accumulation equations for each asset like those presented in section 2.1. These equations provide a perpetual inventory of assets accumulated at different points of time. Prices for different vintages of investment goods are linked to the prices of capital services through a parallel set of asset pricing equations, like those presented in section 2.1. The complete system of vintage accounts gives stocks of assets of each age and the corresponding asset prices. The stocks can be cumulated to obtain quantities of assets, while the prices can be used to value the stocks and derive rental prices for services of the assets and measures of depreciation. Hulten (1992) has formalized the welfare and production approaches to economic performance by means of a model of optimal economic growth. This model includes a production function with output as a function of capital and labor inputs. Output is divided between consumption, which contributes to the welfare of a representative consumer, and saving, which contributes to future consumption through capital accumulation. Income net of depreciation measures the welfare resulting from intertemporal optimization of consumption in Hulten's model of growth, as in a similar model proposed by Weitzman (1976). This measure of welfare summarizes the stream of present and future consumption. Hulten (1992) shows that gross product is the measure of output appropriate for separating the growth of output between productivity growth and the growth of capital and labor inputs, while national income net of depreciation is appropriate for allocating the growth of income between consumption growth and contributions to the growth of future consumption through saving. A complete system of accounts includes a production account, an income and expenditure account, and a wealth account. Measures of depreciation employed in all three accounts can be generated in an internally consistent way from the system of vintage accounts for assets outlined in section 2.1.

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The income account of an integrated system of national accounts can be used in measuring welfare. Hulten (1992) points out that the Haig-Simons definition of taxable income requires that capital cost recovery for tax purposes must equal economic depreciation. Capital cost recovery for tax purposes has differed from economic depreciation whenever capital consumption allowances and the investment tax credit have been used to provide tax incentives for private investment, as in the Accelerated Cost Recovery System during the period 1981 1986. Under U.S. tax law capital recovery is based on the original acquisition price of an asset rather than current replacement cost. During periods of high inflation, the original acquisition price and the current replacement cost have diverged substantially. Making use of the asset classification scheme of the Bureau of Economic Affair's (1987) capital stock study for individual industries, Jorgenson and Yun (1991b) map the economic depreciation rates for the thirty-five asset categories given in table 2.2, into the Bureau's (1987) asset classification scheme to incorporate additional information about asset lives. Jorgenson and Yun (1991b) employ data on economic depreciation and capital cost recovery under U.S. tax law to summarize the tax burden on capital income. For this purpose they employ the marginal effective tax rate introduced by Auerbach and Jorgenson (1980). An effective tax rate represents the complex provisions of tax law in terms of a single ad valorem rate. The production account of an integrated system of income, product, and wealth accounts can be used in measuring productivity. The estimates of depreciation by Jorgenson and Yun (1991b) have been incorporated into price and quantity indices of capital services for productivity measurement by Jorgenson (1990). The underlying estimates of capital stocks and rental prices are classified by four asset classesproducers' durable equipment, nonresidential construction, inventories, and landand three legal forms of organizationcorporate and noncorporate business and nonprofit enterprises. This study is based on annual data for the period 1947 1985 for an average of as many as 156 components of capital services for each of thirty-five industries. These data incorporate investment data from the Bureau of Economic Analysis (1987) study of U.S. national wealth. In constructing data on capital input for each of the thirty-five industrial sectors, Jorgenson (1990) combines price and quantity data for different classes of assets and legal forms of organization by expressing sectoral capital services, say {Ki}, as a translog function of

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its 156 individual components, say {Kki}. The corresponding index of sectoral capital services is a translog quantity index of individual capital services

where weights are given by average shares of each component in the value of sectoral property compensation:

The value shares are computed from data on capital services {Kki} and the rental price of capital services { cross-classified by asset class and legal form of organization.

},

Internal consistency of a measure of capital input requires that the same pattern of relative efficiency is employed in measuring both capital stock and the rental price of capital services. The decline in efficiency affects both the level of capital stock and the depreciation component of the corresponding rental price. Estimates of capital stocks and rental prices that underlie the data presented by Jorgenson (1990) are based on geometrically declining relative efficiencies. The same patterns of decline in efficiency are used for both capital stock and depreciation, so that the requirement for internal consistency of price and quantity indices of capital services is met. The Bureau of Labor Statistics (1983, pp. 57 59) also employs relative efficiency functions estimated by Hulten and Wykoff. However, the Bureau of Labor Statistics does not utilize the geometric relative efficiency functions fitted by Hulten and Wykoff. Instead, it has fitted a set of hyperbolic functions to Hulten and Wykoff's relative efficiency functions. Consistency is preserved between the resulting estimates of capital stocks and rental prices by implementing a system of vintage price accounts for each class of assets. This set of accounts includes asset prices and quantities of investment goods of all ages at each point of time. The Bureau of Labor Statistics (1983, pp. 57 59) shows that measures of capital services based on hyperbolic and geometric relative efficiency functions are very similar.

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The Bureau of Labor Statistics (1991) expands annual productivity estimates reported for private business, private nonfarm business, and manufacturing to include measures of capital services that differ among fifty-seven industries. For each industry, capital service prices and quantities are estimated for seventy-two types of depreciable assets. The Economic Research Service (1991) publishes official productivity estimates for agriculture that include prices and quantities of capital services, based on the approach of Ball (1985). The methodology for depreciable assets, except for breeding livestock, is similar to that employed by Bureau of Labor Statistics. Data on prices and quantities of capital services for breeding livestock have been constructed by the vintage accounting approach developed by Ball and Harper (1990). 12 I conclude that the U.S. National Income and Product Accounts fail to provide internally consistent measures of capital stocks and depreciation. Top priority should be given to replacing the "capital input definition" of depreciation by the "discounted value definition" presented in section 2.1. This has important implications for improving the income, product, and wealth measures now included in the U.S. national accounts. Second, estimates of depreciation should be enhanced by incorporating the information from studies of vintage price functions that are summarized in section 2.2. These two tasks can be accomplished by utilizing concepts and methods already familiar to economic statisticians as a result of the introduction of constant quality prices for computers into the national accounts. 2.4 Conclusion Hall's (1971) "hedonic" model of vintage asset prices, outlined in section 2.1, has proved to be an indispensable guide to empirical research. The applications surveyed here are based on the pioneering studies of Hulten and Wykoff (1981a, 1981b, 1981c), who have constructed vintage price functions covering a sizable proportion of U.S. investment expenditures. Alternative methodologies, such as the rental price approach and the modeling of investment for replacement purposes, have been superseded by the vintage price approach with a correction for sample bias originated by Hulten and Wykoff. I conclude that the results of empirical research on depreciation should be incorporated into the U.S. National Income and Product Accounts. The research of Hulten and Wykoff has been used success-

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fully in two important applications of depreciationthe empirical description of the U.S. tax system for depreciable assets by Jorgenson and Yun (1991b) and the measurement of capital services and rental prices for studies of productivity by Jorgenson (1990) and the Bureau of Labor Statistics (1991). Information from the research summarized in section 2.2 should be combined with the detailed data on investment flows by class of asset and industry produced by The Bureau of Economic Analysis (1987, 1993) to improve measures of U.S. national income, product, and wealth. While the Bureau of Economic Analysis (1987, 1993) provides estimates of capital stocks and economic depreciation, the two sets of estimates are logically inconsistent since they are based, implicitly, on different assumptions about the relative efficiencies of assets of different ages. Gross capital stocks are based on unchanging efficiencies throughout the lifetime of an asset. This implies a declining balance method of depreciation that reflects the time discounting of retirements. However, the Bureau of Economic Analysis's measures of depreciation are based on the straightline method. The first step toward removing these inconsistencies is to adopt the discounted value definition of depreciation presented in section 2.1. My overall conclusion is that top priority in future applications of empirical studies of depreciation should be assigned to developing internally consistent measures of national income and wealth. An equally important priority is to utilize the results of empirical studies of depreciation growing out of the work of Hulten and Wykoff. The appropriate conceptual framework for both of these important tasks is provided by the system of vintage price and quantity accounts introduced by Christensen and Jorgenson. This vintage accounting system is the key to integrating income and product accounts with wealth accounts and incorporating relative efficiencies of assets of different ages based on econometric estimates of vintage price functions. Notes 1. The system of vintage accounts summarized here is discussed in greater generality by Christensen and Jorgenson (1973) and Jorgenson (1980). The model of capital as a factor of production that underlies this system is discussed by Diewert (1980), Hulten (1990), Jorgenson (1973a, 1980, 1989), and Triplett (1975). 2. These vintage accounts are used to generate an integrated system of income, product and wealth accounts for the United States. by Christensen and Jorgenson (1973). This system is extended to the industry level by Jorgenson (1980) and implemented by

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Fraumeni and Jorgenson (1980). Fraumeni and Jorgenson (1986) have incorporated the results of empirical research on depreciation summarized in section 2.2. 3. Surveys of the hedonic technique are given by Triplett (1975, 1987, 1990). 4. Gordon (1989) has presented an alternative constant quality price index for computers. This was incorporated into Gordon's (1990) study of constant quality price indices for all components of producers' durable equipment in the U.S. national accounts. 5. Hulten and Wykoff (1981b) estimated vintage price functions for structures from a sample of 8066 observations on market transactions in used structures. These data were collected by the Office of Industrial Economics of the U.S. Department of the Treasury in 1972 and were published in Business Building Statistics (1975). They estimated vintage price functions for equipment from prices of machine tools collected by Beidleman (1976) and prices of other types of equipment collected from used equipment dealers and auction reports of the U.S. General Services Administration. 6. Detailed histories of U.S. tax policy for capital recovery are presented by Brazell, Dworin, and Walsh (1989) and Jorgenson and Yun (1991b). 7. Joint Committee on Taxation (1986), p. 103. 8. Hulten and Wykoff (1981c) summarize studies of economic depreciation completed prior to their own study. Vintage price functions have provided the most common methodology for such studies. 9. The methodology for constructing estimates of depreciation and the corresponding capital stocks is described by the Bureau of Economic Analysis (1987, 1993). 10. The Bureau of Economic Analysis's methodology is discussed in greater detail by Hulten and Wykoff (1981c) and Wykoff (1989). The incorporation of retirement patterns described in section 2.2, below, results in a decline in efficiency with age after retirements begin. However, this does not affect our conclusion that the Bureau's methodology is internally inconsistent. 11. The capital input definition has been categorically rejected by economists outside the Bureau of Economic Analysis. See the comments on Denison (1957) by Kuznets (1957), comments on Denison (1972) by Jorgenson and Griliches (1972b), and comments on Young and Musgrave (1980) by Faucett (1980). 12. Boskin, Robinson, and Roberts (1989a, 1989b) successfully employed the vintage accounting approach in measuring capital stocks and depreciation for the government sector of the U.S. economy. Jorgenson and Fraumeni (1989) apply this approach to the measurement of stocks and depreciation of human capital.

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3 An Economic Theory of Agricultural Household Behavior Dale W. Jorgenson and Lawrence J. Lau 3.1 Introduction The purpose of our theory of agricultural household behavior is to serve as the basis for a study of household consumption and production patterns for Japan, Korea, and Taiwan. Comparable survey data for all three countries have been assembled for individual agricultural households for the year 1965. Our analysis is confined to behavior patterns for a single year. The accumulation of capital, changes in the distribution of land, and demographic changes are not included in our theory of the farm household. However, our theory may be embedded in a dynamic theory that incorporates patterns of investment in land and reproducible capital. The dynamic theory would require data on the same farm households for several years for implementation. Our objective is to construct a microeconomic model of farm household behavior, relating consumption and production phenomena in one time period to prices of output, variable input and consumption goods and to quantities of fixed input, household wealth, and the internal composition of the household during the period. Our fundamental hypothesis is that the household maximizes welfare under subjective certainty and competitive markets. The welfare function contains leisure and consumption of each of the family members as arguments. An alternative approach would be to assume that the household maximizes an individual utility function, defined in per capita terms. This alternative approach is unsatisfactory for incorporation of the effects of family composition on consumption and labor supply. We assume that household welfare is maximized subject to the production function, to a budget constraint relating sales and purchases of commodities and services, and to a constraint relating labor and leisure time of family members to the total time endowment of the

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household. The household chooses levels of consumption of leisure and goods, levels of family labor supplied to the farm enterprise and to the labor market, and quantities of variable factor inputs, including hired labor, given prices of output, consumption goods, and variables inputs, given quantities of land and reproducible capital, and given financial assets that yield income unrelated to the activities of the farm enterprise. The notion of welfare maximization or ''subjective equilibrium" for the farm household is not new and has been explored in various degrees of generality by Mellor (1963), Schultz (1964), Nakajima (1969), Krishna (1969), Sen (1966a,b), Dean (1966), and Torii (1967). We undertake a detailed analysis of the relationship between the farm enterprise and household economic behavior that arises from the household's relationship to the labor market. If the household buys and sells no labor, equilibrium of the household and the farm enterprise must be analyzed simultaneously; if the household is a buyer or seller of labor, the activities of the farm enterprise may be determined independently of the family's preferences between leisure and goods. This implication of the theory is the major hypothesis to be tested in our econometric work. Secondly, we embed the theory in a dynamic framework that enables us to make explicit the role of capital and land services in the production process. In the following sections the various theoretical constructs used in the modelthe welfare function for the agricultural household, the utility functions for its members, the production function for the farm enterprise, and the specification of the relationship of the household to the economic environment are discussed in detail. In each case we introduce specializations of these constructs necessary for econometric implementation of the model. Finally, we provide a complete formal statement of the model and develop the main theoretical results to be subjected to empirical testing. 3.2 The Welfare Function Each farm household is assumed to possess a well-defined (unique up to a monotonic transformation) and wellbehaved social welfare function. This, of course, presupposes a sufficient degree of consensus among the different members of the household such that the Impossibility Theorem of K. Arrow (1963) does not apply. Such sufficient conditions have been analyzed by Arrow (1963), Black (1958), Inada

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(1964), Sen (1966a,b) and Ward (1965). Here it will merely be assumed that a welfare function exists for the farm household. 1 In its most general form, the social welfare (W) of the farm household may be represented as a function of the individual welfare functions (Wi's).

where W = social welfare of the farm household; Wi = social welfare function of individual i; m = the number of individuals in the farm household. In addition, the following assumptions are made concerning the social welfare function: (a) Social welfare is a strictly increasing, continuous and twice-differentiable function of the individual social welfare functions.2 (b) The social welfare function is strongly quasi-concave with nonpositive second partial derivatives. (c) W = F (0,0, . . . , 0) = 0, that is, at least one individual welfare function must be nonzero for social welfare to be different from zero. Each individual welfare function is in turn a function of the utilities of each member of the household. Thus, it represents the i-th individual's preference ordering over all possible social states. Here the hedonistic assumption of independence of utilities is relaxed, allowing room for all kinds of external economies or diseconomies in consumption. The individual may in general derive satisfaction (or lack of it) not only from commodities consumed by himself, but also from commodities consumed by other members of the household. Examples of these externalities are the consumption anomalies described by Veblen (1925), the relative income hypothesis of Duesenberry (1949), and the "international demonstration effect" so often mentioned in the economic development and international trade literature. Note, however, that the formulation here is not the most general possible for the analysis of these externalities as Wi is weakly separable in the consumption bundles of each individual. Only the utility indices, rather than the quantities of each commodity consumed by each individual, matter.

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where Ui = utility function of individual i. Assumptions similar to those made concerning the social welfare function are also maintained for each individual welfare function. 3 In addition, the following assumption is added: (d) The Wi's are each normalized such that

(e) It is assumed that

for j ¹ i,i,j = 1, . . . , m.4 Following Sen (1967), social consciousness (SCi) and social goodwill (SGi) of individual i are defined respectively as follows:

Social consciousness measures how much an individual member values the utilities of the other members. Social goodwill measures how much an individual member is regarded by the other members. In general, SCi is not equal to SGi. SCi = 1 corresponds to a perfectly altruistic individual and SCi = 1/m to a perfectly self-centered individual. SGi = 1 corresponds to a perfectly popular individual and SGi = 1/m to a perfectly unpopular individual. A more commonly used social welfare function is that of the Bergson-Samuelson type:

This is weakly separable in the groups of commodities consumed by each individual. The complete absence of externalities of consumption will, of course, lead to a Bergson-Samuelson social welfare function. However, observe that since W is a continuous function of the Wi's,

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which are in turn continuous functions of the Ui's from (3.1) and (3.2), any such social welfare function can be put into the Bergson-Samuelson form, that is

It is also obvious that F* possesses properties similar to the assumptions (a) through (c) as stated previously. Of greater interest is the question of what constitutes sufficient restrictions on F such that F* satisfies certain properties. Two of these properties, namely equality and homogeneity, will be examined. 3.2.1 Equality The concept of equality as applied to social welfare functions implies that either the welfare or the utility of each individual is valued equally. The following formal definition may be given DEFINITION. Given a social welfare function of the Bergson-Samuelson type, that is,

where xij = quantity of commodity j consumed by individual i. The social welfare function is said to be absolutely egalitarian if

The following theorem is useful. THEOREM. A social welfare function of the Bergson-Samuelson type is egalitarian if and only if the social welfare function can be written as

where G(t) is a monotonic function of one variable. PROOF. Necessity: It suffices to show that the general solution to the system of differential equations

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Through a transformation of the independent variables

the social welfare function can be written as

The original system of partial differential equation can also be written in terms of the new variables by an application of the chain rule for differentiation.

Hence the transformed system becomes

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The general solution of this alternative system of partial differential equations is immediately seen to be

COROLLARY 3.1. If a social welfare function possesses fixed value scales, that is

where a is a constant. Then

where G(t) is a monotonic function of one variable. COROLLARY 3.2. If a social welfare function is "proportionately egalitarian," that is,

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COROLLARY 3.3. An absolutely egalitarian Bergson-Samuelson social welfare function is homothetic with respect to the individual utilities. COROLLARY 3.4. An absolutely egalitarian Bergson-Samuelson social welfare function is strongly separable with respect to the groups of commodities consumed by each individual. The theorem proved is formally equivalent to the Hicks composition-commodity theorem (Hicks, 1946, pp. 33 34, 312 313). Its conclusions are equally applicable to a social welfare function which allows for the externalities. It should be noted that a fixed valuation scheme in the social welfare functions is neither necessary nor sufficient for the optimality of "shibboleths" or fixed-proportions budgeting procedures, whether intertemporal, allocative, or distributive. Now, given that the social welfare function F is egalitarian with respect to the Fi' s, what additional restrictions are necessary such that F * is absolutely egalitarian, that is,

for all values of Ui' s. The set of conditions, equation (3.13), is clearly necessary and sufficient for equation (3.12). For our purpose, welfare functions for which

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are ruled out. This eliminates the trivial case of dictatorial welfare functions from our considerations. First, observe that if F is egalitarian in the Fi's, then a necessary and sufficient condition is

The class of system of functions {Wi} which satisfy equation (3.14) has been found. 5 They may be written as follows:6

where the Fi's are arbitrary functions and the Wi's satisfy assumptions (a) through (c). Thus a wide class of systems of individual welfare functions admits of an egalitarian Bergson-Samuelson formulation. In particular, the cases of (i)additivity of individual utilities, (ii)identical welfare functions, (iii) perfect altruism, and (iv) perfect self-centeredness7 are obtainable from equation (3.15). Note also that (ii) implies (i). 3.2.2 Homogeneity If F * is a homogeneous function, it implies that

where k is a constant. Now

If F * is egalitarian with respect to the Wi's, equation (3.17) simplifies to

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The necessary and sufficient condition for homogeneity then becomes

The most general system of functions which satisfies equation (3.19) is 8

where the gi's are arbitrary functions and gm is a homogeneous function of degree k. It is also evident that a system which satisfies the equality property does not necessarily satisfy the homogeneity property. In principle, an attempt should be made to construct an explicit theory of intra-household decision-making to analyze the behavior of the household. We assume that this problem has been solved and that the solution is subsumed in the welfare function. In this study, an absolutely egalitarian Bergson-Samuelson social welfare function is employed.9 Hence the welfare function is written as the sum of the individual utilities.10 It should be emphasized that an egalitarian social welfare function does not imply the optimality of an egalitarian distribution unless all utility functions are identical and strictly quasiconcave. The optimal distribution of commodities (and income) depends, in general, on all of the individual utility functions. However, under certain mild conditions, there always exist distribution (or budgeting) schemes which are optimal, including a constant share's distribution scheme as a special case.11 These distribution schemes will be called "shibboleths" following Samuelson (1956), although they are generalized in the sense that price-dependent "shibboleths" are also allowed. The use of a social welfare function which depends on the utility function of the individuals instead of a collective utility function is

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motivated by two considerations: (1) In general, such a collective utility function with the aggregate quantities consumed as arguments does not exist, as has been demonstrated by Samuelson (1956); and (2) the welfare function approach allows the possibility of different utility functions for different individuals (or classes of individuals) which may be of value in an analysis of the economic effects of household composition. Two other problems deserve some discussion before the discussion of the individual utility functions. First, as our model is for activity in a single period, we wish to know the necessary and sufficient conditions on the intertemporal welfare function which will validate the results of our analysis even in a fully dynamic model. A necessary and sufficient condition turns out to be that of "decentralizability"; that is, a condition on the intertemporal welfare function such that given the optimal total expenditure in one period, all the allocation decisions may be undertaken in that period with reference to the prices of that particular period only. 12 Weak separability of the intertemporal welfare function in its direct or indirect form is a sufficient condition. For our purpose, we assume total expenditure as observed, subject to certain adjustments of consumption expenditures of an unforeseeable and nonrecurrent nature, such as wedding expenses, to be given. We proceed to analyze the allocation decisions without reference to the prices of the other periods. In other words, the intertemporal welfare function is assumed to be decentralizable in time periods.13 A very large class of preference orderings are consistent with decentralizability.14 The second problem has to do with our assumption concerning the dependents in the family. One alternative is to assume that the number of dependents is essentially an exogenous affair over which the household does not have any effective control. The other possibility is to embed the model in a theory of family planning. Space does not allow a detailed discussion of a theory of family planning. By the very nature of the problem, it has to be an integer dynamic programming problem. The head of the household at time zero maximizes the intertemporal welfare function with respect to both the number of dependents and the arrival (and departure) times of the dependents. The lag between arrival and departure may be fixed institutionally: for instance, twenty-one years. There may also be a pension value to be added to the budget constraint after departure. The solution to this problem may be complicated. However, we note that as long as no

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arrival or departure is planned in the period under consideration (in this case, one year), for all intents and purposes one may assume that the number of dependents are exogenous. 3.3 The Utility Functions The individuals of the farm household are classified into two functional categories: workers and dependents. The distinction between the two categories is somewhat arbitrary, and in this regard, the definition used by the government ministries of the respective countries is adopted. For instance, all persons between 15 59 inclusive may be classified as workers and all others as dependents. In addition, the individuals are distinguished by sex and age. We let a be the number of workers and b be the number of dependents. All the individuals in the farm household are assumed to possess similar but not identical utility functions. However, all utility functions do have the following properties in common: (a) Utility is a continuous, twice-differentiable, monotonically increasing function of economic leisure, Z, own composition, A, and purchased consumption, C. It has been implicitly assumed that the utility function is weakly separable with respect to own consumption commodities and purchased consumption commodities so that the aggregate indices A and C exist. (b) The individual utility function is strictly quasi-concave. (c) There is nonincreasing marginal utility for Z, A, and C. (d) There is no satiation level. This assumption will be subsequently relaxed to allow the testing of hypotheses of the ''limited aspiration" type.

Specification of individual utility functions which depend on the characteristics of the individual members is one way to treat the effect of family composition on expenditures. Most treatments of this problem, for example, Prais (1953), Prais and Houthakker (1955), and Forsyth (1960), consist of the inclusion of the number of members in

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each age-sex category as an explicit independent variable in the estimation of the reduced form demand equations, 15 with the adult-equivalent consumption scales directly estimated from the data. In this study, we shall make use of extraneous estimates of the adult-equivalent consumption scales. As these scales are, in general, derived from a detailed analysis of consumption patterns by the competent official health authorities, they appear to be more reliable than any scale that we can estimate directly from our data. In addition, it is assumed that, other things being equal, women and children require a higher level of leisure in order to stay on the same utility level as the men. Thus, the utility function of an individual of sex s (m or f) and in age category i is given by

where the adult-equivalent parameters need not be all the same. The function U, however, is common to all individuals. An additional complication is introduced by the distinction between workers and dependents. Dependents, by definition, always enjoy a maximum of leisure. Hence, the utility function for dependents is modified as follows:

where represents the maximum economic leisure that can be enjoyed. It can be taken at twenty-four hours or sixteen hours, but as we shall see, the value of does not affect our analysis at all. It should be noted that the various concepts of "satisfying behavior" of peasant households such as the "achievement standard of living" of Wharton (1963), the "achievement standard of income" of Nakajima (1969), and the "limited aspirations model'' of Mellor (1963) may be regarded as special cases of welfare maximization in which the utility function possesses a nonhomogeneity parameter which corresponds to the satiation level. A more detailed discussion of this point is deferred to the discussion of functional forms. 3.4 The Technology A farm household is also a multi-period, multi-product firm. It is assumed that the technology is characterized by a recursive structure such that the production possibility frontiers of different periods may be represented as follows:

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where Vit = net output of commodity i in period t; Xit = net input of commodity i in period t; and Zit = endowment of input i at the end of period t. In this study, we are not interested so much in the composition of the output as in the aggregate quantity of the output. Hence, it is further assumed that the one-period production possibility frontier of the household farm is weakly separable 16 between outputs and inputs, that is, it can be written as a joint production function.

where V is the output function and F is the input function. Similarly, it is assumed that the input function is weakly separable in inputs classified as the same type, e.g., all kinds of structures. This condition is both necessary and sufficient for the existence of quantity indices of each type of inputs. The value of V may be interpreted as the quantity of total output subject to a normalization rule. The measurement of V is essentially an aggregation problem. It is assumed that such an index can always be calculated so that one can always rewrite the joint production function as a single production function

where the time subscripts are suppressed. The following assumptions are made concerning the production function: (a) It is a strictly nondecreasing continuous and twice-differentiable function of labor, capital (structures and equipment), live capital (animals and trees), fertilizers, other purchased inputs, and land. (b) It possesses diminishing marginal products. (c) It is assumed to be strictly quasi-concave, so that unique supply and demand functions may be derived. (d) Land, structures, and equipment used in production are assumed to be fixed during the production period, that is, they are equal to the initial endowment. As an alternative to assumptions (a) through (d), one may assume that a continuous and twice-differentiable partial profit function exists for the farm household.

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where P is the maximized profit, p is the price of the output, the are the prices of the services of the variable factors, and the Zi's are the quantities of the fixed factors. The ci's will be referred to as the real factor prices. The production model is an annual one so that total output is the quantity of output produced in the year under consideration, and the input refer to services consumed during the same year. 17 It is natural to assume that agricultural production is characterized by a recursive technology. For instance, after the planting season is over, addition of land input is unlikely to contribute to the production of the present period. Under this assumption of a recursive technology, Jorgenson (1968) has shown that for a goodwill or net-worth maximizing firm, the optimization problem (or decision problem) may be decomposed into two parts: optimal production in the present period and optimal production in all future periods, with the result that the present period production plan is independent of the investment plan for the present and all future periods. For our purpose, the inputs that depend on the initial endowments of the period are identified with the fixed factors. The inputs that are not constrained by the initial endowments are identified with the variable factors. The distinction between fixed and variable factors is based on the assumption that amounts of the fixed factors are determined by long-run considerations, while amounts of the variable factors are determined by the household's economic environment during the period of production. In addition to a production plan, giving quantities of output and variable input for given prices and quantities of the fixed factors, the agricultural household has an investment plan in which quantities of land and capital for the following period of production are determined. If data were available on investment by the household, our theory could be extended to incorporate investment as well as production planning. The homogeneity and perfect substitutability of both outputs consumed and marketed and inputs purchased and sold are important assumptions for the present study. For instance, a change in the sale price of an output may result in the substitution of the marketing of outputs of a different quality, although the quantity may remain the

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same. Unfortunately, no independent data are available on the quality of the output sold versus the output consumed, hence it will be assumed that they are of the same uniform quality. The same applies to all the other purchased inputs insofar as they command the same price. An exception is made in the case of labor input, where in some applications hired labor may be of a different quality than family labor; the hypothesis of perfect substitutability will be tested explicitly. In the general model, however, the hypothesis of perfect substitutability of hired labor is maintained. Finally, some inputs that are only utilized seasonally, for instance, draft animals, deserve special attention. While they are only utilized during a very short period in the production year, it may be impossible to rent animals during the peak season. In other words, there will be no market transactions and hence no rental rate that reflects marginal productivity. Whatever transactions that take place during the slack season furnish poor rental rate approximations for the year as a whole. 3.5 The Constraints There are three operational constraints in this model which relate different variables: time, total expenditure and total labor. In addition, there are numerous nonnegativity constraints on the variables. We shall discuss the constraints in turn. (a) The Time Constraint The time constraint applies to each individual. For a worker, the constraint is given by

where Zsi is hours of leisure and LF is hours of work. For the dependents

For the workers, Zsi = zsiZ, where Z has the interpretation of the optimal adult male equivalent leisure. It is also necessary to adjust for sex and age differences in labor efficiency in the aggregation of total labor expended on the farm. It is assumed that the efficiency differentials are reflected by the wage differentials of different types of labor which, in turn, are proportional to the leisure adjustment coefficients in the utility functions, namely, that

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where w is the adult male equivalent wage rate. Hence, total effective labor hours becomes

(b) The Total Expenditure Constraint The total expenditure constraint, in the absence of taxation, 18 may be represented as follows:

where w = wage rate of hired labor; PA = price of the output; Pc = price of the input; P = net profits from farm enterprise, that is, the difference between total revenues and total expenditures, including the imputed value of family labor; I = nonlabor income; and S = saving. This variable can be negative. In our formulation it also includes as negative components unforeseen or nonrecurrent consumption expenditures. Windfall income, such as a lottery winning, is also included in saving; likewise, the net increase in inventory. Both A* and C* are the aggregate consumption quantities obtained by summing over the individual members.

It is clear that the optimal Asi and Csi for all workers will be the same, except for a scalar adjustment factor; likewise, for the dependents. Hence

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As the imputed value of leisure consumed is taken into account explicitly, one may, following a usage initiated by Becker (1965), call the L.H.S. "full consumption" and the R.H.S. "full income." In turn, net profits are given

where L = total quantity of labor input on the farm; ci = rental price of the i-th purchased input; and Xi = quantity of the i-th purchased input. (c) The Total Labor Constraint. It is obvious that

where LH = quantity of hired labor. This variable is allowed to be negative, in which case it represents the net quantity of labor hired out by the farm household. Conceptually, there are two ways to treat nonfarm labor income. The first is to introduce the labor hours expended on the farm and off the farm separately into the utility function, that is,

where LM = quantity of nonfarm labor services performed, and utility decreases with increasing labor hours of either type. The consumer maximizes his utility, subject to the budget constraint and given wage rates w and wM.

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However, this treatment suffers from the defect that there exists no unique shadow price for leisure, and it really does not make sense to speak of two kinds of leisure anyway. In addition, there is no reason to suppose that the peasants intrinsically prefer one type of labor over the other, after all, economically relevant benefits and costs have been taken into account and appropriately adjusted. A rigorous treatment of this problem requires the conversion of all labor hours on and off-the-farm into intensityequivalent units and a revaluation of the farm and nonfarm hired labor wage rates by adjusting for the imputations due to the different benefits and costs apart from physical exertion, such as time lost in commuting, incurred in the two kinds of activities. Unfortunately, data needed for this type of analysis are extremely difficult to obtain, and the problems associated with accurate imputations of benefits and costs are enormous; therefore, a simpler approach is taken; namely, that there is only one wage rate for all labor services, subject to certain adjustments of quality differences. 3.6 The Complete Model The problem of farm management becomes one of maximization of the welfare function

subject to the following constraints: (i) Production Function

(ii) Time Constraint

(iii) Total Expenditure Constraint

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Defining

The budget

The budget constraint may be written as

The maximization is carried out by a Lagrange multiplier method. The Lagrangian is

For the moment, let us consider only interior solutions, so that all the variables are strictly positive at the optimum. The Kuhn-Tucker necessary conditions are

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The set of equations (3.21) may be rewritten as

This is a system of (m + 8) simultaneous equations in the dependent variables Z, Aa, Ab, Ca, Cb, V, L, X1, . . . , Xm, and l. The independent variables are w, PA, Pc, c1, . . . , cm, Z1, . . . , Zn, aa, ba, ac, bc, az, R, I, and S. As we have assumed that both the utility functions and the technology are strictly quasi-concave, there always exists a unique solution of the dependent variables which is optimal. The Hicksian stability conditions are also satisfied by virtue of strict quasi-concavity. Thus, if one specifies the functional forms of U and F, the preceding structural model can be solved for the dependent variables. In this exercise, the relevant wage rate must be that of the adult male equivalent wage rate. To the extent that the seasonally adjusted wage rate is necessary, this should be computed as a geometrically weighted average of the wage rate. The system can also be estimated directly, provided that Aa, Ab, Ca, and Cb are observed directly. Unfortunately, data pertaining to intrahousehold distribution of consumption goods are unavailable. Hence, an alternative method, which consists of computing the reduced form solution explicitly based on the specified functional forms, is used. Given these reduced form solutions, one can compute the ratios Aa/Ab

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and Ca/Cb, which may then be used to calculate Aa, Ab, Ca, and Cb, given the total quantities of own consumption and purchased consumption. In the case of additive homogeneous utility functions, it is clear that the ratios will be constants. Returning to the set of optimality conditions given by eqs. (3.22 3.30), one observes immediately that eqs. (3.28) and (3.29) may be solved jointly for values of V, L, X1, . . . , Xm given Z1, . . . , Zn without reference to the other equations. Labor is applied on the household farm until the value of the marginal product is equal to the wage rate. Likewise, the utilization of the other inputs is governed by the usual marginal productivity conditions. Thus, we note that given a competitive market for the factor inputs, and perfect substitution between family labor and hired labor, the production behavior of the peasant households is completely determined given the price of the output and the factor prices. It is completely independent of the consumption choices of the household. On the other hand, consumption behavior does depend on total net income, which is represented by the budget constraint, and hence, is not independent of the production decisions. As a result, one has a block recursive model in which the production decisions are first made without reference to the consumption decisions. Subsequently, the consumption decisions are made, taking the production decisions as given. Nakajima (1969) also alludes to a "two-phase decision-making process" based on considerations of the production initiation and completion lags of agriculture. However, our result is even stronger: as long as there exists perfect markets for all factors and perfect substitution between family labor and hired labor, the model is always block recursive, independent of the dynamic characteristics of the production process. This independence of the consumption and production plans is analogous to Irving Fisher's result (1930) and has also been pointed out by Jorgenson (1967a,b) in a different context. On the consumption side, the household takes the net profits and nonlabor income as given and proceeds to maximize welfare with respect to leisure, own consumption and purchased consumption. The effective budget constraint of course depends on the optimal income-leisure choice. For the individuals, labor is applied until the marginal utility of leisure divided by the marginal utility of income to the household is equal to the marginal value product of labor. Note that

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the marginal utility of income to the household differs from the marginal utility of income to the individual, in general. Own consumption and purchased consumption are also determined by similar marginal utility considerations. Because of the block recursive nature of the model, one can decompose the model into two parts: production and consumption. In the production part, output and inputs are determined subject to given prices, yielding a net profit on the farm enterprise, which may be considered to be asset or nonlabor income to the farm household. In particular, if the production function is strictly quasi-concave, one can represent this net profit as a profit function which depends only on the ratios of input prices to output price (which will henceforth be called ''real" input prices) and the fixed inputs. In the consumption part, the household maximizes welfare subject to the budget constraint which is given by

where P = the profit function; c* = vector of "real" input prices; and ZF = vector of fixed inputs. Note that both output and input do not enter into the welfare optimization problem. This brings us to a very important conclusion, namely, that given perfect substitutability and competitive markets in labor, variations in the number of family members do not affect the optimal output of the farm household. Withdrawal of working family members lead to their replacement by hired-in agricultural labor. Additions to working family members likewise lead to a displacement of hired labor. However, this does not imply that agricultural labor (as distinct from family members) may be withdrawn from the agricultural sector as a whole, as the "surplus laborites" claim. Any withdrawal from the pool of hired labor will, in general, raise the hired labor wage rate and, ceteris paribus, lower the output of each farm household. One must also study the hired labor supply function to the agricultural sector as well as the marketed output demand function from the nonagricultural sector in conjunction with the microeconomic model of the farm household to evaluate in a meaningful way the full impact of labor withdrawal. Two variants of this welfare maximization model are of some interest. The first case arises when a given household does not make a net purchase of one of the factor inputs. The second case arises when a labor market does not exist for the household, for either institutional

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or locational reasons, that is, there is neither purchase nor sale of labor. These two cases will be considered separately. Zero net purchase of a factor input implies that there exists a corner solution in the suboptimization of the production block of the problem. In other words, the existing own supply of the factor inputs on the farm (which may be zero) is greater than or equal to the derived demand at the prevailing market real rental price. However, in order for a corner solution to be feasible, it is necessary that the production function be of a form such that no net purchased input is absolutely essential, that is,

The Cobb-Douglas production function obviously does not satisfy this criterion. However, a Cobb-Douglas production function with nonhomogeneities does, for instance

where

where the

s are the nonhomogeneities. A more general form of the production function is

s may be zero and may be interpreted as the quantities supplied by the farm household itself.

The profit function to be maximized is then

with the additional requirement that

The Lagrangian function is

The Kuhn-Tucker necessary conditions for a saddle point are

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By the property of complementary slackness of concave programming problems, one has either

In the first case, there is an interior solution. In the second case, there is a corner solution which is characterized by zero net purchase of the i-th input. The optimal behavior of the firm in the second case is governed by the set of equations (3.31) less the marginal condition with µ i > 0, setting Xi = 0 in all the other marginal conditions; that is:

It is immediately clear that the optimal allocation of resources does not depend on cj, the price of the "superfluous" input. It does depend on the level of , which may be zero. In this case, the block recursive structure is still preserved. While the model with the corner solutions can be readily accommodated within the framework of the basic theoretical model, the absence of a labor market completely destroys the block recursive structure of the model. The set of necessary conditions becomes

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Thus, consumption plans can no longer be separated from the production plans. Variations in the composition of the household will have an influence both on output and on labor applied. The behavioral patterns indicated by this model are substantially different from that of the model with a perfectly competitive labor market. Hence it is natural to apply the model described by the system of equations (3.32) to households which do not report any purchase or sale of labor. A test will be made to see whether the parameters of the utility function and the production function of the two categories of households differ significantly. 3.7 An Example of Functional Form Econometric Implementation: Bernoulli Utility Function and Cobb-Douglas Production Function Two general approaches are available for the estimation of the parameters of the model. The first is to specify the functional forms of the utility and production functions and then estimate the parameters from the model. The second is to specify no functional forms but to impose necessary restrictions on the parameters of the linearized reduced form. Note that the latter has the interpretation of being the first-order terms of a Taylor's series expansion around a set of specified values of the exogenous variables.

It is clear that

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Then the per adult-male equivalent demand functions are given by

The production block is represented by the following system of net supply and demand equations:

where X0 = L, c0 = w. Notes 1. Thus game situations are also ruled out. 2. This involves an ethical assumption which implies that social welfare is not increased if the welfare of one individual or a group of individuals is decreased. It corresponds to the postulate D of Fleming's (1952). See also Harsanyi (1955). 3. The positive first partial derivatives effectively rule out phenomena such as "catchling up with the Joneses" which essentially involves a negative first partial derivative with respect to other individuals' utility (and consumption). 4. Again, this involves an ethical assumption that the individual does not value other individuals' utility higher than his own. 5. Note that symmetric goodwill is insufficient unless separable with respect to Ui and Uj's, j ¹ i.

i, j = 1, . . . , m, which in turn implies that Wi is

6. See the mathematical appendix for a proof. 7. The last two cases require the theorem on equality. 8. This will be shown in the mathematical appendix. 9. Note that this does not involve the assumption of a constant marginal utility of money. See Inada (1964). 10. The use of total welfare rather than individual utility as an objective is not uncommon. See, for instance, Cass (1965). 11. See Lau (1969), especially chapter IV. 12. See Lau (1969), chapter IV.

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13. For a definition of decentralizability, see Lau (1969). 14. In fact, one may even weaken this condition further. All that is needed is that the intertemporal welfare function be decentralizable in two categories: the present and all future periods taken together. 15. A.P. Barten (1964) has made an interesting proposal based on the maximization of a family utility function which takes into account the differences in the consumption capacities of different individuals for different goods. However, this proposal does not appear to have been empirically implemented. 16. For al definition of weak separability and its implication, see Goldman and Uzawa (1964). 17. Although there are instances in which a factor used in the present period also contributes to the output of future periods, for example the practice of fallowing and crop rotation, they can, in principle, be treated by an appropriate imputation procedure. This will be discussed in the section on aggregation and measurement. 18. Treatment of taxation will be taken up in a later chapter.

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4 Transcendental Logarithmic Production Frontiers Laurits R. Christensen, Dale W. Jorgenson, and Lawrence J. Lau 4.1 Introduction Additive and homogeneous production possibility frontiers have played an important role in formulating statistical tests of the theory of production. In section 4.2 we characterize the class of production possibility frontiers that are homogeneous and additive. This class coincides with the class of frontiers with constant elasticities of substitution. Constancy of the elasticity of substitution has proved to be a fruitful point of departure for the analysis of production with one output and two factors of production, as in the pioneering study of capital-labor substitution by Arrow, Chenery, Minhas, and Solow (1961). 1 For more than one product or more than two factors of production, constancy of elasticities of substitution and transformation is highly restrictive, as Uzawa and McFadden have demonstrated.2 Our first objective is to develop tests of the theory of production that do not employ additivity and homogeneity as part of the maintained hypothesis. For this purpose we introduce new representations of the production possibility frontier in section 4.3. Our approach is to represent the production frontier by functions that are quadratic in the logarithms of the quantities of inputs and outputs. These functions provide a local second-order approximation to any production frontier. The resulting frontiers permit a greater variety of substitution and transformation patterns than frontiers based on constant elasticities of substitution and transformation.3 A complete model of production includes the production possibility frontier and necessary conditions for producer equilibrium. Under constant returns to scale this model implies the existence of a price possibility frontier, defining the set of prices consistent with zero profits.4 Necessary conditions for producer equilibrium, giving relative prices as a function of relative product and factor intensities,

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imply the existence of conditions determining relative product and factor intensities as a function of relative prices. The price possibility frontier and the conditions determining product and factor intensities are dual to the production possibility frontier and the necessary conditions for producer equilibrium. 5 Our second objective is to exploit the duality between prices and quantities in the theory of production. Our approach is to represent the price possibility frontier by functions that are quadratic in the logarithms of prices, paralleling our treatment of the production possibility frontier. These functions provide a local second-order approximation to any price frontier. The duality between direct and indirect utility functions employed in Houthakker's pathbreaking studies of consumer demand is analogous to the duality between production and price frontiers employed in our study of production.6 We refer to our representation of the production possibility frontier as the transcendental logarithmic production possibility frontier or, more simply, the translog production frontier. The production possibility frontier is a transcendental function of the logarithms of its arguments, the quantities of net outputs. Similarly, we refer to our representation of the price possibility frontier as the transcendental logarithmic price possibility frontier or, more simply, the translog price frontier.7 For many of the production and price frontiers employed in econometric studies of production the translog frontiers provide accurate global approximations.8 The accuracy of the approximation must be determined separately for each application. We present statistical tests of the theory of production in section 4.4. These tests can be divided into two groups. First, we test restrictions on the parameters of the translog production frontier implied by the theory of production. We test these restrictions without imposing the assumptions of additivity and homogeneity. We test precisely analogous restrictions on the parameters of the translog price frontier. Second, we test restrictions on the translog production frontier corresponding to restrictions on the form of the frontier. In particular, we test restrictions on the form of technical change and restrictions implied by the assumption of additivity. Again, we test precisely analogous restrictions on the translog price frontier. We present empirical tests of the theory of production, based on time series data for the United States private domestic economy for 1929 1969 in section 4.5. The data include prices and quantities of investment and consumption goods output and labor and capital

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services input and an index of total factor productivity. For these data we present direct tests of the theory of production, based on the translog production frontier, and indirect tests of the theory, based on the translog price frontier. For both direct and indirect tests our empirical results are consistent with a very extensive set of restrictions implied by the theory of production. Proceeding conditionally on the validity of the theory of production, our empirical results are inconsistent with restrictions on the form of the production frontier implied by the assumption of additivity. 4.2 Additivity and Homogeneity 4.2.1 Introduction Our purpose in this section is to derive the implications of additivity and homogeneity for the representation of the production possibility frontier. The class of additive and homogeneous production possibility frontiers coincides with the class of frontiers with constant elasticities of substitution and transformation. We first represent the production possibility frontier in the form

where yi (i = 1,2, . . . , n) represents the net output of the i-th commodity. The necessary conditions for producer equilibrium take the form of equalities between price ratios and marginal rates of transformation between the corresponding pair of commodities

where qi (i = 1,2, . . . , n) represents the price of the ith commodity. 4.2.2 Commodity-Wise Additivity The production possibility frontier is characterized by constant returns to scale if and only if

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for any l > 0. We refer to production possibility frontiers satisfying this condition as homogeneous. The production possibility frontier is commodity-wise additive if and only if the frontier can be represented in the form

where the functions [Fi] are strictly monotone and depend on only a single variable. 9 The production possibility frontier is homogeneous and commodity-wise additive if and only if the frontier can be represented in the form

But the frontier satisfies this condition if and only if: (1) The functions [Fi] are homogeneous of the same degree, or: (2) The functions [Fi] are logarithmic.10 Any homogeneous function of one variable can be represented in the form

where r is the degree of homogeneity and ai > 0. If the functions [Fi] are homogeneous of the same degree, the production possibility frontier can be represented in the form

For this frontier to have the proper curvature there can be only one output (r < 1) or only one input (r > 1), unless all net outputs are perfect substitutes (r = 1). For only one output the frontier is characterized by constant elasticities of substitution between inputs; for one

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input the frontier has constant elasticities of transformation between outputs. 11 Alternatively, if the functions [Fi] are logarithmic,

where ai > 0 (i = 2, . . . , n), as before. The production possibility frontier can be represented in the form

as before. For this frontier to have proper curvature there is only one output;12 the elasticity of substitution between inputs is constant and equal to unity. We conclude that a commodity-wise additive and homogeneous production possibility frontier is unsuitable for representation of production possibilities with several outputs and several inputs. 4.2.3 Group-Wise Additivity As an extension of our characterization of a production possibility frontier with constant returns to scale, we introduce the concept of additivity in commodity groups. The production possibility frontier is group-wise additive in m mutually exclusive and exhaustive commodity groups if and only if the frontier can be represented in the form

where Sni = n, the number of commodities. For commodity-wise additivity each group consists of a single commodity and the number of groups is the same as the number of commodities. The production possibility frontier is homogeneous and group-wise additive if and only if

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(1) The functions [Fi] are homogeneous of the same degree, or (2) The functions [Fi] are logarithmic functions of functions homogeneous of degree one. If the functions [Fi] are homogeneous of the same degree, the production possibility frontier can be represented in the form

Alternatively, if the functions [Fi] are logarithmic, the production possibility frontier can be represented in the form

In both representations the functions

are homogeneous of degree one, ai > 0 (i = 1,2, . . . , m), and

A homogeneous and group-wise additive production possibility frontier has only one group of outputs (logarithmic or homogeneous with r < 1) or only one group of inputs (r > 1), unless the commodity groups are perfect substitutes (r = 1). In any case the production possibility frontier is additive in inputs and in outputs, considered as commodity groups. We conclude that any test of the implications of additivity should begin with a test of additivity between inputs and outputs. We have defined additivity for individual commodities and for commodity groups. We have obtained explicitly representations for production possibility frontiers characterized by homogeneity and commodity-wise or groupwise additivity. We can extend our characterization by defining two-level additivity as group-wise additivity together with commodity-wise additivity for each commodity group. Two-level additivity is not equivalent to commodity-wise additivity for the production possibility frontier as a whole. We can further

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extend our characterization by extending two-level additivity to any number of levels. Homogeneity and multilevel additivity imply that the production possibility frontier can be represented by means of compositions of power functions and logarithmic functions. To illustrate the representation of additive and homogeneous production possibility frontiers we consider the following examples: (1) For one output and two inputs a homogeneous and commodity-wise additive production possibility frontier can be represented in the form

where y1 is the level of output and [| y2|, | y3|] are the levels of input or, alternatively, in the form

The first representation is the constant elasticity of substitution (CES) production function introduced by Arrow, Chenery, Minhas, and Solow (1961). The second representation is the Cobb-Douglas production function; the elasticity of substitution is constant and equal to unity. 13 (2) For two outputs [y1, y2] and two inputs [| y3|, | y4|], a homogeneous production possibility frontier that is group-wise additive in the outputs and inputs, where each group is commodity-wise additive, can be represented in the form

or, alternatively, in the form

The first representation has constant elasticity of transformation (CET) between the two outputs and constant elasticity of substitution (CES) between the two inputs. In the second the elasticity of substitution is equal to unity. The CET-CES representation was introduced by Powell and Gruen (1968).14 (3) For one output y1 and four inputs [| y2|, | y3|, | y4|, | y5] a homogeneous production possibility frontier that is group-wise additive in output, the first two inputs [| y2, | y3|], and the second two inputs

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[| y4, y5|], where each group is commodity-wise additive, can be represented in the form

or, alternatively

with either form of the functions G2 and G3 given above. The first representation is the two-level constant elasticity of substitution production function introduced by Sato (1967). The second representation was introduced by McFadden (1963) and the third by Uzawa (1962). 15 4.2.4 Price Possibility Frontier Under constant returns to scale the level of profit associated with any set of prices is either zero or positively infinite. We define the set of price possibilities as the set of prices for which profit is equal to zero. We define the price possibility frontier as the frontier of the set of price possibilities. By duality in the theory of production we can characterize the production possibility frontier in terms of the price possibility frontier.16 We first represent the price possibility frontier in the form

where P is the level of profit associated with the set of prices [qi]. The price possibility frontier is homogeneous, so that

for any l > 0. The production possibility frontier and the necessary

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conditions for producer equilibrium are dual to the price possibility frontier and its derivatives

Derivatives of the price possibility frontier, holding the level of profit constant and equal to zero, are equal to the relative product and factor intensities. Under constant returns to scale, if the production possibility frontier is commodity-wise additive, then the price possibility frontier is commodity-wise additive 17 and can be represented in the form

where the functions [Pi] are strictly monotone, depend on only a single variable, and (1) the functions [Pi] are homogeneous of the same degree, or (2) the functions [Pi] are logarithmic. If the functions [Fi] in the representation of the production possibility frontier are homogeneous of degree r the functions [Pi] in the representation of the corresponding price possibility frontier are homogeneous of degree h,18 where

The functions [Fi] can be represented in the form

and the functions [Pi] can be represented in the form

Alternatively, if the functions [Fi] in the representation of the production possibility frontier are logarithmic, the functions [Pi] in the representation of the price possibility frontier are logarithmic and we can write

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in the logarithmic representation of the functions [Fi] given above. Under constant returns to scale, if the production possibility frontier is group-wise additive, the price possibility frontier is group-wise additive in the same commodity groups. We can extend our representations of the price possibility frontier to group-wise additivity and to multi-level additivity in an obvious way. Multi-level additive and homogeneous production possibility frontiers are self-dual in the sense of Houthakker (1965). 19 For our examples of additive and homogeneous production possibility frontiers, the corresponding price frontiers have the same functional form. The parameters of these price frontiers can be determined from the parameters of the corresponding production frontiers, but the two sets of parameters are not identical. The first step in testing additivity is to test group-wise additivity between inputs and outputs, considered as commodity groups. Since a group-wise additive and homogeneous production possibility frontier corresponds to a group-wise additive price possibility frontier, the hypothesis of group-wise additivity of inputs and outputs can be tested directly by means of the production frontier or indirectly by means of the price possibility frontier. Similarly, since a multi-level additive and homogeneous production possibility frontier is self-dual, the hypothesis of multilevel additivity can be tested by means of either frontier. 4.3 Transcendental Logarithmic Frontiers 4.3.1 Introduction Our objective is to develop tests of the theory of production that do not employ additivity and homogeneity as part of the maintained hypothesis. For this purpose we introduce new representations of the production possibility frontier and the price possibility frontier. We

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refer to our representation of the production possibility frontier as the transcendental logarithmic production possibility frontier. Similarly, we refer to representation of the price possibility frontier as the transcendental logarithmic price possibility frontier. Each frontier is a transcendental logarithmic function in its arguments, the logarithms of quantities and prices, respectively. More simply, we refer to our production frontier and price frontier as the translog production and price frontiers. 4.3.2 Translog Production Frontier In presenting the translog production frontier it is useful to specialize to the case of two outputs and two inputs. The basic approach is easily extended to any number of outputs and inputs. We suppose that there are two outputsconsumption C and investment Iand two inputscapital K and labor L. The corresponding prices are qC, qI, qK, qL. The production possibility frontier F can be represented in the form

where A is an index of technology. We approximate the logarithm of the production frontier plus unity by a function of the logarithms of the outputs and inputs, 20

The implications of the theory of production are invariant with respect to transformations of the production possibility frontier equal to zero when the frontier is equal to zero.21 As one example of such a transformation, we add unity and take logarithms. More generally,

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we can transform the production possibility frontier, obtaining f[ln (F + 1)] as the new frontier, where f(0) is equal to zero. We obtain the following coefficients of the translog approximation at ln C = ln I = ln K = ln L = ln A = 0

The function f is equal to zero for F equal to zero, but is otherwise arbitrary. We can choose f' and f'' for convenience in representing the translog approximation to an arbitrary production possibility frontier. A convenient normalization is

The first normalization, aK + aL = 1, is required for estimation of the parameters of equations for the value ratios. The second normalization has the convenient property that a translog approximation to a production possibility frontier group-wise additive in outputs and inputs is group-wise additive. Given this normalization, we can determine one of the four parameters [bCK, bCL, bIK, bIL] from the remaining three. Using the translog form for the production possibility frontier and the necessary conditions for producer equilibrium, we obtain the ratio of the value of investment goods output to the value of capital input:

and yK is similarly defined. Similarly, the ratio of the value of labor input to the value of capital input is

where yL is an expression similar to yI and yK.

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At this point we specialize the discussion to national accounting data for which the value of output is equal to the value of input,

Given any two of the value ratios in the expression,

the third is determined by the accounting identity. This implies that the parameters of the equation for the ratio of the value of consumption goods output to the value of capital input,

where yC is also similar to yI and yK, can be determined from those for the remaining two ratios. In fact,

4.3.3 Translog Price Frontier The translog price possibility frontier can be presented in the same way as the translog production possibility frontier. The price possibility frontier P can be represented in the form

where A is the index of technology. We approximate the price frontier by a function quadratic in the logarithms,

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As before, we can normalize the parameters of the price possibility frontier so that

Differentiating the translog price frontier, while holding the level of profit at zero, we obtain the relative net supply functions; for example, the ratio of the value of investment goods to the value of capital services is

and

is similarly defined.

The form of the functions determining the value ratios is identical to that for the translog production possibility frontier with prices in place of quantities. Given any two of the value ratios, the third is determined by the accounting identity between the value of output and the value of input. 4.4 Testing the Theory of Production 4.4.1 Stochastic Specification The first step in implementing an econometric model of production based on the translog production frontier is to add a stochastic specification to the theoretical model based on the equations for the marginal

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rates of substitution. We add a disturbance term to each of the equations for the ratios of values of consumption goods output, investment goods output and labor input to the value of capital input. From the accounting identity relating these ratios we observe that the disturbance in any one of these equations can be determined from the disturbances in the remaining two. Only two of the three equations are required for a complete econometric model of production. Estimates of the parameters of the remaining equation are implied in the relationships among the parameters given above. 4.4.2 Equality and Symmetry We estimate equations for ratios of the values of investment goods output and labor input to the value of capital input, subject to the normalization aK + aL = 1. If the equations are generated by profit maximization, the six parameters [aK, bCK, bIKbKK, bKLbKA] are the same for both equations. This results in a set of restrictions relating the parameters occurring in both equations. We refer to these as equality restrictions. The production possibility frontier is twice differentiable, so that the Hessian of this frontier is symmetric. This gives rise to a set of restrictions relating the parameters of the cross-partial derivatives. For example, the parameter bIK associated with ln K in the expression for ¶F/¶I must be equal to the same parameter associated with ln I in the expression for three parameters represented explicitly [bIK, bIL, bKL] and three additional parameters entering through the accounting identity between the value of output and the value of input [bCI, bCK, bCL]. We refer to these as symmetry restrictions. Constant returns to scale implies that the production possibility frontier satisfies:

for any l > 0. This implies the following restrictions on the parameters

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Given symmetry restrictions on the six parameters [bIK, bIL, bKL, bCI, bCK, bCL], the first six restrictions are identical to those derived from the accounting identity between the value of output and the value of input. The last restriction is implied in the second through fifth restrictions. We conclude that tests of the symmetry restrictions can also be interpreted as tests of constant returns to scale or homogeneity. Under the accounting identity between the value of output and the value of input, symmetry and homogeneity have precisely the same implications for the parameters of the functions determining the ratios of values of investment goods output and labor input to the value of capital input. The parameters of the translog production possibility frontier can be normalized; only the normalization, bCK + bCL + bIK + bIL = 0, imposes a restriction on the parameters of equations for the value ratios. We refer to this as the normalization restriction. If equations for the value ratios are generated by profit maximization, subject to the translog production possibility frontier, the parameters satisfy equality, symmetry, and normalization restrictions. 4.4.3 Factor Augmentation In the tests of the theory of production we employ the index of total factor productivity as a measure of the technology index A. The index of total factor productivity is invariant and path independent if and only if technical change can be represented by a single index. 22 If technical change is factor-augmenting and depends on a single index, we can write the production possibility frontier in the form

Furthermore, the index of technology A can be taken to be the index of total factor productivity,23 implying the restrictions:

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The first and last of these restrictions can be employed in estimating the parameters [aA, bAA]. Under the normalization suggested above:

Give symmetry, the second restriction is implied by the third, fourth and fifth. We refer to the latter three as factor augmentation restrictions. 4.4.4 Group-wise Additivity Group-wise additivity of the production possibility frontier in the two groups consisting of outputs and inputs implies that the frontier can be represented in the form

Constant returns to scale implies that the functions [Y(C, I, A), X(K, L, A)] can be taken to be homogeneous of degree one in outputs and inputs, respectively. Under our normalization necessary and sufficient conditions for the translog production possibility frontier to be group-wise additive in the inputs and outputs are the following:

We refer to these as group-wise additivity restrictions. Only three of these restrictions are independent. 24 If the production possibility frontier is group-wise additive in inputs and outputs and technical change is factoraugmenting we can write,

Constant returns to scale implies that we can write,

where Y(C, I) is an index of aggregate output, X(K, L) is an index of aggregate input, and the functions [Y(C, I), X(K, L)] are homogeneous of degree one.

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4.4.5 Approximation We have demonstrated that symmetry restrictions can also be interpreted as conditions for homogeneity under the accounting identity between the value of output and the value of input. Similarly, we can provide an alternative interpretation of the factor augmentation and group-wise additivity restrictions by considering the translog approximation to the CET-CES production possibility frontier: 25

This frontier is group-wise additive in two mutually exclusive and exhaustive commodity groups and each group is commodity-wise additive in the two commodities that comprise the group. Of course, the frontier as a whole is not commodity-wise additive. The translog approximation of the CET-CES frontier can be obtained from:

The approximating translog frontier (around the point C = I = K = L = 1) is:

in this approximation:

The parameters of the translog approximation are:

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Following Arrow, Chenery, Minhas, and Solow (1961), 26 the parameter ao is the efficiency parameter, the parameters [aC, aI, aK, aL] are distribution parameters, and the parameters [bCC, bKK] are substitution parameters. This interpretation can be carried over to an unrestricted translog frontier, adding substitution parameters [bII, bLL, bCI, bCK, bCL, bIK, bIL, bKL]. The CET-CES production possibility frontier restricts the substitution parameters to two. Under homogeneity, symmetry, and normalization restrictions the translog production possibility frontier involves five substitution parameters. Imposing group-wise additivity in inputs and outputs, the translog frontier involves only two substitution parameters. A test of the factor augmentation and group-wise additivity restrictions can also be interpreted as a test of the CET-CES frontier or of the CES production function. In the CES production function the index of aggregate output Y(C, I) is measured directly; this index is invariant and path independent if and only if the production possibility frontier is group-wise additive in inputs and outputs. This additivity condition is weaker than the additivity conditions underlying the CET-CES frontier, since the CET-CES frontier also implies commodity-wise additivity of aggregate output and aggregate input in the individual commodities.27 Tests of the CET-CES frontier and the CES production function involve a possible error of approximation if the true production possibility frontier is CET-CES or the true production function is CES. Under constant returns to scale and group-wise additivity between inputs and outputs the only translog production possibility frontier that is commodity-wise additive in capital and labor input involves an index of aggregate input that is Cobb-Douglas in form28

so that

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A similar restriction for the index of aggregate output implies an output index that violates the convexity conditions for the production possibility frontier. 4.4.6 Duality In implementing an econometric model of production based on the translog price possibility frontier the first step, as before, is to add a stochastic specification. Only two of the three equations for ratios of the values of consumption goods output, investment goods output, and labor input to the value of capital input are required for a complete econometric model of production. As before, a convenient normalization is that aK + aL = 1. The restrictions on the translog price possibility frontier are strictly analogous to restrictions on the translog production possibility frontier: Equality, normalization, and symmetry restrictions are the same as before. Tests of the symmetry restrictions can also be interpreted as tests of homogeneity of the price possibility frontier. Factor-augmenting technical change implies that the price possibility frontier can be written

where A can be taken to be the index of total factor productivity. 29 Factor augmentation implies the restrictions:

Group-wise additivity of the production possibility frontier in the two inputs, capital and labor, in the two outputs, consumption and investment, and homogeneity imply that the price possibility frontier can be written in the form

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where the functions [qY (qC,qI,A), qX(qK,qL,A)] can be taken to be homogeneous of degree one in the prices of outputs and the prices of inputs, respectively. Under our normalization necessary and sufficient conditions for the translog price possibility frontier to be groupwise additive in the prices of outputs and inputs are

Only three of these restrictions are independent. As before, the factor augmentation and group-wise additivity restrictions can also be interpreted as restrictions arising from the translog approximation of the CET-CES price possibility frontier. This frontier can be represented in the form

since the CET-CES production possibility frontier is self-dual, that is, the price possibility frontier has the same functional form. Under group-wise additivity between output prices and input prices, commodity-wise additivity of the translog price possibility frontier in the prices of capital and labor inputs implies

as before.

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The translog production possibility frontier and the translog price possibility frontier do not correspond to the same technology. However, these frontiers can be regarded as alternative approximations to the same underlying technology. If, for a given technology, both the production frontier and the price frontier can be represented in closed form, for example with CET-CES frontiers, the error of approximation can be assessed by measuring the discrepancy between the frontiers and their translog approximations. 4.5 Empirical Results 4.5.1 Summary of Tests Our objective has been to develop statistical tests of the theory of production that do not employ the assumptions of homogeneity and additivity. At this point it is useful to summarize the restrictions on the translog production possibility frontier corresponding to the theory of production and to functional forms that are homogeneous and additive. Restrictions on the translog price possibility frontier are analogous. We present these restrictions in a form corresponding to the two equations for ratios of the values of investment output and labor input to the value of capital input: (1) Equality restrictions: The parameters [aK, bCK,bIK,bKK,bKL,bKA] occur in both equations and must take the same value. (2) Symmetry restrictions: The parameters [bIK,bIL,bKL,bCK,bCL,bCI] are the same wherever they occur and must take the same value. (3) Normalization restrictions: bCK + bCL + bIK + bIL = 0. (4) Factor augmentation restrictions: bIA = bIK + bIL, bKA = bKK + bKL, bLA = bKL + bLL. (5) Group-wise additivity restrictions: bCK = bCL = bIK = 0. (6) Commodity-wise additivity in capital and labor input: bKK = 0. The symmetry restrictions, given the identity between the value of output and the value of input, are equivalent to restrictions implied by homogeneity. The factor augmentation and group-wise additivity restrictions are equivalent to restrictions implied by the translog approximation to the CET-CES production possibility frontier. The commodity-wise additivity restriction in capital and labor input implies that aggregate input can be represented in Cobb-Douglas

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form. In addition to the restrictions on the parameters to be estimated from the two behavioral equations we employ the following restrictions to estimate the remaining parameters: (1) Normalization: aK + aL = 1. (2) Constant returns: aC + aI + aK + aL = 0, bCA + bIA + bKA + bAL = 0. (3) Factor augmentation: aA = aK + aL, bAA = bKK + 2bKL + bLL. 4.5.2 Estimation Our empirical results are based on time series data for the United States private domestic economy for 1929 1969. 30 We have fitted the parameters of the translog production possibility frontier, employing the stochastic specification outlined above. Under this specification there are two behavioral equations corresponding to ratios of the values of investment goods and labor services to the value of capital services. Similarly, we have fitted the parameters of the translog price possibility frontier, employing an analogous stochastic specification. The production and price frontiers correspond to two distinct representations of technology. There rate forty-one observations for each behavioral equation, so that the number of degrees of freedom available for either direct or indirect statistical tests is eighty-two. Our maintained hypothesis corresponds to the unrestricted form of the two behavioral equations derived from the production possibility frontier. The unrestricted behavioral equations, estimated under the normalization aK + aL = 1, involve twenty-two unknown parameters or eleven unknown parameters in each equation. Unrestricted estimates of these parameters are presented in the first column of table 4.1.31 The first hypothesis to be tested is that the theory of production is valid; the theory of production implies equality, normalization, and symmetry restrictions on the parameters of the translog production possibility frontier. There are twelve equality, normalization, and symmetry restrictions, so that the theory of production implies that the twenty-two unknown parameters of the translog production possibility frontier can be expressed as functions of only ten. Restricted estimates of the parameters of the production possibility frontier, obtained by imposing the equality, normalization, and symmetry restrictions, are presented in the second column of table 4.1. Analogous estimates of the parameters of the price possibility frontier are presented in the first two column of table 4.2.

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Table 4.1 Estimates of the parameters of the translog production possibility frontier ParametersUnrestricted Equality, normalization Factor augmentation and group-wise additivityc and symmetryb estimatesa Investment Equation aI

0.304

bCI bII bIK bIL bIA bCK bIK bKK bKL bKA

0.305 1.110 0.406 0.491 1.290 1.890 0.476 0.411 0.228 1.050 1.410

0.3010 0.413 0.138 0.115 0.436 0.993 0.497 0.115 0.254 0.129 0.267

Commodity-wise additivityd 0.301

0.1960 0.1960 e e e e e 0.0468 0.0468 e

0.194 0.194 e e e e e e e e

Labor Equation aL 0.618 0.613 0.6170 0.610 0.801 0.054 e e bCL 0.683 0.436 e e bIL 0.614 0.129 0.0468 e bKL 0.519 0.511 0.0468 e bLL 1.740 1.320 e e bLA 0.443 0.497 e e bCK 0.598 0.115 e e bIK 0.410 0.254 0.0468 e bKK 0.655 0.129 0.0468 e bKL 1.320 0.267 e e bKA a Unrestricted estimates under the normalization aK + aL = I. b Restricted estimates, equality, normalization, and symmetry restrictions imposed. c Restricted estimates, factor augmentation and group-wise additivity restrictions together with equality, normalization, and symmetry restrictions imposed. d Restricted estimates, commodity-wise additivity of aggregate input, together with equality, factor augmentation, normalization, group-wise additivity, and symmetry restrictions imposed. e Parameter value constrained to be equal to zero.

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Table 4.2 Estimates of the parameters of the translog price possibility frontier ParametersUnrestricted Equality, normalization Factor augmentation and group-wise additivityc and symmetryb estimatesa Investment Equation aI

0.315

bCI bII bIK bIL bIA bCK bIK bKK bKL bKA

0.305 1.680 0.781 0.235 1.720 1.420 0.435 0.235 0.659 0.317 1.060

0.301 0.099 0.283 0.360 0.544 0.366 0.698 0.360 0.809 0.250 0.983

Commodity-wise additivityd 0.303

0.506 0.506 e e e e e 0.049 0.049 e

0.587 0.587 e e e e e e e e

Labor Equation aL 0.628 0.622 0.613 0.610 6.430 0.515 e e bCL 4.210 0.544 e e bIL 0.317 0.250 0.049 e bKL 5.860 1.310 0.049 e bLL 12.100 1.010 e e bLA 0.826 0.698 e e bCK 0.650 0.360 e e bIK 0.774 0.809 0.049 e bKK 0.553 0.250 0.049 e bKL 0.670 0.983 e e bKA a Unrestricted estimates under the normalization aK + aL = I. b Restricted estimates, equality, normalization, and symmetry restrictions imposed. c Restricted estimates, factor augmentation and group-wise additivity restrictions together with equality, normalization, and symmetry restrictions imposed. d Restricted estimates, commodity-wise additivity of aggregate input, together with equality, factor augmentation, normalization, group-wise additivity, and symmetry restrictions imposed. e Parameter value constrained to be equal to zero.

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Given the validity of the theory of production, the second hypothesis to be tested is that technical change is factor augmenting and the production possibility frontier is group-wise additive in outputs and inputs. There are six factor augmentation and group-wise additivity restrictions, so that the ten parameters of the production possibility frontier implied by the theory of production can be expressed as a function of four parametersa distribution and substitution parameter for aggregate output and corresponding parameters for aggregate input. The efficiency parameter for the frontier determines the units of measurement for the index of factor productivity. Restricted estimates of the parameters of the production possibility frontier, obtained by imposing the factor augmentation and group-wise additivity restrictions together with the equality, normalization, and symmetry restrictions, are presented in the third column of table 4.1. Analogous estimates of the parameters of the price possibility frontier are presented in the third column of table 4.2. Finally, the third hypothesis to be tested is that production possibility frontier is commodity-wise additive in labor and capital input. This hypothesis implies one additional restriction on the four parameters of the frontier. Restricted estimates, imposing this additional restriction, are presented in the fourth column of table 4.1; analogous estimates for the price possibility frontier are presented in the fourth column of table 4.2. 4.5.3 Test Statistics To test the validity of the theory of production and of restrictions on the form of the production possibility frontier we employ a ''nested" series of tests. At each stage in the series we calculate the change in the weighted sum of of squared residuals resulting from restrictions imposed at that stage. We divide this change by the sum of squared residuals at the previous stage. Finally, we divide both numerator and denominator of this ratio by the appropriate number of degrees of freedom. The resulting test statistics are distributed, asymptotically, as F(v1,v2), where v1 is the numerator degrees of freedom and v2 is the denominator degrees of freedom. Of course, each F ratio is distributed, asymptotically, as chi-squared divided by the numerator degrees of freedom. These test statistics are asymptotically equivalent to likelihood ratio test statistics. Critical values of F and chi-squared employed in our tests are given in table 4.3. 32

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Table 4.3 Critical values of F(v1, v2) and c2/v1 Degrees of freedom Level of significance 0.100 0.050 0.025 v1 = 12 F(12, 60) 1.66 1.92 2.17 v2 = 60 c2/12 1.55 1.75 1.94 v1 = 3 F(3, 72) 2.12 2.74 3.32 v2 = 72 c2/3 2.08 2.60 3.12 v1 = 6 F(6, 72) 1.86 2.23 2.61 v2 = 72 c2/6 1.77 2.10 2.41 v1 = 3 F(3, 75) 2.17 2.74 3.31 v2 = 75 c2/3 2.08 2.60 3.12 v1 = 1 F(1, 78) 2.78 3.98 5.22 v2 = 78 c2/1 2.71 3.84 5.02

0.010 2.50 2.18 4.09 3.78 3.08 2.80 4.09 3.78 7.00 6.63

0.005 2.74 2.36 4.69 4.28 3.45 3.09 4.67 4.28 8.39 7.88

To control the overall level of significance for each of our series of tests, direct and indirect, we set the overall level of significance for each series at 0.05. We then allocate the overall level of significance among the various stages in each series of tests. We first assign levels of significance of 0.02 to tests of the theory of production and 0.03 to tests of restrictions on functional form. Given a level of significance of 0.03 for the validity of restrictions on the functional form, we assign 0.02 to group-wise additivity and factor augmentation and 0.01 to commoditywise additivity. We test group-wise additivity and factor augmentation, proceeding conditionally on the validity of the theory of production. Finally, we test commodity-wise additivity in the inputs, proceeding conditionally on the validity of the theory of production and the validity of group-wise additivity and factor augmentation restrictions on functional form. With the aid of the critical levels presented in table 4.3, the reader can evaluate the results of these tests for a range of alternative levels of significance and for alternative allocations of the overall level of significance among stages of the series of tests for either direct or indirect representations of the production possibility frontier. Test statistics for both direct and indirect tests of the theory of production and of restrictions on functional form are given in table 4.4. At a level of significance of 0.02 we accept the hypothesis that restrictions implied by the theory of production are valid for either the direct or the indirect series of tests. The F ratio for the direct test is 0.00; the analogous ratio for the indirect test is 1.83. Proceeding conditionally

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Table 4.4 F ratios for direct and indirect tests of the theory of production and of restrictions on the form of the production and price possibility frontiers Direct Indirect Degrees of freedom Theory of production Equality, normalization, and symmetry (12, 60) 0.00a 1.83 restrictions Functional form 6.39 38.50 Factor augmentation and group-wise (6, 72) additivity 16.80 12.20 Commodity-wise additivity (1, 78) a The estimated change in the sum of squared residuals is negative. on the validity of the theory of production, we can test the validity of restrictions on functional form. These restrictions include factor augmentation and group-wise additivity restrictions. Either set of restrictions can be valid without the other. We test the two sets of restrictions individually; of course, the tests are not "nested" so that the sum of levels of significance for each of the two hypotheses considered separately is an upper bound for the level of significance of tests of the two hypotheses considered simultaneously. Setting the level of significance for each test at 0.01 we obtain an upper bound on the overall level of significance of 0.02. At these levels of significance we reject the hypothesis that restrictions implied by group-wise additivity are valid for either direct or indirect tests. We accept the hypothesis that restrictions implied by factor augmentation are valid for the direct test; however, we reject this hypothesis for the indirect test. We have tested the validity of restrictions on the form of the production possibility frontier, proceeding conditionally on the validity of the theory of production. We have tested restrictions implied by factor augmentation and by group-wise additivity individually. An alternative to our test procedure is to test the validity of factor augmentation and group-wise additivity restrictions jointly. At a level of significance of 0.02 we would reject the joint hypothesis for either direct or indirect tests. The F ratio for the direct test is 6.39; for the indirect test the ratio is 38.5. An additional alternative to our test procedure is to test factor augmentation and then to test group-wise additivity, conditional on factor augmentation. Another alternative is to

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Table 4.5 F ratios for direct and indirect tests of factor augmentation and groupwise additivity DirectIndirect Degrees of freedom Unconditional Factor augmentation (3, 72) 3.23 58.10 Group-wise additivity (3, 72) 8.20 9.12 Conditional Factor augmentation, given group-wise (3, 75) 6.39 55.40 additivity Group-wise additivity, given factor (3, 75) 14.00 20.30 augmentation reverse this procedure. We present test statistics required for each of these alternative procedures in table 4.5. For completeness we present test statistics for the hypothesis of commodity-wise additivity in the two inputs, conditional on the validity of restrictions implied by the theory of production and of factor augmentation and group-wise additivity restrictions. The F ratio for the direct test is 16.8 and the corresponding ratio for the indirect tests is 12.12. Our results for both direct and indirect tests are consistent with the theory of production. Our results for the indirect tests are inconsistent with restrictions on functional form implied by factor augmentation and group-wise additivity; our results for the direct tests are consistent with restrictions implied by factor augmentation but inconsistent with restrictions implied by group-wise additivity. 4.6 Summary and Conclusion Our objective has been to test the theory of production without imposing the assumptions of additivity and homogeneity as part of the maintained hypothesis. We first examine the implications of these assumptions for the form of the production possibility frontier. We conclude that the assumption of commodity-wise additivity that underlies the constant elasticity of substitution production function is unsuitable as a basis for representing a production possibility frontier with several outputs and several inputs. Group-wise additivity implies that inputs and outputs, considered as commodity groups, must be additive. Multi-level additivity implies that the production

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possibility frontier is self-dual; the price possibility frontier has the same functional form with parameters that depend on the parameters of the production possibility frontier. By duality in the theory of production the properties of the production possibility frontier and necessary conditions for producer equilibrium correspond to properties of the price possibility frontier and conditions for relative product and factor intensities. Duality is a natural tool for the construction of tests of the theory of production and tests of hypotheses about the form of the production possibility frontier, including hypotheses about additivity. The first step in testing additivity is to test group-wise additivity between inputs and outputs. This hypothesis can be tested directly by means of the transcendental logarithmic production possibility frontier or indirectly by means of the transcendental logarithmic price possibility frontier. Similarly, the hypothesis of multi-level additivity can be tested by means of either frontier. Our empirical results are summarized in diagrammatic form in figures 4.1 and 4.2. For either direct or indirect series of tests we proceed from an unrestricted form of the behavioral equations to a form implied by equality, normalization, and symmetry restrictions. At this point we can proceed directly to a form implied by separability and factor augmentation restrictions, together with equality, normalization, and symmetry, or we can proceed first to separability and then to factor augmentation or vice versa. Finally, given equality, normalization, symmetry, separability and factor augmentation, we can proceed to a form implied by separability of aggregate input. F ratios along each of these alternative paths of statistical inference are presented in figures 4.1 and 4.2 for the direct and indirect tests, respectively. The results of our direct and indirect tests of the theory of production, based on the transcendental logarithmic production and price frontiers, are consistent with the validity of the extensive set of equality, symmetry, and normalization restrictions implied by the theory. Our results are inconsistent with the hypothesis of group-wise additivity of the production possibility frontier in inputs and outputs for either direct or indirect tests. Production possibility frontiers characterized by additivity and homogeneity have proved useful in representing production with one output and two inputs, as in the study of capital-labor substitution by Arrow, Chenery, Minhas, and Solow (1961). The extension of this approach to production with two outputs and two inputs conflicts sharply with our empirical evidence.

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Figure 4.1 Direct series of tests of the theory of production and the form of the production possibility frontier.

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Figure 4.2 Indirect series of tests of the theory of production and the form of the price possibility frontier.

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Notes 1. See Arrow, Chenery, Minhas, and Solow (1961). A review of the literature on capital-labor substitution is given by Jorgenson (1972). 2. See Uzawa (1962) and McFadden (1963). 3. An alternative representation that also permits a variety of substitution and transformation patterns is the "generalized Leontief" production possibility frontier proposed by Diewert (1971, 1973). 4. The price possibility frontier was introduced by Samuelson (1954). For a single output and many inputs, Samuelson referred to the price possibility frontier as the factor-price frontier. 5. Duality between the production and price possibility frontiers is discussed by Samuelson (1954, pp. 15 20), Bruno (1969), and Burmeister and Kuga (1970). 6. See Houthakker (1960). 7. The translog production and price possibility frontiers were introduced by Christensen, Jorgenson, and Lau (1971); independently, Griliches and Ringstad (1971) and Sargan (1971) proposed a special case of the translog production frontier, the translog production function. The translog production function of Griliches and Ringstad and Sargan has only a single output. 8. Kmenta (1967) employed a special case of the translog production function to approximate the constant elasticity of substitution (CES) production function of Arrow, Chenery, Minhas, and Solow (1961). His approximation is in the space of parameters rather than the space of variables. An approximation to the CES production function that is quadratic in the logarithms of the variables is identical to Kmenta's approximation. See section 4.4.5. Kmenta's approximation to the CES production function, considered as a production function in its own right, is a homogeneous translog production function. Similarly, the log-quadratic production function proposed by Chu, Aigner, and Frankel (1970) can be regarded as a commodity-wise additive but not homogeneous translog production function. 9. The concept of commodity-wise additivity is the same as the concept of additive or strong separability employed by Goldman and Uzawa (1964). 10. This proposition was first derived in the theory of consumer behavior by Bergson (1936); Samuelson (1965b) pointed out the second part of the proposition. Similar results had been obtained earlier in the literature on mean value functions; see Hardy, Littlewood, and Polya (1959), pp. 65 69, and the references given there. 11. In this representation and those that follow, we assume that dimensions of the net outputs are chosen so that the coefficients {(sgn yi)ai} sum to zero. 12. See Mundlak (1964). 13. See Arrow, Chenery, Minhas, and Solow (1961) and Douglas (1948). 14. See Powell and Gruen (1968). 15. See Sato (1967), McFadden (1963), and Uzawa (1962). The relationship between homogeneity and additivity and constancy of elasticity of substitution is discussed in greater detail by Berndt and Christensen (1973a). 16. See the references given above in note 5. Duality in production was first discussed by Hotelling (1932). Duality is also discussed by Diewert (1973), Gorman (1968), Jorgenson and Lau (1974b), Lau (1972), McFadden (1978), and Shephard (1970).

17. An analogous result was obtained by Lau (1972). 18. For the duals of power functions and logarithmic functions, see Rockafellar (1970b, pp. 105 107). 19. See Houthakker (1965); self-duality is also discussed by Samuelson (1965a).

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20. For convenience, we have adopted the convention that outputs and inputs are measured as nonnegative quantities. 21. See Hicks (1946). 22. This is an implication of Hulten's (1973) conditions for invariance and path independence of Divisia index numbers; see Hulten (1973). 23. See Solow (1967). 24. Alternative tests for group-wise additivity have been proposed for versions of the "generalized Leontief" production possibility frontier by Denny (1974) and by Hall (1973). 25. See Powell and Gruen (1968). 26. Arrow, Chenery, Minhas, and Solow (1961), p. 230. 27. See Hulten (1973). 28. See Douglas (1948). Berndt and Christensen (1973a, 1974) have provided a detailed empirical analysis of the internal structure of aggregate input for United States manufacturing. 29. Under constant returns, equal rates of factor augmentation may be interpreted equivalently as equal rates of decline of factor prices. 30. The data are based on the estimates of Christensen and Jorgenson (1969, 1970), extended to 1969. 31. We employ an iterative version of the three-stage least-squares estimator proposed by Zellner and Theil (1962). This estimator is asymptotically equivalent to the maximum likelihood estimator. 32. The values for chi-squared are taken from tables for F with v2 = ¥ degrees of freedom.

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5 The Duality of Technology and Economic Behavior Dale W. Jorgenson and Lawrence J. Lau 5.1 Introduction The usual starting point of the modern theory of production is the set of production possibilities, consisting of all production plans available to the producing unit. 1 Given the properties of the set of feasible production plans, the production function and profit function can be derived and characterized. For any feasible plan the production function gives the maximum net output of any commodity as a function of the net outputs of all other commodities. The profit function gives the maximum value of profit for any feasible production plan as a function of the prices of all commodities. The theory of marginal productivity provides a characterization of the set of production possibilities from the technological point of view. Our first objective is to present a modern approach to the theory of marginal productivity, employing the production function and the marginal productivity correspondence. The marginal productivity correspondence may be identified with the gradient of the production function, wherever the gradient exists. Otherwise, the marginal productivity correspondence takes a set of possible values rather than a single value. Our second objective is to present equivalent specifications of the set of production possibilities, the production function and the marginal productivity correspondence. We take a characterization of any one of the three as a starting point of the theory of production and derive properties of the other two. We characterize the production function in terms of the properties of the marginal productivity correspondence. We also derive properties of the set of production possibilities from properties of the production function, reversing the line of reasoning that has become conventional in the theory of production.

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Our third objective is to characterize the set of feasible production plans from the point of view of economic behavior. For this purpose we present a theory of supply that parallels the theory of marginal productivity. We develop equivalent specifications of the profit function, the supply correspondence, and the set of price and profit possibilities. Any one of the three can be taken as a starting point for the theory of supply. Our final objective is to link the theory of supply with the theory of marginal productivity and to demonstrate the equivalence of the technological and behavioral viewpoints of the theory of production. For this purpose we employ the theory of conjugate convex functions, obtaining equivalent specifications of the production function and the profit function. 2 Conjugate duality implies equivalent specifications of the marginal productivity and supply correspondences and of the sets of production and price possibilities. Any one of the six specifications can be employed as a starting point for the theory of production. 5.2 Technology 5.2.1 Introduction We represent a production plan by a vector with n + 1 components, hi (i = 1, 2, . . . , n + 1), corresponding to net outputs of each of n + 1 commodities. The net output of a commodity is positive if the commodity is an output, negative if the commodity is an input, and zero if the commodity is neither an output nor an input. We assume that all commodities are perfectly divisible, so that components of a production plan can take any real value. 5.2.2 Set of Production Possibilities We begin our study of production with a characterization of the set of technically feasible production plans or the set of production possibilities. Our first assumption is that the producing unit can always shut down, adopting the trivial plan of producing nothing from nothing. This assumption implies that the set of production possibilities is nonempty. (a) Origin. The production plan with all net outputs zero is feasible. We do not require that the set of production possibilities is bounded; however, we rule out the possibility that an unbounded

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level of net output of any commodity can be produced with finite levels of net output of all other commodities. (b) Boundedness. If a production plan is feasible, the level of net output of any commodity corresponding to finite levels of net output of all other commodities is bounded above. The boundary of the set of production possibilities plays an important role in the theory of production, especially in construction of the production function and the development of marginal productivity theory. To rule out the possibility that production plans arbitrarily close to the boundary are feasible while points on the boundary itself are not, we assume that the set of production possibilities contains its boundary. (c) Closure. If a sequence of feasible production plans has a limit, the limiting plan is contained in the set of production possibilities. Convexity is critical to the development of duality in the theory of production. Convexity implies that returns to scale are constant or decreasing and that there are no indivisibilities in the production process. (d) Convexity. If two production plans are feasible every production plan that can be represented as a convex combination of them is feasible. We do not require that the set of production possibilities is characterized by free disposal of all commodities; however, we suppose that at least one of the commodities is freely disposable. (e) Monotonicity. For at least one commodity, if a production plan is feasible, every production plan with less than or the same net output of that commodity and the same net outputs of the other commodities is feasible. 5.2.3 Production Function The starting point of the traditional theory of production is the production function, giving the maximum level of output corresponding to any set of inputs. This notion is easily extended to production planning with more than one output. For this purpose we define the production function in terms of the supremum of feasible values of net output of one commodity, given the values of net outputs of the remaining n commodities. We take the (n + 1)st commodity to be a freely disposable commodity; this can always be done by our

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monotonicity assumption on the set of production possibilities. For convenience we fix the first n components of the production plan and take the supremum of the (n + 1)st

where F is the production function. 3 At this point we introduce notation for elements for Rn; we denote a vector of n net outputs (h1, h2, . . . , hn) by y, so that the production function F(y) satisfies the inequality

We adopt the convention that if there is no feasible production plan with hn+1 finite for given values of the remaining components of the production plan y, then the value taken by the function F(y) is positive infinity, denoted + ¥. With this convention the domain of the function F is all of Rn. We define the effective domain of the function F as the set of all production plans y for which F(y) is finite. The effective domain of the function F is nonempty since the set of production possibilities is nonempty, containing at least the origin with all net outputs zero. The effective domain of F is a convex set. (a) Domain. F(y) is a function with possibly infinite values, defined on Rn. The effective domain of the function F(y) is a convex set containing the origin. The value of F(y) at the origin, say F(0), is nonpositive. The function F(y) can take the value negative infinity only if the net output of the (n + 1)st commodity corresponding to finite values of the remaining n commodities is unbounded above. This would violate the boundedness property of the set of production possibilities. (b) Boundedness. The function F(y) nowhere takes the value negative infinity. Since the set of production possibilities is closed, the function F is lower semicontinuous. Further, since the function F nowhere takes the value negative infinity: (c) Closure. The function F(y) is closed. An implication of the closure of the set of production possibilities is that we can define the function F(y) by

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replacing ''sup" of our original definition by "max" of this definition. As before, if there is no feasible production plan (y, hn+1 with hn+1 finite, the value of the production function is positive infinity. The convexity of the set of production possibilities implies: (d) Convexity. The function F(y) is convex. A convex function F(y) is proper if it is finite for at least one y and not equal to negative infinity for any y. We conclude that the production function is a closed, proper, convex function. We have defined the production function in terms of the set of feasible production plans or the set of production possibilities. It is useful to define the set of production possibilities in terms of the production function. For this purpose we introduce the epigraph of the function F, consisting of all points (y, hn +1) in Rn+1 such that

The epigraph of the function F corresponds to the set of production possibilities. We have characterized the production function in terms of properties of the set of production possibilities. Making use of the correspondence between the epigraph and the set of production possibilities, we find that convexity of the production function implies convexity of the set of production possibilities and closure of the production function implies closure of the set of production possibilities. If the value of the production function at the origin of Rn is finite and nonpositive, the set of production possibilities satisfies our assumption about the origin of Rn+1. The epigraph of the production function has the monotonicity property associated with free disposal of the (n+1)st commodity. Finally, if the production function cannot take the value negative infinity, the set of production possibilities contains no production plan with net output hn+1 equal to positive infinity for finite levels of net output of all other commodities y. We conclude that our specification of the production function is precisely equivalent to our specification of the set of production possibilities. Given the properties of the set of production possibilities and the definition of the production function, we can deduce the properties of the production function. Given the properties of the production function and the definition of the set of production possibilities, we can deduce the properties of the set of production possibilities.

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5.2.4 Marginal Productivity Correspondence We continue our study of production from the technological point of view by introducing the subdifferential of the production function. 4 The subdifferential consists of the systems of marginal products that correspond to a given production plan. After developing the theory of marginal productivity from properties of the production function we reverse the reasoning and characterize the production function in terms of its marginal products. We introduce a vector y*, consisting of marginal products of the first n commodities in terms of the (n+1)st commodity. More precisely, we say that a vector of marginal products y* is a subgradient of the production function F(y) at a point y if

for all z, where . , . represents the inner product of two vectors. If the production function F(y) is finite at y, the linear function H(z), defined by

is a nonvertical supporting hyperplane to the epigraph of the production function F(y) at the point (y, F(y)). The epigraph of the production function corresponds to the set of production possibilities. The system of marginal products need not be unique, so that we may define the marginal productivity correspondence as the set of all vectors of marginal products y * or the set of all subgradients of the production function F(y) at a production plan y. This correspondence is identical to the subdifferential of the production function. If the marginal productivity correspondence is nonempty at a production plan y, we say that the production function is subdifferentiable at y. Where the marginal productivity correspondence is single-valued, the production function is differentiable and the marginal productivity correspondence is identical to the gradient of the production function. The production function is differentiable almost everywhere in the interior of its effective domain. The marginal productivity correspondence may be empty for certain boundary points of the effective domain of the production function F(y). As an illustration, consider the production function,

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At the boundary point h1 = 0, this production function is not subdifferentiable. We have defined the effective domain of the production function F as the set of all production plans y for which F(y) is finite. We define the relative interior of this set as the interior of the set considered as a subset of the smallest linear subspace containing the set. Since the production function is a closed, proper, convex function, the marginal productivity correspondence is nonempty for any production plan y in the relative interior of the effective domain of the production function F(y). If the effective domain of the production function includes any production plan other than the origin, the relative interior of the effective domain is nonempty. If the effective domain of the production function is limited to the origin, the marginal productivity correspondence includes all possible systems of marginal products. We adopt the convention that if there is no subgradient y * of the production function F(y) at the production plan y, the marginal productivity correspondence, say M(y) takes as its value the empty set, denoted f. With this convention the domain of the marginal productivity correspondence M is all of Rn. We define the effective domain of the marginal productivity correspondence M as the set of all production plans y for which M(y) is nonempty. The effective domain of the marginal productivity correspondence M(y) is nonempty and contains the relative interior of the effective domain of the production function F(y). The effective domain of the marginal productivity correspondence is contained in the effective domain of the production function. The origin is contained in the effective domain of the production function F(y). The origin may be contained in the effective domain of the marginal productivity correspondence M(y). Alternatively, the origin may be a relative boundary point of the effective domain of the production function F(y), not contained in the effective domain of the marginal productivity correspondence M(y), as in the example given above. In the latter case a necessary and sufficient condition for the production function to be finite at the origin is that there exists a production plan, say y0, in the effective domain of the marginal

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productivity correspondence M(y) and constants a, such that 0 < a < 1, and b finite, such that

where Ml (ly0) is the subdifferential of F(ly0), considered as a function of l. 5 (a) Domain. M(y) is a set-valued correspondence, possibly empty, from Rn to Rn. The effective domain of the correspondence M(y) is contained in a convex set and contains the relative interior of the convex set. The correspondence M(y) is nonempty at the origin or nonempty for some production plan y0 such that

for constants a and b such that 0 < a < 1 and b is finite. To illustrate the economic interpretation of our condition on the subdifferential Ml (ly) we consider an example where this condition is not satisfied. We let the number of commodities be two (n = 1) and the subdifferential M(y) be

The function which corresponds to M(y) is

where y is a constant. This is a closed, proper, convex function but does not satisfy our assumption that the production function is finite at the origin. A set-valued correspondence from Rn to Rn, say M(y), is a marginal productivity correspondence if it coincides with the subdifferential of a closed, proper, convex production function. A correspondence M(y) is monotone if

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for and such that cyclically monotone if

is contained in M(y0) and

is contained in M(y1). A correspondence M(y) is

for any pairs , (i = 0, 1, . . . , m) such that is contained in M(yi). A cyclically monotone correspondence is monotone (m = 1). If a set-valued correspondence M(y) is a marginal productivity correspondence, it is cyclically monotone. The economic interpretation of the condition of monotonicity is that for any pair of production plans in the effective domain of a marginal productivity correspondence M(y) y0, y1 changes in the production plan from y0 to y1 result in changes in the system of marginal products from to such that the inner product of the changes is positive. If only one component of a production plan increases, all others remaining the same the corresponding marginal product increases or remains the same. Recalling that under our sign convention a decrease in input of a commodity corresponds to an increase in net output of that commodity, this implication of monotonicity is a version of the law of "diminishing" marginal productivity. To illustrate the economic interpretation of cyclical monotonicity, we consider a set of three production plans in the effective domain of a marginal productivity correspondence M(y) y0, y1, y2. From the definition of a subgradient we obtain the inequalities

Combining inequalities for the same pair of production plans, we obtain the three monotonicity conditions

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For n greater than one, the monotonicity conditions do not exhaust the implications of the subgradient inequalities. For example, combining the fourth and fifth inequalities above, we obtain:

from the first inequality

so that

Similarly,

We recognize these inequalities as the two conditions for cyclical monotonicity (m = 2). For two commodities (n = 1), monotonicity implies cyclical monotonicity. A cyclically monotone correspondence is said to be maximal if it is not contained in any other monotone correspondence. To interpret the maximality condition we consider two correspondences, say M1(y) and M2(y), that coincide at every point but one, say y0; suppose that M1(y0) consists of a set of values, while M2(y0) consists of a single value contained in M1(y0); then, M1(y0) is contained in M2(y0) and M1(y) is not maximal. Marginal productivity correspondences are maximal cyclically monotone correspondences from Rn to Rn; a necessary condition that a set-valued correspondence from Rn to Rn is a marginal productivity correspondence is that it is a maximal cyclically monotone correspondence. This property of the marginal productivity correspondence is a generalized law of diminishing marginal productivity. (b) Monotonicity. M(y) is a maximal cyclically monotone correspondence from Rn to Rn.

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We have defined the marginal productivity correspondence in terms of the production function. It is useful to define the production function in terms of the marginal productivity correspondence. Every nonempty, maximal cyclically monotone set-valued correspondence is the subdifferential of a closed, proper, convex function. 6 All functions corresponding to a given correspondence differ only by an additive constant. To define a production function F(y) in terms of a nonempty marginal productivity correspondence we set the correspondence M(y) equal to the subdifferential of a closed, proper, convex function F(y), taking a finite, nonpositive value at the origin. Given the value at the origin, the production function defined by a marginal productivity correspondence M(y) is unique. We may regard the value of the production function F(y) at the origin as part of the definition of this function in terms of the marginal productivity correspondence M(y). The effective domain of the marginal productivity correspondence M(y) contains the relative interior of the effective domain of the production function F(y) and is contained in the relative interior of the effective domain of the production function. Given the value of the production function at every point x in the relative interior of the effective domain of the marginal productivity correspondence M(y), the value of the production function is uniquely determined for any production plan y on the relative boundary by the limit7

The origin may be outside the effective domain of the marginal productivity correspondence M(y), as in the example given above. However, the origin is included in the effective domain of the production function F(y) by our assumption on the domain of the marginal productivity correspondence. We conclude that our specification of the marginal productivity correspondence is precisely equivalent to our specification of the production function. We can take the set of production possibilities, the production function, or the marginal productivity correspondence as the starting point of the theory of production. Given the properties of any one, we can deduce the properties of the remaining two.

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5.3 Economic Behavior 5.3.1 Introduction In the preceding section we have characterized the set of production possibilities from the point of view of technology. In this section we study this set from the point of view of economic behavior. We characterize profitmaximizing production plans in terms of the price systems at which profits are maximized. We represent a price system by a vector q with n + 1 components, li (i = 1,2, . . . , n + 1), corresponding to unit values of each of the n + 1 commodities. We assume that prices of individual commodities may take any real value. The value of a production plan < q, y >, is the sum of values of individual components of the plan. The value of each component is equal to the price times the net output of the commodity. The value of a production plan is equal to profit. 5.3.2 Profit Function We assume that the objective of the producing unit is to maximize profit on the set of production possibilities. Profit is a linear function with coefficients given by the components of the price system. The supremum of a linear function on a convex set, expressed as a function of its coefficients, is the support function of the set. We can characterize the set of production possibilities in terms of properties of its support function. For this purpose, we introduce the profit function, defined by

The profit function gives the supremum of the value of feasible production plans as a function of the price system and is the support function of the set of production possibilities. We adopt the convention that if there is no upper bound on profit for a given price system, the profit function takes the value positive infinity, +¥. With this convention the domain of the profit function includes all of Rn+1. The effective domain of the profit function is a convex set. We observe that the profit function takes the value positive infinity wherever the price of a commodity characterized by free disposal is negative. The profit function is nonnegative, since the zero level of profit can be attained for any price system by shutting down

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or setting all net outputs equal to zero. The value of the profit function is at least zero for any price system. (a) Domain. P(q) is a nonnegative function with possibly infinite values defined on Rn+1. The value of the profit function P(q) for price systems with a negative price for the (n + 1)st commodity is positive infinity, +¥. The effective domain of the profit function P(q) is a convex set. The value of the profit function for a price system with all prices equal to zero is zero

We have defined the profit function in terms of the set of production possibilities. But there is a one-to-one correspondence between profit functions and set of production possibilities. We define the set of production possibilities in terms of the profit function P(q) as the set of production plans (y, hn+1) such that

for all price systems q. The profit function is bounded above for at least one price system not equal to zero. Otherwise, the set of production possibilities includes all of Rn+1, which contradicts the boundedness property of the set of production possibilities. (b) Boundedness. The value of the profit function P(q) is bounded above for at least one price system q not equal to zero. Since the set of production possibilities is nonempty, we conclude immediately that the profit function is lower semi-continuous and: (c) Closure. The function P(q) is closed. In fact, the profit function P(q) is a closed, proper, convex function: (d) Convexity. The function P(q) is convex. Finally, proportional changes in all components of a price system result in a proportional change in the profit function. (e) Homogeneity. The function P(q) is homogeneous of degree one. We have derived the properties of the profit function from the properties of the set of production possibilities. Making use of the correspondence between the profit function and the set of production possibilities, we observe that the set of production possibilities is nonempty, since it includes the origin in Rn+1 by boundedness and nonnegativity of the profit function P(q). The boundedness property

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of the set of production possibilities is an implication of the boundedness property of the profit function. Further, this set is closed and convex, since P(q) is the support function of the set of production possibilities. Finally, the set of production possibilities has the monotonicity property associated with free disposal of the (n + 1)st commodity, since the value of the profit function for price systems with a negative prices of this commodity is positive infinity. We conclude that our specification of the profit function is precisely equivalent to our specification of the set of production possibilities and that either can serve as the starting point for the theory of production. 5.3.3 Supply Correspondence We continue our analysis of production from the behavioral point of view by studying the set of all feasible production plans that maximize profit for a given price system. We first introduce a vector q *, consisting of net outputs of all n + 1 commodities such that q * is a profit-maximizing production plan for the price system q. We say that q * is a subgradient of the profit function P(q) at a point q if

for all price systems p. If the profit function P(q) is finite at q, the linear function

is a nonvertical supporting hyperplane to the epigraph of the profit function P(q) at the point (q, P(q)). The production plan that maximizes profit for a given price system need not be unique. We may define the supply correspondence, say S(q), as the set of all subgradients q* of the profit function P(q) at a price system q in the effective domain of P(q). The supply correspondence is identical to the subdifferential of the profit function. Where the supply correspondence is single-valued, the profit function is differentiable and the supply correspondence is identical to the gradient of the profit function. The profit function is differentiable almost everywhere in the relative interior of its effective domain. The effective domain of the profit function P is the set of all price systems q for which P(q) is finite. The effective domain of the function P is a convex set of Rn+1 containing the origin. Since the profit

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function is a closed, proper, convex function, the supply correspondence is nonempty for any price system q in the relative interior of the effective domain of the profit function P(q). By the boundedness property of the profit function the relative interior of the effective domain is nonempty. We adopt the convention that if there is no subgradient q * for the profit function P(q) at a price system q, the supply correspondence, say S(q), takes as its value of empty set, denoted f. With this convention the domain of the supply correspondence S is all of Rn+1. We define the effective domain of the supply correspondence S as the set of all price systems q for which S(q) is nonempty. The effective domain of the supply correspondence S(q) is nonempty and is contained in the effective domain of the profit function P(q). The origin is included in the effective domain of the supply correspondence S(q). The profit function P(q) is equal to zero at the origin, nonnegative, and homogeneous of degree one. For any price system q included in the effective domain of the profit function P(q) the subdifferential of the profit function P(lq), considered as a function of l, say Sl(lq), is nonnegative. Given the value of the profit function at the origin, this condition is necessary and sufficient for the profit function to be nonnegative. (a) Domain. S(q) is a set-valued correspondence, possibly empty, from Rn+1 to Rn+1. The effective domain of the correspondence S(q) is contained in a convex set and contains the relative interior of the convex set. The correspondence S(q) is nonempty at the origin and for some price systems q not equal to zero. The correspondence S(q) is empty for all price systems with a negative price for the (n + 1)st commodity. For all price systems q for which the correspondence S(q) is nonempty, Sl(lq) is nonnegative. (b) Monotonicity. S(q) is a maximal, cyclically monotone correspondence from Rn+1 to Rn+1. Cyclical monotonicity implies

for any set of price systems in the effective domain of a supply correspondence

In particular, if the value of a profit-maximizing production plan

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corresponding to the price system q0, is less than or equal to the value of this plan at prices q1, then the value of the profit-maximizing plan

at the prices q1 must exceed the value of this plan at the prices q0:

Changes in the price system from q0 to q1 result in changes in the production plan from to so that the inner product of the changes is positive. If only one component of the price system increases, all others remaining the same, the corresponding component of the production plan increases or remains the same. The relationship is valid for any production plans and included in the supply correspondence S(q) for price systems q0 and q1, respectively. If the supply correspondence is single-valued, monotonicity corresponds to the usual law of comparative statics; if the supply correspondence is set-valued, a similar law of comparative statics must hold for every value of the production plan included in the set. Monotonicity is familiar to economists as the weak axiom of revealed preference. Cyclical monotonicity implies that a relationship of this type holds not only between pairs of price systems but among any cycles of m + 1 price systems. The cyclical relationship is, of course, the strong axiom of revealed preference. 8 Since the profit function is homogeneous, the supply correspondence is unaffected by proportional changes in all components of a price system: (c) Homogeneity. S(q) is homogeneous of degree zero. Reversing our line of reasoning, we may define the profit function in terms of a nonempty supply correspondence. We set the correspondence S(q) equal to the subdifferential of a homogeneous, closed, proper, convex function P(q), such that

Monotonicity of the supply correspondence implies the closure and convexity properties of the profit function. The effective domain of the profit function includes no price systems with a negative price for the (n + 1)st commodity, since the supply correspondence is empty for any such price system. Boundedness of the profit function is implied by the existence of a price system q, not equal to zero, such that the supply correspondence S(q) is nonempty. Homogeneity of degree zero of the correspondence implies homogeneity of degree one of the

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profit function. Setting the profit function equal to zero at the origin of Rn+1 suffices to determine the profit function uniquely, given the supply correspondence. Nonnegativity of Sl(lq) implies nonnegativity of the profit function. We conclude that our specification of the supply correspondence is precisely equivalent to our specification of the profit function. 5.3.4 Set of Price and Profit Possibilities The final step in our analysis of production from the behavioral viewpoint is to consider the set of profit levels that exceed those associated with a profit-maximizing production plan for a given price system. For this purpose we introduce the epigraph of the profit function P(q), consisting of all points (q,r) in Rn+2 such that

the profit level r is at least as great as the level P(q) associated with a profit-maximizing production plan at the price system q. We say that such a price system and profit level are possible; we refer to the set of all such price systems and profit levels as the set of price and profit possibilities. The profit level associated with a possible price system is attainable for a feasible production plan only if it is the infimum of all profit levels that are possible for that price system. Since the value of the profit function at the origin of Rn+1 is zero, the set of price and profit possibilities satisfies: (a) Origin. The price system and profit level with all components zero is possible. All possible profit levels are nonnegative and all possible price systems have a nonnegative price for the (n + 1)st commodity. The boundedness property of the profit function implies: (b) Boundedness. There is at least one possible price system, not equal to zero, such that the possible profit level corresponding to that price system is bounded above. Closure of the profit function implies: (c) Closure. If a sequence of possible price systems and profit levels has a limit, the limiting price system and profit level is contained in the set of price and profit possibilities. Convexity of the profit function implies: (d) Convexity. If two price systems and profit levels are possible,

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every price system and profit level that can be represented as a convex combination of them is possible. A convex set is a convex cone if it is closed under positive scalar multiplication. Homogeneity of the profit function implies: (e) Homogeneity. The set of price and profit possibilities is a convex cone. Finally, the definition of the epigraph of the profit function implies monotonicity of the set of price and profit possibilities: (f) Monotonicity. If a price system and profit level is possible, every price system and profit level with the same or higher profit level and the same price system is possible. We have characterized the set of price and profit possibilities in terms of properties of the profit function. Reversing this reasoning, we can define the profit function by

The profit function P(q) is the infimum of possible profit levels r, given the price system q. If, for a given price system, there is no corresponding profit level in the set of price and profit possibilities that is finite, we take the value of the profit function to be positive infinity, +¥. With this convention the domain of the profit function includes all of Rn+1, as before. The value of the profit function corresponding to any price system with a negative price of the (n + 1)st commodity is positive infinity, +¥. The value of the profit function at the origin of Rn+1 is zero. Since all profit levels are nonnegative, the profit function is nonnegative. Convexity of the set of price and profit possibilities implies that the profit function is convex. The boundedness property of the set of price and profit possibilities implies the boundedness property of the profit function. Closure of the set implies that the profit function is lower semi-continuous. Since the profit function is nonnegative and takes the value zero at the origin of Rn+1, this function is a proper convex function and is, therefore, closed. Finally, since the set of price and profit possibilities is a cone, the profit function is homogeneous of degree one. We conclude that our specifications of the set of price and profit possibilities, the profit function, and the supply correspondence are equivalent. Any one of the three can be taken as the starting point of the behavioral approach to the theory of production. In view of the

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one-to-one correspondence between profit functions and sets of production possibilities, the behavioral and technological approaches of the theory of production are equivalent. 5.4 Duality 5.4.1 Introduction We have derived production and profit functions from the underlying set of production possibilities. In view of the equivalence among our specifications of the set of production possibilities, the production function, and the profit function, the technological and behavioral approaches to the theory of production have identical implications. These implications can be derived by first characterizing the set of production possibilities, taking either the production function or the profit function as a starting point. However, a more direct approach is available through the correspondence between production and profit functions implied by conjugate duality. The properties of either can be derived from those of the other through the conjugacy correspondence. 5.4.2 Production and Profit Functions The starting point of our study of conjugate duality is the production function F, a closed, proper, convex function with domain Rn. The conjugate of this function, say F *, is defined as:

where y * is a system of marginal products. We take the domain of F * to be all of Rn. Since the value of the production function F is nonpositive for the production plan y equal to zero, the supremum of the definition of the conjugate function F * is nonnegative. The conjugate function F * is a closed, proper, convex function; accordingly, we refer to this function as the convex conjugate of the production function F. If the supremum of the definition of the convex conjugate F * of the production function F is attained for some production plan y

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for all production plans z, so that y * is a subgradient of the production function F, corresponding to the production plan y. The production function F is subdifferentiable at every point in the relative interior of its effective domain. If the supremum of the definition of the convex conjugate F * is attained, the function F * is equal to the value of the profit-maximizing production plan (y,hn+1) at the price system (y*, 1)

the price of the (n + 1)st commodity is normalized at unity. From the definition of the convex conjugate:

for any production plan (z, zn+1); the value of the convex conjugate function F* (y*); is equal to the maximum value of profit for the normalized price system (y*, 1). Accordingly, we refer to this function as the normalized profit function. The correspondence between the production function F and the normalized profit function F* is symmetric. The convex conjugate of the function F* is defined by

where y is a vector corresponding to the first n components of a production plan and the value of the function F* *, say hn+1, is the value of the (n + 1)st component. We take the domain of F to be all of Rn. Since the conjugate convex function F* is a closed, proper, convex function, the convex conjugate F* * is the production function F and we may write:

The conjugacy correspondence implies a duality between the production function F and the normalized profit function F *. A closed, proper, convex production function F corresponds to a closed, proper, convex normalized profit function F *, and conversely. The origin property of the production function F corresponds to the nonnegativity property of the normalized profit function F *. We can define the production function F in terms of the normalized profit function F * by means of the conjugacy correspondence. Our characterization of

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the production function is implied by the properties of the normalized profit function. We can summarize these properties as follows: (a) Domain. F * (y*) is a nonnegative function with possibly infinite values, defined on Rn. The effective domain of the function F * (y*) is a convex set. (b) Boundedness. F * (y*) is bounded above for some normalized price system y *. (c) Closure. F * (y*) is closed. (d) Convexity. F * (y*) is convex. We have characterized the production function in terms of the normalized profit function. The problem that remains is to characterize the profit function in terms of the normalized profit function. We can define the profit function as 9

The profit function is a closed, proper, convex function, defined on Rn+1 homogeneous of degree one, and equal to zero at the origin of Rn+1. The boundedness property of the profit function is implied by the corresponding property of the normalized profit function. We conclude that the profit function can be defined and characterized in terms of the normalized profit function F * (y*). In defining the profit function P in terms of the normalized profit function F *, we have proceeded from relative prices y * of the first n commodities in terms of the (n + 1)st to absolute prices q for all n + 1 commodities. To define and characterize the normalized profit function in terms of the profit function we normalize the price of the (n + 1)st commodity at unity. 5.4.3 Marginal Productivity and Supply Correspondences We have observed that the system of marginal products y * is a subgradient of the production function F(y) accordingly, the system of

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marginal products y * is contained in the marginal productivity correspondence M(y). Similarly, we may write:

for all systems of marginal products z * by definition of the production function. The production plan y is a subgradient of the normalized profit function F *, so that y is contained in the subdifferential of F *, say M * (y*). The duality between the production function F and the normalized profit function F * implies a duality between the marginal productivity correspondence M(y) and the subdifferential M * (y*) of the normalized profit function F *. A production plan y is contained in the subdifferential M * (y*) if and only if the corresponding system of marginal products y * is contained in the marginal productivity correspondence M(y). Each correspondence is the inverse of the other. We may interpret the production plans included in the subdifferential M * (y*) as the set of plans that maximize profit at the system of prices of the first n commodities relative to the price of the (n + 1)st given by the system of marginal products y *. Accordingly, we refer to the subdifferential M * as the normalized supply correspondence. Similarly, systems of marginal products included in the marginal productivity correspondence M(y) may be interpreted as relative prices at which the production plan y maximizes profit. The properties of the normalized profit function F * imply the following properties of the normalized supply correspondence M *: (a) Domain. M * (y*) is a set-valued correspondence, possibly empty, from Rn to Rn. The effective domain of the correspondence M * (y*) is contained in a convex set and contains the relative interior of the convex set. The correspondence M * (y*) is nonempty for at least one price system y *. For all price systems y * included in the effective domain of the correspondence M * (y*), one of the following conditions must hold for the subdifferential of F* (ly*), considered as a function of l. (i) The effective domain of (ii) (iii) lim min

is bounded above.

is nonnegative for some l > 0. la = b for constants a, such that 1 < a, and b finite. 10

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we consider an example To illustrate the economic interpretation of our condition on the subdifferential where this condition is not satisfied. We let the number of commodities be two (n = 1) and the subdifferential M * (y*) be constant and equal to minus unity

The corresponding function F * (y*) takes the form

Where g is a constant. This function has the boundedness, closure, and convexity properties of a normalized profit function. However, it cannot be made nonnegative for all by adding a suitably chosen constant. (b) Monotonicity. M * (y*) is a maximal cyclically monotone correspondence from Rn to Rn. Properties of the normalized supply correspondence imply our characterization of the normalized profit function. To demonstrate this we let a nonempty, maximal cyclically monotone correspondence be the subdifferential of a closed, proper, convex function F * (y*), taking a finite, nonnegative value for some price system y *. The conditions on the subdifferential given above ensure that F * (y*) is bounded below as components of y * tend to infinity. The function F * (y*) can be made nonnegative by adding a constant to any closed, proper, convex function of which M * (y*) is the subdifferential. 5.4.4 Sets of Production and Price Possibilities Finally, the duality between the production function F and the normalized profit function F * implies a duality between sets of production and price possibilities. The epigraph of the production function corresponds to the set of production possibilities. The value of the normalized profit function is the normalized level of profit; the epigraph of the normalized profit function is the set of normalized price and profit possibilities. This set has the following properties, equivalent to those of the set of price and profit possibilities. (a) Boundedness. There is at least one possible normalized price system and profit level such that the possible normalized profit level corresponding to that normalized price system is bounded above. All possible normalized profit levels are nonnegative.

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(b) Closure. If a sequence of possible systems of normalized prices and profit levels has a limit, the limiting normalized price system and profit level is contained in the set of normalized price and profit possibilities. (c) Convexity. If two normalized price systems and profit levels are possible, every normalized price systems and profit level that can be represented as a convex combination of them is possible. (d) Monotonicity. If a normalized price system and profit level is possible, every normalized price system and profit level with the same or higher normalized profit level and the same normalized price system is possible. The normalized profit function can be defined in terms of the set of normalized price and profit possibilities as the infimum of possible normalized profit levels, given the system or normalized prices. Properties of the set of normalized price and profit possibilities imply our characterization of the normalized profit function. 5.5 Conclusion We have developed the theory of production from six alternative and equivalent starting points. From the technological point of view the theory can be developed from properties of the set of production possibilities, the production function, or the marginal productivity correspondence. Similarly, from the behavioral point of view the theory can be developed from properties of the set of price and profit possibilities, the profit function, or the supply correspondence. The technological and behavioral approaches to the theory of production are precisely equivalent. Moreover, these two approaches are dual to each other through the conjugacy correspondence between the production function and the normalized profit function. We present equivalent specifications of the production function and the marginal productivity correspondence and the normalized profit function and the normalized supply correspondence, resolving the problem of ''integrability" for both approaches to the theory of production. 11 For the supply correspondence S(q) the profit function P(q) can be defined by

where the inner product is the same for any element of the set S(q).

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Integrability for the supply correspondence S(q) requires that the function P(q) has the properties of a profit function and that S(q) is the subdifferential of the profit function P(q). This indirect approach to integrability cannot be applied to the normalized profit function F * (y*) and the normalized supply correspondence M * (y*). We treat inputs and outputs symmetrically, observing the convention that inputs are algebraically negative. We do not require irreversibility of production or even the impossibility of producing positive output from zero input. Our specification of the production function and the normalized profit function is not completely symmetric. The origin assumption for the production function corresponds to the nonnegativity assumption for the normalized profit function. To obtain a completely symmetric formulation of the theory we can replace the boundedness assumption on the production function by the assumption that the production function F(y) is nonnegative. This is equivalent to the assumption that the (n + 1)st commodity is always an output of the production process. This assumption implies that the normalized profit function F * (y*) is nonpositive at the origin. Since this function is also nonnegative, it must be zero at the origin. Under this assumption the boundedness property of the normalized profit function is superfluous, so that the properties of the production function and the normalized profit function are identical. 12 A function with these properties can be employed either as a production function or as a normalized profit function.13 We do not require free disposal of all commodities; our assumption that at least one commodity is freely disposable could be weakened slightly by requiring that there exists some combination of commodities which is freely disposable. This combination could be defined as a new commodity, leaving the theory as we have developed it unchanged. By strengthening the monotonicity assumption for the set of production possibilities, requiring free disposal of all commodities, the production function can be made monotone. All subgradients of the production function are then nonnegative and the effective domains of the normalized profit function and normalized supply correspondence include only nonnegative prices. We do not require that the set of production possibilities is compact. By taking this set to be compact the normalized profit function can be taken to be finite everywhere and "sup" in our definition of the normalized profit function can be replaced by "max", as in our definition of the production function.

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We do not require that the production function and the normalized profit function are differentiable, as in the traditional approach to the theory of production. Subdifferentiability of these functions in the relative interiors of their effective domains is an implication of convexity and closure. We employ the subdifferential of a convex function to characterize marginal productivity and supply correspondences, incorporating the traditional approach to the theory of production in modern form. By strengthening the convexity assumption for the set of production possibilities to strict convexity, we can derive differentiability of the profit function as an implication. Similarly, strengthening the convexity assumption for the set of normalized profit and price possibilities to strict convexity implies differentiability of the production function. By strengthening both convexity assumptions to strict convexity we can reduce the conjugacy correspondence to the Legendre transformation. 14 The theory of production can be further extended in a number of directions. By taking a subset of the commodities as fixed and choosing a variable and freely disposable commodity as the (n + 1)st commodity, the duality between the production function and the normalized restricted profit function, defined as

where x is a vector of fixed commodities, can be established by way of the conjugacy correspondence under the assumption that F is a closed, proper, convex function of z for any x. The properties of the normalized restricted profit function and the normalized restricted supply correspondence can be characterized in the same way as before. Normalized restricted profit functions include normalized revenue functions and the negative of normalized cost functions. By taking the set of production possibilities to be polyhedral, the theory of linear programming can be developed as a specialization of the theory of production.15 Taking the set of production possibilities to be a convex cone with vertex at the origin, the theory of production can be specialized to the case of constant returns to scale employed in general equilibrium theory and the theory of growth.16 Applications of the theory of production to specific functional forms for the production and profit functions can be developed by imposing separability and homotheticity properties of the set of production possibilities.17 Our approach provides a unified framework for applications of the

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theory of production based on either technological or behavioral viewpoints on productive activity. 5.6 Historical Note The production function is the starting point of the traditional theory of production. 18 In this approach the assumption of differentiability is added to the properties of the production function we have employed.19 The origin of the modern theory of production, based on the set of production possibilities, can be traced to the axiomatic treatment of the theory of linear programming by Dantzig (1949). He assumes that the objective of the producing unit is maximization of the value of the production plan. An alternative axiomatic treatment of the theory of linear programming was presented by Koopmans (1951). Koopmans related the technology of linear programming to the older concepts of the production function and marginal productivity.20 Samuelson derived a monotonicity condition for supply correspondences associated with a linear programming technology, using methods he had developed earlier, starting from the production function.21 Debreu (1959) extended Koopmans' analysis of production, taking the set of production possibilities to be either a convex cone or a closed, bounded, convex set. Debreu formalized the concepts of the profit function and the supply correspondence. He proved homogeneity, nonnegativity, and continuity properties of the profit function and homogeneity, monotonicity, and continuity properties of the supply correspondence.22 Debreu (1959, p. 47) employed methods originated by Samuelson in validating the monotonicity of supply correspondences. The concepts of production function and marginal productivity do not appear in the analysis of production by Debreu. The study of duality in the theory of production was initiated by Hotelling (1932, esp. pp. 590 598). His "profit" function can be interpreted as a production function and his "price potential" function can be interpreted as a normalized profit function. He identified marginal productivity functions with the gradient of the production function and supply functions with the gradient of the normalized profit function. Hotelling pointed out the duality between these sets of functions and characterized each in terms of the others. Samuelson developed the duality between a production function with constant returns to scale and the corresponding factor-price frontier (Samuelson,

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1953 1954, pp. 15 20). 23 The factor-price frontier can be interpreted as a normalized profit function equal to zero everywhere in its effective domain. Duality between production functions and normalized profit functions for the differentiable case was analyzed in greater detail by Lau, using the Legendre transformation (Lau, 1973, 1972).24 This transformation is implicit in the treatment of duality by Hotelling and Samuelson. Our approach to the duality between the production and normalized profit functions is based on the conjugacy correspondence between closed, proper, convex functions, which reduces to the Legendre transformation under differentiability. Shephard (1953, 1970) studied the duality between production and cost functions. He subsequently developed a duality between sets of production possibilities and sets of price possibilities. Shephard's approach to duality is based on distance functions associated with convex sets rather than the theory of conjugate convex functions.25 Uzawa (1964) provided an alternative proof of Shephard's correspondence between production and cost functions without using differentiability. Gorman (1968), and McFadden (1966, 1978) introduced the concept of the restricted profit function, generalizing the notion of the cost function, and developed the relationships among sets of production possibilities, production functions and restricted profit functions. Gorman employed the duality between polar cones while McFadden used the duality between gauge functions associated with convex sets. Diewert (1973) provided an alternative proof of the results of Gorman and McFadden. He also developed equivalent characterizations of the production function and the set of production possibilities. Notes 1. Detailed references to the theory of production are given in section 3.6. 2. The theory of convex conjugate functions was originated by Fenchel (1949); see also Fenchel (1953). The definitive treatise on convex analysis by Rockafellar (1970b) provides proofs and references for the results required. 3. The production function thus defined represents the complete nonvertical boundary of the set of production possibilities only under the assumption of monotonicity in the (n + 1)st commodity. Without the monotonicity assumption, there will be another nonvertical boundary, given by

Our assumption of monotonicity corresponds to the assumption of monotonicity in the (n + 1)st commodity required for application of the implicit function theorem to obtain a representation of a transformation function,

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in the form

As a specific example, consider the set of production possibilities defined by

that is, disc with center at the origin and radius equal to 1. This set satisfies our assumptions (a) through (d). However,

does not give the complete nonvertical boundary of the production possibility set. There is another part of the boundary given by

4. A comprehensive review of the theory of the subdifferential is presented by Moreau (1967) and Rockafellar (1970b, pp. 213 226). The terminology "correspondence" for a set-valued mapping is employed by Debreu (1959, pp. 5 7). 5. The subdifferential Ml (ly0) is defined by:

where the inner product is defined by taking a typical element of the set Ml(ly0) to be equal to the inner product of y0 and a typical element of the set M(ly0). Our condition is equivalent to the necessary and sufficient condition for the convergency of improper integrals at a point where the integrand diverges. See, for instance, Courant (1937, pp. 245 256), and Stanaitis (1967, pp. 150 179). 6. See Rockafellar (1966, 1970a) and (1970b, pp. 227 240). 7. See Rockafellar (1970b, p. 57). 8. The weak axiom of revealed preference was introduced into the theory of consumer behavior by Samuelson (1938). The strong axiom was introduced by Houthakker (1950). Note that when n = 1, the weak and strong axioms coincide. 9. See Rockafellar (1970b, pp. 118 119). 10. This condition is equivalent to the necessary and sufficient condition for the convergence of improper integrals at infinity. See Courant (1937, pp. 245 256), and Stanaitis (1967, pp. 150 161). 11. Integrability for the differentiable case is treated by Hotelling (1932); an alternative treatment of this case, based on cyclical monotonicity, is given by Hicks (1956) and by Samuelson (1960). 12. This symmetric duality for conjugate convex functions is discussed by Rockafellar (1970b, p. 109). 13. A similar property of dual gauge functions is employed by Hanoch (1974), and McFadden (1978).

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14. The Legendre transformation is discussed by Courant and Hilbert (1962, pp. 32 35). The Legendre transformation is compared with the conjugacy correspondence by Fenchel (1949) and Rockafellar (1970b, pp. 251 260). 15. A comprehensive exposition of the theory of linear programming is given by Gale (1960). 16. See, for example, Debreu (1959, esp. pp. 39 47), and Samuelson (1953 1954) on the theory of general equilibrium, and Samuelson (1962), Bruno (1969), and the references given there on the theory of growth. 17. See, for example, Christensen, Jorgenson, and Lau (1973) and Diewert (1973). 18. Hicks (1946, pp. 78 111, 319 325), and Samuelson (1983, pp. 57 89). 19. The fact that results of the theory of marginal productivity and the theory of supply do not depend on differentiability is made explicit in Frisch's treatment of "limitational" factors of production. See Frisch (1931). See also Frisch (1965, pp. 225 265), Georgescu-Roegen (1935), and Samuelson (1983, esp. pp. 70 73, 80 81). 20. Koopmans (1951, pp. 33 35, 66 69). See also Samuelson (1958, p. 316); Dorfman, Samuelson, and Solow (1958, pp. 201 203); Koopmans (1957, pp. 93 95). 21. See Samuelson (1949); a supply correspondence for linear programming technologies is derived on pp. 646 647. See also Beckmann (1956), Bailey (1956), and Samuelson (1958). Samuelson's "method" of finite increments (1983, pp. 80 81), is based on the inequality defining the subgradient of a convex (or concave) function. 22. Debreu (1959, Chapter 3, pp. 37 49). This concept of the profit function is employed by Arrow and Hahn (1970), Diewert (1973), Gorman (1968), and McFadden (1966, 1978). 23. For further discussion of duality, see Bruno (1969) and Burmeister and Kuga (1970). 24. See also Samuelson (1960). 25. See Shephard (1970, p. 171). Alternative treatments of duality between production and cost functions are given by Jacobsen (1970, 1972) and McFadden (1966, 1978).

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6 Duality and Differentiability in Production Dale W. Jorgenson and Lawrence J. Lau 6.1 Introduction In a previous paper we have developed the theory of production without requiring global differentiability of the production and profit functions (Jorgenson and Lau, 1974b). The marginal products corresponding to a given production plan and the net supplies corresponding to a given price system need not be unique. In this chapter we consider the implications of global differentiability of the production and profit functions or, equivalently, uniqueness of marginal products and net supplies. By strengthening convexity assumptions for the production and profit functions we can develop the theory of production in terms of properties of differentiable convex functions and their gradients. Under global differentiability the production and profit functions are dual to each other through the Legendre transformation. 1 Using the Legendre transformation we obtain precisely equivalent specifications of the production and profit functions. We employ the gradient of a differentiable convex function to characterize the marginal productivity and supply functions. We characterize production and profit functions in terms of the corresponding marginal productivity and supply functions, resolving the problem of "integrability" for both approaches to the theory of production. After developing the duality between production and profit functions and marginal productivity and supply functions, we consider the duality of the underlying sets of production and price and profit possibilities. We next consider asymmetric forms of duality with a globally differentiable production function corresponding to a profit function that is not globally differentiable and vice versa. Finally, we compare the Legendre transformation with the conjugate correspondence between conjugate convex production and profit functions.2

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The traditional starting point for the theory of production is the assumption that the production function is twice differentiable. This assumption implies uniqueness of the marginal products corresponding to a given production plan and differentiability of the marginal productivity function. Twice differentiability of the production function is neither necessary nor sufficient for uniqueness of the net supplies corresponding to a given price system. Even where net supplies are unique, this assumption is neither necessary nor sufficient for differentiability of the supply function. Precisely analogous results hold for the assumption that the profit function is twice differentiable. 6.2 Differentiability 6.2.1 Introduction We represent a production plan by a vector with n + 1 components hi (i = 1,2, . . . , n + 1), corresponding to net outputs of each of n + 1 commodities. We treat inputs and outputs symmetrically, adhering to the convention that net output is negative for a commodity that is an input and assuming that all commodities are perfectly divisible. The starting point for our discussion of technology is the production function. In our earlier treatment we assumed that the production function is convex, lower semicontinuous, and bounded below (Jorgenson and Lau, 1974b). To consider the implications of uniqueness of marginal products corresponding to a given production plan and net supplies corresponding to a given price system, we replace lower semicontinuity by a differentiability assumption and convexity by an assumption of strict convexity. 6.2.2 Production Function The production function F(y) gives the negative of the maximum level of output of the (n + 1)st commodity corresponding to any set of net outputs of the other n commodities. Denoting a vector of n net outputs (h1,h2, . . . , hn) by y, the production function can be written

We adopt the convention that if there is no feasible production plan with hn+1 finite for given values of the production plan y, the value taken by the production function is positive infinity, denoted + ¥.

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With this convention the domain of the production function F(y) is all of Rn. We define the effective domain of the function F as the set of all production plans y for which the function is finite. We assume that the interior of the effective domain is nonempty and, more specifically, that: a. Domain. The production function F(y) is a function with possibly infinite values, defined on Rn. The effective domain of the function F(y) is a convex set with nonempty interior. The effective domain contains the origin and the value of the function F(y) at the origin, say F(0), is zero. We do not require that production is irreversible or even that production of positive net output from zero input is impossible. To obtain a formulation of the theory that is completely symmetric in production plans and price systems, we assume: b. Nonnegativity. The function F(y) is nonnegative. The assumption of nonnegativity implies that the (n + 1)st commodity is always an input to production. Since the numbering of commodities is arbitrary, this assumption requires that there is at least one nonproducible factor of production. We do not require that this factor is essential to production. The assumption of nonnegativity implies that the production function attains its minimum at the origin. A production function is smooth if it is finite and differentiable throughout its domain. Since the production function can be infinite, we cannot assume that this function is smooth in the usual sense. We have assumed that the effective domain of the production function is a convex set with nonempty interior. Under this assumption the production function is essentially smooth if it is differentiable throughout the interior of its effective domain and

where {yi} is a sequence of production plans in the interior of the effective domain of the production function, converging to a boundary point of the effective domain. The gradient of F(y) is denoted ÑF(y) and | · | is the Euclidean norm of a vector. We assume that: c. Smoothness. The function F(y) is essentially smooth. The assumption that the production function is essentially smooth is weaker than the assumption that the function is smooth in the usual sense and stronger than the assumption that the function is lower

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semicontinuous. If the production function is essentially smooth, the gradient of the function is a vector-valued function defined on the interior of the effective domain of the production function. The gradient is not defined elsewhere. A nonnegative, convex function is closed if it is lower semicontinuous. We conclude that an essentially smooth, nonnegative, convex production function is closed. A production function F(y) is strictly convex on a convex set contained in and possibly equal to its effective domain if

for all y1, y2 in the convex set. An essentially smooth production function is essentially strictly convex if it is convex and strictly convex on the interior of its effective domain. We assume that: d. Convexity. The function F(y) is essentially strictly convex. The assumption that the production function is essentially strictly convex is weaker than strict convexity, since the function is required to be convex but not strictly convex on the boundary of its effective domain. A convex function with nonempty effective domain that nowhere takes the value negative infinity, ¥, is proper. We conclude that the production function is a proper convex function. An example of a production function that is not smooth but satisfies all of our assumptions is the following

6.2.3 Marginal Productivity Function We continue our development of the theory of production from the technological point of view by defining the marginal productivity function, say M(y), as the unique system of marginal products, say y*, corresponding to a given production plan. This function is identical to the gradient of the production function. After characterizing the marginal productivity function in terms of the production function we reverse the reasoning and develop the properties of the production function from properties of the marginal productivity function.

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The domain of the marginal productivity function coincides with the interior of the effective domain of the production function. The origin is contained in the domain of the marginal productivity function M(y), since the production function attains its minimum at the origin. Further, because of essential smoothness: a. Domain. M(y) is a function having an open convex set in Rn with nonempty interior as its domain. The domain contains the origin and the value of the marginal productivity function M(y) at the origin, say M(0), is zero. A marginal productivity function M(y) is cyclically monotone if

for any production plans yi (i = 0, 1, . . . , m) in its domain, where . , . is the inner product of two vectors. Cyclical monotonicity implies monotonicity (m = 1)

The property of cyclical monotonicity is a generalized law of ''diminishing" marginal productivity. Cyclical monotonicity of the marginal productivity function is implied by convexity of the production function: b. Monotonicity. M(y) is a cyclically monotone function from Rn to Rn. If a nonnegative, essentially strictly convex production function F(y) is differentiable at every point in the interior of its effective domain, it is continuously differentiable. 3 c. Continuity. M(y) is a continuous function on its domain. Finally, since the production function F(y) is essentially strictly convex: d. One-to-one. M(y) is a one-to-one function on its domain.4 We have characterized the marginal productivity function M(y) as the gradient of the production function F(y). Every cyclically monotone, continuous, one-to-one, marginal productivity function M(y) is the gradient of a proper, essentially smooth, essentially strictly convex, production function F(y).5 Given the value of the production function at every production plan x in the domain of the marginal productivity function, the value of the production function is uniquely determined for any production plan y on the boundary by the limit

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Since the production functions corresponding to a given marginal productivity function differ by an additive constant, we take the value of the production function at the origin to be zero Given the value at the origin, the production function corresponding to a given marginal productivity function is unique. We conclude that our specifications of the production function and the marginal productivity function are precisely equivalent. 6.3 Duality 6.3.1 Introduction Up to this point we have developed the theory of production from the point of view of technology. We now consider the theory from the point of view of economic behavior. Our study is based on the Legendre transformation. The Legendre conjugate of the production function is the normalized profit function, giving the maximum value of normalized profit for any normalized price system. 6 For a normalized price system all prices are expressed relative to the price of the (n + 1)st commodity; similarly, normalized profit is profit relative to the price of the (n + 1)st commodity. We employ the Legendre transformation to obtain precisely equivalent specifications of the production function and the normalized profit function.7 The Legendre conjugate of the normalized profit function is the production function. The gradient of the production function is the marginal productivity function. Similarly, the gradient of the normalized profit function is the normalized supply function, giving the net outputs as a function of the normalized price system. We employ the gradient of a convex function to characterize the normalized supply function and reverse the reasoning to characterize the normalized profit function in terms of the normalized supply function. We obtain precisely equivalent specifications of the marginal productivity function and the normalized supply function. The normalized supply function and the marginal productivity function are inverse to each other. 6.3.2 Legendre Transformation Given a nonnegative, essentially smooth, and essentially strictly convex production function F(y), having a convex set with nonempty

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interior as its effective domain, the Legendre conjugate of this function, say F* (y*), is defined as

where y* is a system of marginal products. The inverse function M 1(y*), corresponding to the marginal productivity function M(y), is well defined since M(y) is one-to-one. The Legendre conjugate is a nonnegative, essentially smooth, and essentially strictly convex function, having a convex set with nonempty interior as its effective domain. The transformation from the production function F(y) and the interior of its effective domain to the Legendre conjugate, consisting of the function F* (y*) and the interior of its effective domain, is the Legendre transformation. The Legendre transformation gives the value of the Legendre conjugate function F*(y*) on the interior of its effective domain. Given the value of this function at every system of marginal products x* in the interior of its effective domain, the value of the function is determined uniquely for any system of marginal products y* on the boundary by the limit

This limit is, of course, positive infinity for any boundary point not included in the effective domain of the Legendre conjugate function F*(y*). The Legendre transformation between the production function F and its Legendre conjugate function F* is symmetric. The Legendre conjugate of the function F*(y*), having a convex set with nonempty interior as its effective domain, is defined as

where y is a vector with n components consisting of the first n components of a production plan and the negative of the value of the function F* *(y), say hn + 1, is the value of the (n + 1)st component. We conclude that F* *(y) = F(y). The inverse function M* 1(y), corresponding to the gradient of the normalized profit function M*(y*), is well defined, since M*(y*) is one-to-one. For any price system (y*, 1) with y* in the range of the marginal productivity function M(y), the production plans y such that

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M(y) = ÑF(y) = y* are the plans that maximize profit at the price system (y*, 1), so that

for all production plans z. The price of the (n + 1)st commodity is normalized at unity. Accordingly, we refer to the price system (y*, 1) as the normalized price system. The Legendre conjugate function F* is equal to the value of the profit-maximizing plan (y, hn + 1) as the normalized price system

We refer to the Legendre conjugate function F* (y*) as the normalized profit function. The interior of the effective domain of the normalized profit function is the range of the marginal productivity function M(y). 6.3.3 Normalized Profit Function We can summarize the properties of the normalized profit function as follows: a. Domain. The normalized profit function F* (y*) is a function with possibly infinite values, defined on Rn. The effective domain of the function F* (y*) is a convex set with nonempty interior. The effective domain contains the origin and the value of the function F* (y*) at the origin, say F*(0), is zero. b. Nonnegativity. The function F* (y*) is nonnegative. c. Smoothness. The function F* (y*) is essentially smooth. d. Convexity. The function F* (y*) is essentially strictly convex. We have defined the Legendre conjugate of the normalized profit function F* (y*) in terms of the gradient M* (y*) of this function. We continue our development of the theory of production from the point of view of economic behavior by defining the normalized supply function as the unique production plan y corresponding to a given normalized price system y*. This function is identical to the gradient M* (y*) of the normalized profit function F* (y*).

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6.3.4 Normalized Supply Function The duality between the production function F and the normalized profit function F* implies a duality between the marginal productivity function M(y) and the normalized supply function M* (y*). A production plan y is in the range of the normalized supply function M* (y*) if and only if the corresponding normalized price system y* is in the range of the marginal productivity function M(y). Further, M* (y*) = M 1 (y*) and M(y) = M* 1 (y). Production plans given by the normalized supply function can be interpreted as production plans that maximize profits at the system of normalized prices y*. Similarly, normalized prices y* given by the marginal productivity function can be interpreted as normalized prices at which the production plan y maximizes profit. The domain of the normalized supply function coincides with the interior of the effective domain of the normalized profit function and includes the origin, since the normalized profit function attains its minimum at the origin. Further, because of essential smoothness: a. Domain. M* (y*) is a function having an open convex set in Rn with nonempty interior as its domain. The domain contains the origin and the value of the normalized supply function at the origin, say M* (0), is zero. b. Monotonicity. M* (y*) is a cyclically monotone function from Rn to Rn. Cyclical monotonicity of the normalized supply function is familiar to economists as the strong axiom of revealed preference. Monotonicity (m = 1) is the weak axiom of revealed preference. The property of cyclical monotonicity is a generalized law of comparative statics. c. Continuity. M* (y*) is a continuous function on its domain. d. One-to-one. M* (y*) is a one-to-one function on its domain. We have characterized the normalized supply function M* (y*) as the gradient of the normalized profit function F* (y*). We can fix the value of the normalized profit function at the origin at zero, determining the normalized profit function uniquely for any given normalized supply function. We conclude that our specifications of the normalized profit function and the normalized supply function are precisely equivalent.

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6.4 Production Possibilities 6.4.1 Introduction An alternative starting point for development of the theory of production is the set of production possibilities, consisting of all production plans available to the producing unit. There is a one-to-one correspondence between the production function and the set of production possibilities. Similarly, there is a one-to-one correspondence between the normalized profit function and the set of normalized price and profit possibilities, defined by the set of normalized profit levels for a given normalized price system that exceed the level associated with a profit maximizing production plan. The set of production possibilities and the set of normalized price and profit possibilities are dual to each other and have precisely the same properties. 6.4.2 Set of Production Possibilities We can define the set of production possibilities in terms of the production function by introducing the epigraph of the production function, consisting of all production plans (y, hn+1) in Rn+1 such that

The epigraph of the production function corresponds to the set of production possibilities. The properties of the production function F imply the following properties of the set of production possibilities: a. Nonemptiness. The interior of the set of production possibilities is nonempty. The production plan with all net outputs zero is feasible. The origin is contained in the set of production possibilities, since the value of the production function at zero is zero. b. Nonnegativity. There is at least one commodity such that the negative of the net output of that commodity is nonnegative for any feasible production plan. Nonnegativity is implied by the nonnegativity of the production function. Nonpositivity of the net output of a given commodity is a weaker property than irreversibility of the production process as a whole, defined by the condition that (y, hn+1) feasible implies ( y, hn+1) infeasible.

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c. Monotonicity. There is at least one commodity satisfying the nonnegativity assumption such that for a feasible production plan, every production plan with less than or the same net output of that commodity and the same net outputs of the other commodities is feasible. Monotonicity is implied by the definition of the set of production possibilities in terms of the production function. Free disposal of at least one commodity characterized by nonnegativity is a weaker condition than free disposal of all commodities. Since the numbering of commodities is arbitrary, we can take any commodity satisfying our nonnegativity and monotonicity assumptions as the (n + 1)st commodity. d. Closure. If a sequence of feasible production plans has a limit, the limiting plan is contained in the set of production possibilities. Closure of the set of production possibilities is implied by closure of the production function. A nonvertical supporting hyperplane to the set of production possibilities is a set of production plans (z, zn+1) such that

where a is a constant such that

for every production plan (y, hn+1) in the set of production possibilities with equality for some production plan in the set. A vertical supporting hyperplane to the set of production possibilities is a set of production possibilities is a set of production plans (z, zn+1) such that

where a is a constant such that

with equality for some feasible production plan. A feasible production plan (y, hn+1) is exposed if and only if there is a supporting hyperplane H to the set of production possibilities that contains no other feasible production plan. A supporting hyperplane is tangent to the set of production possibilities for a production plan (y, hn+1) if there is no other supporting hyperplane to the set of production possibilities that contains (y, hn+1).

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e. Convexity. If two production plans are feasible, every production plan that can be represented as a convex combination of them is feasible. Every production plan supported by a nonvertical hyperplane is exposed. Our strict convexity assumption on the set of production possibilities corresponds to the assumption of essential strict convexity of the production function. If the production function is strictly convex, every production plan supported by a hyperplane, vertical or nonvertical, is exposed. Under essential strict convexity, we require only that those plans supported by nonvertical hyperplanes are exposed. f. Smoothness. Every supporting hyperplane to the set of production possibilities is tangent. Our smoothness assumption on the set of production possibilities corresponds to the assumption of essential smoothness of the production function. Every production plan in the interior of the effective domain of the production function is supported by a unique nonvertical supporting hyperplane. Every production plan on the boundary of the effective domain of the production function is supported by a unique vertical supporting hyperplane. We have characterized the set of production possibilities in terms of the production function. We can reverse this reasoning by defining the production function in terms of the set of production possibilities as follows:

The production function is the negative of the maximum net output of the (n + 1)st commodity for given values of the first n components of the production plan. As before, we adhere to the convention that if there is no feasible production plan with hn+1 finite for a given value of the production plan y, the value taken by F(y) is positive infinity, denoted +¥. The properties of the production function we have assumed in developing the theory of production are implied by the properties of the set of production possibilities and the definition of the production function. 6.4.3 Set of Normalized Price and Profit Possibilities We can define the set of normalized price and profit possibilities, consisting of all normalized price systems and profit levels such that the

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normalized profit level exceeds that associated with the profit maximizing production plan for the normalized price system, as the epigraph of the normalized profit function. The properties of the set of normalized price and profit possibilities implied by the properties of the normalized profit function F* are the following: a. Nonemptiness. The interior of the set of normalized price and profit possibilities is nonempty. The price system and profit level with all components zero is possible. b. Nonnegativity. The normalized profit level corresponding to any system of normalized prices is nonnegative. c. Monotonicity. For a possible normalized price system and profit level every normalized price system and profit level with the same or higher normalized profit level and the same normalized price system is possible. d. Closure. If a sequence of possible normalized price systems and profit levels has a limit, the limiting normalized price system and profit level is contained in the set of normalized price and profit possibilities. e. Convexity. If two normalized price systems and profit levels are possible, every price system and profit level that can be represented as a convex combination of them is feasible. Every normalized price system and profit level supported by a nonvertical hyperplane is exposed. f. Smoothness. Every supporting hyperplane to the set of normalized price and profit possibilities is tangent. We have characterized the set of normalized price and profit possibilities in terms of the normalized profit function. We can reverse the reasoning, defining the normalized profit function as the minimum level of normalized profit for a given normalized price system. The properties of the normalized profit function and the set of normalized price and profit possibilities are precisely equivalent. The set of production possibilities and the set of normalized price and profit possibilities are dual to each other. Every nonvertical supporting hyperplane to the set of production possibilities H has coefficients y* equal to the marginal products corresponding to the production plan y contained in the supporting hyperplane. Similarly, every nonvertical supporting hyperplane to the set of normalized profit and price possibilities, say H*, has coefficients y equal to the

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net supplies corresponding to the unique normalized price and profit level y* contained in the supporting hyperplane. We can define the set of production possibilities as the intersection of all closed half-spaces that lie above its supporting hyperplanes. We can define the set of normalized profit and price possibilities in a similar manner. The coefficients of a nonvertical supporting hyperplane H to the set of production possibilities correspond to a boundary point of the set of normalized profit and price possibilities. Similarly, the coefficients of a nonvertical supporting hyperplane H* correspond to a boundary point of the set of production possibilities. 6.5 Extensions 6.5.1 Introduction We continue our study of uniqueness by considering unique marginal products corresponding to a given production plan without unique net supplies corresponding to a given normalized price system and vice versa. We first assume that the production function is essentially smooth and convex, but not essentially strictly convex. Under this set of assumptions the normalized profit function can be defined as the Legendre conjugate of the production function, as before. However, the normalized profit function is not differentiable and the Legendre transformation is not symmetric in the production function and the normalized profit function. 8 6.5.2 Convex Conjugacy To characterize the production plans, possibly not unique, corresponding to a given normalized price system, we introduce the subgradient of the normalized profit function. We say that a production plan y is a subgradient of the normalized profit function F*(y*) at a normalized price system y*, if

for all z*. If the normalized profit function F*(y*) is finite at a normalized price system y*, the linear function H*(y*), defined by

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is a nonvertical supporting hyperplane to the set of normalized profit and price possibilities at the normalized price system and profit level (y*,F*(y*)). The production plan y corresponding to the subgradient of the normalized profit function need not be unique. Any production plan that maximizes normalized profit at the normalized price system y* is a subgradient of the normalized profit function for that price system. We can define the normalized supply correspondence, say M*(y*), as the set of all subgradients of the normalized profit function F*(y*) for the normalized price system y* or the set of all profit-maximizing production plans y corresponding to the normalized price system y*. This correspondence is identical to the subdifferential of the normalized profit function. Where the normalized supply correspondence is single-valued, the normalized profit function is differentiable and the subdifferential is identical to the gradient of the normalized profit function. The effective domain of the normalized profit function F* is the set of all normalized price systems y* for which F*(y*) is finite. We can define the relative interior of this set as the interior of the set considered as a subset of the smallest linear subspace containing the set. The normalized supply correspondence is nonempty for any normalized price system in the relative interior of the effective domain of the normalized profit function. We can define the effective domain of the normalized supply correspondence as the set of all normalized price systems y* for which M*(y*) is nonempty. The effective domain of the normalized supply correspondence is nonempty. It contains the relative interior of the effective domain of the normalized profit function F*(y*) and is contained in the effective domain of the normalized profit function. The normalized supply correspondence M*(y*) is nonempty at the origin and includes zero. A correspondence M*(y*) is cyclically monotone if

for any pairs (i=0,1, . . . , m) such that yi is contained in A cyclically monotone correspondence is said to be maximal if it is not contained in any other monotone correspondence. The normalized supply correspondence is a maximal cyclically monotone correspondence.

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We can define the inverse of a normalized supply correspondence M*(y*), say M* 1(y), as the set of all normalized price systems y* such that y Î M*(y*). The inverse of a normalized supply correspondence M*(y*) for an essentially strictly convex normalized profit function is a continuous function, namely, the marginal productivity function M(y). This function is not one-to-one since the normalized supply correspondence M*(y*) is not single-valued. Given a nonnegative, closed, and convex production function defined on a convex set containing the origin, the normalized supply correspondence is one-to-one, or globally univalent if and only if the production function is essentially smooth and essentially strictly convex. Dually, the marginal productivity correspondence is globally univalent if and only if the normalized profit function is essentially smooth and essentially strictly convex. Since the normalized supply correspondence and the marginal productivity correspondence are mutually inverse, the marginal productivity correspondence is globally univalent if and only if the normalized supply correspondence is globally univalent. Essential smoothness and essential strict convexity are necessary and sufficient for global univalence of the marginal productivity correspondence and, dually, the normalized supply correspondence. 9 Since the normalized profit function corresponding to a production function that is convex but not essentially strictly convex is not essentially smooth, the Legendre conjugate is not defined for the normalized profit function. The Legendre transformation is not symmetric in the production function and the normalized profit function. However, we can extend our analysis of the duality of these functions by introducing a generalization of the Legendre transformation, namely, the conjugacy correspondence. A normalized profit function, not necessarily essentially smooth, is essentially strictly convex if it is convex and strictly convex on any convex subset of the effective domain of the normalized supply correspondence M*(y*). If, in addition, the normalized profit function is essentially smooth, this function is convex and strictly convex on the interior of its effective domain. Given a nonnegative, essentially strictly convex, closed, normalized profit function, we can define the convex conjugate function as

where y is a production plan. We take the domain of the convex

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conjugate function F(y) to be all of Rn. The convex conjugate of the normalized profit function is the production function. The production function is nonnegative, essentially smooth, and convex. The effective domain of the production function is a convex set containing the origin with nonempty interior and the value of the production function at the origin is zero. If the supremum of the definition of the convex conjugate function F(y) is attained for some normalized price system y*,

for all normalized price systems z*, so that y is a subgradient of the normalized profit function F*, corresponding to the normalized price system y*. The subdifferential of the normalized profit function F* is nonempty for any normalized price system in the relative interior of the effective domain of the normalized profit function. Convexity of the normalized profit function implies that it is differentiable almost everywhere in the interior of its effective domain. The conjugacy correspondence between the normalized profit function and the production function is symmetric. The convex conjugate of the production function F is defined by

where y* is a normalized price system. We take the domain of the convex conjugate function to be all of Rn. The convex conjugate of the production function is the normalized profit function. The effective domain of the normalized profit function contains the range of the marginal productivity function M(y). The Legendre conjugate of the production function is the convex conjugate restricted to the range of the marginal productivity function. The value of the convex conjugate function on the rest of its effective domain can be obtained by the limiting operation defined above. All of the results we have presented for a nonnegative, essentially smooth, convex production function can be dualized for a nonnegative, essentially smooth, convex normalized profit function. 6.5.3 Twice Differentiability The traditional starting point for the theory of production is the assumption that the production function is twice differentiable on an

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open set of Rn. 10 Under this assumption the production function is convex if and only if the Hessian matrix is positive semidefinite for every production plan y in the open set. We take the open set to be the nonempty interior of a convex set, namely, the effective domain of an essentially smooth production function. Using only the assumption that the production is essentially smooth, we find that a convex production function is differentiable, hence, continuously differentiable on the interior of its effective domain. Essential smoothness is necessary and sufficient for uniqueness of the marginal products corresponding to a given production plan. Twice differentiability of an essentially smooth production function is sufficient but not necessary for uniqueness of the marginal products. Twice differentiability of the production function does not imply that the production function is essentially strictly convex, so that the net supplies corresponding to a given normalized price system need not be unique. A production function can be twice differentiable without being essentially strictly convex. Wherever essential strict convexity fails, the normalized profit function is not differentiable and the net supplies corresponding to some normalized price system are not unique. Even if the production function is essentially strictly convex, so that the normalized supply function is well defined, twice differentiability of the production function does not imply twice differentiability of the normalized profit function or, equivalently, differentiability of the normalized supply function. The production function may be essentially smooth and essentially strictly convex without being twice differentiable. Similarly, twice differentiability of the normalized profit function is neither necessary nor sufficient for essential smoothness and essential strict convexity of the normalized profit function, twice differentiability of this function does not imply differentiability of the marginal productivity function. By strengthening our strict convexity assumption on the production function we obtain a correspondence between twice differentiability of the production function and twice differentiability of the normalized profit function. A production function twice differentiable on an open convex set is strictly convex if and only if the Hessian matrix is positive semidefinite everywhere and positive definite almost everywhere on any line segment in the open convex set. We say that a twice differentiable and essentially smooth production function is strongly convex if the Hessian matrix is positive definite everywhere on the open convex set.11 Similarly, we say that a twice differentiable and essentially

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smooth production function is essentially strongly convex if it is convex and strongly convex on the interior of its effective domain. If the production function is twice differentiable, essentially smooth, and essentially strongly convex, the normalized profit function is twice differentiable, essentially smooth, and essentially strongly convex, and vice versa. 12 Further, if the production function is twice differentiable, essentially smooth, and essentially strictly convex but not essentially strongly convex, the normalized profit function is not everywhere twice differentiable; by duality this proposition holds with the role of the production function and the role of the normalized profit function interchanged.13 Notes 1. The Legendre transformation is discussed by Courant and Hilbert (1962, pp. 32 35) and Rockafellar (1970b, pp. 251 260). The Legendre transformation is employed by Lau (1972, 1973) to study duality in the theory of production. For a review of the literature on duality in production, see Diewert (1974a) and Lau (1974). 2. The theory of conjugate convex functions was originated by Fenchel (1949, 1953). The Legendre transformation is compared with the conjugacy correspondence by Fenchel (1949) and Rockafellar (1970b, esp. pp. 256 257). Rockafellar provides proofs and references for the results we require. 3. See Rockafellar (1970b, Theorem 25.5, p. 246). A convex production function is differentiable almost everywhere on the relative interior of its effective domain. 4. This property is also referred to as global univalence; see section 6.5.2. 5. See Rockafellar (1970b, Theorem 24.8, pp. 268, 251 254.). 6. The normalized profit function is employed by Lau (1972, 1973) and by Jorgenson and Lau (1974b); see also the additional references given by Jorgenson and Lau (1974b). Our analysis of duality is unaffected by the presence of fixed commodities (inputs or outputs) which can be treated as fixed parameters. The results we derive for the profit function can be applied directly to normalized cost functions (outputs fixed) or revenue functions (inputs fixed). 7. Similar properties of dual gauge functions are employed by Hanoch (1974) and McFadden (1978). 8. A closely parallel analysis of differentiability for dual gauge functions is given by McFadden (1978, pp. 111 115). McFadden does not employ the conjugacy correspondence. Results along similar lines are given by Gorman (1968) and Arrow and Hahn (1970). 9. Gale and Nikaido (1965) have given conditions for a differentiable vector-valued function to be globally univalent or one-to-one. See also, Nikaido (1965, pp. 355 392). These conditions are given in terms of the Jacobian of the vector-valued function. The Hessian matrix of the production function is the Jacobian matrix of the marginal productivity function. Differentiability of the marginal productivity function is neither necessary nor sufficient for global univalence of this function. Essential smoothness and essential strict convexity rather than twice differentiability of the production function provide an appropriate starting point for an analysis of global univalence of the marginal productivity function. For further discussion of twice differentiability, see, section 6.5.3.

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10. A function is twice differentiable if a second differential of the function exists. Existence of second-order partial derivatives is necessary but not sufficient for twice differentiability. Existence of continuous second-order partial derivatives is sufficient but not necessary. While a differentiable convex function is continuously differentiable, a twice differentiable convex function is not necessarily twice continuously differentiable. A convex function is twice differentiable almost everywhere in the interior of its effective domain. See Busemann (1958). Hicks (1946, pp. 78 111, 319 325), assumes the existence of a second differential of the production function. Samuelson (1983, pp. 57 89), assumes that the production function has continuous second-order partial derivatives. Arrow and Hahn (1970, p. 72) assume that the profit function has continuous second-order partial derivatives. 11. McFadden (1978) employs the terminology ''differentially strictly convex" to describe strongly convex dual gauge functions in the theory of production. The Gale-Nikaido condition for global univalence of a vector-valued function applied to the marginal productivity function is equivalent to strong convexity. See footnote 9. 12. If the production function is n times differentiable, that is, possesses differentials up to the n-th order, essentially smooth, and essentially strongly convex, the normalized profit function is n times differentiable, essentially smooth and essentially strongly convex. 13. Duality between production functions and normalized profit functions is analyzed by Lau (1972, 1973) under the assumption that the production is twice differentiable and strongly convex. Lau employs the Legendre transformation; this transformation is implicit in earlier treatments of duality by Hotelling (1932) and Samuelson (1953 1954). Samuelson (1960) has discussed the Legendre transformation without specific reference to the theory of production under the assumption that the dual functions related by the Legendre transformation are twice differentiable and strongly convex. Dhrymes (1967) has used the assumption of strong convexity to derive differentiability of the normalized supply function from twice differentiability of the production function.

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7 Efficient Estimation of Nonlinear Simultaneous Equations with Additive Disturbances Dale W. Jorgenson and Jean-Jacques Laffont This chapter develops a theory of CUAN estimation for systems of nonlinear simultaneous equations with additive disturbances. We first derive the Cramer Rao lower bound for the variance of a CUAN estimator. The method of maximum likelihood can be used to generate an estimator that attains this bound. We show that minimum distance and instrumental variables estimators cannot generally attain the Cramer Rao bound. 7.1 Introduction The statistical theory of estimation for systems of linear simultaneous equations is based on the construction of consistent, uniformly asymptotically normal (CUAN) estimators. 1 Within this class it is natural to select estimators that are, in addition, efficient; we refer to such estimators as best consistent uniformly asymptotically normal (Best CUAN) estimators.2 The purpose of this chapter is to develop a theory of CUAN estimation for systems of nonlinear simultaneous equations with additive disturbances.3 The theory of CUAN estimation for systems of linear simultaneous equations can be summarized as follows: estimators can be constructed that attain the Cramer Rao lower bound for the variance of a CUAN estimator.4 The ordinary least-squares estimator for the reduced form is CUAN, but not generally Best CUAN. Best CUAN estimators can be constructed by the method of maximum likelihood, the minimum distance method, and the method of efficient instrumental variables.5 Malinvaud (1970) has developed a theory of CUAN estimation for systems of nonlinear simultaneous equations with an explicit reduced form having additive disturbances.6 For this class of nonlinear systems of ordinary leastsquares estimator for the reduced form is CUAN, but not generally Best CUAN; Malinvaud shows that Best CUAN estimators can be constructed by the method of maximum

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likelihood and the minimum distance method. Hausman (1975) has shown that a Best CUAN estimator can be constructed for a closely related class of models by the method of efficient instrumental variables. 7 Our first step in developing a theory of CUAN estimation for systems of nonlinear simultaneous equations with additive disturbances is to derive the Cramer Rao lower bound to the variance of a CUAN estimator. The method of maximum likelihood can be used to generate a Best CUAN estimator. As for linear systems, the burden associated with the conventional approach to computation of the maximum likelihood estimator, based on the Newton Raphson method or the method of scoring, is very substantial.8 We can distinguish two alternative lines of attack on the problem of reducing the computational burden for Best CUAN estimation of systems of nonlinear simultaneous equations. First, the computation of the maximum likelihood estimator can be simplified. Rothenberg and Leenders (1964) have shown that the first step of the Newton Raphson method is Best CUAN, provided that the initial parameter value is a consistent estimator.9 Although Rothenberg and Leenders apply this result only to systems of linear simultaneous equations, the proposition holds for nonlinear systems as well. In this chapter we concentrate on a second line of attack, namely, construction of estimators by methods, such as minimum distance or instrumental variables, that are easier to compute. Amemiya has proposed a minimum distance estimator for a single equation in a system of nonlinear simultaneous equations.10 We extend his method of estimation to systems of nonlinear simultaneous equations and his proof that the resulting estimator is CUAN. However, we show by means of an example that the minimum distance estimator is not generally Best CUAN. We also develop an instrumental variables estimator for a system of nonlinear simultaneous equations, extending the efficient instrumental variables estimator for linear systems developed by Brundy and Jorgenson (1971, 1973).11 We show that the resulting estimator is CUAN and, in fact, asymptotically equivalent to our minimum distance estimator. Again, the efficient instrumental variables estimator is not Best CUAN. We conclude that minimum distance and instrumental variables estimators can be constructed that are CUAN, but that these estimators are not generally Best CUAN. Further research on Best CUAN

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estimation for systems of nonlinear simultaneous equations should be focused on simplifying the computation of the maximum likelihood estimator. 12 7.2 The Model We consider the following system of simultaneous equations

or in vector form

with

y1t, . . . , yPt are the endogenous variables; for each i = 1, . . . , P,z1t is a Qi-vector of endogenous and (nonrandom) exogenous variables; b is an R-vector of unknown parameters; f1 is a nonlinear function with continuous second derivatives (Ri is the number of elements of b in fi), ut, t = 1, . . . , T are random vectors such that Euit = 0, i = 1, . . . , P; t = 1, . . . , T, Eutu't = W of full rank and Eutu't = 0 if t ¹ t'. This form of the model can be obtained from a model with different parameters bi in each equation. If there are constraints on these parameters, they are solved to obtain a minimal set of parameters b. We assume that the constraints can be solved uniquely at least in a neighborhood of the true value b0. We assume further that the parameter b is identifiable.13 The model is now rewritten differently to use the simplifying Kronecker notation. Let Yi = [yi1, . . . , yiT]', i = 1, . . . , P and Y = [Y1, . . . , YP]'. Let Z be the stacked vector of variables appearing on the right of system (7.1). Let F(Z,b) = [f1(z11,b), . . . , f1(z1t,b), . . . , fP(zP1,b), . . . , fP(zPT,bP)]'

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Then (7.1) can be rewritten

We choose a specific notation for the set of exogenous variables (independent of U) which are K* in number.

X will be in this section a matrix of K variables constructed from X* with maxi Ri £ K £ K *. We refer to assumptions specified in section 7.2 as A0. 7.3 Cramer Rao Bound 7.3.1 Introduction The comparison of different full information methods 14 to estimate nonlinear econometric systems with additive disturbances requires an explicit form of the Cramer Rao bound. In this section, we derive the Cramer Rao bound when the matrix of variances and covariances of errors is unknown, completely known, or known to be diagonal. For this section we rewrite the model more symmetrically

where {xk}k = 1, . . . , K* are predetermined variables {yp}p = 1, . . . , P are endogenous variables {qa} is a Ra vector of parameters a = 1, . . . , P. We assume that the Jacobian of the system is never vanishing (it is clearly a strong assumption) and we assume the multinormal distribution for the errors so that we can derive the logarithm of the likelihood function.

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7.3.2 Unrestricted W Matrix The logarithm of the likelihood function is

and Bt is the matrix of such derivatives. We concentrate the likelihood

Substituting (7.6) in (7.4) we obtain the nonconstant part of the concentrated likelihood.

We want to obtain

which will be the inverse of the Cramer-Rao bound for the parameters q. 15

16

17

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From (7.6)

with the following simplifications (since fit depends only on qi and not on qj with j ¹ i)

with the following simplifications

We can now obtain the second derivatives 18

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We use the general formula

using (7.13) (7.15)

The first element of the right-hand side of (7.20) is then

The third element of the right-hand side of (7.20) becomes

The second terms of the right-hand side of (7.20) becomes

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We compute now

The first term of the right-hand side of 1/T (7.20) is

We assume that

From (7.28) we obtain

since the estimations

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Let

The second term of the right-hand side of (7.20)

The third term of the right-hand side of (7.20)

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The fourth term of the right-hand side of (7.21)

For the special case of the reduced form considered by Malinvaud (1980)

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is nonrandom as well as

If all the derivatives are bounded in the sample space for all i, a = 1, . . . , P

Since there is no endogenous variable on the right-hand side

Malinvaud (1980) shows that the maximum likelihood estimator reaches this Cramer Rao bound. Similarly, the bound is attained by a minimum distance estimator weighted by a matrix S which converge to W. When the model is linear in parameters

C3 is only a function of variables.

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7.3.3 Restricted W Matrix The logarithm of the likelihood is

We first consider the case where W is known so that we have only to differentiate with respect to q.

Using (7.24) and (7.26) we obtain the matrix form

using our previous notation. Asymptotically, the gain represented by the knowledge of W corresponds to C1. Next, we consider the case where W is known to be diagonal. The nonconstant part of the concentrated likelihood function is then

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C3 and C4 will remain as in the general case. We have to compute

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So that using our notation we obtain

7.3.4 Conclusion The general form of the Cramer Rao bound can be decomposed in five parts

where C2 would be the Cramer Rao bound if there were no endogenous variables on the right, represent the modification due to the existence of endogenous variables on the right when W is known, and C1 represents the additional change due to the necessity of estimating W. It is not difficult to specialize the results to the linear case considered by Rothenberg and Leenders. When there is no constraint on W, it is possible in the linear case to obtain the Cramer Rao bound from the bound if there were no endogenous variables on the right, by simply replacing the ''endogenous variables" by the systematic part of the reduced form associated with them. In the nonlinear case, the derivation is much more complicated. 7.4 Minimum Distance We next consider a family of minimum distance estimators of the parameter b in the system of nonlinear equations (7.2). We obtain the minimum distance estimator by minimizing

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with X defined above,

a consistent estimator of order 0(T 1/2) of W.

A1. The parameter space is compact and the matrix X'X is of full rank with probability one. PROPOSITION 7.1. Under A0, A1, a minimum distance estimator exists. A2. ut, t = 1, . . . , T are identical independently distributed random vectors. A3.

A4.

exists and is equal to the nonsingular matrix M.

uniformly in b. Then plim

of rank R uniformly in b, with the notation

PROPOSITION 7.2. Under A1, A2 to A4 a minimum distance estimator is consistent. PROOF:

Note that F(Z, b) = Y U. Multiplying each member of (7.47) by

gives

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which goes to zero in probability when T ® ¥, by Chebychev's theorem (A2 A3). Let

By definition of

Then, 0 £ a'2a2£a'1a1. Since

therefore

The left-hand side of (7.48) converges to 0 in probability when T ® ¥. Also

which by A4 converges to the full rank matrix H when T ® ¥.

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Consequently, we see on (7.48) that

A5.

Q.E.D.

uniformly in b, i = 1, . . . , P, j = 1, . . . , R

PROPOSITION 7.3. Under A1 to A6.

with

between b0 and

Then

with

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To obtain the asymptotic distribution of we will first derive the asymptotic distribution of the pseudoestimator obtained by replacing by By assumption = W + D1 with D1 ~ 0(T 1/2) so

As already shown, the first member of (7.51) converges in probability to zero as T ® ¥, the second member to the matrix W 1 Ä M 1 by A3, and the third member to the matrix

which lies between

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and b0 converges to b0 when T ® ¥ because of the consistency of . Then 19

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Using A2, A3, A6, we can derive

Finally from (7.49) Var plim

so that from A6

Let us now consider

Let us consider an element aj.

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The matrix X(X'X) 1 X'¶2 fh/¶bj¶bi is the projection of ¶2fh/¶bj¶bi on X, so that it is independent of u. Then ×(¶2fh/¶bj¶bi) ~ 0(T1/2). A current element of ¶2J/¶b¶b' is then of the form

The matrix ¶2J/¶b¶b' can then be rewritten symbolically

so that

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By Cramer's theorem (1971, p. 254)

Since is normal, the asymptotic distribution of is the same as that of difference of these two quantities has zero probability limit. Q.E.D.

because the

7.5 Instrumental Variables To generate a family of instrumental variables estimators of the parameter b is the system of nonlinear equations (7.2), we linearize the system around the true value, say b0

with In general fijt which depends on endogenous variables is correlated with errors. Consider the estimation of (7.52) by the instrumental variables method. 20 We denote the set of instrumental variables

where each submatrix Wij has Rj columns. Equation (7.52) can be rewritten Y = F(b b0) with

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The instrumental variables estimator is then

It then appears that the minimum distance estimator is asymptotically equivalent to the pseudoestimator with a specific choice of instrumental variables. If now we can find the best set of instrumental variables

, the best choice of X will be X * such that

at least asymptotically so that F and W can be replaced by consistent estimators. The search for a best set of instrumental variables will reveal the nature of the difficulty. In the linear case an efficient set of instrumental variables is

where is a typical element of (with being a consistent estimator of W) and Wj is a consistent estimator of the systematic part of variables in the jth equation independent of the errors U. So, by

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which analogy, we can attempt to construct consistent estimators of the systematic part of derivatives are not correlated with the U. It is then clear that we want X to be variables independent of U but nevertheless as closed as possible to ¶fi/¶b'i|b0, i = 1, . . . , P. Since there is no constraint on the number of X, as many powers of X as possible seem ideal. However, after some n the powers become probably useless, the n depending on the degree of nonlinearity of the derivatives. Moreover, this method leads to a huge matrix X'X which we have to invert, and leads to X which are collinear. Our suggestion is then the following one: 1. Find a consistent estimator of b using a NL2SLS estimator for example with a minimum set of X (Maxi Ri), taken as a subset (eventually) of exogenous variables. 2. Simulate the model to obtain values of endogenous variables. 3. Use the results of 1 and 2 to approximate the derivatives of fi, i = 1, . . . , P. Malinvaud (1980) restricts himself to the case where a reduced form is available and shows that a minimum distance estimator with S = W 1 or a consistent estimator of W gives the best minimum distance estimator. Moreover, if normality is assumed, it is asymptotically efficient. In his case the

are not correlated with the U, so that if a consistent estimator bi of them to eliminate the dependence on U.

is available, it is not necessary to project

They are the best possible auxiliary variables since they obviously maximize asymptotically

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Malinvaud tells us that asymptotically it is not worth the trouble since X = I does as well. 7.6 Efficiency 7.6.1 Introduction We have developed an explicit form for the Cramer Rao lower bound for the variance of a CUAN estimator of the parameter b in the system of nonlinear simultaneous equations (7.2). This bound is attained by the full information maximum likelihood estimator. We have shown that the minimum distance and efficient instrumental variables estimators are asymptotically equivalent. We next consider the relative efficiency of the minimum distance estimator for a system of nonlinear simultaneous equations and for a single equation in such a system. 7.6.2 Minimum Distance versus Maximum Likelihood We show with an example that the minimum distance estimator of section 7.4 does not generally attain the Cramer Rao bound. It is sufficient to prove that one element in the inverse of the matrix of variances and covariances of the minimum distance estimator is different from the corresponding element in the inverse of the corresponding matrix for the Cramer Rao bound. We consider the system of nonlinear simultaneous equations

Element (1,1) in the inverse of the Cramer Rao bound a) From (7.46*) we have:

Then

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b) From (7.46*) we have: W11 F11

c) From (7.46*) we have:

Then C3(1,1) = 0. Finally, we obtain the element (1,1) of the inverse of the Cramer Rao matrix

Element (1,1) of the inverse of the asymptotic matrix of variances and covariances of the minimum distance estimator is

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Let

The element (1,1) is then

which differs from the corresponding element of the Cramer-Rao bound. 7.6.3 Limited Information versus Full Information We can also consider the relative asymptotic efficiency of the minimum distance estimator for a system of nonlinear simultaneous equations proposed in section 7.4 with the corresponding estimator for a single equation developed by Amemiya (1974). We consider only the case without restrictions across equations. A minimum distance estimator is obtained for each equation i = 1, . . . , P by minimizing

for a given choice of X. To each choice of X corresponds also a minimum distance estimator as defined in section 7.4. We will show that the corresponding estimator for a system of nonlinear simultaneous equations is always asymptotically better (or as good) as the estimator for a single equation. The asymptotic matrix of variances and covariances of the single equation estimator, given by Amemiya (1974), is

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The asymptotic matrix of variances and covariances of the estimator for a system of equations, given above, is

It is then clear that if we replace ¶฀fj/¶b'j|b0 by the familiar Zj (the set of variables on the right-hand side of the jth structural equation), the formal analogy with the classical comparison of 3SLS and 2SLS is complete. Note that here X is not necessarily the set of all exogenous variables. Therefore, the usual proof for linear systems implies our result. We can also deduce that the two estimators coincide when W is diagonal and when

is invertible for i = 1, . . . , P. The invertibility condition means in particular that the matrix is square, i.e., that in each equation there are as many independent variables X as unknowns. We have seen that A4 requires

If the matrix X is restricted to exogenous variables, the condition we find is similar to the just identification in the linear case. However, in the nonlinear case we know that it is not a necessary condition of identification (Fisher, 1966). 21 Since we do not have to restrict ourselves to exogenous variables and can use powers of the X or fitted values, the condition

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is not really a constraint so long as the model is truly nonlinear. The condition

imposes a limit on the number of elements of X to use so that the two estimators are equivalent. We derive directly the result for the two-equations case

The single-equation estimator is obtained by minimizing

and the asymptotic matrix of variances and covariances is

We are led to compare

and the upper left corner of the inverse of

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The upper left corner we look for is then

because

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We inverted a matrix

where

is a positive definite matrix: so is K1; so is

; and

B is definite positive and C is semidefinite positive, B 1 (B + qC) 1 is semidefinite positive. If C is definite positive and q > 0, the minimum distance estimator is strictly better than the corresponding single-equation estimator. If W is diagonal the minimum distance estimator coincides with the single-equation estimator. It is also true if C = 0.

If X'(¶f2/¶b'2)|b is invertiblei.e., square and nonsingularthen C = 0. There are as many exogenous variables as unknowns in the second equation; this yields just identification in the linear case. 7.6.4 Conclusion We conclude that except for the case of linearity in the variables, the minimum distance and efficient instrumental variables estimators are CUAN but not Best CUAN. On the other hand these estimators appear to be an interesting step in the estimation of nonlinear systems

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with constraints across equations, since they provide consistent estimators that incorporate all of the constraints. A consistent estimator can be used to initialize a one-step linearized maximum likelihood estimator. This estimator is asymptotically equivalent to the maximum likelihood estimator, just as in the case of systems of linear simultaneous equations considered by Rothenberg and Leenders (1964). Notes 1. The statistical theory of CUAN estimation is discussed by Rao (1973), pp. 344 351. 2. Best CUAN estimators are discussed by Rao (1973), pp. 350 351. 3. This specification for simultaneous equations models is considered by Eisenpress and Greenstadt (1966). 4. A complete review of the theory of CUAN estimation for systems of linear simultaneous equations models is presented by Malinvaud (1980) and Rothenberg (1974). 5. See Malinvaud (1980) for a discussion of maximum likelihood and minimum distance estimators, and Brundy and Jorgenson (1971) for a discussion of efficient instrumental variables estimators. 6. See Malinvaud (1970), pp. 348 366. 7. See Hausman (1975). 8. The Newton-Raphson method is described by Eisenpress and Greenstadt (1966). 9. See Rothenberg and Leenders (1964). 10. See Amemiya (1974). Minimum distance estimators for a single equation in a system of simultaneous equations are also discussed by Edgerton (1972), Kaleijian (1971), and Zellner, Huang and Chau (1965). 11. See Brundy and Jorgenson (1971, 1973). 12. Important progress along these lines is reported by Berndt, Hall, Hall, and Hausman elsewhere in this issue. 13. Identifiability for systems of nonlinear simultaneous equations is discussed by Fisher (1966), pp. 127 167. 14. We restrict ourselves to cases without constraints across equations to allow comparisons with the RothenbergLeenders (1964) results in the linear case, but there is no substantive difficulty extending these derivations to more general cases. We use the notation of Eisenpress-Greenstadt (1966). 15. See Koopmans and Hood (1953). 16. We adopt the following convention. The differentiation of a numerical function with respect to a column (row) vector of parameters is a column (row) vector. 17. We use the following result. If A = [aij] is a nonsingular matrix with inverse A 1 = [aij], then ¶ log | det A|/¶aij = aij. 18. We do not use ''prime indices," so that the sign' must always be interpreted as a transposition. 19. Amemiya (1973), p. 10, Lemma 4: let fT(w,q) be a measurable function on a measurable space W and for each w in W a continuous function for q in compact set H. If fT(w,q) converges to f(q) a.e. uniformly for all q in H and if qT(w) converges to q0 a.e., then fT(w,qT(w)) converges to f(q0) a.e. 20. It is only a pseudomodel (since b0 is not known), for which we construct a pseudoestimator.

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21. This condition has been obtained also in Edgerton (1972), as a necessary condition for the workability of 2SLS methods suggested by Goldfeld and Quandt (1968). Note that it is not really a constraint for the method since as many powers of the X as necessary can be introduced to satisfy it.

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8 Tests of a Model of Production for the Federal Republic of Germany, 1950 1973 Klaus Conrad and Dale W. Jorgenson The objective of this chapter is to present tests of the theory of production for aggregate time series data from the Federal Republic of Germany for the period 1950 1973. For this purpose we implement two alternative econometric models, based on the translog representations of production and price functions. Our test results are consistent with the theory of production and imply that this theory can be employed as an appropriate starting point for construction of a macroeconomic model of Germany. 8.1 Introduction Our first objective is to develop tests of the theory of production that do not presuppose homogeneity and additivity of the function relating outputs and inputs. For this purpose Jorgenson (1986) employs the translog production function, represented by a functional form that is quadratic is the logarithms of inputs and outputs and in time, considered as an index of technology. 1 For this representation of a production function neither additivity nor homogeneity is part of the maintained hypothesis. The resulting functions permit a greater variety of substitution and transformation patterns than functions based on constant elasticities of substitution and transformation.2 A complete model of production includes the production function and necessary conditions for producer equilibrium. Under constant returns to scale this model implies the existence of a price function, defining the set of prices consistent with the condition that the value of output is equal to the value of input.3 Necessary conditions for producer equilibrium giving relative prices as functions of relative product and factor intensities, imply the existence of conditions determining relative product and factor intensities as functions of relative prices. The price function and the conditions determining product and factor intensities are dual to the production function and the necessary conditions for producer equilibrium. Furthermore, the hypothesis that the system of marginal productivity functions or supply and

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demand functions for outputs is consistent with profit maximization is not an assumption but a hypothesis to be tested. Our second objective is to exploit the duality between production and price functions. 4 For this purpose, Jorgenson (1986) represents the price function by functional forms that are quadratic in the logarithms of prices of inputs and outputs and in time.5 He refers to his representation of a price function as the translog price function. Tests of the theory of production can be carried out by means of a production function as well as a price function. For each of these models we will derive tests of the theory of production based on integrability of the corresponding marginal productivity and supply and demand functions. We will also derive tests based on monotonicity and convexity of the production and price functions. 8.2 Translog Production and Price Functions Specializing to the case of two outputsconsumption C and investment Iand two inputslabor L and capital Kthe production function F can be represented in the form

where t is time, considered an index of technology. The translog production function of Jorgenson (1986) can be derived from a second-order Taylor's series expansion of the logarithm of the production function (8.1) in terms of the logarithms of the outputs and inputs around the point (C, I, K, t) = (1, 1, 1, 0). By neglecting the remainder term in the Taylor's series expansion and by treating the first- and second-order logarithmic derivatives as unknown parameters, we obtain

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If the production function F is homogeneous of degree one, the sum of logarithmic derivatives of F with respect to logarithms of the inputs and outputs is equal to unity,

Second, the sum of second-order logarithmic derivatives with respect to the logarithm of any input or output is equal to zero, for example,

Finally, the sum of second-order logarithmic derivatives with respect to time and all inputs and outputs is equal to zero,

For the translog representation of a production function this implies the restrictions

The restrictions (8.3) imply that the translog production function is homogeneous of degree one. Using the translog representation of the production function, the necessary conditions for producer equilibrium are that elasticities of the production function F with respect to consumption, investment, and capital are equal to the corresponding value shares,

where qC, qI, qK, and qL are the prices of consumption goods,

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investment goods, capital, and labor, respectively. For the translog representation of a production function this implies

If the translog production function is homogeneous of degree one, the sum of the value shares is equal to unity, so that the value of output is equal to the value of input,

Next, we can differentiate the production function F logarithmically with respect to time to obtain the rate of technical change,

The rate of technical change is the rate of growth of labor input with respect to time, holding outputs and capital input constant. If we represent the production function F by the translog production function in (8.2), we obtain

To measure the rate of technical change wt, we differentiate the logarithm of the production function totally with respect to time,

From this expression we get the rate of technical change wt in terms of the value shares and the rates of growth of the inputs and outputs,

From the representation of the value shares as functions of consumption, investment, capital and time we can derive ratios of prices of consumption, investment and capital to the price of labor as functions of the same variables,

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Therefore, Jorgenson (1986) refers to the system of functions giving the three value shares and the rate of technical change wt as the marginal productivity functions. As we have represented the production function F by means of a second-order approximation in the logarithms of outputs, capital and in time, we have derived value shares as loglinear functions in the same variables. Similarly, we can represent the marginal productivity functions as a firstorder approximation in the logarithms of C, I, K and in t. By duality in the theory of production we can characterize the production function in terms of the price function. We represent the price function in the form

The translog price function of Jorgenson (1986) can be derived from a second-order Taylor's series expansion of the logarithm of the price function (8.8) in terms of logarithms of prices and time around the point (qC, qI, qK, t) = (1, 1, 1, 0). By neglecting the remainder term in the Taylor's series expansion and by treating the first and secondorder logarithmic derivatives as unknown parameters we obtain

As before, homogeneity of degree one of the price function P implies that the parameters of the translog price function satisfy restrictions that are precisely analogous to the restrictions on the parameters of the translog production function given under (8.3). Differentiating the price function with respect to prices while holding the level of profit at zero, we obtain the value shares as function of

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the prices of consumption, investment and capital and of time. For example,

by logarithmic differentiation of the translog representation of the price function. Finally, we can differentiate the price function P logarithmically with respect to time to obtain the negative of the rate of technical change,

The negative of the rate of technical change is the rate of growth of the price of labor with respect to time, holding prices of consumption, investment and capital constant. We can express the negative of the rate of technical change wt in terms of relative value shares and rates of growth of the prices of the two outputs and inputs,

Using the fact that the value of outputs equals the value of inputs, we conclude that

Given the price function P, we have derived the three value shares as functions of the prices of consumption, investment, capital and time. These shares can be rewritten as ratios of consumption, investment and capital to labor as functions of the same variables,

Therefore, Jorgenson (1986) refers to the system of functions giving value shares and the rate of technical change wt = ¶ ln P/¶t as functions of the prices qC, qI, qK, and t as the net supply functions.

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Unless the production function F and the price function P are linear logarithmic with a constant rate of technical change, the translog representation of these functions provide alternative second-order approximations to the underlying technology. 6 8.3 Integrability For the preceding section we have derived properties of the marginal productivity functions from the properties of the production function F. To test the theory of production this line of reasoning can be reversed, beginning with properties of the marginal productivity functions. If the marginal productivity functions are generated by a twice continuously differentiable production function, the marginal productivity functions satisfy the symmetry restrictions7

and so on for all cross-partial derivatives of ln F. Conversely, if the marginal productivity functions are continuously differentiable and satisfy the symmetry restrictions, they can be generated by a twice continuously differentiable production function F. For the translog marginal productivity functions the symmetry conditions take the form

These conditions are necessary and sufficient for the marginal productivity functions to be generated from a translog production function. To test the theory of production we test the symmetry restrictions on the unknown parameters. As mentioned in section 8.2, homogeneity of degree one of the production function implies that the value shares wC, wI, and wK sum to unity and that these shares are homogeneous of degree zero in the same variables. Conversely, if the shares are functions of consumption, investment and capital that are homogeneous of degree zero and

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sum to unity, the marginal productivity functions can be generated by a production function that is homogeneous of degree one. Turning next to the translog marginal productivity functions given under (8.5) we observe that homogeneity of degree zero of the value shares and the rate of technical change as functions of C, I, and K implies the restrictions

However, as shown in section 8.2, homogeneity of degree one of the translog representation of a production function of the same degree implies the parameter restrictions given under (8.3). We conclude that if the translog marginal productivity functions satisfy the restrictions (8.11) for homogeneity of degree zero together with the symmetry restrictions (8.10), we obtain the equivalent set of restrictions (8.3) under which the translog production function is homogeneous of degree one. Therefore, a sum of one for the value shares, the restrictions (8.11) and the symmetry restrictions (8.10) provide a test of the hypothesis that the marginal productivity functions can be generated by a production function that is homogeneous of degree one. Similarly, we have derived the properties of the net supply functions from the property of the price function P. If the price function is twice continuously differentiable the net supply functions satisfy the symmetry restrictions

and so on for all cross-partial derivatives of ln P. Conversely, if the net supply functions are continuously differentiable and satisfy the symmetry restrictions, they can be generated by a twice continuously differentiable price function. For the translog representation of the net supply functions the symmetry restrictions are the same as given under (8.10) for the translog marginal productivity functions. Furthermore, the restrictions derived from the

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condition that the net supply functions and the rate of technical change are homogeneous of degree zero are the same as given under (8.11). Together with the restrictions derived from the condition that the value shares sum to unity, the symmetry restrictions (8.10) and the homogeneity restrictions (8.11) imply homogeneity of degree one of the price function in the prices qC, qI, and qk. These restrictions provide a test of the hypothesis that the supply functions of consumption and investment and the demand function for capital can be generated by a price function that is homogeneous of degree one. 8.4 Monotonicity and Convexity of Translog Functions The restrictions on the form of a well-behaved production function (8.1) are that F is a monotone and convex function. Monotonicity requires that

For the translog representation of the production function the first-order partial derivatives with respect to consumption takes the form

where wC is the value share of consumption as before. Similar, we obtain

To check monotonicity at each point, we note that the marginal productivity functions imply wC ³ 0, wI ³ 0, wK £ 0 and wt £ 0. Evaluating the first-order partial derivatives at the point of expansion (C, I, K, t) = (1, 1, 1, 0) we obtain

Thus, necessary conditions for monotonicity are

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To test the hypothesis of monotonicity, we can employ simultaneous one-tailed tests of the hypotheses of the a's. Convexity of the production function F implies that the matrix of second-order partial derivatives with respect to the outputs C and I and input K is positive semi-definite. For the translog representation of the production function the second-order partial derivatives take the form

The translog production function is convex if and only if the corresponding Hessian matrix H is positive semidefinite in each point of observation. 8 Evaluating the derivatives at the point of expansion (C, I, K, t) = (1,1,1,0) we can form the matrix of second-order partial derivatives of the translog production function,

Under convexity of the production function the Hessian matrix is positive semi-definite, so that it can be represented in terms of its Cholesky factorization,9

where L is a unit lower triangular matrix,

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and D is a diagonal matrix with nonnegative elements,

The matrix product takes the form

Given the a parameters, there is a one-to-one transformation between the b parameters of the translog production function and the d and l parameters of the Cholesky-factorization of the matrix exp( a0) H. This transformation takes the form

The symmetry restrictions and the restrictions implied by homogeneity of degree one of the production function imply that the remaining three transformations are linear combinations of the transformations given above and that d3 is equal to zero. Using this transformation, we can express the b parameters in terms of the parameters a, d and l. If the matrix H is positive semi-definite, the diagonal elements of the matrix D must be nonnegative,

We conclude that there are two inequality restrictions under convexity. Similarly, monotonicity of the translog price function requires

These first-order partial derivatives are analogous in form to the

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corresponding derivatives for the translog production function. Necessary conditions for monotonicity are

Monotonicity requires that at is opposite in sign to the corresponding parameter of the production function; to impose monotonicity we proceed as before. Convexity of the price function P implies that the matrix of secondorder partial derivatives with respect to the prices qC, qI, and qK is positive semi-definite. The problem of convexity for the translog price function can be treated in a strictly analogous way to the translog production function. 8.5 Tests of the Theory of Production Our first step in developing tests of the theory of production is to derive tests of restrictions on marginal productivity and supply and demand functions for outputs and capital input implied by the hypothesis that these functions are generated by production and price functions. To test integrability of the marginal productivity functions we add a stochastic component to the translog marginal productivity functions given under (8.5). However, when we specialize to national accounting data for which the value of output is equal to the value of input, that is, for which the value shares sum to unity, only two of the first three equations will have additive error terms that are distributed independently. If the value shares, each consisting of a nonrandom function and a random variable with expectation zero sum to unity for any values of C, I, K, and t, we obtain

so that the random variables are not distributed independently. To estimate the unknown parameters of the marginal productivity functions we combine the first two equations for the shares of consumption and investment goods output with the equation for the rate of technical change to obtain a complete econometric model,

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We estimate the parameters of the equation for the remaining value share of capital, using the restrictions

The complete model involves fifteen unknown parameters. Five additional parameters can be estimated as functions of these parameters. To test integrability we impose six symmetry restrictions

Given these equalities, there are nine unknown parameters to be estimated. As shown in section 8.3 tests of the symmetry restrictions are also tests of homogeneity of degree one of the underlying production function as the symmetry restrictions and the restrictions given under (8.13) imply that the marginal productivity are homogeneous of degree zero. We next turn to tests of monotonicity and convexity. Tests of these restrictions can be carried out in many sequences. We propose to test monotonicity and convexity in parallel, conditional on integrability. Our proposed tests procedure is presented in diagrammatic form in figure 8.1, where double lines indicate our proposed test procedure. For monotonicity of the production function we test four inequality restrictions

We first fit the econometric model based on the translog representation of the marginal productivity functions with symmetry imposed. To test monotonicity we form t ratios for the linear hypothesis corresponding to each of these restrictions. We reject monotonicity if the

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Figure 8.1 Tests of the theory of production. fitted value, say

is significantly negative (aC, aI) or positive (aK, at).

To express the convexity restrictions we reparameterize the equations for the value shares as follows:

The effect of the reparameterization is to replace the parameters bCC, bCI and bII by the parameters d1, d2 and l21. In terms of the new parameters necessary conditions for convexity take the form

We first fit the econometric model with symmetry imposed. To test convexity we require t ratios for the linear hypothesis corresponding

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to each of these inequality restrictions. We reject the hypothesis of convexity if the fitted values, say significantly negative.

are

We can exploit the duality between production and price functions by using the same test procedure for the model based on the translog price function. As before, the value shares sum to unity for any values of qC, qI, qK, t and e, so that the stochastic components of the equations corresponding to these shares must sum to zero. The two stochastic specifications of the translog marginal productivity functions and the translog net supply functions coincide only if the production function and the price function are linear logarithmic with a constant rate of technical change. Unless this condition is satisfied the two stochastic specifications generate the two alternative econometric models of production. To estimate the parameters of the translog net supply functions, we fit the equations

and estimate the parameters of the remaining capital share from (8.13). Tests of integrability, monotonicity, and convexity for the translog price function are strictly analogous to the corresponding tests for the translog production function except that for the parameter at the necessary condition for monotonicity is

8.6 Estimation and Test Statistics Our empirical results are based on annual time series data for the private domestic economy of the Federal Republic of Germany for the period 1950 1973. The data include levels of capital and labor input, levels of consumption and investment output, the corresponding prices, and the rate of technical change for each year. 10 We have fitted the system of eqs. (8.12) and (8.15), respectively.11 Our estimator of the unknown parameters of the two alternative econometric models of production is based on the method of maximum likelihood presented by Malinvaud (1980).

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Estimates of the parameters of the unrestricted form of the three equations are presented in the first column of table 8.1. The second column of table 8.2 gives estimates under the symmetry restrictions associated with integrability of the translog representation of the marginal productivity functions. The third column of table 8.1 provides estimates under the reparameterization of the value shares to test convexity of the translog production function. Corresponding estimates for the translog price function are given in table 8.2. To test the validity of restrictions implied by integrability of the translog representations of the marginal productivity and supply and demand functions we employ test statistics based on the likelihood ratio l, where

The likelihood ratio is the ratio of the maximum value of the likelihood function production w subject to Table 8.1 Parameter estimates, translog production function Parameter (1) Unrestricted 1.0010 (0.0074) aC 0.4440 (0.1750) bCC 0.3520 (0.0800) bCI 0.0440 (0.2420) bCK 0.0032 (0.0172) bCt 0.6380 (0.0035) aI 0.4400 (0.0830) bIC 0.4300 (0.0380) bII 0.0730 (0.1140) bIK 0.0072 (0.0080) bIt 0.6390 (0.0077) aK 0.0037 (0.1810) bKC 0.0780 (0.0830) bKI 0.1180 (0.2490) bKK 0.0104 (0.0180) bKt 0.0483 (0.0074) at 0.3160 (0.1740) btC 0.1220 (0.0790) btI 0.4060 (0.2400) btK 0.0357 (0.0170) btt

(2) Symmetry 0.9990 (0.0049) 0.5070 (0.1230) 0.3850 (0.0510) 0.1220 (0.0940) 0.0079 (0.0024) 0.6400 (0.0026) 0.3850 (0.0510) 0.4080 (0.0280) 0.0230 (0.0360) 0.0021 (0.0009) 0.6390 (0.0054) 0.1220 (0.0940) 0.0230 (0.0360) 0.1450 (0.0830) 0.0100 (0.0017) 0.0580 (0.0052) 0.0079 (0.0024) 0.0021 (0.0009) 0.0100 (0.0017) 0.0019 (0.0007)

d1 d2 l21

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for the econometric model of

(3) Convexity 0.9990 (0.0049) 0.5070 (0.1230) 0.3850 (0.0510) 0.1220 (0.0940) 0.0079 (0.0024) 0.6400 (0.0026) 0.3850 (0.0510) 0.4080 (0.0280) 0.0230 (0.0360) 0.0021 (0.0009) 0.6390 (0.0054) 0.1220 (0.0940) 0.0230 (0.0360) 0.1450 (0.0830) 0.0100 (0.0017) 0.0580 (0.0052) 0.0079 (0.0024) 0.0021 (0.0009) 0.0100 (0.0017) 0.0019 (0.0007) 1.1930 (0.0054) 0.0210 (0.0420) 0.3620 (0.0380)

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Table 8.2 Parameter estimates, translog price function Parameter (1) Unrestricted 0.9870 (0.0127) aC 0.8620 (0.4060) bCC 0.6460 (0.2850) bCI 0.0620 (0.1100) bCK 0.0078 (0.0077) bCt 0.6450 (0.0088) aI 0.8230 (0.2850) bIC 0.3450 (0.1990) bII 0.2340 (0.0770) bIK 0.0180 (0.0050) bIt 0.6330 (0.0085) aK 0.0380 (0.2740) bKC 0.3010 (0.1920) bKI 0.1720 (0.0740) bKK 0.0107 (0.0050) bKt 0.0420 (0.0109) at 0.5760 (0.3510) btC 0.3510 (0.2450) btI 0.1550 (0.0950) btK 0.0130 (0.0066) btt

(2) Symmetry 0.9930 (0.0077) 0.6510 (0.1870) 0.5380 (0.1710) 0.1120 (0.0690) 0.0032 (0.0008) 0.6380 (0.0063) 0.5380 (0.1710) 0.2510 (0.1640) 0.2880 (0.0520) 0.0110 (0.0006) 0.6310 (0.0042) 0.1120 (0.0690) 0.2880 (0.0520) 0.1750 (0.0460) 0.0081 (0.0005) 0.0580 (0.0052) 0.0032 (0.0008) 0.0110 (0.0006) 0.0081 (0.0005) 0.0018 (0.0007)

d1 d2 l21

(3) Convexity 0.9930 (0.0077) 0.6510 (0.1870) 0.5380 (0.1710) 0.1120 (0.0690) 0.0032 (0.0008) 0.6390 (0.0063) 0.5380 (0.1710) 0.2510 (0.1640) 0.2880 (0.0052) 0.0011 (0.0006) 0.6310 (0.0042) 0.1120 (0.0690) 0.2880 (0.0520) 0.1750 (0.0460) 0.0081 (0.0005) 0.0580 (0.0052) 0.0032 (0.0008) 0.0110 (0.0006) 0.0081 (0.0005) 0.0018 (0.0007) 0.8540 (0.0450) 0.0040 (0.1620) 0.1350 (0.0610)

restriction to the maximum value of the likelihood function for the model W without restriction. We have estimated econometric models of production from data on the FRG private domestic economy for 1950 1973. There are twenty-four observations for each of the three behavioral equations, so that the number of degrees of freedom available for statistical tests of the theory of production is seventy-two for either the production or the price specification. For normally distributed disturbances, the likelihood ratio is equal to the ratio of the determinant of the restricted estimator of the variance-covariance matrix of the disturbances to the determinant of the unrestricted estimator, each raised to the power (n/2). Our test statistic is based on minus twice the logarithm of the likelihood ratio, or

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is the restricted estimator of the variance-covariance matrix and is the unrestricted estimator. Under where the null hypothesis, the likelihood ratio test statistic is distributed, asymptotically, as chi-squared with a number of degrees of freedom equal to six, the number of symmetry restrictions to be tested. We employ the asymptotic distribution of the likelihood ratio test statistic for tests of hypotheses. To test the validity of inequality restrictions implied by monotonicity and convexity, we employ test statistics based on the ratio of each inequality constrained coefficient to its standard error. Under the null hypothesis these test statistics are distributed asymptotically as standard normal variables. Under the null hypothesis that the symmetry restrictions are valid, the test statistic for integrability and the test statistics for monotonicity and convexity correspond to 'nested' hypotheses and are distributed independently. To control the overall level of significance of our two series of tests, we set the level of significance for each series at 0.05. We then allocate the overall level of significance among the two stages in each series of tests. We first assign a level of significance of 0.025 to the tests of symmetry. We then assign a level of significance of 0.025 to the tests of monotonicity and convexity. Since the two sets of tests are 'nested', the sum of the levels of significance provides a good approximation to the overall level of significance of these tests considered simultaneously. There are four inequality restrictions associated with monotonicity and two inequality restrictions associated with convexity. We assign a level of significance of 0.0125 to each set of restrictions. The two sets of tests are not 'nested' so that the sum of levels of significance provides an upper bound to the overall level of significance of these tests considered simultaneously. The probability of a false rejection for one test among the collection of all tests we consider is less than or equal to 0.05. With the aid of critical values for our test statistics given in table 8.3, the reader can evaluate the results of our tests for alternative significance levels. Test statistics for each of the hypotheses implied by the theory are given in table 8.4. Our objective has been to develop tests of the theory of production that do not presuppose the restrictions on patterns of substitution implied by the assumptions of additivity and homogeneity. We have presented two alternative econometric models based on translog representations of production and price functions. For each of these models we have derived tests of the theory of production based on

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Table 8.3 Critical values of c2/degrees of freedom and N(0,1) Statistic Level of significance Degrees of freedom 0.100 0.050 0.025 0.010 1 N(0,1) 1.28 1.64 1.96 2.33 6 c2/d.f. 1.77 2.10 2.41 2.80

0.005 2.58 3.09

integrability of the corresponding marginal productivity and supply and demand functions. We have also derived tests based on monotonicity and convexity of the production and price function. The results of our tests of the theory of production as presented in table 8.4 are that the set of restrictions on the parameters of the translog marginal productivity and supply and demand functions, implied by integrability cannot be rejected at the corresponding level of significance for both tests. Our results are consistent with the validity of restrictions implied by the theory. Given symmetry we accept monotonicity and convexity of both the production and the price function. The test statistics for convexity reveal however that the curvature of the function in the outputs is very flat. An important implication of our test results is that the theory of production is an appropriate starting point for construction of a macroeconomic model for the Federal Republic of Germany. A model based on this theory has been constructed for Germany by Conrad Table 8.4 Test statistics for translog production and price functions Hypothesis Degrees of freedom Production 1.77 Symmetry 6

Price 2.37

Given symmetry Monotonicity

ac aI aK at

Convexity

d1 d2

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127.90 101.20 149.30 11.20 19.10

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Table 8.5 Value shares and rate of technical change, translog production function Year (1) wC (2) wI (3) wK (4) wt 1950 1.04445 0.47477 0.51922 0.08142 1951 1.05254 0.48534 0.53788 0.07891 1952 1.04934 0.50277 0.55211 0.07670 1953 1.06553 0.49936 0.56489 0.07462 1954 1.03336 0.54085 0.57420 0.07272 1955 0.99773 0.59007 0.58780 0.07051 1956 1.01283 0.58372 0.59655 0.06871 1957 1.01674 0.58639 0.60313 0.06705 1958 1.02183 0.58616 0.60799 0.06551 1959 1.02342 0.59474 0.61817 0.06360 1960 1.00081 0.62854 0.62934 0.06158 1961 1.00024 0.63508 0.63532 0.05995 1962 0.99927 0.64020 0.63948 0.05845 1963 1.01060 0.63346 0.64407 0.05694 1964 0.97124 0.67974 0.65099 0.05520 1965 0.95818 0.69872 0.65690 0.05356 1966 0.99750 0.66543 0.66293 0.05199 1967 1.06504 0.60372 0.66876 0.05047 1968 1.00354 0.67410 0.67764 0.04855 1969 0.98254 0.70631 0.68886 0.04653 1970 0.98505 0.71217 0.69722 0.04474 1971 0.99914 0.70251 0.70165 0.04325 1972 1.01236 0.69513 0.70749 0.04165 1973 1.02344 0.69365 0.71708 0.03979 (1975). Our results can be regarded as a confirmation of the assumptions from production theory employed by Conrad. 8.7 Economic Interpretation Finally, we provide an economic interpretation of the parameter estimates and analyze patterns of substitution and technical change. We first calculate elasticities of labor input with respect to outputs and capital input. As these elasticities are given by the value shares, they are not constant. The value shares presented in table 8.5 show an elasticity of labor input with respect to consumption output of about one, an increase in the elasticity of labor input with respect to investment output (0.47 in 1950 and 0.69 in 1973) and a steady increase in the elasticity of labor with respect to capital input ( 0.52 in 1950 and 0.72 in 1973).

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We next employ an interpretation introduced by Jorgenson (1986) for translog production models to derive implications of patterns of substitution and technical change to the distribution of the value of the product. For this purpose share elasticities will be defined as changes in the value shares with respect to proportional changes in prices and quantities. 12 They can be employed to derive implications of patterns of substitution for distribution of the value of the product. To interpret the parameters corresponding to time the biases of technical change will be defined as changes in the value shares with respect to time. They can be employed to derive the implications of patterns of technical change for distribution. We first consider share elasticities with respect to quantity. Given the production function F in (8.1) and necessary conditions for producers equilibrium in (8.4), we can differentiate logarithmically a second time with respect to the logarithms of C, I, and K to obtain the share elasticities of C, I, and K with respect to the quantities. The elasticity of the consumption share with respect to the quantity of consumption, say uCC, is

In defining the share elasticity uCC with respect to consumption, we hold the quantities of investment I and capital K fixed and allow the quantity of labor L to vary in accord with the production function. Similarly, the elasticity of the consumption share with respect to the quantity of investment, say uCI, is

that is, the share elasticities are symmetric. Furthermore, the sum of share elasticities for a value share, say wC, is zero,

We evaluate these share elasticities by employing the translog representation of the marginal productivity functions given in (8.5). We obtain from (8.16)

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that is, the share elasticity is a constant and can be evaluated by the estimated parameter of bCC in table 8.1, column 2. The matrix U of share elasticities with respect to quantity is

If an estimated parameter of the matrix U is positive, for instance uCC, the share elasticity with respect to quantity is positive; a proportional increase in the quantity of consumption goods increases the value share of consumption to labor input. If the estimated parameter of U is negative, for instance uIC, the value share decreases with the quantity; a proportional increase in consumption goods decreases the value share of investment to labor input. If an estimated parameter is equal to zero the value share is independent of the quantity. Due to the definition of wK as the negative value share of capital to labor input, a positive uKK implies that a proportional increase in capital decreases the value share of capital to labor input. As the quantity of labor decreases in accord with the production function, the ratio of labor to capital will decrease and therefore the price ratio qK/qL has to drop at an even faster rate. For comparison, a Cobb-Douglas specification in inputs implies a share elasticity of zero and qK/qL and L/K would drop at the same rate. Alternatively, we can consider share elasticities with respect to price. As before, given the price function (8.8) and the value shares as functions of the prices qC, qI and qK, we obtain the share elasticities of consumption, investment and capital with respect to the prices by differentiating the price function P logarithmically a second time, say,

In defining share elasticities with respect to price, say vCC, we hold the prices of investment and capital fixed and allow the price of labor to vary in accord with the price function. The share elasticities with respect to price are also symmetric and the sum of the share elasticities for a given value share is equal to zero. To evaluate the share elasticities with respect to price we employ the translog representation of the price function P (8.9) and the translog supply and demand functions given in (8.15). The elasticity

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of the consumption share with respect to the price of consumption is a constant and can be evaluated by the estimated parameters in table 8.2, column 2. The matrix of share elasticities with respect to the prices, qC, qI and qK, say V, is as follows:

If the estimated parameter of V is positive, for instance vCC, the value share of consumption goods increases with an increase in the price of consumption goods. If the estimated parameter of V is negative, for instance vCI, the value share of consumption goods decreases with the price of investment, due to the higher price for labor. Finally, a negative vKK implies that a proportional increase in the price of capital results in an increase of the value share of capital to labor input. To interpret the parameters of the translog production model corresponding to time, we first consider biases of technical change with respect to quantity, defined by Jorgenson (1986) as changes in the rate of technical change with respect to proportional changes in the quantities. We differentiate the production function F logarithmically with respect to time to obtain the rate of technical change,

The rate of technical change is the rate of decline of the labor input with respect to time, holding the quantities C, I, and K fixed. We present this rate in table 8.5, column 4. By differentiating the rate of technical change with respect to the quantities C, I, and K we obtain the biases of technical change with respect to quantity. The bias of technical change with respect to consumption, say utC, is

In defining the biases of technical change with respect to quantity we hold the quantities C, I, and K fixed and permit the quantity of labor to vary in accord with the production function. Alternatively, we can differentiate the value shares, say wC, with respect to time to obtain the bias the technical change with respect to quantity,

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As the order of differentiation is interchangeable, the two definitions of biases of technical change are equivalent. For the translog production model the bias is a constant parameter, say

By taking the corresponding estimates from table 8.1, column 2, we can evaluate the biases of technical change with respect to quantity,

If a bias of technical change with respect to quantity is positive, for instance uCt, the value share of consumption increases over time, if it is negative, for instance uKt, the negative value share of capital to labor input decreases over time, that is, the value share of capital to labor input increases over time, and if the bias is equal to zero, the value share is independent of time. Alternatively, we can consider biases of technical change with respect to price by differentiating the price function P logarithmically a second time,

In defining biases of technical change with respect to price, say vtC, we hold the prices of investment and capital fixed and allow the price of labor to vary with the price function. By taking estimates from table 8.2, column 2, we can evaluate the biases of technical change with respect to price,

Finally, Jorgenson (1986) defines the rate of change of the rate of technical change by differentiating the rate of technical change (8.18) with respect to time, holding the quantities constant,

For the translog representation the rate of change is constant,

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Alternatively, we can consider the rate of change of the rate of technical change for the translog price function by differentiating

with respect to time. For the translog representation the rate of change is constant,

Notes 1. A form of the translog production function with time separable from quantities of inputs and outputs was introduced by Christensen, Jorgenson, and Lau (1971). The approach to technical change presented below is due to Jorgenson (1986). 2. The equivalence of additive and homogeneous production functions to functions with constant elasticities of substitution and transformation is discussed by Christensen, Jorgenson, and Lau (1973). 3. The price function was introduced by Samuelson (1953 1954) and has been discussed by Burmeister and Kuga (1970), Christensen, Jorgenson, and Lau (1973), and Lau (1972). 4. A review of duality in the theory of production with detailed references to the literature is given by Diewert (1974a) and Lau (1974). See also: Hotelling (1932), Jorgenson and Lau (1974a,b), Samuelson (1953, 1954), Shephard (1970), and Uzawa (1964). 5. A form of the translog price function with time separable from prices of inputs and outputs was introduced by Christensen, Jorgenson, and Lau (1971). For further details on the interpretation of the price function see Christensen, Jorgenson, and Lau (1973, pp. 32 33) and the references given there. 6. For the multiple output case, linear logarithmic production and price functions are not convex. See Mundlak (1964). 7. See Jorgenson (1986). 8. See Jorgenson (1986). 9. This representation was introduced by Lau (1975) and has been discussed by Jorgenson (1980). 10. A detailed description of the data is given by Conrad and Jorgenson (1975, pp. 54 81). See p. 80, table 20, column 4 (L) and column 5 (qL); p. 70, table 14, column 4 (qC), column 5 (C), column 7 (qI) and column 8 (I); p. 51, table 9, column 3 (K) and column 4 (qK) (qK is normalized to one in 1962 and real capital input scaled to real property income). These data have been employed by Conrad (1975). 11. For the translog marginal productivity function we assume that the disturbances are independent of the quantities and time; for the translog net supply function we assume that the disturbances are independent of prices and time. 12. The description of patterns of substitution in terms of share elasticities is due to Samuelson (1973).

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9 The Structure of Technology: Nonjointness and Commodity Augmentation, Federal Republic of Germany, 1950 1973 Klaus Conrad and Dale W. Jorgenson The objective of this chapter is to employ econometric models of production to test restrictions for characterizing the structure of technology and changes in technology empirically. For this purpose we implement two alternative econometric models, based on translog representations of a production function in two outputs and two inputs and of a price function in the corresponding prices. Our test results are consistent with a nonjoint technology. Tests of restrictions implied by types of commodity augmenting technical change reveal that Solow neutrality is a valid specification which we cannot reject with aggregate time series data from the Federal Republic of Germany for the period 1950 1973. 9.1 Introduction The purpose of this chapter is to employ econometric methods introduced by Jorgenson (1986) to characterize the structure of technology and changes in technology over time of the Federal Republic of Germany. In two previous papers (see Conrad and Jorgenson, 1977, 1978) we have employed econometric models of production based on the translog production and price functions 1 to analyze aggregative time series data on production for the Federal Republic of Germany, 1950 1973. We considered the case of two outputs, consumption and investment, and two inputs, capital and labor. An econometric model of production includes a production function and necessary conditions for producer equilibrium giving relative prices as functions of inputs, outputs and time. Our econometric model is based on the translog production function, represented by a functional form that is quadratic in the logarithms of inputs and outputs and time. This representation can be interpreted as a second-order approximation to the underlying production function.2 By exploiting duality in the theory of production we generate a second econometric model of production based on the price function, repre-

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sented by a functional form that is quadratic in the logarithms of prices of inputs and outputs and time. In the first of this series of papers we have tested and accepted the hypothesis that marginal productivity functions and supply and demand functions can be generated form an underlying production function by means of profit maximization. We have also derived tests based on monotonicity and convexity of the production and price function and have accepted monotonicity and convexity of both functions. In a subsequent paper we have imposed the restrictions implied by the theory of production on our econometric models and, proceeding conditionally on the validity of the theory of production, have tested restrictions on the form of production and price functions. We have first tested groupwise separability in commodities, concluding that the production function is groupwise separable in outputs and inputs. The results imply that a production function with separability in inputs and outputs is consistent with German data. We have also employed an approach to the characterization of changes in technology over time based on separability in commodities and time. Our results are consistent with groupwise separability of the two outputs from the group consisting of the two inputs and time. This implies that we can construct an index of real output from price and quantity data on consumption and investment goods output. Output can be represented as a function of the two inputs and time. Our results are not consistent with groupwise separability of the two inputs from the group consisting of the two outputs and time. This implies that we cannot construct an index of real input from price and quantity data on capital and labor input or, equivalently, technical change is not Hicks-neutral. Finally, our results are consistent with groupwise separability of the group consisting of capital and time from the group consisting of the two outputs and labor input. This implies that we can construct an index of the two outputs and labor input from price and quantity data on these commodity groups. Under this restriction technical change is Solow-neutral. Our results are consistent with either of these two simplifications of our representation of technology. The results of our tests of separability in goods and time for the translog price function did not imply simplification of our representation of technology similar to those we have obtained for the translog production function. Our objective in this chapter is to employ econometric models based on translog production and price functions to characterize the

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structure of technology and changes in technology over time for the German economy in more detail. We first consider restrictions on technology associated with nonjoint production. In joint production all outputs are produced jointly from all inputs; in nonjoint production each output is produced separately from the others. Although yearly input-output tables are available for Germany there is no way of separating the primary factors between inputs employed by the consumption goods industry and inputs employed by the investment goods industry. To characterize the production process as nonjoint we derive implications of nonjoint production for substitution patterns among inputs and outputs in a multiple-output technology. An economic implication of a nonjoint technology is that there are no economies or diseconomies of producing all outputs jointly. An equivalent characterization of nonjointness introduced by Hall (1973) is that the total cost of producing all outputs is the sum of the costs of producing consumption and investment separately. We derive tests of nonjointness based on translog production and price functions. In our previous papers we have used an approach to the representation of technical change based on the introduction of time into the production function as an index of technology. An alternative approach to the representation of technical change is to represent technology in terms of augmenting factors for outputs and inputs; this representation also involves time but only through the augmentation factors. Given commodity augmentation, hypotheses concerning the effect of technical change on the various components of the production plan can be formulated and tested. We test the hypothesis that augmentation factors are pairwise proportional, that is, that technical change affects pairs of products and factors at the same proportional rate. We also test the hypothesis that some augmentation factors do not vary over time. By combining these hypotheses, we can discriminate among Hicks-, Harrod- and Solow-neutral technical change. 9.2 Translog Production and Price Functions We consider the case of two outputs, consumption C and investment I, and two inputs, capital K and labor L with their corresponding prices qC, qI, qK and qL. The translog representation of the production functions F,

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then takes the form

The corresponding representation of the marginal productivity functions take the form

is the rate of technical change. The rate of technical change is the rate of decline of the labor input with respect to time, holding quantities C, I and K constant. The parameters of the translog production function can be identified with the coefficients in a Taylor's series expansion to the underlying production function F. They take the values of first- and second-order partial logarithmic derivatives of the underlying production function at the point of expansion (C, I, K, t) = (1, 1, 1, 0). The data have been scaled so that these values are for 1962. Restrictions on the parameters implied by the homogeneity of the production function are as follows:

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restrictions implied by symmetry of cross-price substitution effects are

If a translog production function (9.2) is homogeneous of degree one, profit maximization subject to this production function implies the marginal productivity functions (9.3). These functions satisfy the symmetry restrictions (9.5) and the homogeneity restrictions (9.4). Conversely, if we specialize on national accounting data for which the value of output is equal to the value of input,

the parameters of the marginal productivity functions (9.3) satisfy the parameter restrictions (9.4); if they also satisfy the symmetry restrictions (9.5), these functions are consistent with profit maximization subject to a translog production function (9.2) that is homogeneous of degree one. We impose the parameter restrictions (9.4) and (9.5). The identity between the value of output and input implies that the value shares sum to unity; given the parameters of any two equations for the value shares, the parameters of the third equation can be determined from the parameter restrictions under (9.4). To estimate the unknown parameters we combine the first two equations with the fourth for the rate of technical change. Unrestricted, there are fifteen unknown parameters to be estimated from the three equations. Given the symmetry restrictions (9.5), there are nine unknown parameters to be estimated. Under constant returns to scale this model implies the existence of a price function, defining the set of prices consistent with zero profits and the existence of conditions determining relative product and factor intensities as functions of prices and time. 3 The price function and

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these demand and supply functions are dual to the production function and the marginal productivity functions. 4 By exploiting the duality our second econometric model of production corresponds to the translog supply and demand functions which consist of a system of equations giving the value shares and the rate of technical change as functions of the prices of commodities and time. Again this system can be interpreted as a first-order approximation to the underlying demand and supply functions. Under the restrictions implied by the theory of production, accepted in our previous paper, this system of equations can be integrated to obtain the translog representation of the price function. This representation can be interpreted as a second-order approximation of the underlying price function P. The translog representation of the price function P, here considered,

takes the form

The corresponding representation of the supply and demand functions take the form

We consider restrictions on patterns of substitution implied by nonjointness and on technical change associated with proportional and constant augmentation factors. For each set of restrictions we derive the implications for the translog representation of the marginal productivity functions or supply and demand functions.

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9.3 Nonjointness We next consider restrictions on technology associated with nonjoint production. Necessary and sufficient conditions for a production function to represent a nonjoint technology has been given by Samuelson (1966) and for a price function by Lau (1972). 5 The implications of a production function which is separable and/or nonjoint has been discussed by Hall (1973). A technology characterized by a production function is said to be nonjoint if it can be portrayed in terms of separate production functions and joint, if it can not be so portrayed. We first consider restrictions associated with nonjoint production of {C, I} from {K, L}. The production function F is nonjoint in the outputs {C, I}, given the set of inputs {K, L} if there exist individual production functions,

and L = F(C, I, K, t) is the production function. Under our maintained hypothesis that markets are competitive and the production function is homogeneous of degree one, a necessary and sufficient condition for nonjoint production is that there exists set of price functions

The price of each product depends only on the prices of the factors of production and is independent of the price of the other production. Each of the price functions is homogeneous of degree one in the prices qK and qL, so that we can rewrite these functions

We equate these price ratios to the marginal products as follows:

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where

and so on. Substituting marginal products for price ratios, we obtain the following restrictions on the marginal products

We can differentiate these restrictions with respect to the quantities C, I and K to obtain restrictions on the matrix of second-order partial derivatives of the production function F. Denoting the first-order partial derivatives of the function QC and QI by

we can write

or, in matrix form with the order of differentiation interchangeable

To derive the implications of the restrictions on technology associated with nonjoint production of the outputs in {C, I} from the inputs in {K, L}, we first evaluate the matrix of second-order partial derivatives of the translog representation of the production function F at (C, I, K, t) = (1, 1, 1, 0)

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Evaluating the derivatives of the functions can write

and

at the corresponding value of the marginal product FK, we

and obtain the restriction on the Hessian matrix of the translog function in the form

To test the hypothesis of nonjoint production we transform the matrix H of parameters by means of a Cholesky factorization. 6 The vectors kC and kI are characteristic vectors of the matrix H corresponding to characteristic values equal to zero. They are linearly independent, so that the matrix H has at least two zero characteristic values. This is a necessary condition for nonjoint production; whereas the existence of two zero characteristic values is a sufficient condition for nonjoint production (see Jorgenson, 1986). To construct a test of nonjoint production, we represent the matrix H in terms of its Cholesky factorization,

where L is a unit lower triangular matrix, and D is a diagonal matrix,

The diagonal elements of D are the Cholesky values of matrix e a0 H. The number of positive, negative and zero characteristic values of a symmetric matrix is equal to the number of the positive, negative and zero Cholesky values, so that we can test nonjoint production by testing the hypothesis that the matrix H has two zero Cholesky values. To derive restrictions associated with nonjoint production we express by means of the matrix equation (9.9) the parameters bCC, bCI and bII by the parameters d1, d2 and b21

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After this reparameterization, there are still nine unknown parameters to be tested. Next we note that the symmetry restrictions (9.5) and the restrictions (9.4) implied by homogeneity of degree one of the production function imply the restrictions d3 = 0. This can be shown by using the fact that the first and second row in the Hessian matrix H sum to the third row which must also be the case for the rows of LDL' implying d3 = 0. In terms of the new parameters the necessary conditions for nonjoint production take therefore the form

or both. To test nonjointness we impose each of these restrictions individually. Under the restriction d1 = 0 the parameter l21 cannot be identified, so that we can set this parameter equal to zero, leaving seven unknown parameters to be estimated. Under the restriction d2 = 0 there are eight unknown parameters to be estimated. If both restrictions are imposed, there are three independent restrictions

so that six unknown parameters remain to be estimated. Finally, we consider restrictions on technology associated with nonjoint pricing. The price function P is said to be nonjoint in the output prices {qC, qI} from the input prices {qK, qL} if there exist individual price functions

where

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and

is the price function. A necessary and sufficient condition for nonjoint pricing is that there exists a set of production functions

The production functions FC and FI are homogeneous of degree one in the inputs K and L. Nonjoint pricing is associated with externalities in production. The restrictions on the Hessian matrix of the translog representation of the price function are precisely analogous to those who have derived on the Hessian matrix of the translog production function. To test nonjointness restrictions given symmetry, we proceed therefore exactly as in the model based on the translog representation of the marginal productivity functions. 9.4 Technical Change We next present an approach to the characterization of changes of technology over time studied by Jorgenson (1986). In our previous paper (Conrad and Jorgenson, 1978) time was treated symmetrically with inputs, outputs or prices in the description of the technology. To characterize changes in technology over time we have employed restrictions on the production or price functions corresponding to separability in commodities and time. An alternative approach to technical change is based on a description of technology in terms of augmented inputs, outputs, or prices, defined as the products of these magnitudes and corresponding augmentation factors that are functions of time. We suppose that the outputs C and I and the inputs K and L are augmented by factors aC(t), aI(t), aK(t), and aL(t), respectively, that depend only on time. Under this description of technology the production function F can be written in the form

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where the products of the commodities and the augmentation factors a(t) can be treated as an entity measured in terms of efficiency units. Under homogeneity of degree one of the production function F we consider the equivalent description of technology

where now the products of the commodities and the augmentation factors

are the augmented commodities. Under a description of technology in terms of augmented outputs and inputs the parameters of the translog representation of the production function F can be interpreted as follows:

If the augmentation factors are arbitrary functions of time, a description of technology in terms of augmented commodities imposes no restrictions on the parameters of the translog representation of the production function F. To provide an interpretation of the parameters of the translog representation of the production function F in terms of augmentation factors we introduce the additional parameters

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These parameters are the first-order logarithmic derivatives of the augmentation factors at t = 0. Using these rates of commodity augmentation we can express the parameters of the translog representation, as follows:

To express technical change in terms of commodity augmentation we reparameterize the equations for the value shares in (9.3) by using the expressions for the parameters at, bCt, bIt, and bKt in terms of the a and b parameters and lC, lI, and lK. Given symmetry there are still nine unknown parameters to be estimated. We first consider restrictions on technical change associated with pairwise proportionality between augmentation factors. If two augmentation factors, say AC and AI, are proportional, we can write

In terms of the parameters of the translog representation of the production function F we can derive the restriction

Similarly, we can derive the restriction for {AC, AK} proportionality

and for {AI, AK} proportionality

Given symmetry there are eight unknown parameters to be estimated under each set of pairwise proportionality restrictions. We next consider restrictions on technical change associated with constancy of the augmentation factors A(t). If an augmentation factor is constant, the rate of commodity augmentation is zero. The corresponding restrictions on the parameters of the translog representation can be obtained by setting l equal to zero. Given symmetry we can test the hypothesis of constant augmentation factors as follows:

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AC constant: lC = 0, AI constant: lI = 0, AK constant: lK = 0. As before, there are eight unknown parameters remaining to be estimated under each of these restrictions. To obtain further simplifications we consider restrictions on technical change associated with pairwise constancy of the augmentation factors A (t). The corresponding restrictions can be obtained from the equations (9.11) (9.13) above by setting l equal to zero. Under pairwise constancy of two augmentation factors, say AC and AI the parameters given in (9.10) must satisfy the restrictions

Analogous restrictions must hold for the two remaining possible sets of pairwise constancy of the augmentation factors. Given pairwise proportionality we test the hypothesis of pairwise constancy as follows: pairwise constancy: l = 0, in (9.11), (9.12) and (9.13), respectively. Given pairwise proportionality, there are seven unknown parameters remaining to be estimated under each of these restrictions. Finally, we consider restrictions on technical change based on a description of technology in terms of augmented prices. In this approach of technology the price function P can be written in the form

where the products B(t) · q are the augmented prices. The augmented prices are the corresponding prices for outputs and inputs measured in efficiency units. With AK(t) · K as the capital input, measured in efficiency units qK/AK(t) is the corresponding efficiency price so that BK(t) = 1/AK(t). If the commodity augmenting factor A(t) increases at a proportional rate over time, the price augmenting factor B(t) decreases at that proportional rate. Introducing the first-order logarithmic derivatives of the augmentation factors at t = 0, for example,

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we can reparameterize the translog representation of the supply and demand system (9.8) precisely as done in (9.10). To test restrictions on augmentation factors we proceed exactly as in the model based on the translog representation of the marginal productivity functions. 9.5 Tests We have developed econometric models for characterizing the structure of technology and changes in technology over time. We propose to test restrictions derived from nonjoint production and from commodity augmenting changes in technology. Our proposed test procedure is presented in diagrammatic form in two figures. We first impose the symmetry restrictions implied by the theory of production. We then proceed to test restrictions derived from nonjoint production. Our test procedure is presented diagrammatically in figure 9.1. We test in parallel first the restrictions d1 = 0, l21 = 0; which reduces the nine unknown parameters under symmetry by two. We next test the restriction d2 = 0 which reduces the nine unknown parameters by one. Next, we consider tests of restrictions associated with commodity augmenting changes in technology. First, we test the hypothesis of pairwise proportionality of augmentation factors for all three possible groups of two commodities each. Given these restrictions, we proceed to test the additional restriction implied by pairwise constancy of augmentation factors. All three tests for pairwise proportionality are carried out in parallel (see figure 9.2). Each of the three tests for pairwise constancy is carried out given the corresponding pairwise proportionality restrictions. Second, we test the hypothesis that each of the three augmentation factors is constant. These three tests are carried out in parallel. Each of the restrictions on commodity augmenting technical change involves on equality restriction; given symmetry, eight unknown parameters remain to be estimated under the restrictions of pairwise proportionality and constancy of the augmentation factors and seven under the restrictions of pairwise constancy of the augmentation factors given pairwise proportionality. To dualize this analysis we observe that a precisely parallel test procedure can be developed for the price function with analogous tests of the restrictions on the parameters of the translog representation of the price function.

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Figure 9.1 Tests of nonjointness.

Figure 9.2 Tests of commodityaugmenting technical change.

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9.6 Estimation and Test Statistics Our empirical results are based on the annual time series data for the private domestic economy of the Federal Republic of Germany for the period 1950 1973 employed in our previous papers. 7 We add to each of the value shares and to the rate of technical change an additive disturbance term. Since the value shares sum to unity, the sum of the disturbances across the three equations is zero at each observation. This implies that the disturbance covariance matrix is singular. We drop the disturbance term from one equation for the value shares and specify that the remaining three disturbances are independently and identical normally distributed. We have fitted the two equations for the value shares of consumption and investment and one equation for the rate of technical change generated by translog representation of production and price functions.8 As we impose the symmetry restrictions our estimates of the unknown parameters satisfy these restrictions. In table 9.1 we present estimates of the unknown parameters associated with restrictions implied by nonjoint production and commodity augmenting technical change. Parameter estimates for the translog representation of the price function are given in table 9.2.9 We give the estimates only for those specifications which we will discuss in the last section. To test the validity of restrictions implied by nonjointness and commodity augmenting changes in technology, we employ test statistics based on the likelihood ratio l, where

The likelihood ratio is the ratio of the maximum value of the likelihood function L for the econometric model of production w, subject to restriction to be tested, to the maximum value of the likelihood function for the model W without restriction. There are twenty-four observations for the period 1950 1973 for each behavioral equation so that the number of degrees of freedom available for statistical tests of restrictions on the structure of technology and changes in technology is seventy-two for either model. For normally distributed disturbances, the likelihood ratio is equal to the ratio of the determinant of the restricted estimator of the variance-covariance matrix of the disturbances to the determinant of the

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Table 9.1 Parameter estimates, translog production function Parameter 1. Symmetry 2. Nonjointness 3. Commodity augmentation 4. {AC, AI} proportionality d2 = 0 0.9990 (0.0050) 0.9990 (0.0050) 0.9990 (0.0050) 0.9990 (0.0050) aC 0.5070 (0.1230) 0.4590 (0.0650) 0.5070 (0.1220) 0.4810 (0.0750) bCC 0.3850 (0.0510) 0.3630 (0.0150) 0.3850 (0.0510) 0.3770 (0.0420) bCI 0.1220 (0.0940) 0.0960 (0.0750) 0.1220 (0.0940) 0.1040 (0.0640) bCK 0.0080 (0.0020) 0.0070 (0.0010) 0.0080 (0.0020) 0.0070 (0.0020) bCt 0.6400 (0.0030) 0.6400 (0.0030) 0.6400 (0.0030) 0.6400 (0.0020) aI 0.4080 (0.0280) 0.3980 (0.0200) 0.4080 (0.0280) 0.4090 (0.0280) bII 0.0230 (0.0360) 0.0350 (0.0280) 0.0230 (0.0360) 0.0320 (0.0200) bIK 0.0020 (0.0009) 0.0026 (0.0004) 0.0020 (0.0010) 0.0220 (0.0009) bIt 0.6390 (0.0050) 0.6390 (0.0050) 0.6390 (0.0050) 0.6390 (0.0050) aK 0.1450 (0.0830) 0.1310 (0.0830) 0.1450 (0.0830) 0.1350 (0.0750) bKK 0.0100 (0.0017) 0.0090 (0.0010) 0.0100 (0.0020) 0.0090 (0.0010) bKt 0.0580 (0.0050) 0.0580 (0.0050) 0.0580 (0.0050) 0.0580 (0.0050) at 0.0019 (0.0007) 0.0019 (0.0007) 0.0019 (0.0007) 0.0019 (0.0007) btt 0.1030 (0.0200) 0.1040 (0.0210) lC 0.1020 (0.0200) 0.1040 (0.0210) lI 0.1720 (0.0490) 0.1750 (0.0520) lK Parameter 5. AC constant 6. AI constant 7. Ak constant 8. {AC, AI} constancy 0.9950 (0.0050) 0.9940 (0.0050) 0.9940 (0.0050) 0.9930 (0.0040) aC 0.2690 (0.0680) 0.2740 (0.0800) 0.1800 (0.0580) 0.3600 (0.0180) bCC 0.3570 (0.0520) 0.3680 (0.0530) 0.3330 (0.0530) 0.4180 (0.0200) bCI 0.0880 (0.0190) 0.0940 (0.0300) 0.1520 (0.0150) 0.0560 (0.0070) bCK 0.0034 (0.0010) 0.0035 (0.0010) 0.0016 (0.0010) 0.0050 (0.0006) bCt 0.6400 (0.0030) 0.6400 (0.0030) 0.6400 (0.0030) 0.6400 (0.0020) aI 0.3960 (0.0280) 0.3980 (0.0290) 0.3910 (0.0250) 0.4000 (0.0250) bII 0.0390 (0.0370) 0.0290 (0.0380) 0.0580 (0.0370) 0.0160 (0.0050) bIK 0.0027 (0.0010) 0.0020 (0.0010) 0.0030 (0.0010) 0.0015 (0.0005) bIt 0.6350 (0.0050) 0.6340 (0.0050) 0.6350 (0.0050) 0.6320 (0.0050) aK 0.0480 (0.0210) 0.0650 (0.0130) 0.0930 (0.0330) 0.0720 (0.0080) bKK 0.0060 (0.0007) 0.0060 (0.0008) 0.0050 (0.0006) 0.0060 (0.0006) bKt 0.0590 (0.0054) 0.0590 (0.0050) 0.0590 (0.0050) 0.0580 (0.0050) at 0.0018 (0.0007) 0.0019 (0.0007) 0.0019 (0.0007) 0.0019 (0.0007) btt 0.0120 (0.0100) 0.0420 (0.0040) lC 0.0180 (0.0140) 0.0270 (0.0050) lI 0.1110 (0.0160) 0.0730 (0.0170) 0.0920 (0.0070) lK

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Table 9.2 Parameter estimates, translog price function Parameter 1. Symmetry 2. Nonjointness 3. Commodity augmentation 4. {AC, AI} proportionality d2 = 0 0.9930 (0.0070) 0.9930 (0.0050) 0.9930 (0.0080) 0.9940 (0.0060) aC 0.6510 (0.1800) 0.6470 (0.0880) 0.6510 (0.1870) 0.6190 (0.1560) bCC 0.5380 (0.1700) 0.5340 (0.0350) 0.5380 (0.1710) 0.5430 (0.1490) bCI 0.1120 (0.0700) 0.1130 (0.0600) 0.1120 (0.0690) 0.0770 (0.0210) bCK 0.0030 (0.0008) 0.0030 (0.0040) 0.0030 (0.0008) 0.0030 (0.0007) bCt 0.6380 (0.0060) 0.6380 (0.0040) 0.6380 (0.0060) 0.6390 (0.0060) aI 0.2500 (0.1640) 0.2460 (0.0130) 0.2510 (0.1640) 0.2710 (0.1500) bII 0.2870 (0.0520) 0.2880 (0.0490) 0.2880 (0.0520) 0.2720 (0.0410) bIK 0.0110 (0.0006) 0.0110 (0.0006) 0.0110 (0.0007) 0.0110 (0.0005) bIt 0.0630 (0.0040) 0.6310 (0.0040) 0.6310 (0.0040) 0.6310 (0.0040) aK 0.1750 (0.0460) 0.1750 (0.0560) 0.1750 (0.0460) 0.1950 (0.0300) bKK 0.0080 (0.0005) 0.0080 (0.0005) 0.0080 (0.0006) 0.0080 (0.0005) bKt 0.0580 (0.0050) 0.0580 (0.0050) 0.0580 (0.0050) 0.0580 (0.0050) at 0.0018 (0.0007) 0.0018 (0.0007) 0.0018 (0.0007) 0.0018 (0.0007) btt 0.0860 (0.0070) 0.0840 (0.0060) lC 0.0830 (0.0070) 0.0840 (0.0060) lI 0.1270 (0.0130) 0.1260 (0.0110) lK unrestricted estimator, each raised to the power (n/2). Our test statistic for each set of restrictions is based on minus twice the logarithm of the likelihood ratio, or

where is the restricted estimator of the variance-covariance matrix and is the unrestricted estimator. Under the null hypothesis this test statistic is distributed, asymptotically, as chi-squared with number of degrees of freedom equal to the number of restrictions to be tested. To control the overall level of significance for each series of tests of the production and price representation, we set the level of significance for each series at 0.05. We assign a level of significance of 0.02 to tests of nonjointness. To tests of commodity augmenting technical change we assign a level of significance of 0.03. Within our set of tests for nonjointness, we assign levels of significance of 0.01 to the restrictions d1 = 0, l21 = 0 and to the restriction d2 = 0. Next, we test the six restrictions on the form of commodity augmenting technical change in parallel, i.e., three restrictions on pairwise proportionality and three restrictions on constancy of an augmentation factor. We assign a level of significance of 0.005 to each test. Within our set of tests for pairwise properties of the augmentation factors, we can distinguish two stages:

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pairwise proportionality of augmentation factors and pairwise constance for each of the three groups. We assign a level of significance of 0.0025 to each of these six tests. The reader can also evaluate the results of our tests for alternative significance levels or for alternative allocations of the overall level of significance among stages of the test procedure. In our complete series of tests for econometric models of production based on translog production and price functions, only tests of pairwise proportionality and pairwise constancy of augmentation factors are ''nested," none of the other tests are "nested" so that the sum of levels of significance for all tests provides an upper bound to the overall level of significance for all of these tests considered simultaneously. 9.7 Conclusion Our objective has been to develop tests of restrictions for characterizing the structure of technology and changes in technology over time. For econometric models of production based on translog representations of the marginal productivity and supply and demand functions, we have assigned levels of significance to each of our tests of hypotheses about the structure to technology and changes in technology over time so as to control the overall level of significance for all tests at 0.05. The probability of a false rejection for one test among the collection of tests is less than or equal to 0.05. With the aid of critical values for our test statistics given in table 9.3, we can evaluate the results of our tests given in table 9.4. If the test statistic for one of the hypotheses summarized in table 9.4 is larger than its corresponding critical value, given in table 9.3, we reject the hypothesis at the assigned level of significance. Table 9.3 Critical values of c2/degree of freedom Degrees of freedom Level of significance 0.1000 0.0500 0.0100 0.0050 0.0025 0.0010 2.71 3.84 6.64 7.88 9.14 10.83 1 2.30 3.00 4.61 5.30 5.95 6.91 2 2.08 2.61 3.78 4.28 4.75 5.42 3

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Table 9.4 Test statistics for translog production and price functions Hypothesis Degrees of ProductionPrice freedom Given symmetry Nonjointness 31.78 34.84 d1 = 0, l21 = 0 2 0.27 0.00 d2 = 0 1 21.69 23.23 d1 = 0, l21 = 0, d2 = 0 3 Commodity augmenting technical change 0.10 0.38 {AC, AI} proportionality 1 18.44 12.43 {AC, AK} proportionality 1 13.21 12.80 {AI, AK} proportionality 1 5.86 30.67 AC constant 1 6.51 26.42 AI constant 1 9.10 39.86 AK constant 1 Givent pairwise proportionality Pairwise constancy of augmentation factors 7.10 60.64 {AC, AI} constancy 1 7.66 3.05 {AI, AK} constancy 1 49.94 24.42 {AK, AC} constancy 1 We first test the hypothesis of nonjoint production under the translog representation of a production function. The results of our tests of restrictions associated with nonjointness are that we accept the hypothesis of a nonjoint technology with restrictions on the parameters based on the reparameterization with d2 = 0. The multiple output technology can therefore be portrayed in terms of separate production functions. However, the implication of input-output separability, accepted by Conrad and Jorgenson (1978) with the same set of data, is that output can be produced in joint production because we can construct an index number of output quantities which is produced by the inputs and afterwards split into consumption and investment goods output. 10 We next turn to the interpretation of our results on tests of commodity augmenting technical change. Our first conclusion is that we accept pairwise proportionality between two augmentation factors AC(t) and AI(t). A production function F with pairwise proportionality between the augmentation factors for consumption and investment goods output can be written in the form

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where h is the factor of proportionality. Alternatively, under constant returns to scale technical change is factor augmenting

We do not accept the hypothesis of pairwise proportionality for any two of the three pairs, which would imply pairwise proportionality between all three augmentation factors, that is, Harrod-neutral technical change,

After having accepted {AC, AI} proportionality we can proceed to test pairwise constancy of the augmentation factors AC(t) and AI(t); we accept this hypothesis. A production function F with pairwise constancy of these augmentation factors can be written in the form

that is, technical change is Solow-neutral. With the parameter estimate for lK taken from column 6 in table 9.1 the rate of capital augmentation at the point of approximation is lK = 0.092. To obtain further specifications of technical change in our representation of technology we finally test zero rates of commodity augmenting technical change. From the results in table 9.4 we observe that we accept a constant augmentation factor for consumption and for investment. We note that there is no logical necessity to draw conclusions from combining two or all three of these tests. A production function F with a constant augmentation factor for, say, consumption can be written in the form

As we reject the hypothesis of a constant augmentation factor for capital the production function cannot be written in a form with purely commodity augmenting factors

The results of our tests of restrictions on technology associated with nonjoint pricing and commodity augmenting technical change for the translog price functions are that we accept the hypothesis of nonjoint

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pricing. As before, we accept the hypothesis of pairwise proportionality between the price augmentation factors BC(t) and BI(t). A price function P with pairwise proportionality between augmentation factors for prices of consumption and investment goods can be written in the form

or, in terms of factor price augmenting technical change

By appropriate choice of dimensions the ratios of corresponding commodity and price augmenting factors can always be taken to be equal to unity

Therefore the price augmentation factors are diminishing if the commodity augmentation factors are augmenting and vice versa. Given {AC, AI} proportionality, we reject however pairwise constancy of the corresponding augmentation factors. We finally reject all three types of constancy of the augmentation factors. That the results are not completely similar to those we obtained for the translog production function is no contradiction because the translog price function is not dual to the translog production function. Notes 1. Translog production and price functions were introduced by Christensen, Jorgenson and Lau (1971, 1973). The approach to technical change presented below is due to Jorgenson (1986). An analogous approach to analyzing the structure of consumer preferences and changes in preferences over time is given by Jorgenson and Lau (1975). 2. For more detailed discussion, see Jorgenson (1986) 3. The price function was introduced by Samuelson (1953) and has been discussed by Burmeister and Kuga (1970) and by Christensen, Jorgenson and Lau (1973). 4. A review of duality in the theory of production is given by Diewert (1974a) and Lau (1974). See also: Hotelling (1932), Jorgenson and Lau (1974a, b), Samuelson (1954), Shephard (1953, 1970) and Uzawa (1964). 5. We follow the contribution by Jorgenson (1986). 6. The usage of the Cholesky factorization was proposed by Lau (1978a) to test convexity and by Jorgenson (1986), to test nonjointness. For an empirical application see also Jorgenson (1986).

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7. See Conrad and Jorgenson (1975, p. 80, table 20, column 4 (L) and column 5 (qL); p. 70, table 14, column 4 (qC) column 5 (C), column 7 (qI) and column 8 (I); p. 51, table 9, column 3 (K) and column 4 (qK) (qK is normalized to one in 1962 and real capital input is scaled to real property income)). 8. Our estimator is based on the method of maximum likelihood presented by Malinvaud (1980). 9. For an interpretation of our parameter estimates in terms of share elasticities see Conrad and Jorgenson (1977). 10. The impossibility theorem for separable nonjoint technologies by Hall (1973) states that no multiple output technology with constant returns to scale can be both separable and nonjoint. Combining our results for separability and nonjointness would imply that the individual production functions for consumption and investment are identical except for a scalar multiple.

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10 The Structure of Technology and Changes of Technology over Time, Federal Republic of Germany, 1950 1973 Klaus Conrad and Dale W. Jorgenson In this chapter we use econometric models of production to develop tests of parametric restrictions for characterizing the structure of technology and changes in technology empirically. 10.1 Introduction The purpose of this chapter is to employ econometric methods introduced by Jorgensen (1986) for characterizing the structure of technology and changes in technology over time of the private domestic economy of the Federal Republic of Germany. In a previous paper 1 we have employed econometric models of production based on the translog production and price functions2 to test the theory of production with aggregative time series data for the Federal Republic of Germany, 1950 to 1973. Using these models we have derived tests of the theory of production that do not impose restrictions on patterns of substitution implied by the assumption of additivity and homogeneity. We have accepted the hypothesis that marginal productivity functions and supply and demand functions are generated by profit maximization. Thus we can impose the restrictions implied by the theory of production on our econometric models and can proceed conditionally on the validity of the theory of production to test restrictions on the forms of the production and price functions. Our objective is to employ these econometric models to develop tests of restrictions for characterizing the structure of technology empirically. A complete model of production includes a production function and necessary conditions for producer equilibrium giving relative prices as a function of net outputs and time. Our econometric model of production corresponds to a translog representation of these marginal productivity functions. The model consists of a system of equations giving the value shares and the rate of technical change as

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functions of the quantities and time. The system of translog marginal productivity functions can be interpreted as a first-order approximation to the underlying marginal productivity functions. Under the restrictions implied by the theory of production the system of equations can be integrated to obtain the translog representation of the production function. This representation can be interpreted as a second-order approximation to the underlying production function. 3 We consider the case of two outputs, consumption C and investment I, and two inputs, capital K and labor L. The corresponding prices are qC,qI,qK, and qL. The translog representation of the production function F,

then takes the form:

The corresponding representation of the marginal productivity functions take the form:

is the rate of technical change. The rate of technical change is the rate

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of decline of the labor input with respect to time, holding quantities C, I and K constant. The parameters of the translog production function can be identified with the coefficients in a Taylor's series expansion to the underlying production function F. They take the values of first- and second-order partial logarithmic derivatives of the underlying production function at the point of expansion (C,I,K,t) = (1,1,1,0). The restrictions on the parameters implied by the theory of production are as follows:

Under homogeneity of degree one of the translog production function the parameters of the value shares satisfy the restrictions (10.4). Conversely, if the parameters of the translog marginal productivity functions satisfy (10.4) and (10.5), these functions are homogeneous of degree zero and can be generated by a translog production function with homogeneity of degree one. The identity between the value of output and input,

implies that the value shares sum to unity, so that, given the parameters of any two equations for the value shares, the parameters of the third equation can be determined from the parameter restrictions under (10.4). To estimate the unknown parameters we combine the first two equations with the fourth for the rate of technical change. Unrestricted, there are fifteen unknown parameters to be estimated from the three equations. Given the symmetry restrictions (10.5), there are nine unknown parameters to be estimated. Under constant returns to scale this model implies the existence of a price function, defining the set of prices consistent with zero profits and the existence of conditions determining relative product and

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factor intensities as functions of prices and time. 4 The price function and these net supply functions are dual to the production function and the marginal productivity functions.5 By exploiting the duality our second econometric model of production corresponds to the translog net supply functions which consist of a system of equations giving the value shares and the rate of technical change as functions of the prices of commodities and time. Again this system can be interpreted at a first-order approximation to the underlying net supply functions. Under the restrictions implied by the theory of production, accepted in our previous paper, this system of equations can be integrated to obtain the translog representation of the price function. This representation can be interpreted as a second-order approximation of the underlying price function P. The translog representation of the price function P, here considered,

takes the form:

The corresponding representation of the supply and demand functions take the form:

We consider restrictions on patterns of substitution implied by separability. For each set of restrictions we derive the implications for the translog representation of the marginal productivity functions or

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supply and demand functions. A given set of restrictions on the underlying technology does not necessarily imply the corresponding set of restrictions on the translog representation, so that we distinguish two types of restrictions. First, the translog production and price functions may provide a representation of an underlying technology with a given set of restrictions. Second, the translog representation itself may be characterized by these restrictions. 10.2 Separability We first consider restrictions on technology associated with groupwise separability of the production F given in (10.1). A production function F that is groupwise separable in outputs and inputs, for example, can be represented in implicit form as follows:

We can write this production function in explicit form as follows:

where F is the production function and G the function independent of the inputs K and L. 6 To derive restrictions on the parameters of the translog representation of a production function F that is groupwise separable we can differentiate the logarithm of the production function logarithmically with respect to the two outputs C and I:

Next, we differentiate logarithmically a second time with respect to capital K:

Given groupwise separability, equations (10.9) and (10.10) must hold everywhere; in particular they must hold at the point of approximation (C,I,K,t) = (1,1,1,0), where we can identify the first- and

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second-order partial derivatives with the parameters of the translog production function. We conclude that the parameters of the translog representation satisfy the restrictions:

where r is a constant given by

at the point of expansion. There are two more possible sets of groupwise separability restrictions. In a strictly analogous manner it can be shown that separability of {C,L} from {I,K} implies the following restrictions on the parameters of the translog production function:

and separability of {C,K} from {I,L} implies:

Each set of restrictions involves two restrictions with the introduction of one new parameter. Given symmetry, the number of unknown parameters to be estimated under each set of groupwise separability restrictions is eight, one less than the number without restrictions. Restrictions on the structure of technology do not necessarily imply the corresponding restrictions on the translog function itself. Therefore, the translog representation of a groupwise separable production function F is not necessarily groupwise separable. We distinguish between situations where the translog production function provides an approximation to the underlying production function with a certain property and situations where the translog production function also possesses that property. In the latter case we say that the translog

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production function possesses the property explicitly. For a translog production function to be explicitly groupwise separable in the pair of outputs {C, I} and inputs {K, L}, it is necessary and sufficient that:

Under this restriction the parameters bKC and bKI in (10.11) are zero. An example of a production function groupwise separable in outputs and inputs is a CET-CES production function:

The translog approximation to this separable production function is not separable. A second example of a groupwise separable production function is a CET-Cobb-Douglas production function:

The translog approximation to this separable production function is separable. Similar restrictions must hold for {C, L} separability from {I, K} and for {K, C} separability from {I, L}. Each of these restrictions is imposed, given the corresponding groupwise separability restriction, so that seven unknown parameters remain to be estimated. Similarly, a price function P is groupwise separable in outputs {C, I} and inputs {K, L} with prices {qC, qI} and {qL, qK} if and only if the price function can be represented in implicit form, as follows:

We can represent the price function in explicit form as follows:

Restriction on the parameters of the translog representation of the price function P corresponding to groupwise separability in the prices {qC, qI} and {qK, qL} can be derived in the same way as given in (10.11). The translog representation of a groupwise separable price function P is not necessarily groupwise separable. The jointly necessary and sufficient conditions for groupwise separability of the translog price function are the condition (10.11) and the additional restriction for explicit groupwise separability, r = 0. Similar restrictions must hold for {qC, qL} separability from {qI, qK} and {qC, qK} separability from {qI, qL}.

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If the production function F is homogenous of degree one, groupwise separability of the production function in {C, I} and {K, L}, for example, is equivalent to groupwise separability of the price function in the prices {qC, qI} and {qK, qL}. 7 However, the translog representation of a groupwise separable production function is not necessarily groupwise separable; similarly, the translog representation of a groupwise separable price function, which corresponds to a groupwise separable production function, is not necessarily groupwise separable. We conclude that groupwise separability of the translog representation of the production function F does not imply groupwise separability of the translog representation of the price function P and vice versa. 10.3 Technical Change We next present an approach to the characterization of changes of technology over time studied by Jorgensen (1986) and empirically implemented by them with data for the private domestic U.S. economy. In this approach time is treated symmetrically with inputs, outputs or prices in the description of the technology. To characterize changes in technology over time we employ restrictions on the production or price functions corresponding to separability in commodities and time. We consider restrictions on technical change associated with groupwise separability of the production function F. We begin by considering a production function that is groupwise, separable in the pair of outputs {C, I} and the pair consisting of the dependent variable L and time {L, t}.8 A production function that is separable in these two pairs of variables can be represented, implicity, in the form:

In explicit form, the production function F can be represented as follows:

Proceeding as in our analysis of groupwise separability in two pairs of commodities, we can derive two restrictions on the parameters of the translog representation:

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Groupwise separability in the variables {C, t} and {K, L} implies the restrictions of the form:

We next consider restrictions on the parameters of the translog representation of the production function F, where the dependent variable L is not included in either of the two pairs of variables, say {C, I} and {K, t}. As before, we can represent the production function, implicitly, in the form:

by definition of groupwise separability. By differentiating implicitly two restrictions can be derived on the parameters of the translog representation:

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There are two more sets of restrictions similar to the one given above where L is not included in either one of the two groups considered. Given symmetry, there are twelve possible sets of groupwise separability restrictions, involving time. Besides the three, already given under (10.15), (10.16) and (10.17) we obtain the following: {C, K} separable from {I, t}:

{C, K} separable from {L, t}:

{C, L} separable from {I, t}:

{C, L} separable from {K, t}:

{C, t} separable from {I, K}:

{C, t} separable from {I, L}:

{I, K} separable from {L, t}:

{I, L} separable from {K, t}:

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{I, t} separable from {K, L}:

Each set of restrictions involves a set of two restrictions with the introduction of one new parameter. Under symmetry, there are eight unknown parameters to be estimated. The translog representation of a production function F that is groupwise separable in two pairs of variables involving time is not necessarily groupwise separable in these same variables. For groupwise separability of the translog representation in the pairs of variables {C, I} and {L, t} a necessary and sufficient set of restrictions consists of the groupwise separability restrictions given under (10.15) together with the explicit separability restriction:

This implies that the parameters bCt and bIt are zero, so that the translog function is groupwise separable. Given any set of groupwise separability restrictions involving time and the dependent variable L in the production function, there are nine possible sets of explicit groupwise separability restrictions of this type, corresponding to parts (10.15) (10.26), excluding (10.17), (10.18) and (10.22) which do not involve L. Each of these restrictions is imposed given the corresponding groupwise separability restrictions, so that there are seven unknown parameters to be estimated. We will not focus attention on explicit separability in pairs of variables that exclude the dependent variable L; they can be generated by conjunction of sets of explicit groupwise separability restrictions already discussed. Restrictions on technical change associated with groupwise separability of the price function P can be derived and tested exactly as in the model based on the translog representation of the marginal productivity functions. Groupwise separability of a price function P implies precisely analogous restrictions on the parameters of the translog price function. 10.4 Tests We have developed econometric models for characterizing the structure of technology and changes in technology over time. We propose

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Figure 10.1 Tests of groupwise separability in commodities. to test restrictions derived from groupwise separability in commodities and time. Our proposed test procedure is presented in diagrammatic form in two figures. We first impose the symmetrv restrictions implied by the theory of production. We then proceed to test the restrictions derived from groupwise separability of the production function in commodities. Given these restrictions, we proceed to test the additional restrictions implied by explicit groupwise separability. All three tests for groupwise separability are carried out in parallel. Each of the three tests for explicit groupwise separability is carried out given the corresponding groupwise separability restrictions. The groupwise separability restrictions involve a set of two equality restrictions with the introduction of one new parameter. Given symmetry, this reduces the number of unknown parameters to be estimated by one. The explicit groupwise separability restrictions involve one additional equality restriction, leaving seven unknown parameters to be estimated. Continuing with tests of groupwise separability of the production function in time, our test procedure is presented diagrammatically in

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Figure 10.2 Tests of groupwise separability in time. (There are twelve sets of tests of this type; this diagram gives only two sets of such tests corresponding to the group {C, I}.) figure 10.2. We first test groupwise separability for each of the twelve possible groups consisting of one pair of commodities and one pair of a commodity and time. If we accept groupwise separability for any two pairs, we proceed to test explicit groupwise separability for these two pairs. All twelve tests for groupwise separability in time are carried out in parallel. To dualize this analysis we observe that a precisely parallel test procedure can be developed for the price function with analogous tests of the restrictions on the parameters of the translog representation of the price function. 10.5 Estimation and Test Statistics Our empirical results are based on the same annual time series data for the private domestic economy of the Federal Republic of Germany for the period 1950 1973 as those employed in our paper on tests of the theory of production. 9 As the value shares sum to unity, only two

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of their random variables are distributed independently. We have fitted the two equations for the value shares of consumption and investment and one equation for the rate of technical change generated by translog representation of production and price functions. 10 As we impose the symmetry restrictions our estimates of the unknown parameters satisfy these restrictions. In table 10.1 we present estimates of the unknown parameters associated with restrictions implied by groupwise separability. Parameter estimates for the translog representation of the price function are given in table 10.2. We give the estimates only for those specifications which we will discuss in the last section. To test the validity of restrictions implied by groupwise separability of production and price functions in commodities and in time, we employ test statistics based on the likelihood ratio l, where

The likelihood ratio is the ratio of the maximum value of the likelihood function L for the econometric model of production w, subject to restriction to be tested, to the maximum value of the likelihood function for the model W without restriction. There are twenty-four observations for the period 1950 1973 for each behavioral equation so that the number of degrees of freedom available for statistical tests of restrictions on the structure of technology and changes in technology is 72 for either model. For normally distributed disturbances, the likelihood ratio is equal to the ratio of the determinant of the restricted estimator of the variance-covariance matrix of the disturbances to the determinant of the unrestricted estimator, each raised to the power(n/2). Our test statistic for each set of restrictions is based on minus twice the logarithm of the likelihood ratio, or:

where is the restricted estimator of the variance-covariance matrix and , is the unrestricted estimator. Under the null hypothesis this test statistic is distributed, asymptotically, as chi-squared with number of degrees of freedom equal to the number of restrictions to be tested.

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Table 10.1 Parameter estimates, translog production function 3. {C, I} Parameter 1. Symmetry 2. {C, I} explicitly explicitly separable from separable from {K, t} {K, L} .999 (.005) .995 (.004) .993 (.005) aC .507 (.123) .401 (.029) .362 (.018) bCC .385 (.051) .402 (.029) .418 (.021) bCI .122 .056 bCK (.094) (.007)

4. {C, I} separable from {L, t}

5. {C, L} separable from {K, t}

6. {C, t} separable from {K, L}

.999 (.005) .366 (.047) 3.29 (.025) .036 (.064)

.997 (.005) .30 (.039) .343 (.047) .043 (.010)

.996 (.005) .241 (.052) .339 (.051) .098 (.011)

7. {I, L} explicitly separable from {K, t} .996 (.005) .65 (.091) .478 (.015) .173 (.088)

bCt aI bII bIK

.008 (.002) .64 (.003) .408 (.028) .023 (.036)

.006 (.0008) .639 (.002) .402 (.027)

.005 (.0007) .64 (.002) .401 (.027) .016 (.005)

.005 (.0008) .64 (.003) .387 (.024) .058 (.026)

.004 (.0009) .64 (.003) .396 (.028) .053 (.034)

.0028 (.001) .64 (.003) .396 (.03) .057 (.035)

bIt

.002 (.0009)

.0018 (.0006)

.0015 (.0005)

.003 (.0005)

.003 (.001)

.003 (.001)

aK bKK

.639 (.005) .145 (.083)

.635 (.005)

.632 (.0053) .176 (.069)

.639 (.005) .942 (.073)

.637 (.005) .01 (.043)

.636 (.005) .04 (.039)

.639 (.005) .173 (.084)

bKt at btt r

.01 (.0017) .058 (.005) .0019 (.0007)

.008 (.0007) .058 (.005) .0019 (.0007)

.0065 (.0006) .058 (.005) .0019 (.0007)

.008 (.001) .058 (.005) .002 (.0007) .005 (.0007)

.007 (.0006) .058 (.005) .0018 (.0007) .067 (.015)

.006 (.0006) .059 (.005) .0018 (.0007) .098 (.01)

.011 (.002) .058 (.005) .002 (.0007)

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Table 10.2 Parameter estimates, translog price function Parameter 1. Symmetry 2. {C, K} separable from {I, L} aC bCC bCI bCK

.993 (.007) .651 (.18) .538 (.17) .112 (.07)

.994 (.006) .592 (.118) .470 (.070) .122 (.058)

bCt

.003 (.0008)

.003 (.004)

aI bII bIK bIt aK bKK bKt at btt r

.638 (.006) .25 (.164) .287 (.052) .011 (.0006) .63 (.004) .175 (.046) .008 (.0005) .058 (.005) .0018 (.0007)

.636 (.004) .172 (.025) .298 (.045) .011 (.0006) .631 (.004) .176 (.043) .008 (.0005) .058 (.005) .0018 (.0007) .473 (.072)

3. {C, L} explicitly separable from {K, t} 1.003 (.008) .345 (.169) .345 (1.69) .632 (.006) .125 (.171) .221 (.034) .009 (.170) .635 (.004) .221 (.178) .009 (.0004) .058 (.005) .0016 (.0007) -

4. {C, t} separable from {K, L} .992 (.006) .677 (.149) .541 (.149) .136 (.013) .003 (.006) .638 (.006) .24 (.152) .301 (.036) .011 (.0006) .631 (.004) .165 (.037) .008 (.0005) .058 (.005) .0018 (.0007) .137 (.013)

To control the overall level of significance for each series of tests of the production and price representation, we set the level of significance for each series at 0.05. We assign a level of significance of 0.01 to tests of groupwise separability in commodities. To tests of groupwise separability in time we assign a level of significance of 0.04. Within our set of tests for groupwise separability in commodities, we can distinguish two stages: groupwise separability for each of the three possible groups and explicit groupwise separability for each of these groups. We assign a level of significance of 0.00167 to each of these six tests. Similarly, within our two stages of tests for groupwise separability in time we assign a level of significance of 0.02 to groupwise separability and 0.00167 to each of the twelve tests at that stage. Similarly, we assign a level of significance of .02 to groupwise explicit separability and 0.00167 to each of the twelve tests at that stage. In our complete series of tests for econometric models of production based on translog production and price functions, only tests of groupwise separability and groupwise explicit separability are ''nested," none of the other tests are "nested" so that the sum of levels of significance for all tests provides an upper bound to the overall level of significance for all of these tests considered simultaneously.

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10.6 Conclusion Our objective has been to develop tests of restrictions for characterizing the structure of technology and changes in technology over time. For econometric models of production based on translog representations of the marginal productivity and supply and demand functions, we have assigned levels of significance to each of our tests of hypotheses about the structure of technology and changes in technology over time so as to control the overall level of significance for all tests at 0.95. The probability of a false rejection for one test among the collection of tests is less than or equal to 0.05. With the aid of critical values for our test statistics given in table 10.3, we can evaluate the results of our tests given in table 10.4. If the test statistic for one of the hypotheses summarized in table 10.4 is larger than its corresponding critical value, given in table 10.3, we reject the hypothesis at the assigned level of significance. We first test groupwise separability in commodities under the translog representation of a production function. The results of our tests of groupwise separability in commodities for each possible group are presented in table 10.4. Our first conclusion is that the production function is groupwise separable in outputs {C, I}, and inputs {K, L}. Our test reveals that the usual presentation of a production function with separability in inputs and outputs is a valid specification which we cannot reject with German data. After having accepted groupwise separability we can proceed to test explicit groupwise separability of outputs {C, I} from inputs {K, L}; we accept this hypothesis. To obtain further simplifications in our representation of technology we next test groupwise separability in time. From the results in table 10.4 we observe that we accept groupwise separability in time for the pair {C, I} from {L, t} and explicit groupwise separability in time for the pair {C, I} from {K, t}. Our results are consistent with groupwise separability of the two outputs {C, I} from the group consisting of the two inputs and time {K, L, t}. This implies that we can Table 10.3 Critical values of c2 Degrees of freedom 1

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Level of significance .05 .01 .005 3.84 6.64 7.88

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Table 10.4 Test statistics for translog production and price functions Hypothesis Degrees of freedom Production Given symmetry Groupwise separability in commodities {C, I}

1

.56

Price

15.33

{K, L} {C, K} {C, L}

{I, L} {I, K}

1

49.59

.26

1

15.29

12.13.

Groupwise separability in time {C, I}

1

.84

24.05

{K, t} {C, I} {C, K} {C, K} {C, L} {C, L} {C, t} {C, t} {C, t} {I, K} {I, L} {I, t}

{L, t} {I, t} {L, t} {I, t} {K, t} {I, K} {I, L} {K, L} {L, t} {K, t} {K, L}

1

1.86

52.67

1

37.06

26.99

1

31.37

74.40

1

41.37

11.47

1

3.67

4.79

1

14.32

13.20

1

36.34

21.27

1

6.17

.15

1

10.96

14.09

1

1.17

34.73

1

12.7

32.37]

Given separability in commodities Explicit groupwise separability in commodities {C, I}

1

2.95

10.07

{K, L} {C, K} {C, L}

{I, L} {I, K}

1

7.82

27.27

1

23.39

.06

Given separability in time

Explicit groupwise separability in time {C, I}

1

6.4

13.26

{K, t} {C, I} {C, K} {C, K} {C, L} {C, L} {C, t} {C, t} {C, t} {I, K} {I, L} {I, t}

{L, t} {I, t} {L, t} {I, t} {K, t} {I, K} {I, L} {K, L} {L, t} {K, t} {K, L}

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1

29.6

23.92

1

16.85

10.33

1

.11

0.46

1

.68

7.92

1

17.50

9.29

1

6.86

2.98

1

79.90

53.15

1

43.86

61.27

1

20.52

62.50

1

6.09

35.20

1

6.09

35.30

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construct an index of real output from price and quantity data on consumption and investment goods output. Output can be represented as a function of the two inputs and time. Continuing with our analysis of groupwise separability in time, we accept groupwise separability for the pair {K, L} from {C, t}. However, we reject groupwise separability of the pair {K, L} from {I, t}, so that our results are not consistent with groupwise separability of the two inputs {K, L} from the groups consisting of the two outputs and time {C, I, t}. We conclude that we cannot construct an index of real input from price and quantity data on capital and labor input. Equivalently, we conclude that technical change is not Hicks-neutral. Finally, we accept groupwise separability for the pair {C, L} from {K, t} and explicit groupwise separability for the pair {I, L}, from {K, t}, so that our results are consistent with groupwise separability of the group {K, t} from the group {C, I, L}. This implies that we can construct an index of the two outputs and labor input from price and quantity data on these commodity groups. Under this restriction technical change is Solow neutral. We conclude that our results are consistent with either of two simplifications of our representation of technology, namely, groupwise separability of the group of two outputs {C, I} from the group of two inputs and time {K, L, t} or groupwise separability of the group {C, I, L} from the group {K, t}. The results of our tests of separability in goods and time for the translog price function are consistent with groupwise separability of the group {C, K} from {I, L}, explicit groupwise separability of the group {C, L} from {K, t} and groupwise separability of the group {K, L} from {C, t}. No combination of these restrictions implies simplifications of our representation of technology similar to those we obtained for the translog production function. The result that the price function is not groupwise separable in the input prices from the output prices is no contradiction to the result we have obtained under the translog representation of the production function. Even if the underlying price function is separable, the test results obtained under the translog production function can differ from the test results obtained under the translog price function because the translog price function is not dual to the translog production function. If we know the true specification of the underlying separable production function and the true specification of the underlying price function then both functions must be separable in the same partitioning.

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10.7 Summary The purpose of this chapter is to employ econometric models of production to develop tests of parametric restrictions for characterizing the structure of technology and changes in technology empirically. Our models are based on the translog production function in two outputs and two inputs and the translog price function in the corresponding prices. We consider restrictions on patterns of substitution and technical change implied by separability. We present empirical tests of each set of restriction for time series data of the Federal Republic of Germany for the period 1950 1973. Notes 1. See Conrad and Jorgenson (1977). 2. Translog production and price functions were introducted by Christensen, Jorgenson, and Lau (1971, 1973). The approach to technical change presented below is due to Jorgenson (1986). An analogous approach to analyzing the structure of consumer preferences and changes in preferences over time is given by Jorgenson and Lau (1975). 3. For more detailed discussion, see Jorgenson (1986). 4. The price function was introduced by Samuelson (1953 1954) and has been discussed by Burmeister and Kuga (1970) and by Christensen, Jorgenson, and Lau (1973). 5. A review of duality in the theory of production is given by Diewert (1974a) and Lau (1974). See also: Hotelling (1932), Jorgenson and Lau (1974a, 1974b), Samuelson (1953 1954), Shephard (1953, 1970), and Uzaiva (1964). 6. This definition of groupwise separability was introduced by Jorgenson (1986). The conventional definition of separability, due to Leontief (1947a), is based on the explicit form of the production function. Under this definition the marginal rate of substitution between consumption and investment goods output is independent of capital input:

This definition does not treat labor input symmetrically with the other variables in the production function. See also: Goldman and Uzawa (1964) and the references given there. 7. See Jorgenson (1986). 8. Jorgenson (1986) refers to groupwise separability involving time as groupwise neutrality. 9. See Conrad and Jorgensen (1975), p. 80, Table 20, Column 4 (L) and Column 5 (qL); p. 70, Table 14, Column 4 (qc), Column 5 (C), Column 7 (qI) and Column 8 (1); p. 51, Table 9, Column 3 (K) and Column 4 (qK) qK is normalized to one in 1962 and real capital input is scaled to real property income). These data were employed by Conrad (1975). 10. Our estimator is based on the method of maximum likelihood presented by Mallinvaud (1970).

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11 Statistical Inference for a System of Simultaneous, Nonlinear, Implicit Equations in the Context of Instrumental Variable Estimation A. Roland Gallant and Dale W. Jorgenson Statistical inference for a system of simultaneous, nonlinear, implicit equations is discussed. The discussion considers inference as an adjunct to two- and three-stage least-squares estimation rather than in a general setting. For both of these cases the non-null asymptotic distribution of a test statistic based on the optimization criterion and a test based on the asymptotic distribution of the estimator is found; a total of four. It is argued that the tests based on the optimization criterion are to be preferred in applications. The methods are illustrated by application to hypotheses implied by the theory of demand using a translog expenditure system and data on personal consumption expenditures for durables, non-durables, and energy for the period 1947 1971. 11.1 Introduction Statistical inference for a system of simultaneous, nonlinear, implicit equations is discussed. The discussion considers inference as an adjunct to two- and three-stage least-squares estimation rather than in a general setting; the article is, thus, to be regarded as a continuation of Gallant (1977a). Arguments are put forth to suggest that inference regarding the parameters of more than one equation should be based on three-stage estimators and that inference regarding the parameters of a single equation should be based on two-stage, least-squares estimators. Accordingly, the article considers inference for both bases. The results in three-stage estimation are stated subject to an overall maintained hypothesis. This degree of generality is frequently required in applications and, if not, one need only regard the restrictions as the identity transformation and proceed accordingly. Hypotheses are formulated in one of two forms. The first form is as an explicit functional dependency of a larger set of parameters on a smaller set, H:r = f(b) where b is the smaller set. The second form is as a set of parametric restrictions, H: h(r) = t0. Methods for converting

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from one form to the other are displayed in the appendix but they are awkward except for linear hypotheses. Thus, the form must be that which arises naturally in the application in most instances. The analog of the likelihood ratio test for two- or three-stage least-squares estimation, as appropriate, is the more convenient when the hypothesis is of the first form. The Wald test is the more convenient when the hypothesis is of the second form. Accordingly, both tests are presented for both two- and three-stage analysis; a total of four. The non-null asymptotic distribution of each test statistic is obtained. Neglecting convenience, there arises the question of a choice. Reasoning by analogy with the simulations reported in Gallant (1975, 1976, 1977b) one may conjecture that the analog of the likelihood ratio test will possess the greater robustness of validity; i.e., the nominal significance level computed according to the asymptotic distribution will be more nearly correct in finite samples. Again by analogy, one expects the power curves of these tests to cross and so power considerations will not permit a choice. Thus, the conjecture that the tests based on the optimization criterion possesses the greater robustness of validity must dictate the choice until Monte Carlo evidence is acquired. Confidence intervals and regions are not discussed per se in the paper. However, as is well known a hypothesis of the form H: h(r) = t0 may be inverted to find a confidence region for the (new) parameters t. Typically h is merely a selection function as, for example, h(r1, r2, r3) = (r1, r2). One puts in the confidence region all those t0 for which the hypothesis H: h(r) = t0 is accepted at the a-level of significance. This will result in a (1 a) × 100 percent confidence region. The statistical properties of confidence regions so constructed are discussed by Pratt (1961) in general and by Gallant (1976) in a non-linear single equation context. Simultaneous tests and confidence regions may be constructed by analogy with the methods of Scheffé (1953, 1959) and Gabriel (1964, 1969). A single hypothesis is obtained at the intersection of all parametric restrictions of interest. The critical value for the corresponding test statistic is used as the basis for either simultaneous tests of the individual hypothesis or simultaneous confidence regions. The analog of the likelihood ratio test has the monotonicity property which is useful in simultaneous inference. Data on personal consumption expenditures for durables, non-durables, and energy for the period 1947 1971 are used to illustrate

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the methods proposed here. Three-stage least-squares methods are illustrated by a test of the hypothesis of symmetry in a translog expenditure system. Two-stage least-squares methods are illustrated by a test of the hypothesis of homogeneity in a translog expenditure system. As mentioned above, inference is considered here as adjunct to two- and three-stage least-squares estimation. The competing estimation approach is maximum likelihood (Amemiya, 1977). Results similar to those here are available for inference considered as an adjunct to maximum likelihood estimation in Gallant and Holly (1980). The issues regarding a choice may be addressed as follows. Two- and three-stage least-squares estimators possess the greater robustness of validity. Only first and second moment assumptions are required for their validity whereas maximum likelihood methods require a correct specification of the density function of the error process for validity; the latter is a more difficult task. Maximum likelihood estimation methods are more efficient, granted validity (Amemiya, 1977) and one would presume that the associated tests would, therefore, be more powerful. Statistical package designers appear to prefer two- and three-stage least-squares methods, apparently, because they are easier to implement. As a result, two- and three-stage methods will probably see more frequent use in applications. Presently, the TSP package, Harvard University, Cambridge, Massachusetts and the SAS package, SAS Institute, Raleigh, North Carolina contain two- and three-stage nonlinear least-squares procedures; there may be others. The reader who is interested in the results for use in applications may avoid reading the supporting detail as follows. Skim sections 11.2 11.4, reading only to become familiar with the notation. Then sections 11.3, 11.6, 11.10, and 11.13 may be read independently and only as required for the application. All details needed to perform the tests are found in sections 11.3, 11.6, 11.10, and 11.13. The reader who is interested in proofs will find it necessary to have Gallant (1977a) at hand. 11.I Three-Stage Least-Squares Estimation 11.2 The Statistical Model Subject to a Maintained Hypothesis The observed M vector of endogenous variables yt corresponding to the observed k vector of exogenous variables xt are generated according to the statistical model

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The unknown parameters are pa vectors contained in the parameter spaces Qa, a = 1, 2, . . . , M. The use of the asterisk, throughout, is to emphasize that it is the true, but unknown, value of qa which is meant; it is omitted when qa is treated as a variable for the purpose of, for example, differentiation. The errors

are assumed independent each following the M-variate normal distribution with mean vector zero and positive definite variancecovariance matrix The normality assumption is a convenience in the proofs and experience indicates that it is sound practice in applications to employ transformations f(·) (Kruska, 1968) so that is apparently normally distributed for some preliminary estimate Since the model is already in implicit form such a transformation does not complicate the analysis. If, as in the Box-Cox transformation, f depends on a parameter, this parameter may be included in qa. Instrumental variables consisting of K-vectors zt (t = 1, 2, . . . , n) are assumed available for estimation. The optimal choice of these instrumental variables is discussed in Jorgenson and Laffont (1974) and Amemiya (1977). An estimator is assumed to be available which converges almost surely to S and which has probability. Such a matrix, with typical element may be constructed from residuals as

bounded in

where and are either two- or three-stage least-squares estimators (Gallant, 1977a). It must be emphasized that is fixed throughout. It is not permissible to use one version of for fitting the full model and another for the restricted model when computing the analog of the likelihood ratio test. It is assumed that any a priori within equation restrictions which are not of inferential interest have been eliminated by reparameterization. Also, each normalization rule has been eliminated by parameterization and, thus, the parameters qa above refer to the model in reparameterized form. Estimation and inference are carried out subject to a maintained hypothesis that

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where

and g maps , a subset of r-dimensional Euclidean space with into As formulated here, q = g(r) incorporates only across equation restrictions. This is not strictly necessary; within equation constraints may be included as long as the identification and rank assumptions listed later are satisfied by the unrestricted model. 11.3 The Test Statistics T0: An Analog of the Likelihood Ratio Test Three-stage least-squares estimators are obtained by minimizing

The three-stage least-squares estimator subject to the maintained hypothesis q = g(r) is that value minimizes S[g(r)] over . Let denote the minimum thus obtained;

which

Consider testing

where b is an s vector contained in with s < r. The three-stage least-squares estimator subject to the maintained hypothesis q = g(r) and the null hypothesis r = f(b) is that value which minimizes S{g[f(b)]} over Let

denote this minimum;

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A test statistic which suggests itself is

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When the sample is in accord with the null hypothesis, T0 will be near zero and when it is not T0 will be large. As shown later, T0 is distributed asymptotically as a chi-square with r s degrees of freedom when H is true so that an asymptotically level a test of H against A is: reject H when T0 exceeds the upper a × 100 percentage point of the chi-square distribution with r s degrees of freedom. 11.4 Regularity Conditions Throughout e(·) denotes expectation with respect to the random variables For example, if N(e; 0, S denotes the M-variate normal distribution function and Y(x,e) denotes the reduced form of the systemthat is, yt = Y(xt, et) implies

Some additional notation required is the following:

Assumptions relating to the system. The assumptions listed in section 4 of Gallant (1977a) are in force. In addition, the averages,

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converge uniformly in (qa, qb). These limit assumptions are plausible in applications as shown in detail in Gallant (1977a) and Gallant and Holly (1980). Assumptions relating to the maintained hypothesis. The function g(r) is a twice continuously differentiable mapping of a compact set into the parameter space satisfies g(r) = q *, and r * is contained in an open sphere G(r*) has rank r.

There is exactly one point r * in which * which is, in turn, contained in . The p by r matrix

Assumptions relating to the hypothesis H: r = f(b). The hypothesis H: r = f(b) is to be regarded as one of a sequence of hypotheses indexed by n. The functions fn(b) are twice continuously differentiable mappings of the compact set

into

. As n increases, fn(b) converges to, say,

uniformly in b; Fn(b) = (¶/¶b') fn(b) converges

to, say, uniformly in b; and, (¶2/¶bi¶bj)fkn(b) converges to, say and k = 1, 2, . . . , r. There is exactly one point b * in which satisfies sphere * which is, in turn contained in . Further and by s matrix has rank s.

uniformly in b for i, j = 1, 2, . . . , s, and b * is contained in an open converge to finite limits. The r

The limit assumptions relating to fn, etc. may appear implausible at first glance. Plausible circumstances such that they are satisfied in applications are given in the appendix. In view of the discussion in the appendix, a statement that the null hypothesis is true is taken to mean that and for all n. Note that the choice b0 = b * satisfies the definition of b0 since

when the null hypothesis is true.

11.5 The Asymptotic Distribution of T0 First, three propositions are proved which characterize as linear and quadratic functions of the errors plus a remainder which is negligible in large samples. Next Theorem 11.1, which characterizes

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T0 as a quadratic function of the errors with known sampling distribution plus a negligible residual, is proved using these results. PROPOSITION 11.1. converges almost surely to b * and converges almost sure to r *. PROOF. The argument is entirely analogous to the proof of Theorem 2 of Gallant (1977a). Q.E.D. PROPOSITION11.2.

is bounded in probability and may be characterized as 1

is bounded in probability and may be characterized as

PROOF. By Proposition 11.1, the sequence otherwise in the sense that

is equivalent to the sequence where and converges almost surely to zero. Thus, it may be assumed that

for all n without loss of generality. Let The first-order Taylor series expansion of

etc. about r * may be written as

where H is the nM by r matrix, H = (H'1, H'2, . . . , H'M)'; the t-th row of the n by r submatrix Ha is where varies with t and and lies on the line segment joining typical element of (1/n)' Ha is a finite sum of products each having exactly one term of the form remaining terms selected from among: the first and second partial derivatives of gia(r) evaluated at

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to r *. A and

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and

The latter terms are bounded almost surely as a direct consequence of the assumptions and converges almost surely to zero whence (1/n)Z'Ha and (1/n)(I Ä Z')H converge almost surely to zero. A second Taylor's expansion yields

where

lies on the line segment joining

Since

minimizes S{g[f(b)]} over

to b0 and

and is contained in

After substituting the Taylor's expansion of

Thus

* it follows that

and rearranging terms, the equation may be written as

Lemma A.1 of Gallant (1977a) implies that converges almost surely and its difference with converges almost surely to zero. The same for (1/n)(I Ä Z') (QG + H) and in view of the remarks above regarding (1/n)(I Ä Z')H. By Lemma A.3 of Gallant (1977a), q is bounded in probability as is, by assumption, It follows, then, that is bounded in probability and that

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where an converges in probability to zero. Multiply the left- and right-hand sides by which exists for n sufficiently large, and the first conclusion of Proposition 11.2 is obtained after division by The second conclusion is obtained by a similar argument. Q.E.D. PROPOSITION11.3.

PROOF. As above, expansion about

may be assumed to take on values in

* without loss of generality. A second-order Taylor's

yields

where is on the line segment joining to b0. Now for in * the first-order condition satisfied whence the first-order term of the Taylor's expansion is zero. Now

is

where Dn has typical element

Note that converges almost surely to q * by Proposition 11.1 and the uniform convergence of f(b). By Lemma A.1 of Gallant (1977a) the last term in brackets converges almost surely to

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which is zero, almost surely, by Lemma A.3 of Gallant (1977a). The remaining terms in brackets converge almost surely to finite matrices of finite order by the same argument and by the assumptions regarding and (1/n) Z'Z. Thus, dijn converges almost surely to zero. By the same type of argument, Proposition 11.2 we have

converges almost surely and the difference with

converges almost surely to zero,. Since

is bounded in probability by

and converges in probability to zero. This proves the first assertion. A sketch of the proof of the second assertion follows:

where an converges almost surely to zero and bn converges in probability zero. Q.E.D. THEOREM 11.1. The test statistic,

may be characterized as

where

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The random variable X0 is distributed as the non-central chi-square with r s degrees of freedom and noncentrality parameter

in the notation of Searle (1971, p. 49). 2 PROOF. Observe that: the matrix

is idempotent.

and A' (S 1 Ä PZ)A = P1. Using Proposition 11.2 and these facts, one may write

where an converges in probability to zero. Whence, by Proposition 11.3,

where bn converges in probability to zero. The cross-product term may be decomposed as

It is seen that the cross-product term is the product of matrices of finite order whose elements are bounded in probability save an which converges in probability to zero whence the cross-product term converges in probability to zero. Similarly

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Similar arguments imply

and note that z is distributed as the nM variate normal with mean zero and the identity as variance-covariance matrix. In this notation X0 may be written as

which is a non-central chi-square random variable with

degrees of freedom and non-centrality parameter

by Theorem 2 of Searly (1971, p. 57). Q.E.D. 11.6 The Test Statistic TA: The Wald Test Consider testing

where h(r) is a once continuously differentiable u-vector mapping

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into ; denote its Jacobian by H(r) = (¶/¶r')h(r), a u by r matrix. In the appendix it is shown that this hypothesis is equivalent to a hypothesis of the form H: r = f(b), considered above, provided there is a function h2(r) = b such that defines a one-to-one transformation from onto × . For example, with three parameters the hypothesis

is equivalent to

where the function h2(r) is (b1, b2) = (r1 r2, r3). It is shown in Gallant (1977a) that is distributed asymptotically as the r variant normal with mean zero and a variance-covariance matrix (G'WG) 1 for which

is a strongly consistent estimator. This suggests the use of the test statistic

where As shown later, TA is distributed asymptotically as a chi-square with u degrees of freedom when H is true so that an asymptotically level a test of H against A is: reject H when TA exceeds the upper a × 100 percentage point of the chi-square distribution with u degrees freedom. 11.7 The Asymptotic Distribution of TA The assumptions required to obtain the asymptotic distribution of TA are as in section 11.4 except that the assumptions relating to the hypotheses are replaced by: Assumptions relating to the hypothesis H: h(r) = t0. The hypothesis H: h(r) = t0 is to be regarded as one of a sequence of hypotheses indexed by n. The function h(r) is a once continuously differentiable

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mapping of the compact set onto the compact set . The sequence t* = h(r*). The u by r matrix h(r) = (¶/¶r')H(r) has rank u at r = r*.

converges to a finite limit where

The test statistic TA may be characterized as a quadratic function of the errors which has the non-central chisquare distribution plus a negligible residual. THEOREM 11.2. The test statistic,

may be characterized as

where

The random variable XA is distributed as the non-central chi-square with u degrees of freedom and non-centrality parameter

in the notation of Searle (1971, p. 49). PROOF. As above,

where

may be assumed to take on values in * without loss of generality. Set etc. The first-order Taylor series expansion of about t* may be written as

is on the line segment joining

to r*. The continuity of H(r) and the almost sure convergence of

imply that converges almost surely to H = H(r*). Since 11.2 we may write

to r*

is bounded in probability by Proposition

where an converges in probability to zero. It is seen from this

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expression that is bounded in probability; this and the fact that same limit as the almost sure limit of implies that

where bn converges in probability to zero. Using the previous expression for have

has the

and Proposition 11.2 we

where cn converges in probability to zero. Substituting this expression we have

where dn converges in probability to zero. Now HU + t0 t* is distributed as the u variate normal with mean t0 t* and variance-covariance matrix whence XA is distributed as the non-central chi-square with u degrees of freedom and non-centrality parameter lA by Theorem 2 of Searly (1971, p. 57). Q.E.D. 11.8 An Example: The Hypothesis of Symmetry The section illustrates the application of methods for statistical inference based on the three-stage least-squares estimator for a system of simultaneous nonlinear equations. Consider a model of consumer expenditures based on the direct transcendental logarithmic or translog demand system with time-varying preferences, introduced by Jorgenson and Lau (1975). This system of direct demand functions may be written in the form

where Xj is the quantity consumed of the j-th commodity, Pj is the

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price, ej is the random disturbance, t is time, and M is the value of total expenditure

The parameters {aj,aM,bji,bMi,bjt,bMi} satisfy the restrictions

Normalization of these parameters is required for estimation since this system is homogeneous of degree zero in its parameters. A convenient normalization is

The budget shares yj sum to unity which implies that their variance-covariance matrix has rank one less than the dimension. A regression with such a singular variance-covariance matrix may be estimated without loss of efficiency by dropping an appropriate equation and imposing appropriate parametric restrictions (Kreijger and Neudecker, 1977). In this case it may be verified that any one equation may be deleted from the system and no parametric restrictions are implied in addition to those listed above. The model of consumer expenditures for three commodity groups y1, y2, y3 is, then, given by the system of two equations,

In terms of the notation of the previous sections,

and

The restrictions q = (q'1, q'2)' = g(r) are of the form of a linear transformation q = Gr. The matrix G is of order 18 by 14 with entries 0 or 1 as

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determined by the linear parametric restrictions above. For example, the last row of G is

Tests of the theory of consumer behavior may be formulated as restrictions on the parameters of the translog demand system corresponding to properties of a system derived from the theory of consumer behavior. A critical property of a demand system derived from the theory of consumer behavior is symmetry of the matrix of parameters [bji]. To test the hypothesis, impose the restriction

This hypothesis may be expressed in the notation of the previous sections; put

The restrictions r = f(b) are of the form of a linear transformation r Fb. The matrix F is of order 14 by 11 with entriesd 0 or 1 as determined by the linear parametric restrictions bij = bji. The empirical tests are based on annual time series data for U.S. personal consumption expenditures on three commodity groupsdurables, non-durables, and energyfor the period 1947 1971. 3 The equations for durables and non-durables were the two chosen for fitting. This model is a submodel of a complete system which includes both supply and demand equations. Accordingly, the ratios of prices to total expenditures Pi/M are properly regarded as endogeneous. The complete model implies that the following variables may be taken as instrumental variables: a constant, effective tax rates on labor services and on noncompetitive imports, potential time available for labor with and without an adjustment for quality change, total population, the implicit deflator for the supply of labor services with and without an adjustment for quality change, implicit deflators for government purchases of labor services and noncompetitive imports, income from transfers, lagged private national wealth, total imports, and time. The unrestricted equations involve fourteen unknown parameters. The restricted equations obtained by imposing the symmetry restrictions involve eleven unknown parameters. Estimates of the parameters with and without restrictions are presented in table 11.1. The

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Table 11.1 Three-stage least-squares estimates of the parameters of the translog demand system Parameter (1) Unrestricted (2) Restricted 0.239 0.238 (0.000762) a1 (0.00109) 0.2238 (0.187) 0.110 (0.148) b11 0.168 (0.0840) 0.00929 (0.0306) b12 0.677 (0.454) 0.667 (0.424) b13 0.0262 (0.0242) 0.0136 (0.0225) b1t 0.0500 (0.000357) 0.0500 (0.00316) a2 0.0340 (0.0382) 0.00929 (0.0306) b21 0.0147 (0.0166) 0.0451 (0.00809) b22 0.107 (0.0927) 0.105 (0.0853) b23 0.00142 (0.00488) 0.00102 (0.00454) b2t 0.711 (0.00121) 0.712 (0.000797) a3 1.01 (0.561) 0.667 (0.424) b31 0.566 (0.290) 0.105 (0.0853) b32 1.34 (1.26) 1.37 (1.15) b33 0.0790 (0.0657) 0.0457 (0.0621) b3t entries in parentheses are estimated standard errors, appropriate entries from in the second

in the first column and from

The test statistic for the hypothesis of symmetry of the matrix of parameters [bji] is T0 = 4.88. Under the null hypothesis this test statistic is distributed as chi-squared with three degrees of freedom. Critical values for chisquared are 7.81 for a 0.05 level of significance and 11.3 and a 0.01 level of significance. The hypothesis of symmetry is not rejected. 11.II Two-Stage Least-Squares Estimation 11.9 The Statistical Model In this, the second part, hypotheses involving only the parameters of a single equation of the simultaneous system are considered. In the previous vector notation the model is

It is assumed, as before, that any a priori within equation restrictions which are not of inferential interest and the normalization rule have been eliminated by reparameterization.

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The three-stage least-squares estimator is more efficient than the two-stage least-squares estimator so it would seem that the tests presented in Part I would be more powerful than those in Part II. This may be, but some risk is incurred. The regularity conditions required for the two-stage least-squares estimator are less stringent than for the threestage least-squares estimator. A two equation example in Gallant (1977a) illustrates a consequence of this fact. He shows that if a null hypothesis regarding the parameters of the first equation were true then the second equation would not be identified although the first equation would remain identified. Thus, a test of this hypothesis based on the three-stage least-squares estimator is invalid while a test based on the two-stage least-squares estimator is asymptotically justified. There is always the hazard in nonlinear three-stage least-squares analysis that lack of identification in an equation which is not of interest will invalidate the asymptotic results for tests on the equation of interest. A specification error in an equation which is not of interest will have the same effect. It seems imprudent to assume these risks; thus, one should regard tests for a within equation hypothesis based on two-stage least-squares estimation as preferable to those based on three-stage least-squares estimation. 11.10 The Test Statistic : An Analog of the Likelihood Ratio Test The two-stage least-squares estimators are obtained by minimizing

The two-stage least-squares estimator is that value following previous convention.

which minimizes Sa(qa) over Qa. Let

denote

A strongly consistent estimator of saa, the a-th diagonal element of S, with bounded in probability is necessary. Subject to the regularity conditions stated in the next section, one such estimator is

Hereafter, however,

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Consider testing

where b is an s vector with s < pa and is contained in . Let denote that value which minimizes S[f(b)] over and let denote the minimum thus obtained. One may test H against A using

is distributed asymptotically as a chi-square with pa s degrees of freedom when H is true so that an asymptotically level a test of H against A is: reject H when exceeds the upper a × 100 percentage point of a chi-square distribution with pa s degrees of freedom. 11.11 Regularity Conditions Assumptions relating to the equation. The errors are independent each following the normal distribution with mean zero and variance saa. The moment matrix of the instrumental variables (1/n)Z'Z converges to a positive definite matrix P. The parameter space Qa is compact; the true parameter value is contained in an open sphere which is, in turn, contained in Qa. The function qa(y,x,qa) and its first and second partial derivatives with respect to qa are continuous in qa for fixed (y,x). The averages

converge almost surely uniformly in (qa,q''a); similarly when qa is replaced The averages

and the elements of Qa(qa) by

are bounded almost surely. The pa by pa matrix

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is nonsingular. The equation

is identified by the instruments (Gallant, 1977a).

Assumptions relating to the hypothesis qa = f(b). These are exactly the same as those for H:r = f(b) in section 11.4 excepting that qa replaces r and Qa replaces throughout. 11.12 The Asymptotic Distribution of Results characterizing and section 11.5 as to be redundant. PROPOSITION 11.4. PROPOSITION 11.5.

are stated here but the proofs are omitted as they are so similar to those of

converges almost surely to b* and

converges almost surely to

.

is bounded in probability and may be characterized as

is bounded in probability and may be characterized as

PROPOSITION 11.6.

THEOREM 11.3. The test statistics

may be characterized as

where

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The random variable centrality parameter,

is distributed as the noncentral chi-square with pa s degrees of freedom and non-

in the notation of Searle (1971, p. 49). 4 11.13 The Test Statistics The Wald Test

:

Consider testing

where h(qa) = is a once continuously differentiable u-vector mapping Qa into ; denote its Jacobian by H(qa) = (¶/¶q'a)h(qa), a u by pa matrix. This hypothesis is similar to that of section 11.6 and the comments made there apply here. The two-stage least-squares estimator is distributed asymptotically as the pa-variate normal with mean zero and a variance-covariance matrix for which

is a strongly consistent estimator (Gallant, 1977a). This suggests the use of the test statistic

where is distributed asymptotically as a chi-square with u degrees of freedom when H is true so that an asymptotically level a test of H against A is: reject H when exceeds the upper a × 100 percentage point of the chi-square with u degrees of freedom. 11.14 The Asymptotic Distribution of The assumptions required to obtain the asymptotic distribution of assumptions relating to the hypothesis are replaced by:

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are as in section 11.11 except that the

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Assumptions relating to the hypothesis H: h(qa) = t0. These are exactly the same as those for H: h(r) = t0 in section 11.6 except that qa replaces r and Qa replaces P throughout. Here again, the proof is so similar to the proof of Theorem 11.2 as to be redundant and is omitted. THEOREM 11.4. The test statistics

may be characterized as

where

The random variable parameter,

is distributed as the non-central chi-square with u degrees of freedom and non-centrality

in the notation of Searle (1971, p. 49). 11.15 An Example: The Hypothesis of Homogeneity This section illustrates the application of methods of statistical inference based on the two-stage least-squares estimator for a single equation in a system of simultaneous equations. As before, consider a model of consumer expenditures based on the translog demand system. This system of demand functions is extended by adding a term in the logarithm of total expenditure to obtain

A test of the theory of consumer behavior is that the parameters associated with total expenditure (bjM,bMM) are equal to zero. Under these restrictions the translog demand system is homogeneous of degree zero in prices and total expenditure.

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Table 11.2 Two-stage least-squares estimates of the parameters of the translog demand system Parameter (1)Unrestricted (2)Restricted 0.236 0.236 (0.00159) a1 (0.00217) 0.000873 (0.455) 0.422 (0.269) b11 0.0825 (0.154) 0.0340 (0.113) b12 0.596 (1.62) 1.13 (0.654) b13 0.119 (0.236) b1M 0.0289 (0.0666) 0.00386 (0.0335) b1t 0.604 (1.86) 2.21 (1.14) bM1 0.479 (0.881) 0.326 (0.559) bM2 2.83 (6.38) 3.98 (2.62) bM3 0.159 (1.01) bMM 0.108 (0.258) 0.150 (0.130) bMt As before, the empirical implementation is based on a model of consumer expenditures for three commodity groups. The equation for durables is

The unrestricted equation involves eleven unknown parameters. The restricted equation obtained by imposing the homogeneity restrictions involves nine unknown parameters. Estimates of the parameters with and without restrictions are presented in table 11.2. The test statistic for the hypothesis of homogeneity of degree zero of the equation for durables is Under the null hypothesis this test statistic is distributed as chi-squared with two degrees of freedom. Critical values for chi-squared are 5.99 for a 0.05 level of significance and 9.21 for a 0.01 level of significance. The hypothesis of homogeneity is not rejected.

Appendix A.1 Conditions such that the Assumptions Regarding the Hypothesis Obtain Take as given, for this purpose, all the assumptions except that fn(b), Fn(b), and (¶2/¶bi¶bj)fkn(b) converge uniformly and that

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has a finite limit. Conditions will be set forth such that these limiting assumptions are satisfied. Consider, then, a hypothesis of the form H:h1(r) = t0 and suppose it is possible to find a one-to-one transformation h of the form

mapping into × . If k denotes the inverse transformation, i.e., k(t,b) = k[h(r)] = r, then the hypothesis H:h1(r) = t0 implies H: r = k(t0,b) and the converse. If one sets f(b) = k(t0,b) then the hypothesis H: r = f(b) is equivalent to the hypothesis H:h1(r) = t0. Suppose further that h(r) and k(t,b) are twice continuously differentiable mappings and that the hypothesized sequence tends to t* =h1(r*) at a rate such that has a finite limit. Now × must be compact because it is the continuous image under h(r) of the compact set . Whence the continuous function k(t,b) is uniformly continuous on × . Then, given e > 0 there is a d > 0 such that But there is an N such that for all which establishes the uniform convergence of The same argument applies to Fn(b) and (¶2/¶bi¶bj) fkn(b).

to

An argument analogous to that employed to prove Theorem 2 of Gallant (1977a) implies that n sufficiently large,

whence

provided e(¶/¶q')q(q) = (¶/¶q')eq(q). Recall that (t*,b*) to obtain

where (t#,b#) is on the line segment joining does as well because

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Now for

and expand

in a Taylor's series about

to (t*,b*). Thus, if converges to a finite limit then converges to a positive definite matrix.

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Lastly, it must be verified that

has a finite limit. Expanding

in a Taylor's series one obtains

It is seen at once that converges to a finite limit since it is the product of a matrix of finite order whose elements converge and a vector of finite length whose elements converge. In this context, the statement that the null hypothesis is true means that t0 º t* hence that the determinant of equation above relating

. Then, since

is bounded away from zero for all n larger than, say, N, the implies

A.2 Computation of l0 for a Hypothesis of the Form h(r) = t0 In an application where the hypothesis is of the form H: h1(r) = t0 it may be tedious to find a function h2(r) = b such that

is a one-to-one transformation, and it may be tedious to find the inverse function k(t,b). However, it is not necessary to find these functions in applications, it is only necessary that they exist. The constrained minimum may be obtained by minimizing s[g(r)] subject to h1(r) = t0 using a constrained optimization algorithm. Similarly r0 may be obtained by minimizing subject to h1(r) = t0. It remains to obtain the matrix F0 in order to compute the non-centrality parameter of Theorem 11.1. Let

and choose any s by r matrix H2 such that the r by r matrix

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is nonsingular; the existence of h2 implies the existence of H2. Partition the inverse of H such that

where F is r by s. This choice of F will suffice for use in the formula for Pz of Theorem 11.1. To see that this is so, note that if h2(r) exists then there is an corresponding to it and the Jacobian of the transformation f(b) = k(t0,b) at the point b0 would be given as F above. That is, the above computational procedure would yield F0 if

whence

were chosen as H2. For every choice of H2 with corresponding F it follows that

Thus, the columns of F, which must be linearly independent, comprise a basis for the s-

dimensional linear space Any choice of H2 generates such a basis whence all F so generated must be of the form F = F0 B where B is s by s and nonsingular. It is a well known property of orthogonal projection operators that the matrix XB(B'X'XB) 1 B'X' is invariant to the choice of nonsingular B. Thus P2 must be invariant to the choice of nonsingular B. Thus P2 must be invariant to the choice of B as may be seen by writing

where

Similar remarks apply to the computation of the test based on the optimization criterion for two-stage nonlinear least-squares estimation. Notes 1. The symbols op(1), op(1/n), etc. are used with their customary meaning. Thus, Yn = op(1) means that each element yin of the vector Yn converges in probability to zero; while means that converges in probability to zero.

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2. When the null hypothesis is true, r0 = r* whence l0 = 0 and X0 has the central chisquare distribution with r s degrees of freedom. 3. These results are taken from Jorgenson (1986). 4. When the null hypothesis is true, degrees of freedom.

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whence

has the central chisquare distribution with pa s

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12 Relative Prices and Technical Change Dale W. Jorgenson and Barbara M. Fraumeni 12.1 Introduction Our objective is to analyze technical change and the distribution of the value of output for thirty-six industrial sectors of the U.S. economy. Our most important conceptual innovation is to determine the rate of technical change and the distributive shares of productive inputs simultaneously as functions of relative prices. We show that the effects of technical change on the distributive shares are precisely the same as the effects of relative prices on the rate of technical change. Our methodology is based on a model of production treating substitution among inputs and changes in technology symmetrically. The model is based on a production function for each industrial sector, giving output as a function of time and of capital, labor, energy, and materials inputs. Necessary conditions for producer equilibrium in each sector are given by equalities between the distributive shares of the four productive inputs and the corresponding elasticities of output with respect to each of these inputs. Producer equilibrium under constant returns to scale implies that the value of output is equal to the sum of the values of capital, labor, energy, and materials inputs into each sector. Given this identity and the equalities between the distributive shares and the elasticities of output with respect to each of the inputs, we can express the price of output as a function of the prices of capital, labor, energy, and materials inputs, and time. We refer to this function as the sectoral price function. Given sectoral price functions for all thirty-six industrial sectors, we can generate econometric models that determine the rate of technical change and the distributive shares of the four productive inputs endogenously for each sector. The distributive shares and the rate of technical change, like the price of sectoral output, are functions of

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relative prices and time. We assume that the prices of sectoral outputs are transcendental logarithmic, or more simply, translog functions of the prices of the four inputs and time. While technical change is endogenous in our models of production and technical change, these models must be carefully distinguished from models of induced technical change, such as those analyzed by Hicks (1932), Kennedy (1964), Samuelson (1965b), von Weizsäcker (1962), and many others. 1 In those models the biases of technical change are endogenous and depend on relative prices. In our models the biases of technical change are fixed, while the rate of technical change is endogenous and depends on relative prices. As Samuelson (1965b) has pointed out, models of induced technical change require intertemporal optimization, since technical change at any point of time affects future production possibilities. In our models myopic decision rules are appropriate, even though the rate of technical change is endogenous, provided that the price of capital input is treated as a rental price for capital services (for further discussion of myopic decision rules, see Jorgenson, 1973b). The rate of technical change at any point of time is a function of relative prices but does not affect future production possibilities. This vastly simplifies the modeling of producer behavior and greatly facilitates the implementation of our econometric models. Given myopic decision rules for producers in each industrial sector, we can describe all of the implications of the theory of production in terms of the sectoral price functions. The sectoral price functions must be homogeneous of degree one in the prices of the four inputs, symmetric in the input prices and time, and nondecreasing and concave in the input prices. A novel feature of our econometric methodology is to fit econometric models of sectoral production and technical change that incorporate all of these implications of the theory of production. The translog price functions that we employ to generate our econometric models cannot be monotone and concave for all possible input prices. However, we can assure concavity of these functions for any prices that result in nonnegative distributive shares for all four inputs, so that price effects on demands for productive inputs are nonpositive wherever the inputs themselves are nonnegative. This new methodology is based on the Cholesky factorization of a matrix of constant parameters in our econometric models associated with price effects. We have fitted these matrices subject to the condition that they must be negative semi-definite.

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Under constant returns to scale the rate of technical change for each sector can be expressed as the negative of the rate of growth of the price of output, plus a weighted average of the rates of growth of the prices of the four inputs. We employ indexes of the rate of technical change and the four input prices that are exact for translog price functions. This important innovation assures consistency between the representations of technology that underly our measures of technical change and input prices and the representations that underly our econometric models. Our most striking empirical finding is that the rate of technical change is predominately decreasing in the prices of capital, labor, and energy and increasing in the price of materials. This pattern characterizes nineteen of the industries included in our study. The rate of technical change decreases with the price of capital in twenty-five industries, decreases with the price of labor in thirty-one industries, decreases with the price of energy in twentynine industries, and increases with the price of materials in thirty-three industries. Our econometric models have been fitted to annual data for the period 1958 1974. Since 1973 the relative prices of capital, labor, energy, and materials in the United States have been altered radically as a consequence of the increase in the price of energy relative to the prices of other productive inputs. The sharp increase in the price of energy began with the run-up of world petroleum prices in late 1973 and early 1974 in the aftermath of the Arab Oil Embargo. Our econometric models reveal that slower productivity growth at the sectoral level after 1973 is associated with higher energy prices. Our empirical findings on patterns of substitution are almost as clearcut as our findings on patterns of technical change. We find that the elasticities of the shares of capital with respect to the price of labor are nonnegative for thirty-three of our thirty-six industries, so that the shares of capital are nondecreasing in the price of labor for thirty-three sectors. Similarly, elasticities of the share of capital with respect to the price of energy are nonnegative for thirty-four industries and elasticities with respect to the price of materials are nonnegative for all thirty-six industries. We find that the share elasticities of labor with respect to the prices of energy and materials are nonnegative for nineteen and thirty-six industries, respectively. Finally, we find that the share elasticities of energy with respect to the price of materials are nonnegative for thirty of the thirty-six industries.

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12.2 Econometric Models The development of our econometric model of production and technical change proceeds through two stages. We first specify a functional form for the sectoral price functions, say {Pi}, taking into account restrictions on the parameters implied by the theory of production. 2 Secondly, we formulate an error structure for the econometric model and discuss procedures for estimation of the unknown parameters. Our first step in formulating an econometric model of production and technical change, is to consider specific forms for the sectoral price function {Pi}

For these price functions, the prices of outputs are transcendental or, more specifically, exponential functions of the logarithms of the prices of inputs. We refer to these forms as transcendental logarithmic price functions or, more simply, translog price functions,3 indicating the role of the variables that enter into the price functions. The price functions {Pi} are homogeneous of degree one in the input prices. The translog price function for an industrial sector is characterized by homogeneity of degree one if and only if the parameters for that sector satisfy the conditions

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For each sector the value shares of capital, labor, energy, and materials inputs, say { }, { }, { }, and { }, can be expressed in terms of logarithmic derivatives of the sectoral price function with respect to the logarithms of price of the corresponding inputs

Finally, for each sector the rate of technical change, say , can be expressed as the negative of the rate of growth of the price of sectoral output with respect to time, holding the prices of capital, labor, energy, and materials inputs constant. The negative of the rate of technical change takes the following form

Given the sectoral price functions {Pi}, we can define the share elasticities with respect to price 4 as the derivatives of the value shares with respect to the logarithms of the prices of capital, labor, energy, and materials inputs. For the translog price functions the share elasticities with respect to price are constant. We can also characterize these forms as constant share elasticity or CSE price functions,5 indicating the interpretation of the fixed parameters that enter the price functions. The share elasticities with respect to price are symmetric, so that the parameters satisfy the conditions

Similarly, given the sectoral price functions {Pi}, we can define the biases of technical change with respect to price as derivatives of the value shares with respect to time.6 Alternatively, we can define the biases of technical change with respect to price as derivatives of the rate of technical change with respect to the logarithms of the price of capital, labor, energy, and materials inputs.7 Those two definitions

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of biases of technical change are equivalent. For the translog price functions the biases of technical change with respect to price are constant; these parameters are symmetric and satisfy the conditions

Finally, we can define the rate of change of the negative of the rate of technical change, (i = 1,2, . . . , n), as the derivative of the negative of the rate of technical change with respect to time. 8 For the translog price functions these rates of change are constant. Our next step in considering specific forms of the sectoral price functions {Pi} is to derive restrictions on the parameters implied by the fact that the price functions are increasing in all four input prices and concave in the four input prices. First, since the price functions are increasing in each of the four input prices, the value shares are nonnegative

Under homogeneity these value shares sum to unity

Concavity of the sectoral price functions {Pi} implies that the matrices of second-order partial derivatives {Hi} are negative semi-definite,9 so that the matrices {Ui + vivi' Vi} are negative semi-definite, where10

where

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and {Ui} are matrices of constant share elasticities, defined above. Without violating the nonnegativity restrictions on value shares, we can set the matrices {vivi' Vi} equal to zero, for example, by choosing the value shares

Necessary conditions for the matrices {Ui + vivi' Vi} to be negative semi-definite are that the matrices of share elasticities {Ui} must be negative semi-definite. These conditions are also sufficient, since the matrices {vivi' Vi} are negative semi-definite for all nonnegative value shares summing to unity and the sum of two negative semidefinite matrices is negative semi-definite. To impose concavity on the translog price functions, the matrices {Ui} of constant share elasticities can be represented in terms of their Cholesky factorizations

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Under constant returns to scale the constant share elasticities satisfy symmetry restrictions implied by homogeneity of degree one of the price function. These restrictions imply that the parameters of the Cholesky factorizations must satisfy the following conditions

Under these conditions there is a one-to-one transformation between the constant share elasticities and the parameters of the Cholesky factorizations. The matrices of share elasticities are negative semi-definite if and only if the diagonal elements of the matrices {Di} are nonpositive. This completes the specification of our model of production and technical change. The negative of the average rates of technical change in any two points of time, say, T and T 1, can be expressed as the difference between successive logarithms of the price of output, less a weighted average of the differences between successive logarithms of the prices of capital, labor, energy, and materials inputs, with weights given by the average value shares

and the average value shares in the two periods are given by

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We refer to the expressions for the average rates of technical change { sectoral rates of technical change.

} as the translog price indexes of the

Similarly we can consider specific forms for prices of capital, labor, energy, and materials inputs as functions of prices of individual capital, labor, energy, and materials inputs into each industrial sector. We assume that the price of each input can be expressed as a translog function of the price of its components. Accordingly, the difference between successive logarithms of the price of the input is a weighted average of differences between successive logarithms of prices of its components. The weights are given by the average value shares of the components. We refer to these expressions of the input prices as translog indexes of the price of sectoral inputs. 11 To formulate an econometric model of production and technical change, we add a stochastic component to the equations for the value shares and the rate of technical change. We assume that each of these equations has two additive components.12 The first is a nonrandom function of capital, labor, energy, and materials inputs and time; the second is an unobservable random disturbance that is functionally independent of these variables. We obtain an econometric model of production and technical change corresponding to the translog price function by adding random disturbances to all five equations

where

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are unknown parameters and

are unobservable random disturbances.

Since the value shares sum to unity, the unknown parameters satisfy the same restrictions as before and the random disturbances corresponding to the four value shares sum to zero

so that these random disturbances are not distributed independently. We assume that the random disturbances for all five equations have expected value equal to zero for all observations

We also assume that the random disturbances have a covariance matrix that is the same for all observations; since the random disturbances corresponding to the four value shares sum to zero, this matrix is positive semi-definite with rank at most equal to four. We assume that the covariance matrix of the random disturbances corresponding to the first three value shares and the rate of technical change, say Si, has rank four, where

so the Si is a positive definite matrix. Finally, we assume that the random disturbances corresponding to distinct observations in the same or distinct equations are uncorrelated. Under this assumption that the matrix of random disturbances for the first three value shares and the rate of technical change for all observations has the Kronecker product form,

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Since the rates of technical change can be written

where

are not directly observable, the equation for the rate of technical change

is the average disturbance in the two periods

Similarly, the equations for the value shares of capital, labor, energy, and materials inputs can be written

As before, the average value shares sum to zero

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The covariance matrix of the average disturbances corresponding to the equation for the rate of technical change for all observations, say, W, is a Laurent matrix

The covariance matrix of the average disturbance corresponding to each equation for the four value shares is the same, so that the covariance matrix of the average disturbances for the first three value shares and the rate of technical change for all observations has the Kronecker product form

Although disturbances in equations for the average rate of technical change and the average value shares are autocorrelated, the data can be transformed to eliminate the autocorrelation. The matrix W is positive definite, so that there is a matrix T such that

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To construct the matrix T, we can first invert the matrix W to obtain the inverse matrix W 1, a positive definite matrix. We then calculate the Cholesky factorization of the inverse matrix W 1

where L is a unit lower triangular matrix and D is a diagonal matrix with positive elements along the main diagonal. Finally, we can write the matrix T in the form

where D1/2 is a diagonal matrix with elements along the main diagonal equal to the square roots of the corresponding elements of D. We can transform the equations for the average rates of technical change by the matrix T = D1/2 L' to obtain equations with uncorrelated random disturbances 13

since

The transformation T = D1/2L' is applied to data on the average rates of technical change and data on the average values of the variables that appear on the right-hand side of the corresponding equation. We can apply the transformation T = D1/2L' to the first three equations for average value shares to obtain equations with uncorrelated disturbances. As before, the transformation is applied to data on the average values shares and the average values of variables that appear in the corresponding equations. The covariance matrix of the transformed disturbances from the first three equations for the average value shares and the equation for the average rate of technical change has the Kronecker product form

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To estimate the unknown parameters of the translog price function we combine the first three equations for the average value shares with the equation for the average rate of technical change to obtain a complete econometric model of production and technical change. We estimate the parameters of the equations for the remaining average value shares, using the restrictions on these parameters given above. The complete model involves fourteen unknown parameters. A total of sixteen additional parameters can be estimated as functions of these parameters, given the restrictions. Our estimate of the unknown parameters of the econometric model of production and technical change is based on the nonlinear, three-stage least-squares estimator introduced by Jorgenson and Laffont (1974). 12.3 Empirical Results To implement the econometric models of production and technical change developed in section 12.2 we have assembled a data base for thirty-six industrial sectors of the U.S. economy listed. For capital and labor inputs we have first compiled data by sector on the basis of the classification of economic activities employed in the U.S. National Income and Product Accounts. We have then transformed these data into a format appropriate for the classification of activities employed in the U.S. Interindustry Transactions Accounts. For energy and materials inputs we have compiled data by sector on interindustry transactions among the thirty-six industrial sectors. For this purpose we have used the classification of economic activities employed in the U.S. Interindustry Transactions Accounts. 14 For each sector we have compiled data on the value shares of capital, labor, energy, and materials inputs, annually, for the period 1958 1974. We have also compiled indexes of prices of sectoral output and all four sectoral inputs for the same period. Finally, we have compiled translog indexes of sectoral rates of technical change. For the miscellaneous sector the rate of technical change is equal to zero by definition, so that the econometric model for the sector does not include an equation for the sectoral rate of technical change. There are sixteen observations for each behavioral equation, since two-period averages of all data are employed.

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The parameters can be interpreted as average value shares of capital input, labor input, energy input, and materials input for the corresponding sector. Similarly, the parameters { } can be interpreted as averages of the negative of rates of technical change. The parameters can be interpreted as constant share elasticities with respect to price for the corresponding sector. Similarly the parameters can be interpreted as constant biases of technical change with respect to price. Finally, the parameters { } can be interpreted as constant rates of change of the negative of the rates of technical change. In estimating the parameters of our sectoral models of production and technical change, we retain the average of the negative of the rate of technical change, biases of technical change, and the rate of change of the negative of the rate of technical change as parameters to be estimated for thirty-five of the thirty-six industrial sectors. For the miscellaneous sector the rate of technical change is equal to zero by definition, so that these parameters are set equal to zero. Estimates of the share elasticities with respect to price are obtained under the restrictions implied by the necessary and sufficient conditions for concavity of the price functions presented in section 12.2. Under these restrictions the matrices of constant share elasticities {Ui} must be negative semi-definite for all industries. To impose the concavity restrictions, we represent the matrices of constant share elasticities for all sectors in terms of their Cholesky factorizations. The necessary and sufficient conditions are that the diagonal elements of the matrices {Di} that appear in the Cholesky factorizations must be nonpositive. We present estimates subject to these restrictions for all thirty-six industrial sectors in table 12.1. Our interpretation of the parameter estimates reported in table 12.2 begins with an analysis of the estimates of the parameters The average value shares are nonnegative for all thirty-six sectors included in our study. The negative of the estimated average rate of technical change is negative in sixteen sectors, positive in nineteen sectors, and zero, by definition, in the miscellaneous sector. Negative signs predominate only in transportation, communications, utilities, trade, and services. Positive signs predominate in the extractive industries and durable goods manufacturing industries. Nondurable goods manufacturing industries are almost evenly divided between positive and negative signs.

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Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.1 (continued) Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.1 (continued) Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.1 (continued) Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.1 (continued) Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.1 (continued) Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.1 (continued) Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.1 (continued) Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.1 (continued) Table 12.1 Parameter estimates for sectoral models of production and technical change

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Table 12.2 Classification of industries by biases of technical change Pattern of Industries biases Agriculture, metal mining, crude petroleum and natural gas, Capital using Labor nonmetallic mining, textiles, apparel, lumber, furniture, printing, leather, fabricated metals, electrical machinery, using motor vehicles, instruments, miscellaneous manufacturing, Energy transportation, trade, finance, insurance and real estate, using Material services saving Coal mining, tobacco manufactures, communications, Capital using Labor government enterprises using Energy saving Material saving Petroleum refining Capital using Labor saving Energy using Material saving Construction Capital using Labor saving Energy saving Material using Electric utilities Capital saving Labor saving Energy using Material saving Primary metals Capital saving Labor using Energy saving Material saving Paper, chemicals, rubber, stone, clay and glass, machinery Capital except electrical, transportation equipment and ordinance, gas saving Labor usingutilities Energy using Material

saving Capital saving Labor saving Energy using Material using

Food

The estimated share elasticities with respect to price describe the implications of patterns of substitution among capital, labor, energy, and materials inputs for the relative distribution of the value of output among these four inputs. Positive share elasticities imply that the corresponding value shares increase with an increase in price; negative share elasticities imply that the value shares decrease with an increase in price; share

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elasticities equal to zero imply that the value shares are independent of price. It is important to keep in mind that we have fitted these parameters subject to the restrictions implied by concavity of the price function. These restrictions require that all the share elasticities be set equal to zero for eleven of the thirty-six industries. For eleven of the thirty-six industries the share elasticities for all inputs with respect to the prices of energy and materials inputs are set equal to zero. For thirteen of the thirty-six industries the share elasticities with respect to the price of labor input are set equal to zero. Finally, for thirty-three of the thirty-six industries the share elasticities for all inputs with respect to the price of capital input are set equal to zero. Of the three hundred and sixty share elasticities for the thirty-six industries included in our study, two hundred and four are set equal to zero and one hundred and fifty-six are fitted without constraint. Our interpretation of the parameter estimates given in table 12.1 continues with the estimated elasticities of the share of each input with respect to the price of the input itself . Under the necessary and sufficient conditions for concavity of the price function for each sector, these share elasticities are nonpositive. The share of each input is nonincreasing in the price of the input itself. This condition together with the condition that the sum of all the share elasticities with respect to a given input is equal to zero implies that only two of the elasticities of the shares of each input with respect to the prices of the other three inputs can be negative. All six of these share elasticities can be nonnegative, and this condition holds for twelve of the thirty-six industries. The share elasticity of capital with respect to the price of labor { } is nonnegative for thirty-three of the thirtysix industries. By symmetry this parameter can also be interpreted as the share elasticity of labor with respect to the price of capital. The share elasticity of capital with respect to the price of energy { } is nonnegative for thirtyfour of the thirty-six industries. Finally, the share elasticity of capital with respect to the price of materials { } is nonnegative for all thirty-six industries. These parameters can also be interpreted as the share elasticities of energy and materials with respect to the price of capital. Considering the elasticities of the share of labor with respect to the prices of energy and materials inputs, we find that the share elasticity with respect to the price of energy is nonnegative for nineteen of the thirty-six industries, while the share elasticity with respect to the price of materials is nonnegative for all thirty-six industries. These parame-

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ters can also be interpreted as share elasticities of energy and materials with respect to the price of labor. Finally, the share elasticity of energy with respect to the price of materials input is nonnegative for thirty of the thirty-six industries. This parameter can also be interpreted as the share elasticity of materials with respect to the price of energy. We continue the interpretation of parameter estimates given in table 12.1 with the estimated biases of technical change with respect to the price of each input . These parameters can be interpreted as the change in the negative of the rate of technical change with respect to the price of each input or, alternatively, as the change in the share of each input with respect to time. The sum of the four biases of technical change with respect to price is equal to zero, so that we can rule out the possibility that all four biases are either all negative or all positive. Of the fourteen remaining logical possibilities, only eight actually occur among the results presented in table 12.1. Of these, only three patterns occur for more than one industry. It is important to note that the biases of technical change are not affected by the concavity of the price function, so that all four parameters are fitted for thirty-five industries. For the miscellaneous sector the rate of technical change is zero by definition, so that all biases of technical change are set equal to zero. We first consider the bias of technical change with respect to the price of capital input. If the estimated value of this parameter is positive, technical change is capital using. Alternatively the rate of technical change decreases with an increase in the price of capital. If the estimated value is negative, technical change is capital saving, and the rate of technical change increases with the price of capital. Technical change is capital using for twenty-five of the thirty-five industries included in our study of the biases of technical change; it is capital saving for ten of these industries. We conclude that the rate of technical change decreases with the price of capital for twenty-five industries and increases with the price of capital for ten industries. The interpretation of the biases of technical change with respect to the prices of labor, energy, and materials inputs is analogous to the interpretation of the bias with respect to the price of capital input. If the estimated value of the bias is positive, technical change uses the corresponding input; alternatively, the rate of technical change decreases with an increase in the input price. If the estimated value is negative, technical change saves the corresponding input; alternatively the rate of technical change decreases with the input price.

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Considering the bias of technical change with respect to the price of labor input, we find that technical change is labor using for thirty-one of the thirty-five industries included in our study and labor saving for only four of these industries. The rate of technical change decreases with the price of labor for thirty-one industries and increases with the price of labor for four industries. Considering the bias of technical change with respect to the price of energy input, we find that technical change is energy using for twenty-nine of the thirty-five industries included in our study and energy saving for only six of these industries. The rate of technical change decreases with the price of energy for twenty-nine industries and increases for six industries. Finally, technical change is materials using for only two of the thirty-five industries included in our study and materials saving for the remaining thirty-three. We conclude that the rate of technical change increases with the price of materials for thirty-three industries and decreases with the price of materials for two industries. A classification of industries by patterns of the biases of technical change is given in table 12.2. The pattern that occurs with greatest frequency is capital-using, labor-using, energy-using, and materials-saving technical change. This pattern occurs for nineteen of the thirty-five industries included in our study. For this pattern the rate of technical change decreases with increases in the prices of capital, labor, and energy inputs and increases with the price of materials input. The pattern that occurs next most frequently is capital-saving, labor-using, energy-using, and materials-saving technical change. This pattern occurs for seven industries. The third most frequently occurring pattern is capital-using, labor-using, energy-saving, and materials-saving technical change. This pattern occurs for four industries. No other pattern occurs for more than one industry. Our interpretation of the parameter estimates given in table 12.1 concludes with rates of change of the negative of the rate of technical change { }. If the estimated value of this parameter is positive, the rate of technical change is decreasing; if the value is negative, the rate is increasing. For twenty-four of the thirty-five industries included in our study of the rate of change of the rate of technical change the estimated value is positive; for eleven industries, the value is negative; for the remaining industry, this parameter is set equal to zero, since the rate of technical change is zero by definition. We conclude that the

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rate of technical change is decreasing with time for twenty-four industries and increasing with time for eleven industries. 12.4 Conclusion Our empirical results for sectoral patterns of production and technical change are very striking and suggest a considerable degree of similarity across industries. However, it is important to emphasize that these results have been obtained under strong simplifying assumptions. First, for all industries we have employed conditions for producer equilibrium under perfect competition; we have assumed constant returns to scale at the industry level; finally, we have employed a description of technology that leads to myopic decision rules. These assumptions must be justified primarily by their usefulness in implementing production models that are uniform for all thirty-six industrial sectors of the U.S. economy. The most important simplification of the theory of production and technical change employed in the specification of our econometric models is the imposition of concavity of the sectoral price functions. By imposing concavity on the sectoral price functions, we have reduced the number of share elasticities to be fitted from three hundred sixty, or ten for each of our thirty-six industrial sectors, to one hundred fifty-six, or less than five per sector on average. All share elasticities are constrained to be zero for eleven of the thirty-six industries. The concavity constraints contribute to the precision of our estimates but require that the share of each input be nonincreasing in the price of the input itself. Although it might be worthwhile to weaken each of the assumptions we have enumerated above, a more promising direction for further research appears to lie within the framework provided by these assumptions. First, we can provide a more detailed model for allocation among productive inputs. We have disaggregated energy and materials into thirty-six groupsfive types of energy and thirty-one types of materialsby constructing a hierarchy of models for allocation within the energy and materials aggregates. For this purpose we have assumed that each aggregate is homogeneously separable within the sectoral production function. We assume, for example, that the share of energy in the value of sectoral output depends on changes in technology, while the share of, say, electricity in the value of energy input does not.

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The second research objective suggested by our results is to incorporate the production models for all thirty-six industrial sectors into a general equilibrium model of production in the U.S. economy. An econometric general equilibrium model of the U.S. economy has been constructed for nine industrial sectors by Hudson and Jorgenson (1974). This model is currently being disaggregated to the level of the thirty-six industrial sectors included in our study. A novel feature of the thirty-six sector general equilibrium model will be the endogenous treatment of the rate of technical change for thirty-five of the thirty-six industries we have analyzed. A general equilibrium model will make it possible to analyze the implications of sectoral patterns of substitution and technical change for the U.S. economy as a whole. Notes 1. A review of the literature on induced technical change is given by Binswanger (1978a). Binswanger distinguishes between models, like ours and those of Ben-Zion and Ruttan (1978), Lucas (1967a) and Schmookler (1966) with an endogenous rate of technical change and models, like those of Hicks (1932), Kennedy (1964), Samuelson (1965b), von Weizsäcker (1962), and others, with an endogenous bias of technical change. Additional references are given by Binswanger (1978a). 2. The price function was introduced by Samuelson (1953 1954). 3. The translog price function was introduced by Christensen, Jorgenson, and Lau (1971, 1973). The translog price function was first applied at the sectoral level by Berndt and Jorgenson (1973) and Berndt and Wood (1975). References to sectoral production studies incorporating energy and materials inputs are given by Berndt and Wood (1979). 4. The share elasticity with respect to price was introduced by Christensen, Jorgenson, and Lau (1971, 1973) as a fixed parameter of the translog price function. An analogous concept was employed by Samuelson (1973). The terminology is due to Jorgenson (1986). 5. The terminology ''constant share elasticity price function" is due to Jorgenson and Lau (1981), who have shown that constancy of share elasticities with respect to price, biases of technical change with respect to price, and the rate of change of the negative of the rate of technical change are necessary and sufficient for representation of the price function in translog form. 6. The bias of technical change was introduced by Hicks (1932). An alternative definition of the bias of technical change is analyzed by Burmeister and Dobell (1969). Binswanger (1974a) has introduced a translog cost function with fixed biases of technical change. Alternative definitions of biases of technical change are compared by Binswanger (1978b). 7. This definition of the bias of technical change with respect to price is due to Jorgenson (1986). 8. The rate of change of the negative of the rate of technical change was introduced by Jorgenson (1986). 9. The following discussion of share elasticities with respect to price and concavity follows that of Jorgenson (1986). Representation of conditions for concavity in terms of the Cholesky factorization is due to Lau (1978a).

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10. The following discussion of concavity for the translog price function is based on that of Jorgenson and Lau (1981). 11. The price indexes were introduced by Fisher (1922) and have been discussed by Tornquist (1936), Theil (1965), and Kloek (1966). These indexes were first derived from the translog price function by Diewert (1976). The corresponding index of technical change was introduced by Christensen and Jorgenson (1970). The translog index of technical change was first derived from the translog price function by Diewert (1980) and by Jorgenson (1986). Earlier Diewert (1976) had interpreted the ratio of translog indexes of the prices of input and output as an index of productivity under the assumption of Hicks neutrality. 12. The following formulation of an econometric model of production and technical change is based on that of Jorgenson and Lau (1981). 13. The Cholesky factorization is used to obtain an equation with uncorrelated random disturbances by Jorgenson and Lau (1981). 14. Data on energy and materials are based on annual interindustry transactions tables for the United States, 1958 1974, compiled by Jack Faucett Associates (1977) for the Federal Preparedness Agency. Data on labor and capital are based on estimates by Fraumeni and Jorgenson (1980).

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13 Relative Price Changes and Biases of Technical Change in Japan Masahiro Kuroda, Kanji Yoshioka, and Dale W. Jorgenson 13.1 Introduction Simon Kuznets in his laborious work, Economic Growth of Nations: Total Output and Production Structure (1971), pointed out the following dominant characteristic of modern economic growth: " . . . the important point is that the acceleration in the rate of growth in per capita product, characteristic of modern economic growth, was accompanied by an equally conspicuous acceleration in the rate of change in production structure. . . . " The average growth rate of gross products estimated by Kuznets during one and a quarter centuries was around 3 percent annually. Japan experienced more than a 10 percent annual growth rate in real GNP successively during the period 1960 1970 which was much larger than the historical standard level in the modern economic growth period. We can thus expect that there existed dramatic structural changes, illustrated by the shift of the allocation among industries in the total gross output, total labor force and total capital formation. We can summarize sources of economic growth in Japan during the period 1960 1979 in table 13.1. The last three columns, (9) through (11)indicate labor and capital quantity indices (1960 = 1.0) measured by simply adding up man-hour labor inputs and capital stock. The simple adding-up quantity index of labor input represents an annual growth rate of 1.08 percent and that of capital input represents an annual growth rate of 6.15 percent during the period 1960 1979. Consequently the annual growth rate of total factor productivity is 4.85 percent. If we use these indices in order to evaluate sources of economic growth in Japan, we have to conclude that the contribution to economic growth of technological progress is more than 50 percent, while labor input contributes 8 percent and capital input 35 percent.

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Table 13.1 Source of economic growth in Japan

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 Average annual growth rate 1960 1979 Contribution(%)

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971

Aggregated divisa quantity index (direct estimation) (1) (2) (3) (4) V.A. L K TFP 1 1.0000 1.0000 1.0000 1.0000 1.1277 1.1142 1.1192 1.0095 1.2198 1.1121 1.2825 1.0169 1.3411 1.1369 1.4222 1.0490 1.5328 1.2367 1.5881 1.0872 1.6427 1.3230 1.7591 1.0692 1.8268 1.3647 1.9530 1.1081 2.0672 1.4222 2.1240 1.1756 2.4128 1.3889 2.4267 1.2880 2.7257 1.4443 2.7413 1.3360 3.1091 1.5530 3.0806 1.3829 3.3635 1.5959 3.5297 1.3728 3.7178 1.6360 3.8465 1.4327 4.0240 1.6759 4.3079 1.4451 3.9943 1.7175 4.7486 1.3493 4.2063 1.6444 5.1417 1.3989 4.4561 1.7073 5.3596 1.4247 4.6918 1.7012 5.6183 1.4706 4.7828 1.7164 5.8583 1.4630 5.0518 1.8431 6.1551 1.4543 0.0852 0.0322 0.0956 0.0218 100.0 19.21 54.99 25.80 Aggregated divisa quantity index (dual estimation) (5) (6) (7) (8) V.A. L K TFP 2 1.0000 1.0000 1.0000 1.0000 1.1360 1.0389 1.1249 1.0478 1.2337 1.0722 1.2836 1.0453 1.3669 1.1167 1.4296 1.0745 1.5712 1.1687 1.6021 1.1389 1.6924 1.1510 1.7856 1.1664 1.8906 1.1990 1.9825 1.2088 2.1503 1.2581 2.1541 1.2855 2.5246 1.2542 2.4633 1.4032 2.8669 1.3057 2.7829 1.4623 3.2892 1.3620 3.1261 1.5433 3.5756 1.3391 3.5648 1.5484

(table continued on next page)

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Table 13.1 (continued) Table 13.1 Source of economic growth in Japan Aggregated divisa quantity index (dual estimation) (5) (6) (7) (8) V.A. L K TFP 2 1972 3.9763 1.4223 3.86121.6324 1973 4.3401 1.4303 4.27911.6844 1974 4.3264 1.4406 4.08181.7134 1975 4.6262 1.4283 5.02071.6661 1976 4.9263 1.4945 5.23171.6988 1977 5.2153 1.5307 5.34191.7585 1978 5.3033 1.5479 5.92921.6927 1979 5.6201 1.6144 6.26181.7096 Average annual growth rate 1960 1979 0.0909 0.0252 0.09660.0282 14.10 52.25 33.65 Contribution(%) 100.0 Simple adding aggregation quantity index (11) (10) (9) (man-hour)(K-stock) TFP 3 K L 1960 1.0000 1.0000 1.0000 1961 1.0128 1.0471 1.0899 1962 1.0253 1.1175 1.1323 1963 1.0341 1.1482 1.2226 1964 1.0475 1.2144 1.3498 1965 1.0643 1.2992 1.3852 1966 1.0865 1.3802 1.4771 1967 1.1074 1.4586 1.6078 1968 1.1213 1.6044 1.7703 1969 1.1298 1.7267 1.9143 1970 1.1419 1.8640 2.0826 1971 1.1464 2.0428 2.1398 1972 1.1453 2.1988 2.2762 1973 1.1789 2.3908 2.3262 1974 1.1740 2.5936 2.2539 1975 1.1708 2.7773 2.3000 1976 1.1816 2.8599 2.3918 1977 1.1975 2.9934 2.4482 1978 1.2123 3.1113 2.4349 1979 1.2282 3.2193 2.5129 Average annual growth rate 1960 1979 0.0108 0.0615 0.0485 7.58 35.50 56.92 Contribution(%)

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However, as Kuznets suggested, high economic growth should be accompanied by an equal acceleration in the rate of change in the production structure. Structural change in production should be represented not only by changes of distribution of various types of intermediate inputs, labor and capital used in each industry, but also by changes of the distribution of them among industries. Changes in distribution of intermediate inputs, labor and capital in each industry can be evaluated through output and input divisia index as will be clarified later. Figure 13.1 shows the average annual growth rate in gross output, capital, labor and material inputs measured by divisia aggregates in each industry during the period 1960 1979. The number on the horizontal line stands for the industry number in table 13.2, while the height of each pole represents the average annual growth rate for each measure. This figure depicts the remarkable differences in the growth rate among industries. Aggregate measures of output and input should reflect such sorts of differences in growth rates among industries. Now return to table 13.1. We reevaluated aggregate measures of output and inputs by using divisia indices which reflect not only changes in distribution of output and input in each industry, but also changes in distribution of these input and output among industries. The first four columns, (1) through (4) represent aggregate quantity divisia indices estimated directly and the next four columns, (5) through (8) represent aggregate quantity divisia indices indirectly imputed from price divisia indices. According to directly estimated divisia indices, the annual growth rate of labor input is 3.22 percent, which is higher than the growth rate of man-hour measured labor input by more than 2 percent. Also, the annual growth rate of capital input is 9.56 percent, which is higher than the growth rate of the adding-up measure of capital stock by more than 3 percent. Consequently, the contribution of technical progress to economic growth is evaluated at only in 25 percent, which is almost half of that from the simple adding-up measure. The increases in growth rates for labor and capital inputs imply that during the period structural changes in production should not be ignored when evaluating sources of economic growth. Indeed, the

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Figure 13.1 Annual growth rate among industries 1960 1979. Note: The number on the horizontal line stands for the industry number in table 13.2, while the height of each pole represents the average annual growth rate of gross output (X), capital (K), labor (L) and material (M).

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Table 13.2 List of industries Japan United States 1. Agriculture, forestry & 1. Agricultural production & service fisheries 2. Mining 2. Metal mining 3. Coal mining 4. Crude petroleum & natural gas 5. Nonmetallic mining 3. Construction 6. Contract construction 7. Food & kindred products 4. Food & kindred products 5. Textile mill products 9. Textile mill products 10. Apparel & other fabricated textile products 6. Apparel & other fabricated textile products 11. Lumber & wood products 7. Lumber & wood products, except furniture 8. Furniture & fixtures 12. Furniture & fixtures 13. Paper & allied products 9. Paper & allied products 10. Printing, publishing & 14. Printing & publishing allied products 15. Chemical & allied products 11. Chemical & allied products 12. Petroleum refinery & 16. Petroleum & coal products related industries 17. Rubber & misc. plastic products 13. Rubber & misc. plastic products 18. Leather & leather products 14. Leather & leather products 15. Stone, clay & glass 19. Stone, clay & glass products products 16. Iron & Steel 20. Primary metal products 17. Nonferrous metal 18. Fabricated metal 21. Fabricated metal products products 19. Machinery 22. Machinery except electrical 20. Electric machinery 23. Electric machinery equipment & supplies 21. Motor vehicles & 24. Motor vehicles & equipment equipment 22. Transportation equip. 25. Transportation equipment except motor except motor 23. Precision instruments 26. Prof. photo equipment & watches 24. Miscellaneous 27. Miscellaneous manufacturing manufacturing 8. Tobacco manufacturing 25. Transportation & 28. Railroad & rail express service Street rail, bus lines & taxi Trucking communication services & warehousing Water transportation Air transportation 29. Telephone, telegraph, misc. radio broadcasting & TV 26. Electric utility, gas 30. Electric utility supply & water supply 31. Gas utility, water supply & sanitary service

27. Wholesale & retail trade

32. Wholesale trade 33. Retail Trade Finance, insurance & real estate

28. Finance & insurance 29. Real estate 30. Service 34. Service except private households 31. Government service 35. Government service U.S. classification was referred from Jorgenson and Fraumeni (1983).

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analysis of structural change in capital input was remarkably important as a source of Japanese economic growth. During the high economic growth period from 1960 to 1973, the growth rate of capital input was higher than that of gross output by more than 1 percentage point. On the other hand, the growth rate in labor input was much lower than that of gross output. This kind of input growth pattern was expected to be highly related to relative prices among inputs, where price increase of labor inputs were remarkably higher than those of capital and energy inputs. After the oil crisis, as we can expect, the annual increase in the price of energy input was ten times what it had been before, while the increasing rate of capital price decreased gradually. Through changes in the relative price system between inputs and the slowdown in economic growth itself, input structures in production were influenced remarkably. Here we aim to describe producer behavior econometrically in terms of a price possibility frontier function. We principally focus on the measurement of gross substitutability among inputs and analysis of technical biases, which are induced from changes in the relative price of inputs. 13.2 Theoretical Framework 13.2.1 Measurement of Technical Change Our production model aims to explain producer behavior in each industrial sector of Japan. We assume that there exists a twice differentiable production function, by which the gross flow of output in the jth sector is related to the service flows of various inputs under the certain state of technology. Under competitive market conditions, producer behavior can be described equivalently in terms of production functions or price possibility frontiers. This means that we can deduce a price possibility frontier dual to the production function. As a definition of the accounting balance in the jth sector, the following equation can be introduced

Differentiating by time,

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Now we will define the growth rate of aggregated inputs and input prices as the following indices

Using these definitions we can deduce the growth rate of the total efficiency of production in the jth sector as follows

where y is Yj/xj a divisia index of total factor productivity. Equation (13.6) means that the growth rate of total factor productivity as an integrated measure of the efficiency of production is defined as the difference of the growth rates between output and the aggregated inputs. Under the condition of producer's equilibrium in a competitive market, the linear homogeneous production function assures that the value of output is fully distributed as the returns to inputs. Thus, the growth rate of total factor productivity can alternatively be rewritten as the difference in growth rates between the aggregated input price and output price. Now, let us suppose that the industry with a state of technology, T (t) at the time period t is described by a linear homogeneous production function with n inputs,

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where the function is twice-differentiable, concave and monotonic. Under competitive market conditions, the producer is alternatively described by a price possibility frontier function dual to (13.7).

From equations (13.7) and (13.8), the growth rate in technical efficiency in the production function and the growth rate of output price reduction derived from technical change are defined respectively as follows

Using the well-known Konyus-Byushgen's lemma under the condition of producers' equilibrium in competitive market, we obtain

Symmetrically applying Shephard's lemma to the dual price possibility frontier function gives

Inserting (13.11) and (13.12) into (13.9) and (13.10) respectively, and comparing them to (13.6), we obtain

This means that the growth rate of total factor productivity is conceptually equal to the growth rate of technical change and also to the growth rate of output price reduction for the certain time period t.

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13.2.2 Specification of Price Possibility Frontier Next, we concentrate on the specification of the price possibility frontier function as a description of the state of technology. We assume that factor inputs in the jth sector are commonly separated into the four major input categories of capital input, labor input, energy input and material input. These four major input categories are presumably mutually separable according to following expression

We suppose that this price function is specified in terms of a transcendental logarithmic form as follows

where In Pk = the vector whose elements are composed of the logarithmic value of four major input prices, a0, aT, and BTT = scalar parameters and A, BkT, and Bk1 = vectors and matrix of parameters defined as follows

We refer to this specification as the KLEM translog price possibility frontier model or simply the KLEM model. We assume that each aggregator function for the four major input categories is specified also as a transcendental logarithmic function. Thus we obtain a discrete divisia growth rate of four major input prices from Diewert's quadratic approximation lemma and Shephard's lemma as follows

where wkit = the value share of the ith input in the certain subcategorized aggregator function in (13.14), (k = KLEM) and Pkit = the ith

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input price in the certain subcategorized aggregator function (k = KLEM). Finally we can deduce a discrete formulation of the rate of technical change,

where wkt = the value share of each four major input in the total input of the KLEM model (k = KLEM), and wTt = the discrete divisia growth rate of technical change. 13.2.3 Stochastic Formulation and Estimation of KLEM Model We begin with an explanation of share functions and technical change in the KLEM model. Shephard's lemma implies that each partial derivative of In qj with respect to In pk and T is equal to the respective input share, wkt and the negative rate of technical change, wTt at time period t.

We have to introduce some theoretical constraints in the above share functions of the translog formulation, so that the translog price function becomes consistent with producer's equilibrium. (a) Symmetry

and

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(b) Adding-up and Linear Homogeneity

(c) Monotonicity

(d) Concavity The concavity condition for a linear homogeneous price function is that its Hessian matrix is negative semidefinite. The Hessian matrix of the translog price function H is denoted as follows

In equation (13.23), the matrix , is negative definite, because all of the value shares are less than unity. Therefore if Bkl is negative semi-definite, the Hessian, H might be sufficiently negative semi-definite. On the other hand, when setting the matrix to converge to zero as a special case that wkt approximately becomes unity, while other shares go to zero, the matrix Bkl must be negative semi-definite as a necessary condition for the negative semi-definiteness of H. Necessary and sufficient conditions for the negative semi-definiteness of H can be expressed as nonpositive conditions for Cholesky values after the Cholesky factorization of matrix Bkl as follows

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where there is a one-to-one correspondence between the share elasticity matrix Bkl and the parameters of the Cholesky factorization lkl and dk. The negative semi-definiteness of Bkk simply implies the Cholesky values, dk (k, l = KLEM) are all nonpositive. After the imposition of symmetry conditions and reparameterization, we can rewrite the share functions with additive stochastic components, u as follows

where the additive stochastic components, u are assumed to have a stochastic process such that

In the above formulation of the translog share function, the negative rate of technical change, wTt is not directly observable, but its average value in subsequent time, 1/2[wTt + wTt 1] is obtainable through the discrete divisia index of technical change shown in

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(13.17). In order to estimate the KLEM model using this discrete approximation of wTt, we need to rearrange (13.25) with the average of each variable between the successive time periods t and t 1.

Since the new disturbance is now serially correlated, the data for shares, prices and technical change must be transformed again to eliminate this serial correlation. That is to say, the covariance matrix of the average disturbance corresponding to each equation is a Laurent matrix

Thus, the next relation obtains

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The matrix W 1 is positive definite, so that W 1 is Cholesky factorizable,

where L is a unit lower triangular matrix and D is a diagonal matrix whose diagonal elements are all positive. Then, defining the matrix F such the F = D1/2 L', we get the next relation

On the other hand, each share function of (13.27) is rewritten as follows

By multiplying the matrix F from the left-hand side in (13.28), we get the next relation

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where,

At this stage, the new disturbance e is no longer serially correlated.

Therefore, using this transformed data Yk and X, we can estimate the parameters of the KLEM model. Our estimation technique is based on the nonlinear three-stage least-square method, where we suppose the existence of several instrumental variables. 13.3 Design of Experiment and Data Sources Annual observations for the period 1960 1979 were prepared by estimating the translog price possibility frontier for 30 industrial sectors excluding government service in Japan shown in table 13.2. The discrete divisia index is consistent with a translog aggregator functional form. Here we will summarize briefly the compiling procedures for price indices of output and input in each industrial sector. 13.3.1 Output, Energy and Intermediate Material Input Divisia price indices of output, energy and intermediate material input were constructed from estimated time-series input-output tables. In Japan, input-output tables classified on a commodity basis

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have been compiled every five years since 1955. At first we tried to estimate the time-series input-output tables compiled using consistent concepts and definitions over the period 1960 1979 by using published tables as benchmarks. We applied here the Lagrangian method instead of the well-known RAS method. Secondly we tried to convert estimated tables classified on a commodity basis into transaction tables on an industry basis by using information of product mix in each industrial sector. Each column vector in the estimated transaction table represents the intermediate input structure classified by commodity and the primary input structure in each industrial sector. Commodity transaction in each cell includes both domestic and imported goods. We tried to sum up noncompetitive imports like crude oil in each sector and reclassify them as a directly allocated import. For instance, the almost totally imported crude oil was allocated directly to mining, electric utility and petroleum refinery industries. Next secondary energy products fabricated in the petroleum refinery industry or electric utility sector are provided to other industrial sectors. With such dependency between intermediate input transactions, energy inputs in each industry except mining, petroleum refinery and electric utility are secondary energy produced in the latter two industries while directly allocated energy import is one of the intermediate inputs in mining, petroleum refinery industry and electric utility sector. The energy input price index in the ith industry is defined by the aggregated price of both petroleum refinery (the 12th industry) and electric utility products (the 26th industry) as follows

where Poi(t) represents the overall price of the ith commodity and wi(t) denotes the input share of ith energy input to total energy input in jth sector. Poi(t) is defined by a similar divisia aggregation of both ith domestic and imported commodity prices.

where Pci1(t) denotes the price of the domestically produced ith commodity and Pcj2 denotes the price of the imported ith commodity. wi1(t) represents the share of the ith domestically produced commodity and wi2(t) is the share of the ith imported commodity. The price

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index of other intermediate material inputs is defined by similar divisia aggregation.

where Poi(t) is an overall price of the ith commodity. The price index of the jth industrial output is also defined by the divisia aggregation of domestically produced commodity prices.

where wi(t) represent the nominal share of the ith commodity product to total output of the jth industry. 13.3.2 Labor Input Similarly we can formulate a discrete divisia price index for labor service as follows

where Pklmnkj(t) denotes the price of the klmnth labor service type in the jth sector and wklmnkj stands for the income share of the klmnth labor type in total labor compensation of the jth sector. Labor type is classified as follows: (1) Employment status (1. Ordinary employee, 2. Temporary worker, 3. Daily worker, 4. Self-employed, 5. Unpaid family worker); (2) Sex (1. Male, 2. Female); (3) Occupation (1. Blue-collar worker, 2. White-collar worker); (4) Education (1. Elementary and Junior High, 2. High school, 3. Junior college and Technical School, 4. College and University); (5) Age (1. less 17 years old, 2. 18 19 years old, 3. 20 24 years old, 4. 25 29 years old, 5. 30 34 years old, 6. 35 39 years old, 7. 40 44 years old, 8. 45 49 years old, 9. 50 54 years old, 10. 55 59 years old, 11. 60 64 years old, 12. more than 65 years old); (6) Industry (shown in table 13.2). Data for ordinary workers in the

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nonagricultural sectors were principally obtained from the Basic Wage Structure Survey (BWSS). While estimates for ordinary workers in agricultural and government service sectors were deduced from the Labor Force Survey (LFS). Data for temporary workers, daily workers, self-employed and unpaid family workers were estimated from LFS, Manufacturing Census, Establishment Census and Employment Status Survey. 13.3.3 Capital Input We can define the divisia price index of capital service input as follows

where Cklmnkj(t) represents the capital service price of the klmnth type of capital service input and wklmnkj(t) stands for the income share of the jth sector. Unlike the prices of labor service, we cannot directly observe the price of capital service, Cklmnkj(t). According to well-known procedures for imputation, a relationship in which the capital service price is regarded as a function of the price of klmnth investment goods, qklmnkj(t), rate of return on capital gjt, economic rate of replacement, µ j and certain tax variables can be obtained. Ignoring tax variables for simplicity, the next well-known relationship holds.

On the other hand, data for business profit adjusted for compensation of capital in the jth sector, Bjt are available from the estimation of time-series input-output tables. Under the assumptions of perfect market competition and linear homogeneity of the price possibility frontier function, Bjt must be equal to the total capital service cost of the jth industry.

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where Kklmnkj(t) denotes klmnth type capital asset in constant price. Regarding this relation as the equation in the unknown variables gjt and putting the observed data of Bjt, qklmnkj(t), µ klmnkj and Kklmnkj(t) into this equation, we can solve for the rate of return gjt in jth sector and hence impute the capital service price, Cklmnkj(t) in (13.35). Substituting estimated time series of Cklmnkj(t) into (13.34), we estimate the divisia price index for capital service. 13.4 Estimated Results According to the above stochastic formulation we estimated translog price frontier functions applying time series data for the period 1960 1979 to thirty industrial sectors in Japan. We can summarize the empirical results according to the substitutability among inputs and the bias of technical change. Jorgenson and Fraumeni (1983) estimated translog price frontier functions applying U.S. time-series data for the period 1960 1973 to thirty-eight industrial sectors. As similar methodology was used, we can compare their results to our estimates to observe differences in technological properties between the United States and Japan. Parameters, Bkk in the translog price function represent share elasticities as a measure of own and cross substitutability of inputs. The share elasticity, bij indicates the value share change of the ith input (or the jth input), when the price of only the jth input (or the ith input) increases infinitesimally. We may regard a positive elasticity as an indicator that a price increase in the ith or jth input price cause an increase in the value share of the jth or ith input because of high mutual substitutability. We may also regard a zero share elasticity as an indicator of neutrality which means that a price increase in the ith or jth input does not cause any changes of value shares of another input. On the other hand, a negative elasticity means that a price increase in the ith or jth input causes a decrease in the value share of the jth or ith input because of less substitutable or complementary relationships. Alternatively we can define the well-known Allen partial elasticity of substitutability among inputs as follows

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where a negative, zero or positive value of sij indicates a complementary, neutral or competitive relationships between the ith and jth inputs. Also we can define the price elasticity of demand for each input as follows

Next we can use the estimates of biT as an indicator for bias of technical change with respect to the ith input. When biT (= ¶wi/¶T = ¶wT/¶ ln Pi = bTi) is equal to zero for the ith input, this means that changes in the state of technology do not affect the value share and alternatively that changes in the ith input price do not induce any changes in the state of technology. When biT is positive (negative), this means that the value share on the ith input increases (decreases) through technical change and alternatively that increases in the ith input price induces technical changes positively (negatively). Results we obtained are summarized as follows: (1) We estimated the price function under conditions of symmetry, a concave and linear homogeneous translog form, which includes the Douglas form as a special case. The results show that the Douglas-type price function was applicable only in five of thirty industries; fabricated textile, paper products, chemical products, nonferrous metal products and transportation and communication. In order to impose a concavity condition upon the translog price function, we had to restrict some of the Cholesky values, di (i = KLEM) to zero as follows: dK = 0; mining, construction, foods and kindreds, fabricated metal, trade and real estate; dE = 0; rubber, electric machinery and other service; dK = dE = 0; lumber and wood products, furniture and fixture, printing, petroleum refinery, stone and clay, machinery, transportation equipment except motor vehicle, precision, miscellaneous manufacturing and finance and insurance; dL = dE = 0; agriculture, textile, iron and steel and motor vehicle; dK = dL = 0; leather products and utility industry. (2) Own price elasticities of inputs are shown in table 13.3 next to the U.S. estimates of Jorgenson and Fraumeni. Own price elasticity was calculated from (13.38) in which estimated parameter, ai (i = KLEM) were substituted to obtain the value share wi as an average. Price elasticities of capital input are less than unity in all industries except

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rubber in Japan and lumber and wood products and communication in the United States. Values of price elasticities less than unity are quite similar between Japanese and the U.S. industries. As for labor input, price elasticities are more than unity in 16 industries of Japan and 22 industries of the United States. Value of price elasticities are scattered over a wide range both in Japan and the United States. Thirdly, price elasticities of energy input are more than unity in 20 industries of Japan and 19 industries of the United States. In energy intensive industries such as chemical products, petroleum refinery, stone and clay, iron and steel, nonferrous metal product, transportation and communication, and utility, price elasticities of energy input are less than unity in all industries except iron and steel and utility in Japan and stone and clay in the United States. Fourthly, price elasticities of material input in Japan are somewhat different from those in the United States. Elasticities in Japan are less than unity in all industries except agriculture, mining and real estate. On the other hand those in the United States are more than unity in 20 industries. (3) Table 13.4 shows Allen partial elasticities of substitution for each input in Japan, which is calculated from (13.37). Relationships between capital and labor, energy and capital, material and capital, material and labor and material and energy indicate ''substitutable relations" in almost all industries with only a few exceptions. Energycapital for textile, rubber and iron and steel and material-energy for machinery and finance are those few exceptional cases. The relationship between energy and labor indicates a "complementary" relation in 14 sectors and a "substitutable" relation in 16 sectors. (4) Table 13.5 shows an international comparison of cross-share elasticities between Japan and the United States. According to the U.S. estimates, cross-share elasticities between capital and labor, capital and energy and capital and material indicate "neutrality" with respect to value share. These properties are also observed in Japanese industries with a few exceptions. On the other hand, patterns of cross-share elasticities between labor and material, energy and material and labor and energy are rather different between Japan and the United States. Especially, cross-share elasticities between labor and energy indicate "complementarity" in more than half of the Japanese industries which means increases in the price of energy induce decreases in the labor share. On the other hand, this property is not necessarily observed in the United States.

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Table 13.3 Own price elasticity of input: International comparison between Japan and the United States hK hL hE hM Japan U.S. Japan U.S. Japan U.S. Japan U.S. 1 1 0.6770 0.8135 1.2907 3.9892 1.2647 1.0043 1.3432 2.6492 2 2 0.6561 0.7680 1.1020 5.2635 2.1989 0.9652 2.2452 3.9797 3 0.7523 0.4852 1.9650 2.1085 4 0.5144 0.8867 0.9421 0.6568 5 0.7097 0.6672 0.9202 0.7029 3 6 0.8662 0.9265 0.8698 1.4876 1.7314 1.2088 0.4139 1.8577 4 7 0.7581 0.9405 1.1237 0.8325 1.5101 0.9884 0.4221 0.2386 5 9 0.8435 0.9261 1.3215 0.7792 1.4162 0.9836 0.4869 0.3111 6 10 0.8982 0.9554 0.8026 0.9894 0.9894 1.0389 0.3097 0.7721 7 11 0.9360 1.3725 1.5549 1.9008 1.2028 1.0740 0.5117 2.1694 8 12 0.9349 0.9296 0.8054 4.3798 1.1446 1.2252 0.3743 3.8173 9 13 0.8475 0.8807 0.8893 0.7243 0.9526 0.9672 0.3107 0.4278 10 14 0.9071 0.8863 0.7626 1.6095 1.2366 1.2007 0.5138 1.4749 11 15 0.8294 0.8616 0.8875 1.3754 0.9122 0.9057 0.3709 0.6313 12 16 0.8016 0.8842 1.7038 4.8516 0.9264 0.3618 0.3669 2.6791 13 17 1.5225 0.8936 1.2522 3.0465 1.1769 2.0710 0.5829 2.0424 14 18 0.9167 0.9493 0.8259 1.2070 1.7092 1.0987 0.2817 0.8600 15 19 0.8475 0.8719 1.3352 5.5413 0.9195 1.0714 0.5719 6.1820 16 20 0.9059 0.9059 0.9373 0.6994 1.4828 0.9568 0.4428 0.4379 17 0.7773 0.9192 0.9466 0.3568 18 21 0.8997 0.9031 1.0559 2.2586 2.5149 1.2274 0.7540 1.8082 19 22 0.8622 0.8931 1.3198 6.8585 1.2320 1.5713 0.4623 4.6359 20 23 0.9203 0.8981 1.0948 2.3374 0.9943 1.1014 0.4702 1.7552 21 24 0.8674 0.8792 1.2412 2.5112 0.9921 0.9988 0.4453 0.8705 22 25 0.8556 0.9589 1.1950 0.6044 1.0890 0.9901 0.5798 0.4466 23 26 0.8699 0.8609 0.8934 4.1258 1.4070 1.0530 0.4451 2.9885 24 27 0.9162 0.9005 1.5248 3.5741 1.0189 1.4668 0.5199 2.6503 8 0.8206 1.3488 1.0255 0.4243 25 28 0.7819 0.8161 0.5624 0.5429 0.9250 0.9439 0.7308 0.6972 29 2.4740 4.1630 1.3433 28.1671 26 30 0.6422 0.6474 0.8305 0.7555 1.2159 0.7406 0.9699 0.8564 31 0.7712 0.8079 0.4210 1.0000 27 32 0.7276 0.8419 0.8436 2.7272 2.0205 0.9870 0.9591 8.7936 28 33 0.6166 0.8604 0.8996 1.1897 1.0459 1.0751 0.9698 1.1065 29 0.1668 4.1048 4.3257 1.9583 30 34 0.7152 0.8938 1.2005 0.4423 0.9738 0.9786 0.8278 0.6852

35 0.8883 1.2058 1) All above figures are represented by absolute value. 2) U.S. figures are referred from Jorgenson-Fraumeni estimates.

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Table 13.4 Allen partial substitutability among inputs L-K E-K M-K E-L M-L M-E 0.6627 0.0800 0.6813 4.7970 3.0078 6.4795 1. Agriculture 1.0 1.0 1.0 1.2507 3.4077 8.2795 2. Mining 1.0 1.0 1.0 2.5733 1.0954 1.6463 3. Construction 1.0 1.0 1.0 0.3560 1.3952 2.0565 4. Foods 0.1019 1.2489 1.2202 0.4870 2.0863 3.7209 5. Textile 1.0 1.0 1.0 1.0 1.0 1.0 6. Fab. textile 1.0 1.0 1.0 6.9325 2.1564 3.3143 7. Lumber 1.0 1.0 1.0 2.8552 1.0900 0.4441 8. Furniture 1.0 1.0 1.0 1.0 1.0 1.4108 9. Paper 1.0 1.0 1.0 2.1422 1.2029 3.2555 10. Printing 1.0 1.0 1.0 1.0 1.0 1.0 11. Chemical 1.0 1.0 1.0 0.2721 2.2308 1.0779 12. Petroleum 2.9545 2.0017 1.5344 0.4655 1.3725 2.5361 13. Rubber 1.0 1.0 1.0 1.0 1.0 1.9877 14. Leather 1.0 1.0 1.0 1.1740 1.8629 0.9478 15. Stone & clay 0.9661 0.4212 1.2611 0.8538 1.0267 2.1207 16. Iron & steel 1.0 1.0 1.0 1.0 1.0 1.0 17. Nonferrous metal 1.0 1.0 1.0 4.2083 1.7948 5.8328 18. Fab. metal 1.0 1.0 1.0 7.3405 1.6533 0.6048 19. Machinery 0.1240 1.5507 1.3155 1.9910 1.5652 0.6426 20. Elec. machinery 0.1664 0.7638 1.2157 0.1673 1.7555 1.2155 21. Motor vehicle 1.0 1.0 1.0 2.7748 1.7678 2.5594 22. Trans. eqpt. 1.0 1.0 1.0 3.0785 1.2586 3.0567 23. Precision 1.0 1.0 1.0 1.6976 2.0926 1.7872 24. Misc. mfg. 1.0 1.0 1.0 1.0 1.0 1.0 25. Trans. & comm. 1.0 1.0 1.0 1.0 1.0 2.3931 26. Utilities 1.0 1.0 1.0 0.2177 1.5974 5.0366 27. Trade 1.0 1.0 1.0 3.0801 1.7889 1.3477 28. Finance 1.0 1.0 1.0 303.2023 23.6396 92.8415 29. Real estate 0.6042 0.9280 1.2212 0.6302 2.1308 1.2062 30. Other service Notes: 1) Allen partial elasticity of substitution was calculated by equation (12.37), in which each income share of input, wi (i = KLEM) is substituted by estimated parameter, ai (i = KLEM). 2) "Positive," "zero," and "negative" values of Allen partial elasticity mean that the ith and the jth inputs have "substitutable,'' "neutral" and "complementary" relationships, respectively. 3) Unity of elasticity indicates bij = 0. It means that increases or decreases of the ith and jth input prices do not induce any changes of the value shares of the jth and ith input mutually, while technologically the ith and the jth inputs have a "substitutable" relationship.

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Table 13.5 Cross share elasticity and biases of technical change between U.S. and Japanese sectors A. Cross Share Elasticity: Capital-Labor U.S. Japan bKL < 0 bKL = 0 Neutral bKL > 0 Complementary Competitive Complementary Lumber Rubber Neutral Agriculture, textile, Mining, construction, foods, fab. text., furniture, paper, printing, chemicals, petroleum, leather, stone iron steel, elec. clay, fab. metal, machinery, trans. eqpt, precision, machinery, motor misc. mfg., trade vehicle, services Competitive B. Cross Share Elasticity: Capital-Energy U.S. Japan bKE < 0 bKE = 0 Neutral Complementary Complementary Neutral Agriculture, rubber Mining, construction, foods, fab. text., lumber, furniture, paper, printing, chemicals, petroleum, iron steel, textile, leather, stone clay, fab. metal, machinery, trans. motor vehicle, eqpt., precision, misc. mfg., trade services Competitive

bKE > 0 Competitive Elec. machinery

(table continued on next page)

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Table 13.5 (continued) Table 13.5 Cross share elasticity and biases of technical change between U.S. and Japanese sectors C. Cross Share Elasticity: Labor-Material U.S. Japan bLM < 0 bLM = 0 bLM > 0 Competitive Complementary Neutral Complementary Chemicals Printing Mining, textile, lumber, rubber, iron steel, elect. machinery, Neutral Fab. misc. mfg. textile, paper, leather Competitive Agriculture, construction, foods, furniture, petroleum, stone clay, machinery, trans. eqpt., precision, fab. metal, motor vehicle, trade, services D. Cross Share Elasticity: Energy-Material U.S. Japan bEM < 0 bEM = 0 bEM > 0 Competitive Complementary Neutral Complementary Foods, petroleum, trans. eqpt., services Neutral Elec. Paper Mining, textile, lumber, rubber, leather, iron steel, misc. machinery mfg. Agriculture, construction, printing, motor vehicle, fab. Competitive Furniture, stone Fab. metal, precision, trade clay, machinery textile, chemicals (table continued on next page)

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Table 13.5 (continued) Table 13.5 Cross share elasticity and biases of technical change between U.S. and Japanese sectors E. Cross Share Elasticity: Capital-Material U.S. Japan bKM < 0 bKM = 0 Neutral bKM > 0 Complementary Competitive Complementary Neutral Mining, construction, foods, fab. text., lumber, Agriculture, textile, furniture, paper printing, chemicals, petroleum rubber, iron, steel, leather, stone clay, fab. metal, machinery, trans. elec. machinery, motor vehicle, eqpt., precision, misc. mfg., trade services Competitive F. Cross Share Elasticity: Labor-Energy U.S. Japan bLE < 0 bLE = 0 Neutral bLE > 0 Competitive Complementary Paper, chemicals Construction ComplementaryAgriculture, furniture, machinery, printing, motor stone, clay vehicle, fab. metal, Precision, services Elec. machinery Neutral Mining, textile, Fab. textile, leather lumber rubber, iron steel, misc. mfg. Competitive Foods, petroleum, trans. eqpt., trade Notes: 1. Evaluation of bij in the U.S. economy depends upon Jorgenson and Fraumeni estimates (1983). 2. Column of the above table represents the classification of bij (< 0, = 0 and > 0, that is, "complementary," "neutral" and "competitive") in Japan. Row of the table represents the classification of bij in the U.S.

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Figure 13.2 Biases of Technical Change: Capital, Labor, Energy, and Materials.

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(5) Finally, we will discuss estimates of the bias of technical change shown in figure 13.2. According to (13.20), a positive bias in technical change means that changes in the technological state induce increases in the value share of the input and alternatively that increases in the input price induce positive changes in the technological state. Technical bias for labor input is positive in all industries and that for material input is negative in all industries except food and kindreds and petroleum. This implies that in the Japanese economy increases in labor input price and material input price induce technological changes in which the ratio of value added and the value share of labor input is increased dominantly. Technical bias for energy is positive in 26 industries and significantly negative only in 2 industriesmachinery and finance. Technical bias for capital is negative in 22 industries and positive in 7 industries. The dominant pattern of bias of technical change in Japan is "positive for labor and energy and negative for material and capital." Nomenclature Pc/1= the price of the domestically produced jth commodity, Pc/2= the price of the imported jth commodity, Pij = the price of the ith input in the jth sector, Pj = the aggregate input price of the jth sector, Poj = the overall price of the jth sector, qj = the price of output of the jth sector, T(t) = a state of technology at the time period t, wij = share of the ith input in the value of output of the jth sector, xj = the aggregate input of the jth sector, xij = the ith input of the jth sector, Yj = output of the jth sector, sij = Allen partial elasticity of substitution between the ith and jth input, Y = divisia index of total factor productivity or the total efficiency of the production.

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Subscripts i= the ith input, specifically i indicates capital (K), labor (L), energy (E) and material (M) respectively, j= the jth industry classified in thirty-one industrial sectors shown in table 13.2, t= the time period.

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14 The Role of Energy in Productivity Growth Dale W. Jorgenson 14.1 Introduction The objective of this chapter is to analyze the role of energy in the growth of productivity. The special significance of energy in economic growth was first established in the classic study, Energy and the American Economy 1950 1975, by Schurr and his associates (1960) at Resources for the Future. 1 For the period from 1920 to 1955, Schurr noted that energy intensity of production had fallen while both labor and total factor productivity were rising. The simultaneous decline in energy intensity and labor intensity of production ruled out the possibility of explaining the growth of productivity solely on the basis of substitution of less expensive energy for more expensive labor. Since the quantity of both energy and labor inputs required for a given level of output had been reduced, technical change is also a critical explanatory factor. An alternative explanation for growth of output with declining energy and labor intensity required an examination of the character of technical change. This examination was motivated by the fact that from 1920 to 1955 the utilization of electricity had expanded by a factor of more than ten, while consumption of all other forms of energy only doubled. Two key features of technical change during this period were that the thermal efficiency of conversion of fuels into electricity increased by a factor of three and that "the unusual characteristics of electricity had made it possible to perform tasks in altogether different ways than if the fuels had to be used directly" (Schurr, 1983). Schurr illustrated this point by the impact of electrification on industrial processes, which led to much greater flexibility in the application of energy to industrial production. The importance of electrification in productivity growth has also been documented by Rosenberg (1983):

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Increasingly, the spreading use of electric power in the 20th century has been associated with the introduction of new techniques and new arrangements which reduce total costs through their saving of labor and capital. Perhaps the most distinctive features of these new techniques are (1) that they take so many forms as to defy easy categorization, and (2) that they occur in so many industries that they defy a simple summary. Rosenberg has illustrated this point with examples drawn from the production of iron and steel, the making of glass, and the production and utilization of aluminum. Rosenberg, like Schurr and his associates, has drawn attention to the significance of electrification of industrial processes that took place during the first several decades of the twentieth century. Electrical motors have provided greater flexibility in the supply of power to industrial processes and in the organization and layout of production processes. Rosenberg (1983) reaches the overall conclusion: It seems obvious that there has been a very wide range of labor saving innovations throughout industry which have taken an electricity using form. As a consequence, greater use of electricity is, from an historical point of view, the other side of the coin of a labor saving bias in the innovation process. Schurr (1982) has recently extended the analysis of Energy and the American Economy 1950 1975 through 1981. For the period through 1969 his conclusions are as follows (Schurr, 1982): Although the inverse relationship between total factor productivity and energy intensity virtually disappeared during the 1953 1969 period, it is still noteworthy that high rates of improvement in total factor productivity were essentially not associated with increases in energy intensity. Schurr has analyzed the experience of the U.S. economy in the aftermath of the oil embargo of 1973. He points out that energy intensity of production has fallen steadily since 1973 and that the rate of decline accelerated sharply after the second oil price shock in 1979, following the Iranian revolution. He goes on to point out that (Schurr, 1982): While energy productivity has been improving at a very high rate during the past decade, the overall productivity efficiency side of the story has been highly unfavorable, and has become a matter of great concern. The post-1979 years that witnessed a new high in the rate of growth of national energy productivity also saw a decline in productive efficiency with a fall in total factor productivity of about 0.3 percent per year between 1979 and 1981.

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We can summarize the evidence on energy intensity and productivity growth by saying that energy intensity was falling while productivity was rising for the period 1920 1953. Between 1953 and 1969 energy intensity was relatively stable, while productivity continued to rise. After 1973 energy intensity has resumed its downward trend with a sharp acceleration after 1979, while productivity growth has fallen off since 1973 and has given way to productivity decline since 1979. In exploring the determinants of trends in energy intensity and in productivity growth a useful framework is provided by Schurr and his associates (1979) in the study Energy in America's Future. This study emphasizes the role of change in the composition of the national output, trends in energy intensity within industrial sectors, the significance of changing energy forms, and the role of price developments. Focusing on historical experience through 1975 and 1977, Schurr and his associates conclude that the perspective provided by change in the composition of the national output "offers a useful but, at best, limited insight" (Schurr et al., 1979). Similarly, they find that energy intensity has declined in some sectors and risen in others (Schurr et al., 1979). They find that the transformation of energy forms, especially toward greater electrification and the use of fluid forms of energy such as petroleum and natural gas, has played an important role (Schurr et al., 1979): "The changes have made possible shifts in production techniques and locations within industry, agriculture, and transportation that greatly enhanced the growth of national output and productivity," and finally, they argue that "quite apart from energy prices, technology developed its own momentum." The framework suggested by Schurr and his associates and the historical evidence on trends in energy intensity and productivity growth suggests that an explanation of these trends must encompass a wide range of determinants. First, the gradual decline in real energy prices through the early 1970s and the sharp increases in energy prices that have followed the oil shocks of 1973 and 1979 suggest an important role for substitution between energy and other productive inputs, especially labor input. While the real price of labor input rose steadily through the early 1970s, this price has been declining since that time. These price trends would suggest the possibility of substitution of energy for labor through the early 1970s and substitution of labor for energy afterward. Second, technical change is an important component of any explanation of trends in energy intensity and productivity. Schurr (1982)

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has suggested a possible role for technical change in reviewing U.S. experience through 1969, as follows: The net result, then, was that strong improvements in both energy productivity and overall productive efficiency were achieved without any special efforts being made to bring about this desirable combination of circumstances. Energy was abundantly available, and its price was low and, for the most part, falling during this period. Simple economic reasoning would tell us that the intensity of energy use should have risen because favorable energy prices would have encouraged energy consumption. But even though energy use rose relative to labor inputs, it fell in relationship to the final output of the economy. Did this decline in energy intensity take place in spite of low energy prices, or somehow because of them? The mechanisms of technical change, as described above by Schurr and by Rosenberg, indicate a specific role for electrification. For this reason it is essential to analyze the role of both the price of electricity and the price of nonelectrical energy in the determination of productivity growth. Another potential determinant of changes in energy intensity and changes in productivity growth in the U.S. economy is the change in the composition of the national output. The role of this change in stimulating and retarding U.S. productivity growth is suggested by a review of postwar U.S. economic history. The growth of the U.S. economy in the postwar period has been very rapid by historical standards. The rate of economic growth reached its maximum during the period 1960 to 1966. Growth rates have slowed substantially since 1966 and declined further since 1973. A decomposition of the growth of output into contributions of capital input, labor input, and productivity growth shows that capital input has made the most important contribution to postwar growth of output, while growth in productivity and labor input are less significant. Focusing on the period since 1973, the fall in the rate of economic growth is due to a dramatic decline in productivity growth. Declines in the contributions of capital and labor input are much less significant in explaining the slowdown. 2 To analyze the potential sources of the decline in productivity growth since 1973 it is useful to decompose productivity growth into components that can be identified with productivity growth at the sectoral level and with reallocations of output, capital input, and labor input among sectors. For the period up to 1973 the contribution of

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reallocations of output and inputs among sectors is insignificant relative to sectoral productivity growth. For the period since 1973 the contributions of these reallocations were positive rather than negative, but relatively small. Declines in productivity growth for the individual industrial sectors of the U.S. economy must bear the full burden of explaining the slowdown in productivity growth for the economy as a whole. The overall conclusion from our review of postwar U.S. economic history is that the key to understanding the role of energy in the advance of productivity is to analyze the growth of productivity at the level of individual industrial sectors in the United States. As Rosenberg and Schurr have indicated, special attention must be devoted to the substitution of electricity for other forms of energy. An important force in electrification has been identified by Schurr as the cheapening of electricity relative to other forms of energy that has accompanied the dramatic increases in the thermal change through innovations that have occurred throughout the whole range of industrial activity. Second, our review of postwar history shows that growth in productivity at the level of individual industrial sectors is the primary explanation of productivity growth for the economy as a whole. Furthermore, the decline in economic growth that has taken place since 1973 can be attributed almost entirely to a decline in productivity growth at the level of individual industrial sectors. To revive the growth of sectoral productivity it will be necessary to revive the process of innovation at the sectoral level. Given the importance of electrification in stimulating innovation, it is essential to assess the potential of electrification for generating a revival in innovation and in productivity growth. In order to assess the role of energy in stimulating productivity growth, it will be necessary to go behind trends in energy utilization and productivity. For this purpose we employ an econometric model of sectoral productivity growth for individual industrial sectors in the United States. In order to assess the significance of changing forms of energy, we divide inputs in each sector among capital, labor, electricity, nonelectrical energy, and materials. Our econometric model encompasses substitution among productive inputs in response to changes in relative prices. Our model also determines the growth of sectoral productivity as a function of relative prices.

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14.2 Econometric Models For each industry our model of production will be based on a sectoral price function that summarizes both possibilities for substitution among inputs and patterns of technical change. Each price function gives the price of output of the corresponding industrial sector as a function of the prices of capital, labor, electricity, nonelectrical energy, and materials inputs and time, where time represents the level of technology in the sector. 3 Obviously an increase in the price of one of the inputs, holding the prices of the other inputs and the level of technology constant, necessitates an increase in the price of output. Similarly, if productivity of a sector improves and the prices of all inputs into the sector remain the same, the price of output must fall. Price functions summarize these and other relationships among the prices of output, capital, labor, electricity, nonelectrical energy, and materials inputs, and the level of technology. The sectoral price functions provide a complete model of production patterns for each sector, incorporating both substitution among inputs in response to changes in relative prices and technical change in the use of inputs to produce output. To characterize both substitution and technical change it is useful to express the model in an alternative and equivalent form. First, we can express the shares of each of the five inputscapital, labor, electricity, nonelectrical energy, and materialsin the value of output as functions of the prices of these inputs and time, again representing the level of technology.4 Second, we can add to these five equations for the value shares an equation that determines productivity growth as a function of the prices of all five inputs and time. This equation is our econometric model of sectoral productivity growth.5 Like any econometric model, the relationships determining the value shares of capital, labor, electricity, nonelectrical energy, and materials inputs and the rate of productivity growth involve unknown parameters that must be estimated from data for the individual industries. Included among these unknown parameters are biases of productivity growth that indicate the effect of change in the level of technology on the value shares of each of the five inputs. For example, the bias of productivity growth for capital input gives the change in the share of capital input in the value of output in response to changes in the level of technology, represented by time. We say that productivity growth is capital using if the bias of productivity growth

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for capital input is positive. Similarly, we say that productivity growth is capital saving if the bias of productivity growth for capital input is negative. For the purposes of a study of the role of electrification in productivity growth the key parameter in our econometric model is the bias of productivity growth for electricity. This bias gives the change in the share of electricity in the value of output in response to changes in the level of technology. We say that productivity is electricity using, as suggested by Rosenberg, if the bias of productivity growth for electricity is positive. Similarly, we say that productivity growth is electricity saving if the bias of productivity growth for electricity input is negative. To test the hypothesis that technical change is electricity using for an individual industrial sector, we fit the bias of productivity growth for electricity, together with other parameters that describe substitution and technical change in that sector. We then test the hypothesis that the bias of productivity growth is positive. It is important to observe that the sum of the biases of all five inputs must be precisely zero, since the changes in all five shares with any change in technology must sum to zero. To put this another way, if productivity growth is electricity using, then productivity growth must be input saving in some other input. For example, productivity growth could be labor saving and electricity using, as suggested by Rosenberg. This would correspond to a positive bias of productivity growth for electricity and a negative bias of productivity growth for labor. Our econometric model will make it possible to classify each of the thirty-five industries included in our study of technical change among the thirty logically possible patterns of productivity growth that correspond to positive or negative values of each of the five biases. Only the possibility that all five biases are negative or all five are positive can be ruled out on the basis of purely analytical considerations. We have pointed out that our econometric model for each industrial sector of the U.S. economy includes an equation giving the rate of sectoral productivity growth as a function of the prices of the five inputs and time. The biases of technical change with respect to each of the five inputs appear as the coefficients of time, representing the level of technology, in the five equations for the value shares of all five inputs. The biases also appear as coefficients of the prices in the equation for the negative of sectoral productivity growth. This feature of our econometric model makes it possible to use information about

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changes in the value shares with time and information about changes in the rate of sectoral productivity growth with prices in determining estimates of the biases of technical change. The biases of productivity growth express the dependence of the value shares of the five inputs on the level of technology and also express the dependence of the rate of productivity growth on the input prices. We say that capital-using productivity growth, associated with a positive bias of productivity growth for capital input, implies that an increase in the price of capital input decreases the rate of productivity growth. Similarly, capital-saving productivity growth, associated with a negative bias for capital input, implies that an increase in the price of capital input increases the rate of productivity growth. Analogous relationships hold between the biases of labor, electricity, nonelectrical energy, and materials inputs on the one hand and the impact of changes in the prices of each of these inputs on the other. The dual role of the bias of productivity growthexpressing the impact of a change in technology on the value share of an input and the impact of a change in the price of that inputis the key to an assessment of the role of electrification in productivity growth. Historical evidence, as summarized by Rosenberg, suggests that much of the innovation in the twentieth century is electricity using so that it increases the share of electricity in the value of output for a given set of input prices, including the price of electricity. Entirely different evidence, analyzed by Schurr and his associates, has linked the reduction in the cost of electricity resulting from increased thermal efficiency in electricity generation to enhanced productivity growth. Within our econometric model these two pieces of historical evidence are both consistent with the hypothesis that the bias of productivity growth for electricity is positive. Electricity-using productivity growth, associated with a positive bias of productivity growth for electricity, implies that the utilization of electricity increases as technology changes. This is precisely what happened over the first several decades of this century, according to the evidence reviewed by Rosenberg. Electricity-using productivity growth also implies that the overall rate of productivity growth increases as the price of electricity declines, which is consistent with historical evidence on the growth of output, productivity, and energy consumption analyzed by Schurr. Our overall conclusion is that this historical evidence suggests the hypothesis that technical change at

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the level of individual sectors of the U.S. economy is electricity using. This implies a central role for electrification in the growth of productivity. 14.2.1 Empirical Results We provide a detailed description of our econometric models of production and technical change in the appendix. To implement these models we have assembled a data base for the thirty-five industrial sectors of the U.S. economy listed in table 14.1. 6 The thirty-five industries encompass all sectors of the U.S. economy. Manufacturing is subdivided among twenty-one two-digit industries. These industries differ greatly in their relative importance to the economy and in their energy intensity. The thirty-five industries also include the primary production sectors of agriculture and mining, the energy intensive transportation and public utilities industries, and the construction, communications, trade, and service industries. For capital and labor inputs we have first compiled data by sector on the basis of the classification of economic activities employed in the U.S. National Income and Product Accounts. We have then transformed these data into a form appropriate for the classification of activities employed in the U.S. Inter-industry Transactions Accounts. For electricity, nonelectric energy, and materials inputs we have compiled data by sector on inter-industry transactions among the thirty-five industrial sectors. For this purpose we have used the classification of economic activities employed in the U.S. Inter-industry Transactions Accounts.7 For each sector we have compiled data on the value shares of capital, labor, electricity, nonelectrical energy, and materials inputs, annually, for the period 1958 1979. We have also compiled indexes of prices of sectoral output and all four sectoral inputs for the same period. Finally, we have compiled translog indexes of sectoral rates of technical change. There are twenty-one observations for each behavioral equation, since unweighted two-period averages of all data are employed. The parameters can be interpreted as average value shares of capital, labor, electricity, nonelectrical energy, and materials inputs for the corresponding sector. Similarly, the parameters can be interpreted as averages of the negative of rates of technical change. The parameters

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Table 14.1 Industrial sectors 1. Agriculture, forestry, and fisheries 2. Metal mining 3. Coal mining 4. Crude petroleum and natural gas 5. Nonmetallic mining and quarring, except fuel 6. Construction 7. Food and kindred products 8. Tobacco manufactures 9. Textile mill products 10. Apparel and other fabricated textile products 11. Lumber and wood products 12. Furniture and fixtures 13. Paper and allied products 14. Printing, publishing and allied industries 15. Chemicals and allied products 16. Petroleum refining 17. Rubber and miscellaneous plastic products 18. Leather and leather products 19. Stone, clay and glass products 20. Primary metal industries 21. Fabricated metal products 22. Machinery, except electrical 23. Electric machinery 24. Motor vehicles and motor vehicles equipment 25. Transportation equipment and ordinance, except motor vehicles 26. Instruments 27. Miscellaneous manufacturing industries 28. Transportation 29. Communication 30. Electric utilities (including federal, state, and local) 31. Gas utilities 32. Trade 33. Finance, insurance, and real estate. 34. Service (including water and sanitary services) 35. Government enterprises (excluding electric utilities) can be interpreted as constant share elasticities with respect to price for the corresponding sector. Similarly, the parameters technical change with respect to price. Finally, the parameters { of the negative of the rates of technical change.

can be interpreted as constant biases of } can be interpreted as constant rates of change

In estimating the parameters of our sectoral models of production and technical change we retain the average of the negative of the rate of technical change, biases of technical change, and the rate of change

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of the negative of the rate of technical change as parameters to be estimated for all thirty-five industrial sectors. Estimates of the share elasticities with respect to price are obtained under the restrictions implied by the necessary and sufficient conditions for concavity of the price functions presented in the appendix. Under these restrictions the matrices of constant share elasticities {Ui} must be negative semi-definite for all industries. To impose the concavity restrictions we represent the matrices of constant share elasticities for all sectors in terms of their Cholesky factorizations. The necessary and sufficient conditions are that the diagonal elements of the matrices {Di} that appear in the Cholesky factorizations must be nonpositive. We present estimates subject to these restrictions for all thirty-five industrial sectors in table 14.2. 14.2.2 Substitution Our interpretation of the parameter estimates reported in table 14.2 begins with an analysis of the estimates of the parameters . It is useful to recall that if the sectoral price functions are increasing in the prices of capital, labor, electricity, nonelectrical energy, and materials inputs, the average value shares are nonnegative for each sector. These conditions are satisfied for all thirty-five sectors included in our study. The sectoral price functions are not universally decreasing in time. The negative of the estimated average rate of technical change is negative in thirty-one sectors and positive in four sectors. Negative signs characterize the construction, apparel, printing and publishing, and motor vehicles industries. The estimated share elasticities with respect to price describe the implications of patterns of substitution among capital, labor, electricity, nonelectrical energy, and materials inputs for the relative distribution of the value of output among these five inputs. Positive share elasticities imply that the corresponding value shares increase with an increase in price; negative share elasticities imply that the value shares decrease with an increase in price; share elasticities equal to zero imply that the value shares are independent of price. It is important to keep in mind that we have fitted these parameters subject to the restrictions implied by concavity of the price function. These restrictions imply that all share elasticities be set equal to zero for ten of the thirty-five industries listed in table 14.2. For all industries the share

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Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Agriculture, forestry and fisheries Metal mining AK AL AE AN AM AT BKK BKL BKE BKN BKM BKT BLL BLE

.179 .243 .00341 .0215 .553 .0201

.00362 .0586 .00189

(.00301) (.00587) (.000469) (.00105) (.00374) (.0318)

(.000430) (.0307) (.00466)

.226 .328 .0253 .0143 .407 .0681

.00206 .422 .00217

(.00770) (.0134) (.00141) (.000567) (.0133) (.107)

(.00110)

Coal mining .242 .494 .0140 .102 .148 .0213

.00923

(.00743) (.00537) (.000306) (.00120) (.00684) (.0512)

(.00106)

Crude petroleum and natural gas (.00299) .471 (.00247) .109 (.000764) .00870 (.00150) .0492 (.00388) .362 (.0489) .00510

.00203

(.000428)

(.247) (.0246)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Agriculture, forestry and fisheries Metal mining BLN BLM BLT BEE BEN BEM BET BNN BNM BNT BMM BMT BTT

.0168 .0735 .00271 .00886 .000661 .00630 .0000677 .00484 .0210 .000714 .101 .00155 .000868

(.00592) (.0350) (.00149) (.00528) (.00193) (.00317) (.000216) (.00274) (.00816) (.000277) (.0406) (.00149) (.00455)

.0294 .394 .0178 .0000111 .000151 .00203 .000765 .00205 .0275 .000884 .369 .0197 .00835

Coal mining

Crude petroleum and natural gas

(.00827) (.219) (.00688) (.000259) (.00177) (.0231) (.000669) (.000570) (.00720) (.000238) (.194) (.00615) (.0153)

.00803 .000240 .00321 .00345 .000830 .0429 .0461 .00612 .0495 .0103 .00130

(.000767) (.000158) (.000981) (.00114) (.0000522) (.00522) (.00502) (.000237) (.00500) (.000990) (.00731)

.000124

(.000353) (.0112)

.0231 .00475

(.00590) (.00588)

.0184 .000151 .000977 .00378 .000880

(.000169) (.00198) (.00393) (.000224) (.00341)

.0146 .00264 .000571

(.000555) (.00699)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Nonmetallic mining Construction Food and kindred products AK (.00318) (.000650) (.000656) .281 .0715 .0578 AL (.00539) (.00427) (.00192) .314 .449 .159 AE (.000894) (.000176) (.000306) .0296 .000294 .00441 AN (.000392) (.000767) (.000151) .0486 .0261 .00679 AM (.00482) (.00474) (.00243) .327 .453 .772 AT (.114) (.0482) (.0279) .0600 .00585 .0152 BKK BKL BKE BKN BKM (.000455) (.0000929) (.0000938) BKT .00329 .000667 .000900 BLL (.109) (.0333) (.0165) .377 .311 .0656 BLE (.0146) (.000915) (.00462) .0441 .00622 .00209

Tobacco manufacturers .171 .130 .00211 .00121 .696 .00138

.00224 .133 .0000237

(.00381) (.00462) (.000246) (.0000986) (.00620) (.0912)

(.000544) (.0279) (.00155)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Nonmetallic mining Construction Food and kindred products BLN (.00938) (.00378) (.00214) .0131 .0563 .00138 BLM (.109) (.0355) (.0167) .346 .361 .0663 BLT (.00294) (.000998) (.000487) .0102 .00377 .000638 BEE (.00401) (.0000405) (.00563) .00514 .000124 .10973 BEN (.00163) (.000213) (.00253) .00153 .00112 .00167 BEM (.0119) (.00107) (.00334) .0404 .00722 .0101 BET (.000418) (.0000297) (.000154) .000624 .0000649 .0000834 BNN (.000728) (.00125) (.000689) .000454 .0102 .000302 BNM (.00861) (.00465) (.00149) .0120 .0654 .00275 BNT (.000262) (.000120) (.0000650) .00185 .00152 .0000285 BMM (.109) (.0378) (.0178) .318 .420 .0737 BMT (.00291) (.00108) (.000535) .0147 .00455 .000150 BTT (.0162) (.00688) (.00399) .00657 .00113 .00162

Tobacco manufacturers .000447 .133 .0108 .000601 .000962 .00154 .000127 .00154 .00295 .0000938 .138 .0132 .00464

(.000626) (.0282) (.00142) (.000756) (.000577) (.00187) (.0000787) (.000212) (.000652) (.0000314) (.0287) (.00156) (.0130)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Textile mill products Apparel and other fabricated textile products AK AL AE AN AM AT BKK BKL BKE BKN BKM BKT

.0730 .210 .00977 .00586 .701 .0215

.00000139

(.000969) (.00206) (.000120) (.000102) (.00254) (.00742)

(.000138)

BLL BLE

.0429 .315 .00414 .00221 .636 .00542

.000551 .0775 .00159

(.000668) (.00318) (.0000818) (.0000676) (.00366) (.0301)

(.0000954) (.0321) (.00124)

Lumber and wood products (.00260) .160 (.00574) .287 (.000203) .00790 (.000377) .0109 (.00760) .534 (.0485) .0308

.00586

(.000372)

Furniture and fixtures .0675

(.000516) (.00725)

.295 .00351 .00262

(.000207) (.000214) (.00739)

.631 .00895

.000150

(.0158)

(.0000738) (.00564)

.0160 .00808

(.00122)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Textile mill products Apparel and other fabricated textile products BLN

.00365

BLM BLT

.0796 .00104

(.000294)

BEE

.0000324

BEN

.0000747

BEM BET

.00163 .000215

(.0000172)

BNN

.00398 .000159

(.0000146)

BMM BMT BTT

.000105 .000402

BNM BNT

.000839

.000202 .0820

.00142 .00136

(.000363) (.00105)

.00170 .000187

Lumber and wood products

(.000668)

.000440

(.0330) (.000831)

.00750 .00146

(.000820)

(.0000618)

.000222

(.00126)

.00378 .000280

(.0000290)

(.000259)

.000206 .000483

(.0000538)

(.0341) (.000889) (.00430)

.00358 .0000121

(.00812) (.0000215)

.00783 .00408

(.0000880)

(.0000308)

Furniture and fixtures

.0000593 .00351

.00808 .00245

(.00109) (.00692)

.00840 .00109

(.00117) (.00426) (.00104) (.00130) (.000540) (.00116) (.0000427) (.0000617) (.000576) (.0000425) (.00279) (.00106) (.00226)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Paper and allied products Printing publishing and allied industries AK AL AE AN AM AT BKK BKL BKE BKN BKM BKT

.116 .266 .0117 .0215 .584 .0327

.00137

(.000498) (.00178) (.0000739) (.000185) (.00193) (.0528)

(.0000712)

BLL BLE

.109 .384 .00478 .00351 .499 .0103

.00116 .360 .00338

(.00192) (.00532) (.000139) (.000119) (.00703) (.0191)

(.000274) (.0742) (.00242)

Chemicals and allied products (.00976) .135 (.00335) .209 (.000788) .0191 (.000440) .0741 (.00386) .562 (.0527) .0321

.00449 .121 .0884

(.000139) (.0249) (.0114)

Petroleum refining (.00299)

.111 .0393 .00311

(.00302) (.000202) (.00378)

.635

(.00172)

.211 .0433

.000913

(.0839)

(.000427) (.0151)

.350 .00722

(.00293)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Paper and allied products Printing publishing and allied industries BLN

.0170

BLM BLT

.373 .000649

(.000254)

BEE

.00955

BEN

.00369

BEM BET

.00249 .000267

(.0000106)

BNN

.0154 .000351

(.0000264)

BMM BMT BTT

.0000269 .00211

BNM BNT

.0134

.000688 .391

.000104 .00183

(.000276) (.00755)

.0153 .000566

(.00300) (.0765) (.00249) (.00186) (.000673) (.00243) (.0000813) (.000332) (.00323) (.000101) (.0791) (.00265) (.00272)

Chemicals and allied products (.0116) .0179 (.0162) .0504 (.00105) .00246 (.0122) .0646 (.00989) .0131 (.00899) .0369 (.000435) .00301 (.00361) .00264 (.00555) .00745 (.000436) .00257 (.00982) .0210 (.000821) .00247 (.00753) .000930

Petroleum refining (.00894)

.145

(.0173)

.198 .00246

(.000377) (.00309)

.0216 .0149 .000549 .000392

(.00106) (.00367) (.0000516) (.00818)

.0746

(.00303)

.0848 .00436

(.000453) (.0155)

.113 .00243 .00343

(.000283) (.0120)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Rubber and miscellaneous plastic products Leather and leather products AK (.00131) (.000770) .104 .0491 AL (.00398) (.00471) .269 .356 AE (.000318) (.000101) .00983 .00451 AN (.000243) (.000114) .0139 .00346 AM (.00486) (.00477) .603 .587 AT (.0585) (.0144) .0320 .00547 BKK BKL BKE BKN BKM (.000187) (.000110) BKT .00284 .0000830 BLL (.0624) .337 (.00622) BLE .0215

Stone, clay, and glass products (.00134) .125 (.00382) .373 (.000368) .0156 (.000379) .0302 (.00497) .456 (.0215) .00767

.00323 .440 .120

(.000192)

Primary metal industries (.000960) .0930 (.00262) .291 (.000158) .0154 (.000277) .0270 (.00285) .574 (.0168) .0135

.00179

(.000137)

(.0698) (.00716)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Rubber and miscellaneous plastic products Leather and leather products BLN (.00568) .0416 (.0685) BLM .400 (.00145) (.000673) BLT .00578 .000452 BEE (.00465) .00750 (.00290) BEN .00228 (.00745) BEM .0313 (.000153) (.0000144) BET .000732 .000110 BNN (.00164) .00516 (.00752) BNM .0490 (.000146) (.0000163) BNT .00116 .000113 BMM (.0759) .481 (.00163) (.000681) BMT .00484 .000592 BTT (.00836) (.00206) .00110 .00128

Stone, clay, and glass products (.00587) .0202 (.0629) .339 (.00186) .0152 (.00393) .0330 (.00208) .00554 (.00711) .0930 (.000193) .00303 (.000614) .000930 (.00450) .0156 (.000171) .000347 (.0558) .262 (.00177) .00933 (.00307) .000437

Primary metal industries

.00348

.000186

.00365

.00151 .00151

(.000375)

(.0000226)

(.0000396)

(.000406) (.00240)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Fabricated metal Machinery except Electrical products electrical machinery AK (.00231) (.00232) (.00220) .0926 .103 .0978 AL (.00569) (.00312) (.00211) .356 .305 .348 AE (.000102) (.000144) (.0000610) .00659 .00441 .00528 AN (.0000956) (.000236) (.000194) .00602 .00622 .00491 AM (.00796) (.00244) (.00210) .538 .581 .544 AT (.0170) (.00916) (.0119) .00781 .00623 .00979 BKK BKL BKE BKN BKM (.000329) (.000331) (.000314) BKT .000890 .000751 .000426 BLL (.0260) (.0843) .441 .240 BLE (.00122) (.00288) .00667 .0156

Motor vehicles and equipment (.00396) .115 (.00176) .205 (.0000720) .00317 (.000107) .00424 (.00383) .673 (.0250) .0127

.000385 .00230 .00350

(.000566) (.00207) (.00151)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Fabricated metal Machinery except Electrical products electrical machinery BLN (.00121) (.00271) .0214 .0131 BLM (.0264) (.0887) .469 .237 BLT (.000898) (.00203) (.000302) .0135 .00176 .00189 BEE (.0000349) (.000690) .000101 .00102 BEN (.0000657) (.000315) .000323 .000864 BEM (.00132) (.00251) .00709 .0155 BET (.0000237) (.0000747) (.00000872) .000297 .00378 .0000817 BNN (.000140) (.000178) .00103 .000716 BNM (.00139) (.00297) .0227 .0130 BNT (.0000236) (.0000723) (.0000277) .000485 .000401 .0000873 BMM (.0270) (.0931) .499 .235 BMT (.00120) (.00212) (.000300) .0151 .00103 .00130 BTT (.00243) (.00130) (.00169) .000675 .000201 .00170

Motor vehicles and equipment (.000680) .00164 (.00289) .00416 (.000258) .00501 (.000935) .00532 (.000314) .00249 (.00167) .00633 (.0000542) .000134 (.000165) .00116 (.000650) .00296 (.0000286) .0000755 (.00379) .00753 (.000553) .00561 (.00357) .00305

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Transportation equipment Instruments Miscellaneous manufacturing industries and ordnance AK (.00138) (.00541) (.00195) .0400 .132 .0955 AL (.00335) (.00607) (.00248) .384 .355 .333 AE (.0000796) (.0000939) (.000177) .00473 .00414 .00502 AN (.000121) (.0000659) (.000130) .00500 .00278 .00567 AM (.00386) (.0106) (.00327) .567 .505 .561 AT (.0217) (.0468) (.0177) .00680 .0179 .0161 BKK BKL BKE BKN BKM (.000197) (.000773) (.000278) BKT .00135 .000224 .000794 BLL (.0643) (.0430) .674 .375 (.00148) (.00350) BLE .000538 .00507

Transportation (.00195)

.178

(.00288)

.444 .00374 .0519

(.0000450) (.00130) (.00365)

.322 .00181

.000354

(.0163)

(.000278)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Transportation equipment Instruments Miscellaneous manufacturing industries and ordinance BLN (.00108) (.00243) .0101 .0317 (.0653) (.0447) BLM .685 .401 (.000478) (.00116) (.00104) BLT .000162 .0140 .00845 BEE (.00000233) (.0000979) .000000429 .0000686 (.0000218) (.000322) BEN .00000802 .000429 (.00150) (.00372) BEM .000546 .00543 (.0000114) (.0000227) (.0000876) BET .0000710 .0000842 .0000444 BNN (.0000346) (.000480) .000150 .00269 (.00112) (.00276) BNM .0102 .0340 (.0000173) (.0000169) (.0000620) BNT .000110 .000199 .000901 BMM (.0664) (.0468) .695 .430 (.000552) (.00171) (.00112) BMT .00133 .0141 .0101 BTT (.00310) (.00668) (.00253) .000433 .000652 .000307

Transportation

.00123

.0000108

.000784

(.000411)

(.00000642)

(.000186)

(.000521)

.00239 .00173

(.00232)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Communications Electric utilities Gas utilities AK (.00296) (.00487) (.00211) .353 .342 .220 AL (.00357) (.00289) (.00363) .364 .241 .167 AE (.000209) (.00473) (.000140) .00844 .104 .00347 AN (.000379) (.00211) (.00264) .00667 .149 .546 AM (.00568) (.00793) (.00340) .268 .164 .0644 AT (.0255) (.0309) (.0398) .0355 .0159 .0188 BKK BKL BKE BKN BKM (.000423) (.000696) (.000302) BKT .00420 .00367 .00599 BLL (.0508) (.0403) .0901 .161 BLE (.0401) (.00164) .00168 .00897

.154 .586 .0129 .0105 .238 .00507

.000916 1.032 .0188

Trade (.00101) (.00698) (.000278) (.000434) (.00738) (.0116)

(.000144) (.217) (.109)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Communications Electric utilities Gas utilities BLN (.0165) (.0122) .00984 .0728 BLM (.0385) (.0322) .0819 .0970 BLT (.000509) (.00162) (.000876) .00306 .00353 .0100 BEE (.0689) (.00118) .115 .00266 BEN (.0315) (.000802) .0355 .00220 BEM (.0581) (.00120) .0816 .00942 BET (.0000298) (.00155) (.0000460) .000134 .00216 .000407 BNN (.000708) (.0144) (.00584) .00108 .0119 .0345 BNM (.000708) (.0240) (.0100) .00108 .0335 .0405 BNT (.0000450) (.000666) (.000413) .0000148 .00592 .000724 BMM (.000708) (.0599) (.0249) .00108 .130 .0660 BMT (.000811) (.00152) (.000739) .00712 .000874 .00516 BTT (.00364) (.00441) (.00568) .00164 .00198 .00173

.0145 .999 .0355 .000343 .000264 .0182 .000708 .000203 .0140 .000833 .966 .0349 .00129

Trade (.0113) (.218) (.00672) (.000457) (.000144) (.0104) (.000343) (.000287) (.0109) (.000354) (.219) (.00676) (.00166)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Finance insurance and Services real estate AK (.00596) (.000468) .260 .103 AL (.00121) (.00368) .254 .538 AE (.000229) (.000168) .00711 .0129 AN (.000166) (.000126) .0115 .00804 AM (.00663) (.00367) .468 .338 AT (.0354) (.0231) .0233 .00244 BKK BKL BKE BKN BKM (.000851) (.0000668) BKT .00215 .00125 BLL BLE

Government enterprises .105 .597 .0149 .0168 .266 .0339

.00567 .435 .0512

(.00403) (.00796) (.000822) (.00100) (.00668) (.0420)

(.000575) (.0174) (.0114)

(table continued on next page)

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Table 14.2 (continued) Table 14.2 Parameter estimates for sectoral models of production and technical change Parameter Industry Finance insurance and Services Government enterprises real estate BLN (.00553) .0290 BLM (.0143) .0213 BLT (.000174) (.000525) (.00152) .00255 .00714 .0173 BEE (.0161) .0602 BEN (.00692) .0341 BEM (.106) .0251 BET (.0000327) (.0000240) (.000679) .000106 .000225 .00346 BNN (.00324) .0194 BNM (.00645) .0142 BNT (.0000237) (.0000180) (.000328) .0000380 .0000313 .00140 BMM (.0103) .0104 BMT (.000947) (.000525) (.00129) .000560 .00859 .0209 BTT (.00504) (.00330) (.00600) .00238 .0000208 .00310

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elasticities of all inputs with respect to the price of capital input are set equal to zero. For thirteen of the thirty-five industries the share elasticities of all inputs with respect to the price of labor input are set equal to zero. For eleven industries all share elasticities with respect to the price of electricity input are set equal to zero. Finally, for ten industries all share elasticities with respect to the price of nonelectrical energy input are set equal to zero. Of the five hundred twenty-five share elasticities for the thirty-five industries included in our study 234 are fitted without constraint and 291 are set equal to zero. Our interpretation of the parameter estimates given in table 14.2 continues with the estimated elasticities of the share of each input with respect to the price of the input itself . Under the necessary and sufficient conditions for concavity of the price function for each sector, these share elasticities are nonpositive. The share of each input is nonincreasing in the price of the input itself. This condition together with the condition that the sum of all the share elasticities with respect to a given input is zero implies that only two of the elasticities of the shares of each input with respect to the prices of the other four inputs can be negative. All ten of these share elasticities can be nonnegative, and this condition holds for eleven of the thirty-five industries included in our study. We have already pointed out that the share elasticities of all inputs with respect to the price of capital input are equal to zero. The share elasticity of labor with respect to the price of electricity { } is nonnegative for twentyseven of the thirty-five industries. By symmetry this parameter can also be interpreted as the share elasticity of electricity with respect to the price of labor. The share elasticity of labor with respect to the price of nonelectrical energy {

} is nonnegative for twenty-one of the thirty-five industries. Finally, the share elasticity of labor with

respect to the price of materials { } is nonnegative for all thirty-five industries. Considering the share elasticities of electricity, nonelectrical energy, and materials inputs, we find that the share elasticity of electricity with respect to the price of nonelectrical energy { } is nonnegative for twenty-eight of the thirty-five industries, while the share elasticity of electricity with respect to the price of materials { } is nonnegative for twenty-four of these industries. Finally, the share elasticity of nonelectrical energy with respect to the price of materials { } is nonnegative for twenty-six of the thirty-five industries.

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14.2.3 Technical Change We continue the interpretation of parameter estimates given in table 14.2 with the estimated biases of technical change with respect to the price of each input . These parameters can be interpreted as the negative of the change in the rate of technical change with respect to the price of each input or, alternatively, as the change in the share of each input with respect to time. The sum of the five biases of technical change with respect to price is equal to zero, so that we can rule out the possibility that the five biases are all negative or all positive. Of the thirty remaining logical possibilities, only fifteen actually occur among the results presented in table 14.2. Of these, only nine occur for more than one industry, and only four occur for more than two industries. It is important to note that the biases of technical change are not affected by the concavity of the price function. All five parameters are fitted for thirty-five industries, subject to the constraint that their sum is equal to zero. We first consider the bias of technical change with respect to the price of capital input. If the estimated value of this parameter is positive, technical change is capital using. Alternatively, the rate of technical change decreases with an increase in the price of capital input. If the estimated value is negative, technical change is capital saving, and the rate of technical change increases with the price of capital. Technical change is capital using for twenty of the thirty-five industries included in our study; it is capital saving for fifteen of these industries. We conclude that the rate of technical change decreases with the price of capital input for twenty industries and increases with this price for fifteen industries. The interpretation of the biases of technical change with respect to the prices of labor, electricity, nonelectrical energy, and materials inputs is analogous to the interpretation of the bias with respect to the price of capital input. If the estimated value of the bias is positive, technical change uses the corresponding input; alternatively, the rate of technical change decreases with an increase in the input price. If the estimated value is negative, technical change saves the corresponding input; alternatively, the rate of technical change decreases with the input price. Considering, the bias of technical change with respect to the price of labor input, we find that technical change is labor using for twenty-six of the thirty-five industries and labor saving for nine of these industries. The rate of technical change decreases

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with the price of labor input for twenty-six industries and increases with this price for nine industries. Considering the bias of technical change with respect to the price of electricity input, we find that technical change is electricity using for twenty-three of the thirty-five industries included in our study and electricity saving for twelve of these industries. The rate of technical change decreases with the price of electricity for twenty-three industries and increases with this price for twelve industries. Turning to the bias of technical change with respect to the price of nonelectrical energy input we find that technical change is nonelectrical energy using for twenty-eight of the thirty-five industries and nonelectrical energy saving for only seven of these industries. We conclude that the rate of technical change increases with the price of nonelectrical energy for twenty-eight industries and decrease with this price for the remaining seven. Finally, technical change is materials using for eight of the thirty-five industries included in our study and materials saving for the other twenty-seven, so that the rate of technical change increases with the price of materials for twenty-seven industries and decreases with this price for the remaining eight. 14.2.4 Patterns of Technical Change A classification of industries by patterns of the biases of technical change is given in table 14.3. The pattern that occurs with the greatest frequency is capital using, labor using, electricity using, nonelectrical energy using and materials saving technical change. This pattern occurs for eight of the thirty-five industries included in our study. For this pattern the rate of technical change decreases with the prices of capital, labor, electricity, and nonelectrical energy inputs and increases with the price of materials input. The pattern that occurs next most frequently is capital saving, labor using, electricity using, nonelectrical energy using, and materials saving technical change. This pattern occurs for five industries. For this pattern the rate of technical change decreases with the prices of labor, electricity, and nonelectrical energy inputs and increases with the prices of capital and materials inputs. These two patterns of technical change differ only in the role of the price of capital input. Our interpretation of the parameter estimates given in table 14.2 concludes the rates of change of the negative of the rate of technical change { }. If the estimated value of this parameter is positive, the

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Table 14.3 Classification of industries by biases of technical change Industries Pattern of Biases Capital using Tobacco, textiles, apparel, lumber and wood, printing and Labor using publishing, fabricated metal, motor vehicles, transportation Electricity using Nonelectrical energy using Materials saving Capital using Eletrical machinery Labor saving Electricity using Nonelectrical energy using Materials using Capital using Metal mining, services Labor using Electricity using Nonelectrical energy saving Materials saving Capital using Nonmetallic mining, miscellaneous manufacturing, Labor using government enterprises Electricity saving Nonelectrical energy using Materials saving Capital using Construction Labor saving Electricity using Nonelectrical energy saving Materials using Capital using Coal mining, trade Labor using Electricity saving Nonelectrical energy saving Materials saving Capital using Agriculture, crude petroleum and natural gas, petroleum

Labor saving refining Electricity saving Nonelectrical energy using Materials saving Food, paper Capital saving Labor using Electricity using Nonelectrical energy using Materials using (table continued on next page)

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Table 14.3 (continued) Table 14.3 Classification of industries by biases of technical change Pattern of Biases Industries Rubber, leather, instruments, gas utilities, finance, Capital saving insurance and real estate Labor using Electricity using Nonelectrical energy using Materials saving Chemicals Capital saving Labor using Electricity saving Nonelectrical energy using Materials using Transportation equipment and ordnance, Capital saving communications Labor saving Electricity using Nonelectrical energy using Materials using Stone, clay and glass, machinery Capital saving Labor using Electricity saving Nonelectrical energy using Materials saving Primary metals Capital saving Labor using Electricity using Nonelectrical energy saving Materials saving Electric utilities Capital saving Labor saving Electricity using Nonelectrical energy using Materials saving Furniture Capital saving Labor saving Electricity saving Nonelectrical energy saving Materials using rate of technical change is decreasing; if the value is negative the rate is increasing. For fifteen of the thirty-five industries included in our study the estimated value is positive and the rate of technical change is decreasing; for twenty industries the fitted value is negative, so that the rate of technical change is increasing.

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14.3 Summary and Conclusion In this chapter we have analyzed the role of electrification in the growth of productivity. For this purpose we have developed and implemented a new econometric model of productivity growth. We have estimated the unknown parameters of this model from data for thirty-five individual industries of the United States for the period 1958 1979. Our econometric model determines the growth of sectoral productivity as a function of the relative prices of sectoral inputs. To capture the impact of electrification we have divided inputs for each sector among capital, labor, electricity, nonelectrical energy, and materials inputs. To represent the impact of the relative prices of sectoral inputs on productivity growth we have defined biases of technical change as the negative of changers in the rate of technical change with respect to proportional changes in input prices. Biases of technical change can also be defined in an alternative and equivalent way as changes in the value shares of each input with respect to time. Biases of technical change can be employed to derive the implications of changes in the relative prices of sectoral inputs on the rate of technical change. They can also be used to derive the implications of patterns of technical change for the distribution of the value of sectoral output among the inputs. 14.3.1 The Role of Electrification We have estimated biases of technical change with respect to prices of capital input, labor input, electricity input, nonelectrical energy input, and materials input. These biases are unknown parameters of the econometric models for the thirty-five industrial sectors included in this study. In order to test the hypothesis advanced by Schurr and Rosenberg about the importance of electrification in productivity growth, we can focus on the bias of technical change with respect to electricity input. If this bias is positive, then technical change is electricity using; if the bias is negative, technical change is electricity saving. If technical change is electricity using, the share of electricity input in the value of output increases with technical change, while the rate of technical change increases with a decrease in the price of electricity.

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We have found that technical change is electricity using for twenty-three of the thirty-five industries included in our study. Our first and most important conclusion is that electrification plays a very important role in productivity growth. A decline in the price of electricity stimulates technical change in twenty-three of the thirty-five industries and dampens productivity growth in only twelve. Alternatively and equivalently, we can say that technical change results in an increase in the share of electricity input in the value of output, holding the relative prices of all inputs constant, in twenty-three of the thirty-five industries included in our study. Technical change results in a decrease in the share of electricity input in only twelve of these industries. Our empirical results provide strong confirmation for the hypothesis advanced by Schurr and Rosenberg about the interrelationship of electrification and productivity growth in a wide range of industries. Schurr and his associates in the study Energy in America's Future have shown that the price of electricity fell in real terms through 1971. This decline in real electricity prices has promoted electrification through the substitution of electricity for other forms of energy and through the substitution of energy for other inputs, especially for labor input. In addition, the decline in the real price of electricity has stimulated the growth of productivity in a wide range of industries. The spread of electrification and the rapid growth of productivity through the early 1970s are both associated with a decline in real electricity prices. This decline was made possible in part by advances in the thermal efficiency of electricity generation. Beginning in the early 1970s the downward trend in the real price of electricity was reversed. The reversal in the trend of real electricity prices has been associated with a marked slowdown in advances in the thermal efficiency of electricity generation that can be traced back to the late 1960s. However, the reduction in the rate of technical change in the electric generating industry is only part of the explanation of the reversal in the trend of real electricity prices. In addition, prices of primary energy sources employed in electricity generation have risen sharply in the aftermath of the oil price shocks of 1973 and 1979. Rising electricity prices have slowed the growth of productivity in U.S. industries throughout the 1970s. These price increases have an important role to play in the explanation of the slowdown of U.S. productivity growth since 1973. In linking electrification and productivity growth, Schurr has advanced an important subsidiary hypothesis. This hypothesis is that

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electrification is especially significant in stimulating the growth of productivity in the manufacturing industries. Schurr's hypothesis is supported by the fact that technical change is electricity using in fifteen of the twenty-one manufacturing industries included in our study, while technical change is electricity using in only eight of the fourteen nonmanufacturing industries. Schurr's explanation for this phenomenon is that electrification of industrial processes led to much greater flexibility in the application of energy. Rosenberg's examples of the importance of electrificationiron and steel, glass, and aluminum productionare also drawn from manufacturing. Rosenberg has advanced a second subsidiary hypothesis in analyzing the link between electrification and productivity growth. This hypothesis is that electricity using technical change is the ''other side of the coin" of labor saving technical change. We have been unable to find support for this hypothesis in our empirical results. In fact, technical change is labor saving for only nine of the thirty-five industries included in our study and labor using for the remaining twenty-six of these industries. However, we have pointed out that the sum of biases of technical change for all five inputs must be equal to zero. The predominance of technical change that uses electricity must be balanced by technical change that saves other inputs. We have found that technical change is materials saving for twenty-seven of the thirty-five industries included in our study and materials using for the remaining eight. For all other inputs, including labor and electricity, technical change is predominantly input using. We conclude that technical change that uses electricity input and inputs of capital, labor, and nonelectrical energy is balanced by technical change that saves materials. 14.3.2 Utilization of Nonelectrical Energy We have found that electrification plays an important role in productivity growth and we have also examined the utilization of nonelectrical energy. Our findings are that technical change is nonelectrical energy using for twentyeight of the thirty-five industries included in our study and energy saving for seven of these industries. Our second conclusion is that greater utilization of nonelectrical energy plays an even more significant role in productivity growth than electrification. A decline in the price of nonelectrical energy stimulates technical change in twentyeight of the thirty-five industries and dampens

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productivity growth in only seven. Alternatively, we can say that technical change results in an increase in the share of nonelectrical energy input in the value of output in twenty-eight of the thirty-five industries and results in a decrease in the nonelectrical energy input share in only seven. Again considering the evidence on energy price developments presented by Schurr and his associates, we find that the price of nonelectrical energy has fallen in real terms through the early 1970s, reaching a minimum for natural gas and fuel oil in 1970 and for gasoline in 1972. This decline in nonelectrical energy prices in real terms has promoted greater utilization of nonelectrical energy through the substitution of these forms of energy for capital, labor, and materials inputs. In addition, the decline in the real price of nonelectrical energy, like the decline in electricity prices we have examined earlier, has stimulated the growth of productivity in a wide range of industries. We conclude that the greater utilization of nonelectrical energy in relation to other inputs such as labor and the rapid growth of productivity through the early 1970s are associated with the decline in the real price of nonelectrical energy. Beginning in the early 1970s the downward trend in the real price of nonelectrical energy has been reversed and the increase in utilization of nonelectrical energy relative to other inputs in U.S. industries has slowed dramatically. The reversal in the trend of nonelectrical energy prices, as well as an important part of the reversal in the trend of electricity prices we have examined above, has been associated with the oil price shocks of 1973 and 1979. Rising prices of nonelectrical energy have reinforced the negative impact of rising prices of electricity on the growth of productivity throughout the 1970s. Increases in the real prices of both electricity and nonelectrical energy have a role to play in the explanation of the slowdown in U.S. productivity growth since 1973. In linking greater utilization of nonelectrical energy and productivity growth, Schurr and his associates in the study Energy in America's Future have advanced an important subsidiary hypothesis. This hypothesis is that greater utilization of fluid forms of energy has enhanced productivity in agriculture, transportation, and manufacturing. We find that technical change is nonelectrical energy using in agriculture and transportation, as suggested by Schurr and his associates. We also find that technical change is nonelectrical energy using

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for nineteen of the twenty-one manufacturing industries included in our study. Technical change is nonelectrical energy using for only seven of the twelve industries other than agriculture, manufacturing, and transportation. We conclude that greater utilization of nonelectrical energy has a significant role in productivity growth for an even wider range of industries than electrification. 14.3.3 Conclusion We have now completed our analysis of the role of electrification and the utilization of nonelectrical energy in productivity growth. For this purpose we have employed an econometric model of production and technical change. Within the framework provided by our model we can offer a tentative explanation of the disparate trends in energy intensity and productivity growth. These trends first drew the attention of Schurr and his associates to the special role of electrification. Over the period 1920 1953 energy intensity of production was falling while productivity was rising. While the fall in real prices of electricity and nonelectrical energy resulted in the substitution of energy inputs for other inputs, especially for labor input, these price trends also generated sufficient growth in output per unit of energy input that the energy intensity of production actually fell. This explanation is completely in accord with the explanation advanced by Schurr and his associates. Between 1953 and 1973 energy intensity was stable, while productivity continued to grow. During this period real energy prices continued to fall, but at slower rates than during the period 1920 1953. As before, the fall in real prices of electricity and nonelectrical energy resulted in the substitution of energy inputs for other inputs. But this was almost precisely offset by the growth in output per unit of energy input, leaving the energy intensity of production unchanged. Finally, real energy prices began to rise in the early 1970s, increasing dramatically after the first oil shock in 1973 and again after the second oil shock in 1979. These price trends resulted in the substitution of capital, labor, and materials inputs for inputs of electricity and nonelectrical energy, thereby reducing energy intensity of production. At the same time, the energy price trends contributed to a marked slowdown in productivity growth. Although much research remains to be done before we obtain a complete understanding of the role of energy utilization in productiv-

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ity growth, it is important to emphasize that we have made important progress toward this goal. We have analyzed the character of technical change in a wide range of industries covering the whole of the U.S. economy. We have confronted hypotheses advanced in earlier research by Schurr and his associates and by Rosenberg with new empirical evidence. We have found support for the hypothesis that electrification and productivity growth are interrelated. Somewhat surprisingly, we have found that the utilization of nonelectrical energy and productivity growth are even more strongly interrelated. Finally, we have identified new research objectives that will make it possible to obtain a deeper understanding of the interrelationship between energy utilization and productivity change. Given the support for the hypothesis that technical change is electricity using and nonelectrical energy using, we can assess the potential for electrification and greater utilization of nonelectrical energy in reviving the growth of productivity at the level of individual industries in the United States. Schurr (1983) has summarized this potential as follows: If this line of theorizing is correct, one of the keys to reconciling the future growth of energy productivity and labor and total factor productivity would be (a) through the vigorous pursuit of these energy supply technologies which assure the renewed future availability, on favorable terms, of those energy forms which possess the highly desirable flexibility features that have characterized liquid fuels and electricity, and (b) through the search for counterpart energy consumption technologies that can put these characteristics to efficient use in industrial, commercial, and household application. 14.4 Appendix The development of our econometric model of production and technical change proceeds through two stages. We first specify a functional form for the sector price functions, say {Pi}, taking into account restrictions on the parameters implied by the theory of production. Secondly, we formulate an error structure for the econometric model and discuss procedures for estimation of the unknown parameters. Our first step in formulating an econometric model of production and technical change is to consider specific forms for the sectoral price function {Pi}

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For these price functions, the prices of outputs are transcendental or, more specifically, exponential functions of the logarithms of the prices of capital (K), labor (L), electricity (E), nonelectrical energy (N), and materials (M) inputs. We refer to these forms as transcendental logarithmic price functions or, more simply, translog price functions, indicating the role of the variables that enter into the price functions. 14.4.1 Homogeneity and Symmetry The price functions {Pi} are homogeneous of degree one in the input prices. The translog price function for an industrial sector is characterized by homogeneity of degree one if and only if the parameters for that sector satisfy the conditions

For each sector the value shares of capital, labor, electricity, nonelectrical energy, and materials inputs, say can be expressed in terms of logarithmic derivatives of the sectoral

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price functions with respect to the logarithms of price of the corresponding input

Finally, for each sector the rate of technical change, say , can be expressed as the negative of the rate of growth of the price of sectoral output with respect to time, holding the prices of capital, labor, electricity, nonelectrical energy, and materials inputs constant. The negative of the rate of technical change takes the following form

Given the sectoral price functions {Pi}, we can define the share elasticities with respect to price 8 as the derivatives of the value shares with respect to the logarithms of the prices of capital, labor, electricity, nonelectrical energy, and materials inputs. For the translog price functions the share elasticities with respect to price are constant. We can also characterize these forms of constant share elasticity or CSE price functions indicating the interpretation of the fixed parameters that enter the price functions. The share elasticities with respect to price are symmetric, so that the parameters satisfy the conditions

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Similarly, given the sectoral price functions {Pi}, we can define the biases of technical change with respect to price as derivatives of the value shares with respect to time. 9 Alternatively, we can define the biases of technical change with respect to price in terms of the derivatives of the rate of technical change with respect to the logarithms of the price of capital, labor, electricity, nonelectrical energy, and materials inputs. Those two definitions of biases of technical change are equivalent. For the translog price functions the biases of technical change with respect to price are constant; these parameters are symmetric and satisfy the conditions

Finally, we can define the rate of change of the negative of the rate of technical change (i = 1, 2, . . . , n) as the derivative of the negative of the rate of technical change with respect to time.10 For the translog price functions these rates of change are constant. 14.4.2 Concavity Our next step in considering specific forms of the sectoral price functions {Pi} is to derive restrictions on the parameters implied by the fact that the price functions are increasing in all five input prices and the concave in the five input prices. First, since the price functions are increasing in each of the five input prices, the value shares are nonnegative

Under homogeneity these value shares sum to unity

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Concavity of the sectoral price functions {Pi} implies that the matrices of second-order partial derivatives {Hi} are negative semi-definite. This implies, in turn, that the matrices {Ui + vivi' Vi} are negative semi-definite: 11

and {Ui} are matrices of constant share elasticities, defined above. Without violating the nonnegativity restrictions on value shares we can set the matrices {vivi' Vi} equal to zero, for example, by choosing the value shares

Necessary conditions for the matrices {Ui + vivi' Vi} to be negative semi-definite are that the matrices of share elasticities {Ui} must be negative semi-definite. These conditions are also sufficient, since the matrices {vivi' Vi} are negative semi-definite for all nonnegative value shares summing to unity and the sum of two negative semidefinite matrices is negative semi-definite.

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To impose concavity on the translog price functions the matrices {Ui} of constant share elasticities can be represented in terms of their Cholesky factorizations:

Under constant returns to scale the constant share elasticities satisfy symmetry restrictions and restrictions implied by homogeneity of degree one of the price function. These restrictions imply that the parameters of the Cholesky factorizations must satisfy the following conditions:

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Under these conditions there is a one-to-one transformation between the constant share elasticities and the parameters of the Cholesky factorizations. The matrices of share elasticities are negative semi-definite if and only if the diagonal elements of the matrices {Di} are nonpositive. This completes the specification of our model of production and technical change. 14.4.3 Index Numbers The negative of the average rates of technical change in any two points of time, say T and T 1, can be expressed as the difference between successive logarithms of the price of output, less a weighted average of the differences between successive logarithms of the prices of capital, labor, electricity, nonelectrical energy, and materials inputs, with weights given by the average value shares

and the average value shares in the two periods are given by

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We refer to the expressions for the average rates of technical change { sectoral rates of technical change.

} as the translog price index of the

Similarly, we can consider specific forms for prices of capital, labor, electricity, nonelectrical energy, and materials inputs as functions of prices of individual capital, labor, electricity, nonelectrical energy, and materials inputs into each industrial sector. We assume that the price of each input can be expressed as a translog function of the price of its components. Accordingly, the difference between successive logarithms of the price of the input is a weighted average of differences between successive logarithms of prices of its components. The weights are given by the average value shares of the components. We refer to these expressions of the input prices as translog indexes of the price of sectoral inputs. 12 14.4.4 Stochastic Specification To formulate an econometric model of production and technical change we add a stochastic component to the equations for the value shares and the rate of technical change. We assume that each of these equations has two additive components. The first is a nonrandom function of capital, labor, electricity, nonelectrical energy, and materials inputs and time; the second is an unobservable random disturbance that is functionally independent of these variables. We obtain an econometric model of production and technical change corresponding to the translog price function by adding random disturbances to all six equations

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unknown parameters and

unobservable random disturbances.

are

Since the value shares sum to unity, the unknown parameters satisfy the same restrictions as before and the random disturbances corresponding to the four value shares sum to zero

so that these random disturbances are not distributed independently. We assume that the random disturbances for all six equations have expected value equal to zero for all observations

We also assume that the random disturbances have a covariance

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matrix that is the same for all observations; since the random disturbances corresponding to the five value shares sum to zero, this matrix is positive semi-definite with rank at most equal to five. We assume that the covariance matrix of the random disturbances corresponding to the first four value shares and the rate of technical change, say Si, has rank five, where

so that the Si is a positive definite matrix. Finally, we assume that the random disturbances corresponding to distinct observations in the same or distinct equations are uncorrelated. Under this assumption that the matrix of random disturbances for the first four value shares and the rate of technical change for all observations has the Kronecker product form,

Since the rates of technical change can be written

where

are not directly observable, the equation for the rate of technical change

is the average disturbance in the two periods

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Similarly, the equations for the value shares of capital, labor, electricity, nonelectrical energy, and materials inputs can be written

As before, the average value shares sum to zero

sum to unity, so that the average disturbances

The covariance matrix of the average disturbances corresponding to the equation for the rate of technical change for all observations, say W, is a Laurent matrix

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The covariance matrix of the average disturbance corresponding to each equation for the four value shares is the same, so that the covariance matrix of the average disturbances for the first four value shares and the rate of technical change for all observations has the Kronecker product form

14.4.5 Estimation Although disturbances in equations for the average rate of technical change and the average value shares are autocorrelated, the data can be transformed to eliminate the autocorrelation. The matrix W is positive definite, so that there is a matrix T such that

To construct the matrix T we can first invert the matrix W to obtain the

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inverse matrix W 1, a positive definite matrix. We then calculate the Cholesky factorization of the inverse matrix W 1,

where L is a unit lower triangular matrix and D is a diagonal matrix with positive elements along the main diagonal. Finally, we can write the matrix T in the form

where D1/2 is a diagonal matrix with elements along the main diagonal equal to the square roots of the corresponding elements of D. We can transform the equations for the average rates of technical change by the matrix T = D1/2 L' to obtain equations with uncorrelated random disturbances

since

The transformation T = D1/2 L' is applied to data on the average rates of technical change { } and data on the average values of the variables that appear on the right-hand side of the corresponding equation. We can apply the transformation T = D1/2 L' to the first four equations for average value shares to obtain equations with uncorrelated disturbances. As before, the transformation is applied to data on the average value shares and the average values of variables that appear in the corresponding equations. The covariance matrix of the transformed disturbances from the first four equations for the average value shares and the equation for the average rate of technical change has the Kronecker product form

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To estimate the unknown parameters of the translog price function we combine the first four equations for the average value shares with the equation for the average rate of technical change to obtain a complete econometric model of production and technical change. We estimate the parameters of the equations for the remaining average value shares, using the restrictions on these parameters given above. The complete model involves twenty unknown parameters. A total of twenty-two additional parameters can be estimated as functions of these parameters, given the restrictions. Our estimates of the unknown parameters of the econometric model of production and technical change will be based on the nonlinear three-stage least-squares estimator introduced by Jorgenson and Laffont (1974). Notes 1. This summary is based on Schurr (1983). 2. This summary of the sources of postwar economic growth in the United States is based on Jorgenson (1983a). 3. The price function was introduced by Samuelson (1953 1954). 4. Our sectoral price functions are based on the translog price function introduced by Christensen, Jorgenson, and Lau (1971, 1973). The translog price function was first employed at the sectoral level by Berndt and Jorgenson (1973) and Berndt and Wood (1975). References to sectoral production studies incorporating energy and materials inputs are given by Berndt and Wood (1979). 5. This model of sectoral productivity growth is based on that of Jorgenson (1986). A useful survey of studies of energy prices and productivity growth is given by Berndt (1982). 6. These industries have been employed by Jorgenson and Fraumeni (1983). 7. Data on energy and materials are based on annual inter-industry transactions tables for the United States, 1958 1974, compiled by Jack Faucett Associates (1977). Data on capital and labor input are based on estimates by Fraumeni and Jorgenson (1983). 8. The share elasticity with respect to price was introduced by Christensen, Jorgenson, and Lau (1971, 1973) as a fixed parameter of the translog production function. An analogous concept was employed by Samuelson (1973). The terminology is from Jorgenson (1986). 9. The bias of productivity growth was introduced by Hicks (1932). An alternative definition of the bias of productivity growth was introduced by Binswanger (1974a,b). The definition of bias of productivity growth to be employed in our econometric model is based on that of Jorgenson (1986). 10. The rate of change was introduced by Jorgenson (1986). 11. The following discussion of share elasticities with respect to price and concavity follows that of Jorgenson (1986). Representation of conditions for concavity in terms of the Cholesky factorization is due to Lau (1978a). 12. The price indexes were introduced by Fisher (1922). These indexes were first derived from the translog price function by Diewert (1976). The corresponding index of technical change was introduced by Christensen and Jorgenson (1970). The translog index of technical change was first derived from the translog price function by Diewert (1980) and by Jorgenson (1986).

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15 Bilateral Models of Production for Japanese and U.S. Industries Dale W. Jorgenson, Masahiro Kuroda, Hikaru Sakuramoto, and Kanji Yoshioka 15.1 Introduction The purpose of this chapter is to present bilateral models of production for twenty-eight Japanese and U.S. industries for the period 1960 1979. We treat data on production for the two countries as separate sets of observations. However, we assume that these observations are generated by an econometric model with common parameters. This model determines the distribution of the value of output among capital, labor, and intermediate inputs in each country. It also determines rates of technical change for both countries and the difference between the level of technology in the two countries. Our methodology is based on the economic theory of production. The underlying model of production is the same as that employed in a companion paper by Jorgenson and Kuroda (1990). We utilize this model in generating an econometric model of producer behavior for individual industries. Jorgenson and Kuroda employ the model to generate index numbers of productivity growth and differences in productivity between Japan and the United States. We use these indices as data, together with prices and quantities of output and inputs for each industry in modeling production. Our models of production are based on the bilateral translog model introduced by Jorgenson and Nishimizu (1978). The point of departure for these models is a production function for each industry, giving output as a function of capital, labor, and intermediate inputs, a dummy variable equal to zero for the United States and one for Japan, and a time trend. The dummy variable allows for productivity differences between the two countries, while the time trend permits technology in each country to change from period to period. In analyzing differences in each industry's production patterns between the two countries, we combine the production function with

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necessary conditions for producer equilibrium. We express these conditions as equalities between shares of input in the value of output and the elasticity of output with respect to that input. The elasticities depend on input levels, the dummy variables for each country, and time. For given input intensities and given levels of technology, we find that U.S. industries have higher rates of labor renumeration than the corresponding Japanese industries. Japanese industries have higher rates of renumeration for capital and intermediate inputs. Technical change is predominantly capital saving, labor saving, and intermediate input using in both countries. An important focus for our bilateral models of production is the difference between rates of technical change in Japanese and U.S. industries. For six of the twenty-eight industries, we find that rates of technical change are higher in the United States than in Japan at given relative input intensities. Rates of technical change are higher in Japan for the remaining twenty-two industries. An alternative and equivalent interpretation of these results can be given in terms of the difference in technology between the two countries or the ''technology gap." The technology gap between Japan and the United States is increasing for twenty-two of the industries included in our study and decreasing for only six industries. For industries where the United States has an advantage, the gap is closing; for industries where Japan has an advantage, the gap is widening. Our industry classification is based on that of Jorgenson and Kuroda (1990). The Japanese industries are classified among thirty-one industries, while the U.S. industries are classified among thirty-five industries. In estimating our bilateral production model, we have consolidated the two classifications to twenty-eight industries. In section 15.2, we provide a theoretical framework for our bilateral models of production. Section 15.3 outlines the empirical results and section 15.4 provides a brief summary and conclusion. We present additional details on the constraints that must be satisfied by the parameters of our econometric models in order to meet the requirements imposed by the theory of production presented below in appendix A. The detailed empirical results are presented in appendix B. 15.2 Theoretical Framework We treat data on production patterns for Japan and the United States as separate sets of observations. We assume that these observations are generated by an econometric model with common parameters.

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We describe the implications of the theory of production in terms of a bilateral production function for each sector. These functions are homogeneous of degree one, nondecreasing, and concave in capital, labor, and intermediate inputs. In representing our bilateral models of production we employ the same notation as Jorgenson and Kuroda (1990). To characterize producer behavior in greater detail we introduce share elasticities with respect to quantity, 1 defined as derivatives of the vectors of value shares (vi) with respect to vectors of logarithms of the inputs (ln Xi)

where ln Zi (i = 1, 2, . . . , I) is the logarithm of output in the ith sector. For the translog production functions the share elasticities { } are constant. We can also characterize these production functions as constant share elasticity (CSE) production functions, indicating the role of fixed parameters.2 If a share elasticity is positive, the value share increases with the input. If a share elasticity is negative, the value share decreases with the input. Finally, if a share elasticity is zero, the value share is independent of the input. Continuing with a detailed characterization of producer behavior we define biases of technical change with respect to quantity as derivatives of the value shares with respect to time T:3

Alternatively, we can define these biases as derivatives of the rates of technical change ( ) with respect to logarithms of the inputs. These two definitions are equivalent. For translog production functions the biases of technical change ( ) are constant. If a bias of technical change is positive, the corresponding value share increases with technology; we say that technical change is input-using. If a bias is negative, the value share decreases with technology and technical change is input-saving. Finally, if a bias is zero, the value share is independent of technology and we say that technical change is neutral. Alternatively, biases of technical change contain the implications of changes in inputs for the rate of technical change. If a bias is positive, the rate of technical change increases with the corresponding input. If a bias is negative, the rate of technical change decreases

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with the input. Finally, if a bias is zero, the rate is independent of the input. Similarly, we define biases of the difference in technology with respect to quantity as derivatives of the value shares with respect to the dummy variable D, 4 equal to zero for the United States and one for Japan

Alternatively, we can define these biases as derivatives of the differences in technology ( ) between Japan and the United States with respect to logarithms of the inputs. These two definitions are equivalent. For the translog production functions the biases of differences in technology ( ) are constant. If a bias of the difference in technology is positive, the corresponding value share increases between the United States and Japan and we say that the difference in technology is input-using. If a bias is negative, the value share decreases between the United States and Japan and the difference in technology is input-saving. Finally, if a bias is zero, the value share is the same in the United States and Japan and we say that the difference in technology is neutral. Alternatively, the vectors of biases of differences in technology contain the implications of changes in inputs for the difference in technology between the United States and Japan. If a bias of the difference in technology is positive, the difference in technology increases with the input. If a bias is negative, the difference in technology decreases with the input. Finally, if a bias is zero, the difference in technology is independent of the input. Finally, we can define the biases of technical change with respect to the difference in technology between Japan and the United States as the derivatives of the rates of technical change ( ) with respect to the dummy variable5

Alternatively, we can define these biases as the derivatives of the differences in technology ( ) with respect to technology. The two definitions are equivalent. For the translog production functions the biases ( technology increases with tech-

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nology; correspondingly, the rate of technical change increases between the United States and Japan. If the bias is negative, the difference in technology decreases with technology; correspondingly, the rate of technical change decreases between the United States and Japan. Finally, if the bias is zero the difference in technology is independent of technology and the rate of technical change is the same for the United States and Japan. To complete the description of technical change we can define the acceleration of technical change as the derivative of the rate of technical change with respect to technology

If the acceleration is positive, negative, or zero, the rate of technical change is increasing, decreasing, or independent of the level of technology. Similarly, we can define the difference of the difference in technology as the derivative of the difference in technology between Japan and the United States with respect to the dummy variable 6

If this difference is positive, negative, or zero, the difference in technology is increasing, decreasing, or independent, respectively, of the dummy variable. For the translog production functions both the accelerations ( ) and the differences ( ) are constant. This completes the detailed characterization of producer behavior in terms of the parameters of our bilateral translog models of production. To estimate the unknown parameters of the bilateral translog production function we combine the first two equations for the average value shares in Japan and the United States, the equations for the average rates of technical change in the two countries, and the equation for the average difference in technology to obtain a complete econometric model of production. We estimate the parameters of the equations for the remaining average value shares in the two countries, using the restrictions on these parameters given in appendix A. The complete model involves fourteen unknown parameters. A total of sixteen additional parameters can be estimated as functions of these parameters, given the restrictions. Our estimates of the unknown

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parameters of the econometric model of production is based on the nonlinear three-stage least-squares estimator introduced by Jorgenson and Laffont (1974). 15.3 Empirical Results To implement the bilateral econometric models of production developed in section 15.2, we employ a database for twenty-eight United States and Japanese industrial sectors compiled by Jorgenson, Kuroda and Nishimizu (1987). For each sector they have assembled data on the value shares of capital, labor, and intermediate inputs, for both countries, annually, for the period 1960 1979. They have also compiled quantity indices of output and all three inputs for both countries for the same period. Finally, they have developed translog indexes of technical change for both countries and a translog index of the difference in technology between the two countries. There are nineteen observations for each country, since two-period averages of all data are employed. The parameters can be interpreted as average value shares of capital input, labor input, and intermediate input, respectively, for the corresponding industrial sector in Japan and the United States. Similarly, the parameters are averages of rates of technical change and the parameters are averages of differences in technology between the two countries. The parameters can be interpreted as constant share elasticities with respect to quantity for the corresponding sector in Japan and the United States. Similarly, the parameters are constant biases of technical change with respect to quantity for the corresponding sector in the two countries and the parameters are constant biases of differences in technology between the two countries. Finally, the parameters are constant accelerations of technical change in Japan and the United States, the parameters { ) are constant biases of technical change with respect to the difference in technology between the two countries, and the parameters ( ) are constant differences in the difference in technology. In estimating the parameters of our bilateral models of production, we retain the average value shares, the average rate of technical change, and the average difference in technology between the two countries as parameters to be estimated for all twenty-eight industrial

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sectors. Similarly, we estimate the biases of technical change, the biases of differences in technology, and the biases of technical change with respect to the difference in technology. Finally, we estimate the accelerations of technical change and the differences in the difference in technology for all twenty-eight sectors. Estimates of the share elasticities with respect to quantity are obtained under the restrictions implied by the necessary and sufficient conditions for concavity of the bilateral production function described in appendix A below. Under these restrictions the matrices of constant share elasticities must be nonpositive definite for all industries. To impose the concavity restrictions, we represent the matrices of constant share elasticities for all sectors in terms of their Cholesky factorizations. The necessary and sufficient conditions are that the diagonal elements of the matrices (Di) that appear in the Cholesky factorizations must be nonpositive. The estimates presented below in appendix B incorporate these restrictions for all twenty-eight industries. Our interpretation of the parameter estimates reported in appendix B begins with an analysis of the estimates of the parameters The average value shares are nonnegative for all twenty-eight industries included in our study. The estimated average rates of technical change are positive in nineteen sectors and negative in nine sectors. The estimated average differences in technology between the United States and Japan are positive in seventeen sectors and negative in eleven. For given input levels, differences in technology favor Japan in seventeen of the twenty-eight industries. The industries with positive and negative estimates of these parameters are listed in table 15.1. The estimated share elasticities with respect to quantity describe the implications of patterns of substitution among capital, labor, and intermediate inputs for the relative distribution of the value of output among these three inputs. Positive share elasticities imply that the value shares increase with the quantity of the corresponding input; negative share elasticities imply that the value shares decrease with the input; share elasticities equal to zero imply that the value shares are independent of the input. It is important to keep in mind that we have fitted these parameters subject to the restrictions implied by concavity of the bilateral production functions. These restrictions require that all share elasticities be set equal to zero for six of the twenty-eight industriesconstruction, food processing, stone, clay, and glass, machinery, transportation equipment, and precision instruments.

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Table 15.1 Rates of technical change and differences in technology Rates of technical change aT > 0 aT < 0 (1) Agriculture, forestry & fisheries (2) Mining (4) Food & kindred products (3) Construction (5) Textile mill products (7) Lumber & wood products (except furniture) (6) Apparel & other fabricated textile products (12) Petroleum refinery (8) Furniture & fixtures (15) Stone, clay & glass products (9) Paper & allied products (16) Iron & steel (10) Printing & publishing (23) Miscellaneous manufacturing (11) Chemical (25) Electric utility, gas supply, & water supply (13) Rubber & miscellaneous plastic (28) Service (14) Leather (17) Fabricated metal products (18) Machinery (19) Electric machinery (20) Motor vehicles & equipment (21) Transportation equipment (except motor) (22) Precision instruments (24) Transportation & communication (26) Wholesale & retail trade (27) Finance & insurance Differences in technology aD > 0 aD < 0 (3) Construction (1) Agriculture, forestry & fisheries (4) Food & kindred products (2) Mining (5) Textile mill products (6) Apparel & other fabricated textile products (7) Lumber & wood products (except furniture) (9) Paper & allied products (8) Furniture & fixtures (12) Petroleum refinery (10) Printing & publishing (14) Leather & leather products (11) Chemical & allied products (15) Stone, clay & glass (13) Rubber & miscellaneous plastic products (16) Iron & steel (17) Fabricated metal (18) Machinery (20) Motor vehicles & equipment (19) Electric machinery (21) Transportation equipment (except motor) (22) Precision instruments (23) Miscellaneous manufacturing (24) Transportation & communication (table continued on next page)

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Table 15.1 (continued) Table 15.1 Rates of technical change and differences in technology Differences in technology aD > 0 (25) Electric utility, gas supply & water supply (26) Wholesale & retail trade (27) Finance & insurance (28) Service a The detailed appendix from which this table is drawn is available from the authors

aD < 0

Our interpretation of the parameter estimates given in appendix B continues with the estimated elasticities of the share of each input with respect to the quantity of the input itself Under the necessary and sufficient conditions for concavity of the bilateral production functions, these share elasticities are nonpositive. The share of each input is nonincreasing in the quantity of the input itself. This condition together with the condition that the sum of all the share elasticities with respect to a given input is equal to zero implies that only one of the elasticities of the share of each input with respect to the quantities of the other two inputs can be negative. All three of these share elasticities can be nonnegative, and this condition holds for seventeen of the twenty-eight industries. The share elasticity of capital with respect to the quantity of labor is nonnegative for all twenty-eight industries. By symmetry this parameter can also be interpreted as the share elasticity of labor with respect to the quantity of capital. This share elasticity is positive for the fifteen industries listed in table 15.2 and zero for the remaining thirteen industries. The share elasticity of capital with respect to the quantity of intermediate input is negative for the five industries listed in table 15.2, zero for thirteen industries and positive for ten industries. This parameter can also be interpreted as the share elasticity of intermediate input with respect to the quantity of capital. Finally, the share elasticity of labor with respect to the quantity of intermediate input is negative for the six industries listed in table 15.2, zero for six industries, and positive for sixteen industries. This parameter can also be interpreted as the share elasticity of intermediate input with respect to the quantity of labor.

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Table 15.2 Share elasticities Capital-Labor bKL < 0 bKL = 0 (3) Construction (4) Foods (5) Textiles (11) Chemical (14) Leather (15) Stone, clay (16) Iron & steel (18) Machinery (21) Transportation equipment (22) Precision instruments (23) Miscellaneous manufacturing (25) Utilities (27) Finance

bKL > 0 (1) Agriculture (2) Mining (6) Apparel (7) Lumber (8) Furniture (9) Paper (10) Printing (12) Petroleum (13) Rubber (17) Fabricated metal (19) Electric machinery (20) Motor vehicles (24) Transportation & communication (26) Trade (30) Other services

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Table 15.2 (continued) Table 15.2 Share elasticities Capital-Intermediate bKM < 0 (8) Furniture (13) Rubber (19) Electric machinery (24) Transportation & communication (30) Other services

bKM = 0 (3) Construction (4) Foods (5) Textiles (11) Chemical (14) Leather (15) Stone, clay (16) Iron & steel (18) Machinery (21) Transportation equipment (22) Precision instruments (23) Miscellaneous manufacturing (25) Utilities (27) Finance

bKM > 0 (1) Agriculture (2) Mining (6) Apparel (7) Lumber (9) Paper (10) Printing (12) Petroleum (17) Fabricated metal (20) Motor vehicles (26) Trade

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Table 15.2 (continued) Table 15.2 Share elasticities Labor-Intermediate bLM < 0 (1) Agriculture (2) Mining (6) Apparel (7) Lumber (20) Motor vehicles (26) Trade

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bLM > 0 (5) Textiles (8) Furniture (9) Paper (10) Printing (11) Chemical (12) Petroleum (13) Rubber (14) Leather (16) Iron & steel (17) Fabricated metal (19) Electric machinery (23) Miscellaneous manufacturing (24) Transportation & communication (25) Utilities (27) Finance (30) Other services

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We continue the interpretation of parameter estimates given in appendix B with the estimated biases of technical change with respect to the quantity of each input . The estimated biases describe the implications of technical change for the relative distribution of the value of output among capital, labor, and intermediate inputs. Alternatively, they give the implications of patterns of substitution among these three inputs for the rate of technical change. Positive biases imply that the value shares increase with the level of technology; negative biases imply that the value shares decrease with technology. If a bias is positive, we say that technical change uses the corresponding input; if a bias is negative, we say that technical change saves the input. Input-using change implies that the rate of technical change increases with the quantity of the corresponding input, while input-saving change implies that this rate decreases with the input. The sum of the three biases of technical change with respect to quantity is equal to zero, so that we can rule out the possibility that the three biases are either all negative or all positive. Of the six remaining logical possibilities, only capital-saving and intermediate input-saving and labor-using technical change fails to occur among the results for individual industries presented in table 15.3. The biases of technical change are not affected by the concavity restrictions on the bilateral production functions, so that all three parameters are fitted for each of the twenty-eight industries included in our study. We first consider the bias of technical change with respect to the quantity of capital input. If the estimated value of this parameter is positive, technical change is capital-using. Alternatively, the rate of technical change increases with an increase in the quantity of capital input. If the estimated value is negative, technical change in capitalsaving and the rate of technical change decreases with the quantity of capital input. Technical change is capitalusing for twelve of the twenty-eight industries included in our study and capital-saving for the remaining sixteen. The interpretation of biases of technical change with respect to the quantities of labor and intermediate inputs is analogous to the interpretation of the bias for capital input. Technical change is labor-using for ten of the twentyeight industries and labor-saving for the eighteen remaining industries. Technical change is intermediate inputusing for twenty-three of the twenty-eight industries and intermediate input-saving for the remaining five. We conclude that technical

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Table 15.3 Biases of technical change bTK > (3) Construction, (5) Textiles, (26) Trade 0 bTL > 0 . . . bTM < . . . 0 bTK > (1) Agriculture, (8) Furniture, (10) Printing 0 bTL < 0(17) Fabricated metal, (19) Electric machinery, (21) Transportation equipment bTM > (24) Transportation & communication 0 bTK > (6) Apparel, (7) Lumber 0 bTL < 0 . . . bTM < . . . 0 bTK < (4) Foods, (9) Paper, (16) Iron & steel 0 bTL > 0(20) Motor vehicles, (22) Precision instruments, (27) Finance bTM > (30) Other services 0 bTK < . . . 0 bTL > 0 . . . bTM < . . . 0 bTK < (2) Mining, (11) Chemical, (12) Petroleum 0 bTL < 0(13) Rubber, (14) Leather, (15) Stone, clay bTM < (18) Machinery, (23) Miscellaneous manufacturing, (25) Utilities 0 change is predominantly capital-saving, and intermediate input-using for Japanese and United States industries. We next consider the interpretation of the estimated biases of the difference in technology with respect to the quantity of each input . The estimated biases describe the implications of the difference in technology between the United States and Japan for the relative distribution of the value of output among capital, labor, and intermediate inputs. Alternatively, they give the implications of patterns of substitution among these three inputs for the difference in technology. Positive biases imply that the value shares increase from the United States and Japan; negative biases imply that the value shares decrease from the United States to Japan. If a bias is positive, we say that the difference in technology between the United States and Japan uses the corresponding input; if a bias is negative, we say that technical change saves the input. An input-using difference in technology implies that the difference in technology increases with the quantity of the corresponding input, while input-saving change implies that this rate decreases with the input. The sum of the three biases of the difference in technology with respect to quantity is equal to zero, so that we can rule out the possibility that the three biases are either all negative or all positive. All six

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Table 15.4 Biases of the differences in technology bDK > (2) Mining, (11) Chemical, (12) Petroleum 0 bDL > (25) Utilities, (27) Finance 0 bDM < 0 bDK > (3) Construction, (5) Textiles, (9) Paper, (15) Stone, clay 0 bDL < (16) Iron & steel, (19) Machinery, (22) Transportation equipment 0 bDM > (27) Trade 0 bDK > (1) Agriculture, (4) Foods, (14) Leather 0 bDL < . . . 0 bDM < . . . 0 bDK < (8) Furniture, (24) Miscellaneous Manufacturing, (30) Other services 0 bDL > . . . 0 bDM > . . . 0 bDK < (10) Printing, (25) Transportation & communication 0 bDL > . . . 0 bDM < . . . 0 bDK < (6) Apparel, (7) Lumber, (13) Rubber, (17) Fabricated metal 0 bDL < (19) Electric Machiner, (20) Motor vehicles (22) Precision instruments 0 bDM < . . . 0 of the remaining logical possibilities occur among the results for individual industries presented in table 15.4. The biases of the difference in technology, like the biases of technical change, are not affected by the concavity restrictions on the bilateral production functions, so that all three parameters are fitted for each of the twenty-eight industries included in our study. We first consider the bias of the difference in technology between the United States and Japan with respect to the quantity of capital input. If the estimated value of this parameter is positive, the difference in technology is capitalusing. Alternatively, the difference in technology increases with an increase in the quantity of capital input. If the estimated value is negative, the difference in technology is capital-saving and the difference in technology decreases with the quantity of capital input. The difference in technology is capital-using for sixteen of the twentyeight industries included in our study and capital-saving for the remaining twelve. The interpretation of biases of the difference in technology with respect to the quantities of labor and intermediate inputs is analogous to the interpretation of the bias for capital input. The difference in technology between the

United States and Japan is labor-using for ten of the twenty-eight industries and labor-saving for the eighteen

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remaining industries. The difference in technology is intermediate input-using for eighteen of the twenty-eight industries and intermediate input-saving for the remaining ten. We conclude that for given input prices and a given level of technology, production is more capital intensive and intermediate input intensive in Japanese industries and more labor intensive in U.S. industries. We continue with the interpretation of the estimated biases of technical change with respect to the difference in technology between the United States and Japan (bTD). The estimated biases describe the implications of the difference in technology for the rate of technical change. Alternatively, they give the implications of the level of technology for the difference in technology. A positive bias implies that the rate of technical change increases from the United States to Japan; a negative bias implies that the rate of technical change decreases from the United States to Japan. Alternatively, a positive bias implies that the difference in technology between the United States and Japan increases with the level of technology, while a negative bias implies that the difference in technology between the two countries decreases with the level. The rate of technical change increases from the United States to Japan for twenty-two of the twenty-eight industries included in our study; the rate of technical change decreases for only six of the twenty-eight industries. More detailed results are given in figure 15.1. Our interpretation of the parameter estimates given in appendix B concludes with the accelerations of technical change and the differences in the difference in technology ( ). A positive acceleration corresponds to a rate of technical change that is increasing with the level of technology, while a negative acceleration implies that the rate of technical change is decreasing with the level of technology. The estimated accelerations given in table 15.5 are positive for ten industries and negative for the eighteen remaining industries. A positive difference in the difference in technology corresponds to a difference in technology that is increasing between the United States and Japan, while a negative difference implies that the estimated difference given in table 15.5 are positive for eleven industries and negative for the seventeen remaining industries. 15.4 Conclusion Our empirical results on bilateral models of production in Japan and the United States reveal some striking differences between the two

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Figure 15.1 Biases of technological change with respect to the difference in technology.

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Table 15.5 Accelerations of technical change and differences in the difference in technology Accelerations of technical change bTT > 0 bTT < 0 (1) Agriculture (2) Mining (4) Food (3) Construction (6) Apparel (5) Textile (10) Printing (7) Lumber (13) Rubber (8) Furniture (16) Iron & steel (9) Paper (18) Machinery (11) Chemical (19) Electric machinery (12) Petroleum (20) Motor vehicles (14) Leather (27) Finance (15) Stone, clay (17) Fabricated metal (21) Transportation equipment (22) Precision instruments (23) Miscellaneous manufacturing (24) Transportation & communication (25) Electric utilities, gas supply, & water supply (26) Wholesale & retail trade (28) Service Differences in the difference in technology bDD > 0 bDD < 0 (1) Agriculture (3) Construction (2) Mining (4) Food (6) Apparel (5) Textile (9) Paper (7) Lumber (12) Petroleum (8) Furniture (14) Leather (10) Printing (15) Stone, clay (11) Chemical (16) Iron & steel (13) Rubber (18) Machinery (17) Fabricated metal (19) Electric machinery (20) Motor vehicles (21) Transportation equipment (22) Precision instruments (23) Miscellaneous manufacturing (24) Transportation & communication (25) Electric utilities (26) Wholesale & retail trade (27) Finance (28) Service

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countries. With identical relative quantities of all inputs, Japanese industries have higher rates of compensation for capital and intermediate inputs than their U.S. counterparts. By contrast U.S. industries have higher rates of labor compensation than the corresponding Japanese industries. It is important to emphasize that these differences in technology would prevail under identical input proportions in the two countries. The observed patterns of production also reflect differences in these proportions. High rates of technical change in Japan relative to the United States, have been revealed by the results of Jorgenson, Kuroda, and Nishimizu (1987). Our finding that rates of technical change increase from the United States to Japan is, therefore, not surprising. This increase characterizes twenty-two of the twenty-eight industries included in our study. An alternative and equivalent interpretation of these results is that the difference in technology between the United States and Japan increases with the level of technology. The technology gap between the two countries is closing for most industries at given relative quantities of all inputs. Observed changes in the technology gaps also reflect changes in input proportions. Our bilateral models of production are based on strong simplifying assumptions. Although we allow for differences in the value shares of the three inputscapital, labor, and intermediate inputsthe rate of technical change, and the difference in technology between the two countries, we require that share elasticities and the biases and accelerations of technical change are the same for each industry in the two countries. In addition, we have employed conditions for producer equilibrium under perfect competition and we have assumed constant returns to scale at the industry level for both countries. These assumptions must be justified primarily by their usefulness in implementing production models that are uniform for all twenty-eight industrial sectors in Japan and the United States. Our important simplification of the theory of production is the imposition of concavity of the sectoral production function for Japan and the United States. By imposing concavity we have reduced the number of share elasticities to be fitted from one hundred sixty-eight, or six for each of our twenty-eight industrial sectors, to ninety-three, or somewhat more than three per sector on average. All share elasticities are constrained to be zero for six of the twenty-eight industries. The concavity constraints have contributed to the precision of our estimates but require that the share of each input be nonincreasing in the quantity of the input itself.

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Appendix A Our objective is to describe restrictions on the parameters of our econometric models. If a system of equations, consisting of value shares, the rate of technical change, and the difference in technology can be generated from a production function, we say that the system is integrable. A complete set of conditions for integrability is given below. 1. Homogeneity. The value shares, rate of technical change, and difference in technology are homogeneous of degree zero in the inputs. We can write the value shares, the rate of technical change, and the difference in technology in the form

where the parameters parameters must satisfy

are constant. Homogeneity implies that these

where i is a vector of ones. Five restrictions are implied by homogeneity for three inputs. 2. Product exhaustion. The sum of the value shares is equal to unity

The three inputs exhaust the value of the product. This implies that the parameters must satisfy the restrictions

Six restrictions are implied by product exhaustion for three inputs.

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3. Symmetry. The matrix of share elasticities, biases, acceleration, and the difference in the difference in technology must be symmetric. Imposing homogeneity and product exhaustion restrictions we can represent the system of value shares, the rate of technical change, and the difference in technology without imposing symmetry. A necessary and sufficient condition for symmetry is that the matrix of parameters must satisfy the restrictions

For three inputs the number of symmetry restrictions is ten. 4. Nonnegativity. The value shares must be nonnegative

By product exhaustion the value shares sum to unity, so that we can write

where vi ³ 0 implies

and vi ¹ 0.

Nonnegativity of the value shares is implied by monotonicity of the production functions

For the translog production functions, the conditions for monotonicity take the form

Since the production functions are quadratic in the logarithms of inputs In Xi (i = 1, 2, . . . , I), we can always choose inputs so that monotonicity is violated. Accordingly, there are no restrictions on the parameters that would imply nonnegativity of the value shares for all inputs. Instead we consider restrictions that imply concavity of the production functions for all nonnegative value shares.

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5. Monotonicity. The matrix of share elasticities must be nonpositive definite. Concavity of the production functions implies that Hessian matrices, say (Hi), are nonpositive definite, so that the matrices are nonpositive definite

where

the production functions are positive, so that Zi > 0, (i = 1, 2, . . . , I), and ( elasticities defined above.

) are matrices of constant share

Without violating the product exhaustion and nonnegativity restrictions we can set the matrices ( ) equal to zero, for example, by choosing one of the value shares equal to unity and the others equal to zero. Necessary conditions for the matrices to be nonpositive definite are that the matrices of constant share elasticities ( ) must be nonpositive definite. These conditions are also sufficient, since the matrices ( nonpositive definite for all nonnegative value shares summing to unity. The sum of two nonpositive definite matrices is nonpositive definite. 7 To impose concavity on the translog production functions the matrices of constant share elasticities ( represented in terms of the Cholesky factorizations

) are

) can be

where the matrices (Ti) are unit lower triangular and the matrices (Di) are diagonal. For three inputs we can write the matrices ( ) in terms of their Cholesky factorizations as follows

where

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The matrices of constant share elasticities ( ) must satisfy symmetry restrictions and restrictions implied by product exhaustion. These imply that the parameters of the Cholesky factorizations must satisfy the conditions

Under these conditions there is a one-to-one transformation between the share elasticities ( ) and the parameters of the Cholesky factorizations (Ti, Di). The matrices of share elasticities are nonpositive definite if and only if the diagonal elements of the matrices (Di), the so-called Cholesky values, are nonpositive. Our econometric models are generated from translog production functions for each industrial sector. To complete these models we add a stochastic component to the system of equations. We associate this component with unobservable random disturbances. Producers maximize profits for given prices of inputs, but the value shares, the rates of technical change, and the difference in technology are subject to random disturbances. These disturbances result from errors in implementation of production plans, random elements in technologies not reflected in the production functions, or errors of measurement. We assume that each equation has two additive components. The first is a nonrandom function of the inputs, time, and the dummy variable; the second is a random disturbance that is functionally independent of these variables. 8

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Appendix B Table 15.B Parameter estimates for bilateral models of production in Japan and the United States Parameter Industry Mining Construction Agriculture forestry & fisheries 0.124 0.281 0.064 aK (16.503) (36.381) (38.959) 0.296 0.184 0.391 aL (40.709) (79.135) (99.969) 0.580 0.535 0.545 aM (119.647) (84.111) (164.769) 3940.953 40714.641 9865.922 aD ( 0.644) ( 1.588) (1.620) 0.017 0.023 0.007 aT (1.868) ( 0.649) ( 0.992) bKK bLL bMM 7881.535 81429.375 19732.504 bDD (0.644) (1.588) ( 1.620) 0.001 0.003 0.001 bTT (0.729) ( 0.464) ( 1.031) bLK bMK bML 0.216 0.060 0.070 bDK (24.660) (3.458) (11.021) 0.035 0.080 0.165 bDL ( 4.223) (10.218) ( 24.873) 0.181 0.139 0.095 bDM ( 39.107) ( 10.382) (22.882) 0.003 0.001 0.001 bTK (3.393) ( 1.518) (4.059) 0.004 0.003 0.003 bTL ( 6.082) ( 9.162) (11.479) 0.001 0.004 0.004 bTM ( 4.064) (11.288) ( 13.935) 0.023 0.060 0.012 bTD ( 11.723) (8.201) ( 4.577)

Food & kindred products 0.065 (26.212) 0.164 (54.226) 0.771 (442.898) 1218.600 (0.125) 0.001 (0.037)

2437.716 ( 0.125) 0.001 (0.272)

0.176 (18.730) 0.057 ( 8.670) 0.119 ( 14.538) 0.002 ( 6.429) 0.0002 ( 1.373) 0.002 (11.688) 0.0002 (0.088)

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Table 15.B (continued) Table 15.B Parameter estimates for bilateral models of production in Japan and the United States Parameter Industry Lumber & Apparel & Textile wood products other fabricated mill except furniture textile products products 0.085 0.040 0.145 aK (37.141) (15.397) (44.960) 0.268 0.326 0.341 aL (72.509) (105.669) (49.424) 0.646 0.634 0.514 aM (133.167) (192.393) (71.545) 10900.395 17440.969 7526.488 aD (0.699) ( 1.923) (0.802) 0.027 0.010 0.012 aT (1.228) (1.316) ( 0.566) 0.102 bKK ( 11.373) 0.004 0.018 bLL ( 0.253) ( 2.745) 0.004 0.034 bMM ( 0.253) ( 3.490) 21801.141 34881.598 15053.172 bDD ( 0.699) (1.923) ( 0.802) 0.001 0.0002 0.006 bTT ( 0.347) (0.181) ( 1.615) 0.043 bLK (5.159) 0.059 bMK (6.045) 0.004 0.25 bML (0.253) ( 10.367) 0.106 0.003 0.077 bDK (41.254) (0.215) ( 12.093) 0.168 0.104 0.155 bDL ( 6.149) ( 9.498) ( 11.236) 0.062 0.101 0.233 bDM (2.241) (16.295) (20.756) 0.001 0.004 0.0007 bTK (3.236) (9.256) (1.781) 0.001 0.0002 0.003 bTL

Furniture & fixtures 0.065 (33.621) 0.356 (49.171) 0.578 (73.132) 26555.227 (3.487) 0.004 (0.496) 0.073 ( 6.001) 0.111 ( 5.107) 0.004 ( 0.855) 53111.727 ( 3.487) 0.002 ( 1.368) 0.901 (7.377) 0.017 ( 1.826) 0.021 (1.502) 0.144 ( 7.172) 0.098 (2.493) 0.046 (1.678) 0.002 (3.50) 0.005

bTM bTD

(1.160) 0.002 ( 2.258) 0.015 ( 3.172)

(0.410) 0.005 ( 7.021) 0.006 (2.127)

( 7.560) 0.002 (6.928) 0.033 (12.214)

( 4.068) 0.003 (2.809) 0.012 (4.223)

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Table 15.B (continued) Table 15.B Parameter estimates for bilateral models of production in Japan and the United States Parameter Industry Petroleum Chemical & Printing, Paper & refinery & allied publishing & allied related industries products allied products products 0.129 0.109 0.159 0.091 aK (80.912) (52.086) (87.053) (16.958) 0.255 0.394 0.202 0.103 aL (52.148) (96.059) (50.691) (22.254) 0.616 0.497 0.639 0.805 aM (109.489) (96.991) (141.025) (158.978) 1779.908 9356.211 1655.133 35802.32 aD ( 0.405) (0.611) (0.406) ( 1.633) 0.001 0.006 0.004 0.044 aT (0.033) (0.362) (0.455) ( 1.109) 0.010 0.165 bKK ( 1.639) ( 7.715) 0.0563 0.182 0.124 bLL ( 2.890) ( 8.618) ( 17.996) 0.027 0.053 0.124 bMM ( 1.033) ( 2.508) ( 17.996) 3559.253 18714.41 3309.924 71604.250 bDD (0.404) ( 0.611) ( 0.406) (1.633) 0.002 0.0002 0.301 0.003 bTT ( 0.738) (0.091) ( 1.827) ( 0.450) 0.020 0.147 bLK (2.010) (11.620) 0.010 0.018 bMK ( 0.913) (0.973) 0.036 0.034 0.124 bML (1.814) (1.994) (17.996) 0.006 0.234 0.012 0.111 bDK ( 0.388) ( 11.604) (1.414) (6.796) 0.060 0.216 0.048 0.068 bDL ( 1.859) (6.104) (5.017) ( 12.542) 0.066 0.018 0.060 0.042 bDM (1.911) (0.601) ( 6.175) ( 2.894) 0.0002 0.003 0.004 0.004 bTK (0.621) (7.843) ( 12.956) ( 6.831) 0.002 0.004 0.006 0.001 bTL

bTM bTD

( 1.694) 0.002 (1.506) 0.007 (3.495)

( 6.568) 0.001 (0.985) 0.002 ( 0.503)

( 11.492) 0.010 (16.974) 0.024 (15.90)

( 8.951) 0.005 (9.426) 0.003 (0.507)

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Table 15.B (continued) Table 15.B Parameter estimates for bilateral models of production in Japan and the United States Parameter Industry Stone, clay Leather & Rubber & & glass leather miscellaneous products products plastic products 0.982 0.054 0.130 aK (34.947) (21.223) (48.845) 0.370 0.356 0.342 aL (59.380) (69.442) (96.358) 0.532 0.590 0.528 aM (75.830) (92.986) (139.675) 2931.092 9925.094 0.167 aD (0.178) ( 1.704) ( 0.020) 0.014 0.004 0.004 aT (0.416) (0.145) ( 0.309) 0.016 0.004 bKK ( 2.169) ( 0.988) 0.115 0.092 bLL ( 5.818) ( 3.517) 0.045 0.056 bMM ( 3.509) ( 1.914) 5862.695 19849.809 333.760 bDD ( 0.178) (1.704) (0.020) 0.001 0.003 0.002 bTT (0.128) ( 0.569) ( 0.977) 0.043 0.020 bLK (3.561) (2.065) 0.027 0.016 bMK ( 5.047) ( 2.803) 0.072 0.072 bML (4.979) (2.565) 0.022 0.003 0.0212 bDK ( 1.442) (0.189) (2.783) 0.025 0.071 0.155 bDL ( 1.056) ( 2.180) ( 30.647) 0.047 0.068 0.134 bDM (3.063) (1.964) (23.953) 0.0003 0.001 0.003 bTK ( 0.602) (1.470) ( 6.632) 0.001 0.002 0.0001 bTL

Iron & steel 0.092 (44.968) 0.239 (59.960) 0.669 (159.034) 3754.465 ( 0.512) 0.002 ( 0.172)

0.008 ( 1.401) 0.008 ( 1.401) 7508.832 (0.512) 0.005 (2.178)

0.008 (1.401) 0.046 (10.845) 0.165 ( 16.727) 0.119 (11.360) 0.003 ( 8.980) 0.0002

bTM bTD

( 0.700) 0.001 (1.065) 0.003 (0.566)

( 1.608) 0.001 (0.890) 0.012 (5.844)

( 1.543) 0.003 (6.406) 0.016 (6.500)

(0.320) 0.003 (5.074) 0.018 (7.748)

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Table 15.B (continued) Table 15.B Parameter estimates for bilateral models of production in Japan and the United States Parameter Industry Machinery Electric Motor Fabricated machinery vehicle & metal equipment products 0.094 0.120 0.098 0.112 aK (33.979) (44.149) (23.830) (20.575) 0.376 0.380 0.371 0.206 aL (126.698) (66.547) (105.181) (50.784) 0.531 0.500 0.531 0.682 aM (182.333) (75.034) (134.562) (104.216) 5048.535 10990.098 10809.703 0.0001 aD (1.033) ( 1.173) ( 1.665) (0.501) 0.009 0.003 0.018 0.00482 aT (0.864) (0.178) (2.119) (0.225) 0.011 0.080 bKK ( 0.786) ( 4.960) 0.036 0.019 0.030 bLL ( 2.571) ( 1.000) (1.913) 0.035 0.019 0.057 bMM ( 1.737) ( 1.000) ( 1.956) 10097.117 21979.551 21618.820 13130.848 bDD ( 0.1033) (1.172) (1.665) ( 0.501) 0.0003 0.00103 0.0001 0.004 bTT ( 0.153) (0.396) (0.050) (1.101) 0.006 0.027 bLK (0.592) (2.527) 0.005 0.535 bMK (0.336) (2.856) 0.030 0.0194 0.0036 bML (2.481) (1.000) (0.200) 0.001 0.020 0.064 0.024 bDK ( 0.076) (3.396) (12.958) (1.410) 0.060 0.169 0.170 0.030 bDL ( 2.644) ( 13.343) ( 5.908) ( 1.190) 0.061 0.149 0.105 0.006 bDM (3.093) (13.741) (3.699) (0.199) 0.001 0.002 0.001 0.004 bTK (1.477) ( 5.314) ( 1.616) ( 4.591) 0.002 0.831 0.0003 0.001 bTL

bTM bTD

( 3.201) 0.001 (2.336) 0.015 (9.390)

( 0.516) 0.002 (4.983) 0.022 (7.503)

(0.390) 0.001 (0.767) 0.026 (10.492)

(0.824) 0.003 (3.328) 0.002 ( 5.37)

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Table 15.B (continued) Table 15.B Parameter estimates for bilateral models of production in Japan and the United States Parameter Industry Transportation equipment, Precision Miscellaneous manufacturing Transportation except motor instruments & communication 0.043 0.149 0.153 0.171 aK (10.994) (26.604) (46.026) (78.090) 0.345 0.498 0.294 0.319 aL (50.987) (65.066) (61.543) (149.582) 0.612 0.353 0.553 0.510 aM (70.859) (43.445) (84.550) (265.597) 0.0003 5894.293 1729.229 8332.473 aD (0.016) ( 0.856) (0.212) (1.087) 0.006 0.012 0.003 0.009 aT (0.497) (0.543) ( 0.129) (1.310) 0.048 0.020 0.00494 0.902 bKK ( 4.37) ( 1.841) ( 1.036) ( 2.701) 0.002 0.020 0.088 0.119 bLL ( 0.518) ( 1.034) ( 4.293) ( 3.922) 0.069 0.000002 0.052 0.002 bMM ( 2.989) ( 0.009) ( 2.181) ( 0.188) 0.002 11788.879 3458.634 16665.172 bDD ( 0.017) (0.856) ( 0.212) ( 1.087) 0.002 0.002 0.001 0.002 bTT ( 0.969) ( 0.576) ( 0.357) ( 1.383) 0.010 0.020 0.021 0.104 bLK ( 1.057) (1.721) (2.088) (3.569) 0.058 0.0002 0.016 0.013 bMK (4.267) (0.017) ( 2.901) ( 0.801) 0.011 0.0002 0.068 0.016 bML (0.903) ( 0.017) (3.057) (1.194) 0.117 0.051 0.094 0.092 bDK (8.341) ( 3.172) ( 6.266) ( 2.206) 0.113 0.257 0.012 0.280 bDL ( 7.114) ( 9.358) (0.456) (6.475) 0.005 0.308 0.081 0.188 bDM ( 0.186) (13.231) (2.764) ( 10.091) 0.001 0.006 0.001 0.003 bTK ( 1.368) ( 5.394) (1.435) (1.876) 0.001 0.0002 0.003 0.003 bTL ( 2.689) ( 0.139) ( 3.076) ( 2.339)

bTM bTD

0.002 (2.381) 0.029 (5.079)

0.006 (5.757) 0.015 (5.238)

0.003 (2.058) 0.019 (5.296)

0.001 (0.682) 0.022 (6.976)

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Table 15.B (continued) Table 15.B Parameter estimates for bilateral models of production in Japan and the United States Parameter Industry Finance Wholesale Electric utility, & & retail gas supply & insurance trade water supply 0.278 0.149 0.186 aK (76.882) (58.100) (59.765) 0.150 0.520 0.211 aL (44.945) (77.690) (51.148) 0.562 0.331 0.603 aM (112.380) (48.761) (130.635) 18942.898 9951.582 44612.750 aD (2.081) (1.213) (3.295) 0.004 0.010 0.006 aT ( 0.344) (1.739) (1.487) 0.081 bKK ( 3.472) 0.122 0.022 bLL ( 8.713) ( 1.531) 0.122 0.019 bMM ( 8.713) ( 1.172) 37888.145 19903.762 89225.500 bDD ( 2.081) ( 1.213) ( 3.295) 0.001 0.0004 0.002 bTT ( 0.512) ( 0.430) (2.929) 0.042 bLK (2.675) 0.039 bMK (1.906) 0.122 0.020 bML (8.713) ( 3.387) 0.066 0.071 0.196 bDK (4.991) (3.338) (27.958) 0.172 0.162 0.123 bDL (9.670) ( 7.036) (17.046) 0.238 0.090 0.319 bDM ( 11.395) (5.770) ( 35.387) 0.006 0.002 0.004 bTK ( 10.327) (3.018) ( 9.536) 0.008 0.002 0.002 bTL

service 0.430 (129.970) 0.250 (119.346) 0.320 (81.072) 9306.051 (2.046) 0.002 ( 0.367)

18612.191 ( 2.046) 0.001 ( 1.103)

0.127 ( 15.222) 0.051 ( 9.888) 0.178 (22.825) 0.002 ( 3.170) 0.003

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Notes 1. The share elasticity was introduced by Christensen, Jorgenson, and Lau (1971, 1973) and by Samuelson (1973). 2. Share elasticities were first employed as constant parameters of an econometric model of producer behavior by Christensen, Jorgenson, and Lau (1971, 1973). Constant share elasticities and biases of technical change are employed by Jorgenson and Fraumeni (1983), Jorgenson (1983b, 1984), and Kuroda, Yoshioka, and Jorgenson (1984). Binswanger (1974a,b,c, 1978a) uses a different definition of biases of technical change in parameterizing an econometric model with constant share elasticities. 3. Alternative definitions of biases of technical change are compared by Binswanger (1978b). 4. Biases of the difference in technology were introduced by Binswanger (1974a,c, 1978a). 5. Biases of technical change with respect to the difference of technology were introduced by Jorgenson and Nishimizu (1978) in the context of a bilateral model of production for Japan and the United States at the aggregate level. This model was extended to the sectoral level by Jorgenson and Nishimizu (1981). 6. The difference of the difference of technology was introduced by Jorgenson and Nishimizu (1978, 1981). 7. This approach to global concavity was originated by Jorgenson and Fraumeni (1983). The Cholesky factorization was first employed in imposing local concavity restrictions by Lau (1978a). 8. Alternative stochastic specifications for econometric models of production are discussed by Fuss, McFadden, and Mundlak (1978). Additional detail on econometric methods for modeling producer behavior is given by Jorgenson (1986).

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References Ackerman, Susan. 1973. Used Cars as a Depreciating Asset. Western Economic Journal 11, no. 4 (December): 463 474. Afriat, S.N. 1972. Efficiency Estimation of Production Function. International Economic Review 13, no. 3 (October): 568 598. Aigner, Dennis J., Takeshi Amemiya, and Dale J. Poirier. 1976. On the Estimation of Production Frontiers: Maximum Likelihood Estimation of the Parameters of a Discontinuous Density Function. International Economic Review 17, no. 2 (June): 377 396. Amemiya, Takeshi. 1973. Regression Analysis When the Dependent Variable is Truncated Normal. Econometrica 41, no. 6 (November): 997 1016. . 1974. The Nonlinear Two-Stage Least-Squares Estimator. Journal of Econometrics 2, no. 2 (July): 105 110. . 1977. The Maximum Likelihood Estimator and the Nonlinear Three-Stage Least-Squares Estimator in the General Nonlinear Simultaneous Equation Model. Econometrica 45, no. 4 (May): 955 968. . 1983. Nonlinear Regression Models. In Handbook of Econometrics, vol. 1, eds. Zvi Griliches and Michael D. Intriligator, 333 389. Amsterdam: North Holland. Anderson, Richard G. 1981. On The Specification of Conditional Factor Demand Functions in Recent Studies of U.S. Manufacturing. In Modeling and Measuring Natural Resource Substitution, eds. Ernst R. Berndt and Barry C. Field, 119 144. Cambridge, MA: MIT Press. Applebaum, Elie. 1978. Testing Neoclassical Production Theory. Journal of Econometrics 7, no. 1 (February): 87 102. . 1979a. On the Choice of Functional Forms. International Economic Review 20, no. 2 (June): 449 458. . 1979b. Testing Price Taking Behavior. Journal of Econometrics 9, no. 3 (February): 283 294. Arrow, Kenneth J. 1963. Social Choice and Individual Values, 2nd ed. New York: John Wiley and Sons, Inc.

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Index A Accelerated Cost Recovery System, 85, 91 ACMS, 2 3, 125, 143, 154 Acquisition prices, 77, 86 Ad valorem rate, 91 Additivity. See also Nonlinear simultaneous equations with additive disturbances, efficient estimation of background information, 127 commodity-wise, 127 129, 143, 146, 150 151, 153 group-wise, 129 132, 134, 141 144, 151, 152 154 multi-level, 131, 134, 153 154 price possibility frontier and, 132 134 two-level, 130 131 Agricultural household behavior background information, 97 98 complete model, 115 122 constraints, 112 115 time, 112 113, 115 total expenditure, 113 114 total labor, 114 115 functional form econometric implementation and, 122 123 Japanese patterns, 97 Korean patterns, 97 labor and, 108 109, 119 Taiwanese patterns, 97 technology, 109 112 utility functions, 99 100, 107 109, 122 123, 126 welfare function, 98 108

decentralization and, 107 egalitarian, 101 106 family dependents and, 107 109 family workers and, 108 109, 119 formulation of, 98 101 homogeneous, 105 108, 111 utility function and, 106 107 welfare maximization, 97 98, 109, 115 122 Allen partial substitutability, 392, 396 Amemiya, Takeshi, 5 6, 210, 234, 314 Analog of likelihood ratio test, 315 316, 330 331 Anderson, Richard G., 33 Applebaum, Elie, 36, 41 42 Approximation, 142 144 Arab Oil Embargo, 343 Arrow, Kenneth J., 2 3, 98 99, 125, 143, 154 Asset classification scheme, 83, 91 Asset Depreciation Range System, 84 85 Asset prices model, 74 81, 93 94 Asymptotic distribution of TA, 324 326 of TA/a, 333 334 of TO, 317 323 of TO/a, 332 333 Atkinson, Scott E., 60 Auerbach, Alan J., 91 Augmentation factors. See Factor augmentation Autocorrelation, 22 24 Averch, Harvey, 55, 59 Averch-Johnson model, 55, 59 B Bailey, Martin J., 84

Ball, V.E., 93 Barquin-Stolleman, Joan A., 79 80 Basic Wage Structure Survey (BWSS), 391 BEA, 79 80, 84 85, 87 89, 91, 94 Becker, G.S., 114

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Beidleman, Carl R., 85 Belgium, 36 Bell Canada, 62 63 Bell Laboratories, 62 Bergson-Samuelson social welfare function, 100 101, 104, 106 Berndt, Ernst R., 31 34, 36 37, 41 42, 65, 68 69 Berndt-Jorgenson model, 31 33, 43 44 Bernoulli utility function, 122 123 Best CUAN estimator, 209 211, 238 Best geometric rate (BGA), 82 BGA, 82 Biases of productivity, 409 410 Biases of scale, 48 49, 51, 56 Biases of technical change Binswanger and, 42 capital inputs and, 367 368, 433 classification of industries by, 365, 368, 376, 435 436 defined, 11, 263 264, 459 460 electricity inputs and, 434 energy inputs and, 368 Hicks and, 4 interpretation of, 39 41 patterns of, 39 41 quantity input and, 469 470, 480 486 vector of, 11 12, 15, 459 460 Bilateral models of production for Japanese and U.S. industries appendix, 476 486 homogeneity, 476 monotonicity, 478 479 nonnegativity, 477

parameter estimates, 480 486 product exhaustion, 476 477 symmetry, 477 background information, 457 458 conclusions, 472 475 framework, theoretical, 458 462 results, empirical, 462 472 Binswanger, Hans P., 41 42 Bjorn, E., 80 81 Black, D., 98 99 Boundedness, 161 162, 171 173, 175, 179, 181 Box-Cox transformation, 41 42, 60, 78, 82 Bräutigan, Ronald R., 61 Brazell, D.W., 84 85, 87, 89 Brown, G.M., 36 Brown, Randall, 41, 60 Brundy, James M., 210 Budget constraint, 116 Bureau of Economic Analysis (BEA), 79 80, 84 85, 87 89, 91, 94 Bureau of Labor Statistics, 92 94 Burgess, David F., 43 BWSS, 391 C Cameron, Trudy A., 34 Canada, 35, 37, 44 45, 62 63 Capital accumulation equation, 76 77 asset pricing equation, 77 cost recovery, 91 goods and services, 75 81, 86, 91 92, 391 392 input definition, 88 89 inputs, 36 37, 88, 91 92, 139, 143 144, 150, 292 295, 367, 379, 391 392, 433

outputs, 379 price of, 367 368 as production factor, 67 68 share elasticities of, 366 367 stock, 84, 87, 91 92, 94 Capital Stock Study (BEA), 85 Capital-labor substitution, 154 Capital-saving productivity growth, 410 Capital-using technical change, 469 Caves, Douglas W., 41 42, 60, 62, 66 67 CES, 3 5, 125, 131 132 CET, 131 132 CET-CES representation, 131 132, 142 143, 145 146, 297 Chebychev's theorem, 224 Chen, Y.C., 79 80 Chenery, B., 2 3, 125, 143, 154 Chiang, S. Judy Wang, 61 62 Cholesky factorization, 19, 27, 250, 275, 342, 353, 355, 384 385, 387, 393, 413, 463, 478 479 Chow, Gregory C., 79 Christensen, Laurits R., 4 5, 16, 36 37, 41 43, 55 56, 58, 60, 62, 66 67, 73 74, 76, 89 90, 94 Christensen-Greene model, 55 56 Class I railroads in United States, 60 61, 67 Closure production functions and, 162 163, 179, 199

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profit functions and, 171 172, 179 set of price possibilities and, 175 176, 182, 201 set of production possibilities and, 161, 182 set of profit possibilities and, 175 176, 201 Cobb, Charles W., 2 Cobb-Douglas production function, 4, 64, 120, 122 123, 143 144, 146 147 Coen, Robert M., 86 Cole, Rosanne, 79 80 Commodity inputs and outputs, 160 Commodity tests, 281 282, 285 Commodity transaction, 389 Commodity-wise additivity, 127 129, 143, 146, 150 151, 153 Communications industry, 55, 62 63 Complementary inputs, 12 13, 49 Composition-commodity theorem, 104 Computer Price Guide, 85 86 Computers, 80, 85 86 Concavity of cost functions, 47, 49 KLEM model and, 384 of price functions, 10, 12 13, 18 19, 346, 367, 445 448 of translog price functions, 19, 347 348 Conrad, Klaus, 43, 260, 287 Constancy, 279 281 Constant elasticity of substitution (CES), 3 5, 125, 131 132 Constant elasticity of transformation (CET), 131 132 Constant quality price index, 73 Constant returns to scale, 9, 56, 139 141, 143, 147, 343, 348 Constant share elasticity (CSE), 4 5, 15, 51, 345, 444, 459 Constraints

budget, 116 role of, 6 time, 112 113, 115 total expenditure, 113 114 total labor, 114 115 Consumption anomalies, 99 consumer behavior theories and, 328 consumer demand and, 126 outputs, 292 295 Continuity, 193, 197 Convexity convex cone and, 176 convex conjugacy and, 160, 177 178, 202 205 of production functions, 163, 177 179, 184, 192, 196, 202 205 of profit functions, 171 172, 179, 184 restrictions, 254 255 set of price possibilities and, 175 176, 182, 201 set of production possibilities and, 161, 175 176, 182, 200 set of profit possibilities and, 201 of translog production functions, 249 252 Cost exhaustion, 52 Cost flexibility, 47 48, 50 52, 56, 60 Cost functions. See also specific properties applications of, 55 63 economies of scale, 56 60 multiple outputs, 60 63 overview, 55 concavity of, 47, 49 constant share elasticity, 51 cost flexibility and, 47 48

defined, 47 degree of return to scale and, 47 48 duality in, 46 48 economies of scale and, 48 50, 66 homogeneous, 47, 50, 52 homothetic, 50 integrability and, 51 53 Leontief, 59 Monotonicity of, 47, 53 nonnegativity of, 53 overview, 45 46 parameterization and, 51 53 properties of, 47 stochastic specification and, 53 55 substitution and, 48 50 translog, 51, 59 63 Court, Andrew T., 79 Covariance matrix, 352 Cowing, Thomas G., 58 59 Cramer-Rao bound for variance of CUAN estimation background information, 209 210, 212 restricted W matrix, 220 222 unrestricted W matrix, 213 219 CSE, 4 5, 15, 51, 345, 444, 459

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CUAN estimation for systems of nonlinear simultaneous equations with additive disturbances background information, 209 211, 212 Best CUAN estimator and, 209 211, 238 Cramer-Rao bound for variance of, 212 222 background information, 209 210, 212 restricted W matrix, 220 222 unrestricted W matrix, 213 219 efficiency and, 232 239 background information, 232 conclusions about, 238 239 limited information versus full information, 234 238 minimum distance versus maximum likelihood, 232 234 instrumental variables and, 229 232 minimum distance and, 222 229 model, 211 212 Cummings, Dianne, 62 Cyclical monotonicity, 167 169 D Dargay, J., 34, 44 Daugherty, Andrew F., 61 Dean, E., 98 Debreu, Gerard, 185 Deceleration of technical change, 12 Decentralization, welfare function and, 107 Degree of return to scale, 47 48, 50 Demand and supply functions, 2, 4 5, 194, 197, 268, 272 Denison, Edward F., 88 89 Denmark, 36 Denny, J.G.S, 34 35, 42 43, 63, 68 Department of Commerce, 80

Department of the Treasury, 74 Depreciation acquisition prices and, 77, 86 Asset Depreciation Range System and, 84 85 defined, 76 econometric models of, 75 81 measures, 87 93 rates of economic, 82 84 studies, empirical, 73 74, 86 87 applications of, 87 93 asset prices model, 74 81, 93 94 background information, 73 74 in future, 93 94 Hulton-Wykoff model, 74, 78, 81 87, 93 94 Oliner, 85 87 vintage accounts and, 74 81, 85 86, 93 94 tax law and, 85 Diewert, W. Erwin, 16, 42, 186 Differentiability in production background information, 189 190 extensions and, 202 207 marginal productivity functions and, 192 194 price functions and, 10 production functions and, 190 192 production possibilities and, 198 202 twice, 205 207 Dillon, John L., 16 Discounted value definition, 88 89, 93 Distance estimators, minimum, 222 229 maximum likelihood versus, 232 234 Divisia price indices of capital service input, 391

of labor input, 391 392 output and input aggregate measures and, 376 of outputs, 388 390 Domain effective of marginal productivity correspondence, 165 of normalized supply correspondence, 203 of production function, 162, 191, 206 of supply correspondence, 173 marginal productivity correspondence and, 165 168, 180 181 marginal productivity functions and, 193 normalized profit functions and, 196 normalized supply functions and, 197, 203 production functions and, 162, 179, 191 profit functions and, 171, 179, 196 supply correspondence and, 173, 180 181 Doms, M.E., 86 Douglas, Paul H., 2 Douglas-type price function, 393

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Duality conjugate, 160 in cost functions, 46 48 in price functions, 8 11 in producer behavior theory, 6 7 in production, 194 207 background information, 189 190, 194 extensions and, 202 207 Legendre transformation and, 194 196 normalized profit functions and, 196 normalized supply functions and, 197 production possibilities and, 198 202 of production function and normalized restricted profit function, 184 of production and price functions, 255 in production theory, 1, 3, 194, 245 in production theory testing, 144 146, 154 of technology and economic behavior, 177 182 background information, 170 conclusions about, 182 185 historical note about, 185 186 marginal productivity correspondence and, 164 169, 179 181 production functions and, 177 179 profit functions and, 170 172, 177 179 set of production and price possibilities and, 181 182 supply correspondence and, 172 175, 179 181, 184 in utility functions, 126 Duesenberry, J. S., 99 Dulberger, Ellen, 79 80 Durable goods, 76 78, 84 Dworin, L., 84 85, 87, 89

Dynamic models of production, 67 69 E Econometric models. See also specific types of depreciation, 75 81 energy in productivity growth and, 408 436 background information, 408 411 estimation, 453 455 results, empirical, 411 413 stochastic specification, 449 453 substitution, 413 432 technical change, 405 406, 433 436 of substitution, 43 44 technology structure in Federal Republic of Germany (1950 1973) and, 281 282 technology structure over time in Federal Republic of Germany (1950 1973) and, 301 303 unknown parameters of, 5 Economic behavior and technology, duality of background information, 170 conclusions about, 182 185 historical note about, 185 186 profit functions and, 170 172, 177 179 set of price and profit possibilities and, 181 182 supply correspondence and, 172 175, 177 179 Economic Growth of Nations (Kuznets), 373 Economic performance measures, 89 93 Economic Research Service, 93 Economies of density, 67 Economies of scale cost functions and, 48 50, 66 applications of, 56 60 restrictions on, 54 Effective domain of marginal productivity correspondence, 165

of normalized supply correspondence, 203 of production functions, 162, 191, 206 of supply correspondence, 173 Efficiency background information, 232 conclusions about, 238 239 limited information versus full information and, 234 238 minimum distance versus maximum likelihood and, 232 234 Egalitarian welfare function, 101 106 Ehud, Ron I., 45 Elbadawi, Ibraham A., 16, 34 Electricity-using productivity growth, 410 411 Electrification, 42, 59, 409, 436 439 Energy, 388 390, 394, 405. See also Energy in productivity growth Energy and the American Economy (Schurr), 403 404

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Energy in America's Future (Schurr), 405, 438, 440 Energy prices, 81 Energy in productivity growth appendix, 442 455 concavity, 445 448 estimation, 453 455 homogeneity, 443 445 index numbers, 448 449 overview, 442 443 stochastic specification, 449 453 symmetry, 443 445 background information, 404 407 conclusions about, 441 442 econometric models, 408 436 background information, 408 411 conclusions about, 182 185 estimation, 453 455 results, empirical, 411 413 stochastic specification, 449 453 substitution, 413 432 technical change, 405 406, 433 436 electrification and, 436 439 nonelectrical energy and, 439 441 Epigraph of production function, 198 Epstein, L.G., 68 69 Equality restrictions, 30, 139 140, 144, 146 149 welfare function and, 101 106 Equipment prices, 86 Estimation.

See also Nonlinear simultaneous equations with additive disturbances, efficient estimation of, specific methods as econometric model of energy in productivity growth and, 453 455 production theory testing and, 147 150, 255 260 statistical method of, 24 31 technology structure in Federal Republic of Germany (1950 1973) and, 283 286 technology structure over time in Federal Republic of Germany (1950 1973) and, 303 306 three-stage least-squares, 313 329 analog of likelihood ratio test and, 315 316 asymptotic distribution of TA and, 324 326 asymptotic distribution of TO and, 317 323 hypothesis of symmetry and, 326 329 regularity conditions and, 316 317 statistical model subject to a maintained hypothesis and, 313 315 statistical package and, 313 Wald test and, 323 324 two-stage least-squares, 329 335 analog of likelihood ratio test, 330 331 asymptotic distribution TA/a and, 333 334 asymptotic distribution TO/a and, 332 333 hypothesis of homogeneity and, 334 335 regularity conditions and, 331 332 statistical model, 329 330 statistical package, 313 Wald test and, 333 Evans, David S., 62 Everson, C., 63 Exposed production plan, 199 Extensions, differentiability in production and background information, 202 convex conjugacy, 202 205 twice differentiability, 205 207 F

Factor augmentation, 140 141, 147, 151, 278 281 restrictions, 141, 145, 146, 150, 152, 154, 277 278 Family dependents, 107 109 Family workers, 108 109, 119 Federal Republic of Germany (FRG) manufacturing in, 34, 36, 42, 45 production in (1950 1973), 241 265 background information, 241 242 convexity of translog production functions and, 249 252 economic interpretation of, 260 265 estimation statistics and, 255 260 integrability and, 247 249 monotonicity of translog production functions and, 249 252 production theory testing and, 252 255

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test statistics and, 255 260 translog price functions and, 242 247 translog production functions and, 242 247 technology structure in (1950 1973), 267 289 background information, 267 269 conclusions about, 286 289 estimation and, 283 286 nonjointness and, 273 277, 281, 283, 285, 287 statistical methods and, 283 286 technical change and, 268, 277 281 test restrictions and, 281 282 translog price functions and, 269 272 translog production functions and, 269 272 technology structure over time in (1950 1973), 291 310 background information, 291 295 conclusions about, 307 310 estimation and, 303 306 separability and, 295 298 statistical methods and, 303 306 technical change and, 298 301 test restrictions and, 301 303 Field, Barry C., 36 37 First-order partial differential equations, 15, 51, 251 252 Fisher, Irving, 118 Fixed coefficients assumptions, 64 Fixed Reproducible Tangible Wealth in the United States, 1925 1989 (BEA), 87 Ford, J., 45 Forsyth, F.G., 108 109 Foundations of Economic Analysis (Samuelson), 2 Fourier functional form, 16, 33 34

France, 36, 45 Fraumeni, Barbara M., 31, 37 38, 65, 393 Frenger, Petter, 35 FRG. See Federal Republic of Germany Friede, Gerhard, 34, 44 Friedlaender, Ann F., 35, 61 62 Frisch, Ragnar, 47 Fullerton, Daniel, 64 Functional form econometric implementation, 122 123 Fuss, Melvyn, 34, 44, 59, 62 63, 68 Fuss's model, 59 G Gabriel, K.R., 312 Gallant, A. Ronald, 16, 30 31, 33 34, 311 313, 319, 321, 324, 330, 336 Gallant's lemmas, 319 321 Gauss-Newton method, 27 General equilibrium modeling, 64 65 Generalized Leontief parametric form, 16 Germany. See Federal Republic of Germany (FRG) Globally univalent normalized supply correspondence, 204 Gollop, Frank M., 58 60 Gorman, William M., 186 Grant, James, 37 Grebenstein, Charles, 36 Greene, William H., 55 56, 58, 66 67 Gregory, Paul R., 36 Griffin, James M., 35 36, 58 Griliches, Zvi, 79, 86, 88 89 Group-wise additivity, 129 132, 134, 141 144, 151 154 restrictions, 141, 145 146, 150, 152 Groupwise separability, 268, 295 303, 306 307 restrictions, 299 301

Gruen, F.H.G., 131 Grumbaugh, Stephen S., 34 H Haig-Simons definition of taxable income, 91 Hall, Robert E., 43, 60, 73, 78 80, 93 94, 269, 273 Hall's ''hedonic" model of asset prices, 74 75, 78 81, 93 94 Halvorsen, Robert F., 45, 58 Hamermesh, Daniel S., 37 Hansen, Lars P., 69 Harmatuck, Donald J., 61 Harper, Michael J., 42 Harper, M.J., 93 Hausman, Jerry, 210 Heady, Earl O., 16 Heckman, James J., 62 Helvacian, Nurhan, 79 80 Henderson, Yolanda K., 64 Hessian matrix, 206, 275 276, 384 Hicks, John R., 2 4, 37, 104, 342 Hodge, James H., 79 80 Holly, Alberto, 30, 313

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Homogeneity. See also Translog production frontiers, additivity and homogeneity and of cost flexibility, 52 of cost functions, 47, 50, 52 of cost shares, 52 hypothesis of, 334 335 integrability and, 17, 476 linear, 384 of price functions, 10, 13, 443 445 of production functions, 251 of production possibility frontier, 128, 130 of profit functions, 171 rate of technical change and, 17 set of price and profit possibilities and, 176 of supply correspondence, 174 test statistics for, 56 57 of welfare function, 105 108, 111 Homotheticity, 13 14, 49 50, 54 56 Hotelling, Harold S., 3, 185 186 Household consumption. See Agricultural household behavior Houthakker, Henrik S., 108 109, 126, 134 Hudson, Edward A., 33, 65 Hulten, Charles R., 74, 78, 80 87, 89 94 Humphrey, David Burras, 35 36 Hyperplane, nonvertical supporting, 164, 172, 199, 201 202 I IBM, 79 80, 85 Identification, 24 31 Impossibility Theorem, 98 99 Inada, K., 98 99

Income marginal utility of, 118 119 standard, achieving, 109 taxable, 91 Independent inputs, 12 13, 49 50 Index numbers, 448 449 Index of returns, 47 Inference, 311 313. See also Statistical inference for a system of simultaneous, nonlinear, implicit equations Information, limited versus full, 234 238 Input-saving technical change, 11, 459 Input-using technical change, 11, 459 Inputs. See also specific types aggregate measures, 11 14, 31, 49 50, 143 144, 376, 380 capital, 36 37, 88, 91 92, 139, 143 144, 150, 292 295, 367, 379, 391 392, 433 commodity, 160 complementary, 12 13, 49 electricity, 434 energy, 368 fixed coefficients assumption and, 64 independent, 12 13, 49 50 labor, 36 37, 68, 139, 143 144, 146, 150, 292 295, 390 391, 405 pairs of, 35 36 prices, 4 5, 8 11, 18, 20, 51, 145, 346, 380 quantities of, 8 9 shares of, 9 substitute, 42 43, 49 for Swedish manufacturing study, 35 value of, 9 11 Instrumental variables, 229 232 Integrability conditions for, 17 19

cost functions and, 51 53 homogeneity and, 17, 476 monotonicity and, 18 19, 478 479 nonnegativity and, 18, 477 price functions and, 17 19 product exhaustion and, 476 477 production in Federal Republic of Germany (1950 1973) and, 247 249 symmetry and, 17 18, 52, 477 Intermediate material input, 388 390 International Data Corporation, 85 International demonstration effect, 99 Investment and investment goods, 75 77, 93, 139, 292 295 Italy, 36, 45 J Japan. See also Agricultural household behavior bilateral models of production in, 457 486 appendix, 476 486 background information, 457 458 conclusions about, 472 475 framework, theoretical, 458 462 results, empirical, 462 472 economic growth in, source of, 373 375 manufacturing in, 34, 45 relative price changes and biases of technical change in, 373 402

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background information, 373 379 data sources and, 388 392 design of experiment and, 388 392 KLEM model estimation and, 383 388, 393 measurement of technical change and, 379 381 results and, estimated, 392 402 specification of price possibility frontiers and, 382 383 stochastic specification and, 383 388 technology in, 460 461, 464 465, 470 472 Jara-Diaz, S., 61 Jenrich, R.I., 5 6 Johansen, Leit, 64 65 Johnson, Leland L., 55, 59 Jorgenson, Dale W., 4 6, 16, 25, 30 33, 36 38, 42 43, 45, 65, 73 74, 76, 84, 88 92, 94, 111, 118, 210, 242, 245 246, 264, 267, 277, 287, 291, 298, 314, 326, 342, 354, 393, 457 459, 462, 475 Jorgenson-Fraumeni model, 37 38, 41 K Kang, H., 36 Karabadjian, S., 63 Karlson, Stephen M., 59 Kennedy, Charles, 37, 342 Khaled, Mohammed, 41 42 Kiss, F., 63 KLEM model estimation, 382 388, 393 Kohli, Ulrich R., 43 Koopmans, Tjalling C., 185 Kopp, Raymond J., 35, 42 Korea. See Agricultural household behavior Krishna, R., 98 Kronecker product form, 23 24, 54, 211, 350 351, 353

Kuh, Edwin, 86 Kuhn-Tucker necessary conditions, 28, 116 117, 120 121 Kuroda, Masahiro, 457 459, 462, 475 Kuznets, Simon, 373, 376 L Labor agricultural household behavior and, 108 109, 119 divisia price index for, 390 391 inputs, 36 37, 139, 143 144, 146, 150, 292 295, 390 391, 405 price of, 366 367 share elasticities of, 366 367 working family members and, 108 109, 119 Labor Force Survey (LFS), 391 Laffont, Jean-Jacques, 5 6, 25, 314, 354, 462 Lagrangian function/method, 28, 116, 120, 389 Lau, Lawrence J., 4 5, 16, 43, 186, 273, 326 Laurent matrix, 22 23, 352 Leenders, C.T., 210, 222, 239 Lefebvre, B.J., 63 Legendre transformation, 184, 186, 189, 194 196, 202, 204 205 Leontief cost functions, 59 Leontief functional form, 16, 35 Leontief price functions, 37 Leontief, Wassily W., 64 65 LFS, 391 L.H.S. full consumption, 114 Liew, Chong K., 28 Likelihood ratio test statistics, 56, 150, 304, 315 316, 330 331 Limited aspirations model, 109 Linear homogeneity, 384 Linear logarithmic production functions, 64 Long Lines, 62

Long, Thomas V. III, 34 Longva, Svein, 34 Lucas, Robert E., 68 M McFadden, Daniel L., 3, 125, 132, 186 Machinery Dealers National Association, 85 McRae, Robert N., 34 Magnus, Jan R., 45 Malinvaud, Edmond, 5 6, 25, 31, 209, 219, 231, 255 Malpezzi, Stephen, 86 Manufacturing, 34 37, 41 42, 44 45 Marginal productivity correspondence domain and, 165 168, 180 181 duality of technology and economic behavior and, 164 169, 179 181 monotonicity and, 166 169 production theory and, 159 functions differentiability in production and, 192 194

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(cont.) Marginal productivity domain and, 193 monotonicity and, 193 profit maximization and, 268 translog, 248 set of production possibilities and, 159 theory, modern, 159 160 Matrix W average rate of technical change and, 23 restricted, 220 222 unrestricted, 213 219 Maximal cyclically monotone normalized supply correspondence, 203 Maximal normalized supply correspondence, 203 May, J. Douglas, 34 35 Meese, Richard, 69 Mellor, J.W., 98, 109 Melnik, Arie, 45 Meyer, John, 86 Minhas, S., 2 3, 125, 143, 154 Minimum distance estimators, 222 229 maximum likelihood versus, 232 234 Monotonicity of cost functions, 47, 53 cyclical, 167 169, 193, 203 of distributive shares, 27 integrability and, 18 19, 478 479 KLEM model and, 384 marginal productivity correspondence and, 166 169 of marginal productivity function, 193

normalized price and profit possibilities and, 182 normalized supply correspondence and, 181 of normalized supply function, 197 of price functions, 10, 13 of production functions, 253 254 of set of price and profit possibilities, 176, 201 of set of production possibilities, 161, 199 supply correspondence and, 173 174 of translog price functions, 251 of translog production functions, 249 252 Moroney, John R., 36 Morrison, Catherine J., 33, 68 Multi-level additivity, 131, 134, 153 154 Multiple outputs, 60 63 Mundlak, Yair, 67 Musgrave, John C., 88 N Nadiri, Mohammed Ishaq, 62 Nakajima, C., 98, 109, 118 Nakamura, Shinichiro, 42, 45 Netherlands, 34, 36, 45 Neutral technical change, 11 Newton-Raphson method, 210 Nishimizu, Mieko, 457, 462, 475 NL2SLS method, 5 6, 231. See also Two-stage least-squares estimation NL3SLS method, 6, 25 31. See also Three-stage least-squares estimation Nonelectrical energy, 42, 439 441 Nonemptiness, 198, 201, 203 Nonjointness, 273 277, 281, 283, 285, 287 Nonlinear simultaneous equations with additive disturbances, efficient estimation of, 5, 209 211. See also CUAN estimation for nonlinear simultaneous equations with additive disturbances Nonlinear three-stage least-squares (NL3SLS) method, 6, 25 31. See also Three-stage least-squares estimation Nonlinear two-stage least-squares method (NL2SLS), 5 6, 231. See also Two-squares least-squares estimation

Nonlinearity, 78 79 Nonnegativity of cost functions, 53 integrability and, 18, 477 of normalized profit functions, 196 of production function domain, 191 restrictions, 347 of set of price and profit possibilities, 201 of set of production possibilities, 198 of share elasticities, 465 supply correspondence and, 175 Nonvertical supporting hyperplane, 164, 172, 199, 201 202 Normalization of price and profit possibilities, 181 182 of price system, 196 of profit functions, 178, 196 restrictions, 140, 144, 146 149, 147 of supply correspondence defined, 180, 203 globally univalent, 204

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maximal, 203 maximal cyclically monotone, 203 of supply functions, 197 Norsworthy, J. Randolph, 42 Norway, 34 36, 45 O Office of Industrial Economics, 84 Office of Tax Analysis (OTA), 74, 80, 85, 87, 89 Oil prices, 343 Oliner, S.D., 74, 79 80, 85 87, 89 Olsen, Oystein, 34 One-to-one function, 193 194, 197 Optimal production planning, 68 Optimization, 68 69 Origin, 160 161, 175 OTA, 74, 80, 85, 87, 89 Outputs. See also specific types aggregate measures, 144, 376, 380 capital, 379 commodity, 160 consumption, 292 295 divisia price indices of, 388 390 fixed coefficients assumption and, 64 investment, 292 295 multiple, 60 63 prices, 4 5, 15, 145, 380 quantity of, 9 random effects of, 67 value of, 9 10

Ozanne, Larry J., 86 Ozatalay, Savas, 34 P Pairwise constancy, 281, 285 286, 288 289 Pairwise proportionality, 279, 281, 285 286, 288 289 Pakes, A., 86 Panel data techniques, 66 67 Parametric form, 3 5 Parameterization cost functions and, 51 53 price functions and, 14 16 Parks, Richard W., 35 Peterson, H. Craig, 59 Pindyck, Robert S., 45, 69 Pinto, Cheryl, 34, 43 Positivity, 10, 13, 47 Powell, Alan A., 131 Prais, S.J., 108 109 Pratt, J.W., 312 Price of capital, 366 367 elasticities, 393 395 of labor, 366 367 Price functions. See also specific properties applications of, 31 45 Berndt-Jorgenson model and, 31 33, 43 44 industrial sectors of U.S. economy, 31 37 Jorgenson-Fraumeni model and, 37 38, 41 overview, 31 37 technical change, 37 43 two-stage allocation, 43 45

concavity of, 10, 12 13, 18 19, 346, 367, 445 448 constant share elasticity and, 15 defined, 9 10 differentiability of, 10 domain and, 255 Douglas-type, 393 duality in, 8 11 first-order partial differential equations and, 15 Fourier functional form for, 33 34 homogeneity of, 10, 13, 443 445 integrability and, 17 19 Leontief, 37 logarithm of, 11 monotonicity of, 10, 13 nonjointness and, 276 277 overview, 6, 8 parameterization and, 14 16 production functions and, 8, 10 properties of, 10 11 separability of, 13, 36 substitution and, 11 14, 36 symmetry and, 443 445 technical change and, 11 14 transcendental logarithms and, 15 translog, 16 Price index of aggregate inputs, 11 14, 49 50 of capital goods and services, 76 of computers, 80, 85 86 constant quality, 73 defined, 13 14 divisia

of capital service input, 391 of labor input, 391 392

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(cont.) Price index output and input aggregate measures and, 376 of outputs, 388 390 equipment, 86 properties of, 13 14 of property, residential, 86 quality-corrected, 80 translog, 349 Price possibilities frontiers, 132 134, 382 383 set of, 175 177, 181 182, 200 202 normalized, 200 202 Producer behavior, econometric methods for modeling applications of cost functions, 55 63 economies of scale, 56 60 multiple outputs, 60 63 overview, 55 applications of price functions, 31 45 Berndt-Jorgenson model and, 31 33, 43 44 industrial sectors of U.S. economy, 31 37 Jorgenson-Fraumeni model and, 37 38, 41 overview, 31 37 technical change, 37 43 two-stage allocation, 43 45 background information, 1 2 cost functions, 45 55 cost flexibility, 47 48 degree of returns to scale, 47 48 duality, 46 48

economies of scale, 48 50 integrability, 51 53 overview, 45 46 parameterization, 51 53 properties, 47 stochastic specification, 53 55 substitution, 48 50 dual formulation of theory of, 6 7 in future, 63 69 dynamic models of production, 67 69 general equilibrium modeling, 64 65 panel data, 66 67 parametric form, 3 5 price functions, 8 19 defined, 9 10 duality, 8 11 integrability, 17 19 overview, 6, 8 parametrization, 14 16 properties, 10 11 substitution, 11 14 technical change, 11 14 production theory, 2 3 statistical methods, 20 31 autocorrelation, 22 24 estimation, 24 31 identification, 24 31 overview, 5 6, 20 stochastic specification, 20 21 traditional approach to, 2 Producer equilibrium, 8 Product exhaustion, 17, 476 477

Production bilateral models of, for Japanese and U.S. industries, 457 486 appendix, 476 486 background information, 457 458 conclusions, 472 475 framework, theoretical, 458 462 results, empirical, 462 472 differentiability in, 190 194 background information, 189 190 extensions and, 202 207 marginal productivity functions and, 192 194 production functions and, 190 192 production possibilities and, 198 202 twice, 205 207 duality in, 194 207 background information, 189 190, 194 extensions and, 202 207 Legendre transformation and, 194 196 normalized profit functions and, 196 normalized supply functions and, 197 production possibilities and, 198 202 dynamic model of, 67 69 extensions, 202 207 background information, 202 convex conjugacy, 202 205 twice differentiability, 205 207 in Federal Republic of Germany (1950 1973), 241 265 background information, 241 242

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convexity of translog production functions and, 249 252 economic interpretation of, 260 265 estimation statistics and, 255 260 integrability and, 247 249 monotonicity of translog production functions and, 249 252 production theory testing and, 252 255 test statistics and, 255 260 translog price functions and, 242 247 translog production functions and, 242 247 Koopmans' analysis of, 185 optimal planning, 68 parameter estimates for sectoral models of, 355 364, 414 431 plan, 68, 199 possibilities, 198 202 background information, 198 normalized price and profit, 200 202 set of, 198 200 Production functions closure and, 162 163, 179, 199 Cobb-Douglas, 4, 64, 120, 122 123, 143 144, 146 147 constant elasticity of substitution and, 3 convexity of, 163, 177 179, 184, 192, 196, 202 205 differentiability in production and, 190 192 domain and, 162, 179, 191 duality of technology and economic behavior and, 177 179 epigraph of, 198 essentially smooth, 191 192 essentially strongly convex, 206 207 general form of, 120 homogeneity of, 251

linear logarithmic, 64 marginal productivity correspondence and, 159, 164 169 monotonicity of, 253 254 nonjointness and, 273 277 normalized restricted profit function and, 184 price functions and, 8, 10 production theory and, 161 163, 185, 205 206 smoothness of, 191 192, 206 subdifferential of, 164, 184 subgradient of, 164 165 supply and demand functions, 2 welfare maximization and, 115 Production possibilities frontiers, 128, 130 set of, 159 161, 181 182, 198 202 Production theory classic formulation of, 3 4 duality in, 1, 3, 194, 245 in future, 63 64 marginal productivity correspondence and, 159 producer behavior and, modeling, 2 3 production functions and, 159, 161 163, 185, 205 206 set of production possibilities and, 159 161 traditional approach, 2 3 Production theory testing approximation, 142 144 background information, 125 127, 134 135 conclusions, 153 156 direct, 151 155 duality, 144 146, 154 equality, 139 140 estimation, 147 150, 255 260

factor augmentation, 140 141 group-wise additivity, 141 indirect, 151, 153 154, 156 production in Federal Republic of Germany (1950 1973) and, 252 255 results, empirical, 146 154 significance and, levels of, 151 152 statistical methods, 150 153 stochastic specification, 138 139 summary, 146 147 symmetry, 139 140 translog price frontiers and, 137 138 translog production frontiers and, 135 137, 154 Productivity. See also Energy in productivity growth biases of, 409 410 capital-saving, 410 electricity-using, 410 411 estimates, 91 93 growth, 406, 410 411 U.S., 406

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Profit functions, 120, 170 172 closure and, 171 172, 179 convexity of, 171 172, 179, 184 domain and, 171, 179, 196 duality of technology and economic behavior and, 170 172, 177 179 homogeneity of, 171 normalized, 178, 184, 194, 196 smoothness of, 196 subdifferential of normalized, 203 subgradient of, 202 203 Profit maximization, 268 Profit possibilities, set of, 175 177, 200 202 normalized, 200 202 Property, residential, 86 Q Quality-corrected price index, 80 Quantity index, 13 14, 49 50, 76 R Railroads, Class I U.S., 60 61, 67 Random disturbances / effects, 20 21, 54, 67, 349 354 RAS method, 389 Rasche, Robert, 86 Rate of technical change average, 23 capital and, 367 368 defined, 9, 263 265 differences in technology and, 464 465 homogeneity and, 17 negative, 10, 346, 348, 355 pattern of, 343

random disturbances to, 20 21, 351 354 translog, 16 Regularity conditions three-stage least-squares estimation and, 29, 316 317 two-stage least-square estimation and, 331 332 Reifschneider, D., 58 59 Relative efficiencies of vintages of capital, 75 Relative income hypothesis, 99 Relative interior of set, 165, 203 Relative prices in Japan, 373 402 background information, 373 379 data sources and, 388 392 design of experiment and, 388 392 KLEM model estimation and, 383 388, 393 measurement of technical change and, 379 381 results, empirical, 392 402 specification of price possibility frontiers and, 382 383 stochastic specification and, 383 388 supply and demand functions and, 2 technical change and, 341 370 background information, 341 343 conclusions about, 369 370 econometric models, 344 354 results, empirical, 354 369 Rental prices, 76, 86, 92 93 Restricted W matrix, 220 222 R.H.S. full income, 114 Roberts, Mark J., 58, 60 Robertson, J.W., 80, 84, 87, 89 Rosenberg, Nathan, 403 404, 409 410, 437 439, 442 Rotemberg, Julio J., 69

Rothenberg, Thomas J., 210, 222, 239 Russell, Clifford S., 35 S Saddlepoint of Lagrangian function, 28 Samuelson, Paul A., 2 4, 37, 65, 106, 107, 273, 342 Sargent, Thomas J., 69 SAS package, 313 Sato, Kazuo, 132 Schankerman, Mark, 62 Scheffé, H., 312 Schoech, P.E., 62 Schultz, T.W., 98 Schurr, Sam, 403 407, 437 442 Schwartz, Sandra L., 34 Searle, S.R., 325, 334 Sen, A.K., 98 100 Separability groupwise, 268, 295 303, 306 307 homothetical, 13 14, 49 50 of labor inputs, 36 37 of price functions, 13, 36 restrictions, 154 technology structure over time in Federal Republic of Germany (1950 1973) and, 295 298 Set of normalized price and profit possibilities, 200 202

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Set of price possibilities, 175 177, 181 182, 200 202 Set of production possibilities, 159 161, 181 182, 198 202 Set of profit possibilities, 175 177, 200 202 Share elasticities, 4, 11 12, 15, 18 19, 38 39, 51, 261 263, 345, 366 367, 444, 459, 463, 465 468 Shares of inputs, 9 Shephard, Ronald W., 3, 186 Shephard's lemma, 383 Shibboleths (distribution schemes), 106 Shoven, John B., 64 Significance, levels of, 151 152, 258, 306 Smith, V. Kerry, 35, 42, 58 Smoothness of production functions, 191 192, 206 of profit functions, 196 set of price and profit possibilities and, 201 set of production possibilities and, 200 Social consciousness, 100 Social goodwill, 100 Social welfare function, 89 90, 98 100. See also Welfare function Solow neutrality, 267 Solow, Robert M., 2 3, 125, 143, 154 Souza, Geraldo, 16, 34 Spady, Richard H., 35, 61 62 Standard income, achieving, 109 Standard of living, achieving, 109 Statistical inference for a system of simultaneous, nonlinear, implicit equations appendix, 335 339 background information, 311 313 three-stage least-squares estimation, 313 329 analog of likelihood ratio test, 315 316

asymptotic distribution of TA, 324 326 asymptotic distribution of TO, 317 323 hypothesis of symmetry, 326 329 regularity conditions, 316 317 statistical model subject to a maintained hypothesis, 313 315 statistical package and, 313 Wald test, 323 324 two-stage least-squares estimation, 329 335 analog of likelihood ratio test, 330 331 asymptotic distribution TA, 333 334 asymptotic distribution TO/a, 332 333 hypothesis of homogeneity, 334 335 regularity conditions, 331 332 statistical model, 329 330 statistical package and, 313 Wald test, 333 Statistical methods. See also specific types autocorrelation, 22 24 estimation, 24 31 identification, 24 31 overview, 5 6 production theory testing and, 150 153, 255 260 stochastic specification, 20 21 test statistics technology structure in Federal Republic of Germany (1950 1973) and, 283 286 technology structure over time in Federal Republic of Germany (1950 1973) and, 303 306 Stevenson, Rodney E., 58 59 Stochastic specification cost functions and, 53 55 as econometric model of energy in productivity growth and, 449 453 production theory testing and, 138 139

relative price changes and biases of technical change in Japan and, 383 388 statistical method of, 20 21 Subdifferential of production functions, 164, 184 of profit functions, 203 Subgradient of production functions, 164 165 of profit functions, 202 203 Subjective equilibrium, 97 98, 115 112 Substitute inputs, 49 Substitution Binswanger and, 41 capital-labor, 154 constant elasticity of, 3 5 cost functions and, 48 50 econometric models of, 43 44

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(cont.) Substitution elastiticity of, 3 4 energy in productivity growth and, 413 432 inputs and, 12 13 Jorgenson and, 42 Kopp and Smith and, 35, 42 measures of, 3 4, 7, 11 patterns of, 343 price functions and, 11 14, 36 restrictions on, 13 share elasticities and, 12 13 Wills and, 42 Sullivan, M.A., 84 Supply correspondence domain and, 173, 180 181 duality of technology and economic behavior and, 172 175, 179 181, 184 homogeneity and, 174 monotonicity and, 173 174 nonnegativity and, 175 normalized defined, 180, 203 globally univalent, 204 maximal, 203 maximal cyclically monotone, 203 Supply and demand functions, 2, 4 5, 194, 197, 268, 272 Support function of set, 170 Swanson, J.A., 60 Sweden, 34 35, 44 45 Symmetry

hypothesis of, 326 329 integrability and, 17 18, 52, 477 KLEM model and, 383 384 price functions and, 443 445 production theory testing and, 139 140 restrictions, 139 140, 144, 146 149, 251, 253, 276, 281 T Taiwan. See Agriculture household behavior Tangent hyperplane, 199 Taubman, Paul, 86 Tax law, 85, 91 Tax Reform Act (1986), 85 Taylor's series expansion, 16, 122, 242, 245, 270, 318 320 Technical change biases of Binswanger and, 42 capital inputs and, 367 368, 376, 433 classification of industries by, 365, 435 436 defined, 11, 263 264, 459 460 electricity inputs and, 434 energy inputs and, 368 Hicks and, 4 interpretation of, 39 41 patterns of, 39 41 quantity of input and, 469 470, 480 486 vector of, 11 12, 15, 459 460 Binswanger and, 41 capital-using, 469 deceleration of, 12 energy in productivity growth and, 405 406, 433 436 input-saving, 11, 459 input-using, 11, 459 in Japan, 373 402

background information, 373 379 data sources and, 388 392 design of experiment and, 388 392 KLEM model estimation and, 383 388, 393 measurement of technical change and, 379 381 results and, estimated, 392 402 specification of price possibility frontiers and, 382 383 stochastic specification and, 383 388 Jorgenson and, 42 measures of, 3 4, 7, 11, 379 381 negative, 39 neutral, 11 parameter estimates for sectoral models of, 355 364, 414 431 patterns of, 434 436 price functions and, 11 14 application of, 37 43 rate of average, 23 capital and, 367 368 defined, 9, 263 265 differences in technology and, 464 465 homogeneity and, 17 negative, 10, 346, 348, 355 pattern of, 343

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random disturbances to, 20 21, 351 354 translog, 16 relative prices and, 341 370 background information, 341 343 conclusions about, 369 370 econometric models, 344 354 results, empirical, 354 369 restrictions on, 13, 279 technology structure in Federal Republic of Germany (1950 1973) and, 268, 277 281 technology structure over time in Federal Republic of Germany (1950 1973) and, 298 301 translog price index and, 22 Wills and, 42 Technology agricultural household behavior and, 109 112 economic behavior and, duality of, 160 169 background information, 159 160, 160 conclusions about, 182 185 historical note about, 185 186 marginal productivity correspondence and, 164 169 production functions and, 161 163 set of production possibilities and, 160 161 index of, 135 Japanese versus United States, 460 461, 464 465, 470 472 random elements in, 20 21, 24 structure in Federal Republic of Germany (1950 1973), 267 289 background information, 267 269 conclusions about, 286 289 estimation and, 283 286 nonjointness and, 273 277, 281, 283, 285, 287 statistical methods and, 283 286

technical change and, 268, 277 281 test restrictions and, 281 282 translog price functions and, 269 272 translog production functions and, 269 272 structure over time in Federal Republic of Germany (1950 1973), 291 310 background information, 291 295 conclusions about, 307 310 estimation and, 303 306 separability and, 295 298 statistical methods and, 303 306 technical change and, 298 301 test restrictions and, 301 303 Terborgh, George, 81 Test statistics for homogeneity, 56 57 for homotheticity, 56 57 TA, 323 325 TA/a, 333 334 TO, 315 316 TO/a, 330 331 technology structure in Federal Republic of Germany (1950 1973) and, 283 288 technology structure over time in Federal Republic of Germany (1950 1973) and, 303 306 for translog production and price functions, 258 259 Testing. See Production theory testing; specific tests Theory of Wages (Hicks), 3 Thibodeau, Thomas G., 86 Thompson, R.G., 35 Three-stage least-squares estimation analog of likelihood ratio test and, 315 316 asymptotic distribution of TA and, 324 326 asymptotic distribution of TO and, 317 323 framework of bilateral models of production for Japanese and U.S. industries and, 461 462

hypothesis of symmetry and, 326 329 regularity conditions and, 29, 316 317 statistical model subject to a maintained hypothesis and, 313 315 statistical package and, 313 Wald test and, 323 324 Time constraint, 112 113, 115 Toevs, Alden L., 36 Torii, Y., 98 Total expenditure constraint, 114 115 Total labor constraint, 114 115

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Transcendental logarithmic price frontiers. See Translog price frontiers Transcendental logarithmic price functions. See Translog price functions Transcendental logarithmic production frontiers. See Translog production frontiers Translog cost functions, 51, 59 63 Translog marginal productivity functions, 248 Translog price frontiers approximations and, 126 described, 126 production theory testing and, 137 138 representation of, 137 138 Translog price functions concavity of, 19, 347 348 defined, 15 16 econometric models of relative prices and technical change and, 344 349 monotonicity of, 251 parameter estimates for, 256 257 production in Federal Republic of German (1950 1973) and, 242 247 random disturbances and, 349 354 technology structure in Federal Republic of Germany (1950 1973), 269 272 test statistics for, 258 259 Translog price index, 349 Translog production frontiers additivity and homogeneity and, 127 134 background information, 127 commodity-wise additivity, 127 129, 143 144, 146, 151, 153 group-wise additivity, 129 132, 134, 141, 143 144, 151, 153 154 multi-level additivity, 131, 134, 153 154 price possibility frontier and, 132 134 background information, 125 127 described, 126

production theory testing and, 135 137, 154 representation of, 135 137 Translog production functions convexity of, 249 252 monotonicity of, 249 252 parameter estimates for, 256, 263, 283 285, 304 306 production in Federal Republic of Germany (1953 1970) and, 242 247 technology structure in Federal Republic of Germany (1950 1973), 269 272 test statistics for, 258 259 Translog rate of technical change, 16 Transportation industry, 55, 60 61, 67 Trethaway, Michael W., 41 42, 60, 62, 66 67 Triplett, Jack E., 80 Trucking industry, 61 62 TSP package, 313 Turnquist, Mark A., 61 Twice differentiability, 205 207 Two-level additivity, 130 131 Two-phase decision-making process, 118 Two-stage allocation, 43 45 Two-stage least-squares estimation analog of likelihood ratio test and, 330 331 asymptotic distribution of TA/a and, 333 334 asymptotic distribution of TO/a and, 332 333 hypothesis of homogeneity and, 334 335 regularity conditions and, 331 332 statistical model, 329 330 statistical package, 313 Wald test and, 333 U United Kingdom, 36, 45 United Nations System of National Accounts, 90

bilateral models of production in, 457 458 United States agriculture in, 41 bilateral models of production in appendix, 476 486 background information, 457 458 conclusions about, 472 475 framework, theoretical, 458 462 Class I railroads in, 60 61, 67 communications industry in, 55 manufacturing in, 34 36, 41 42, 45 productivity, 406

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steel industry in, 42 technology in, 460 461, 464 465, 470 472 transportation industry in, 55, 61 62, 67 trucking industry in, 61 62 Unrestricted W matrix, 213 219 U.S. Bell System (1947 1977), 62 U.S. Inter-industry Transactions Accounts, 354, 411 U.S. National Income and Product Accounts, 73 74, 80 81, 86 90, 93, 354, 411 Utility functions, 99 100, 107 109, 122 123, 126 Uzawa, Hirofumi, 3, 68, 125, 132, 186 V Value and Capital (Hicks), 2 Variables, instrumental, 229 232 Vaughn, William J., 35 Veblen, T., 99 Vintage accounts, system of, 74 81, 85 86, 93 94 Von Weizsacker, Carl Christian, 37, 342 W Wald test, 312, 323 324, 333 Walsh, M., 84 85, 87, 89 Ward, B., 99 Watkins, Gordon C., 68 Waugh, F.V., 79 Waverman, Leonard, 62 63, 68 Webster, Alan R., 34 Welfare function decentralization and, 107 egalitarian, 101 106 family dependents and, 107 109 family workers and, 108 109, 119

formulation of, 98 101 homogeneous, 105 108, 111 inter-temporal, 107 utility function and, 106 107 Welfare maximization, 97 98, 109, 115 122 West Germany. See Federal Republic of Germany (FRG) Western Electric, 62 Wharton, C.R., Jr., 109 Wills, John, 42 Windle, Ralph, 67 Winfrey, R.C., 84 Winston, Clifford, 61 62 Wolkowitz, Benjamin, 35 Wood, David O., 31, 33 34, 36 37, 42, 69 Woodland, Alan D., 37, 45 Woodward, G. Thomas, 42 Wykoff, Frank C., 74, 78, 80 87, 89, 92 94 Y Yatchew, Adonis, 69 Young, Allan, 88 Yun, Kun-Young, 89, 91, 94

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