AN INTERNATIONAL STUDY OF MANUFACTURING PRODUCTION FUNCTIONS: ESTIMATION AND IMPLICATIONS “Thesis for the Degree of Ph.‘ D. MICHIGAN STATE UNIVERSITY KYE WOOLEE . ' 1972 LIBRARY E Michigan State 5, University This is to certify that the thesis entitled AN INTERNATIONAL STUDY OF MANUFACTURING PRODUCTION FUNCTIONS: ESTIMATION AND IMPLICATIONS presented by Kye Woo Lee has been accepted towards fulfillment of the requirements for Ph.D. degree in Economics @942} W- {m Major professor Date W ”27/ ”72 I m: V BINDING IY IIIIAB I: SUNS’ BIIIII'. BIIIIIUIY INC. ‘ 1:52.52? SINGERS OIIIIIDIIT IICIIBII ABSTRACT AN INTERNATIONAL STUDY OF MANUFACTURING PRODUCTION FUNCTIONS: 'ESTIMATION AND IMPLICATIONS BY Kye Woo Lee The purpose of this study is to provide an addi— tional contribution to the inquiry of empirical relevance of some of the fundamental assumptions of the modern inter- national trade theory (with special reference to the theory of Heckscher—Ohlin and international factor-price equaliza- tion) which goes along with the comparative advantage. Without exception, all the critical assumptions in these theories concern the production relations in each industry. Therefore, this thesis attempts to estimate production functions for three-digit manufacturing indus- tries of the International Standard Industry Classifica- tion (ISIC) using combined international cross—sectional and time-series data obtained from the official reports of each country sampled. Kye Woo Lee Some theoretical consideration of the models most .frequently used in the past is followed by the development of an extended CES (Constant Elasticity of Substitution) model, which allows for nonunitary returns to scale and by the examination of more generalized production functions. The models used in the estimation are four general- ized production functions discussed recently in the literature as well as the extended CES model developed in this thesis. To make a choice among the alternative forms of production functions, some specification error tests are performed, in addition to the classical methods of model discrimination. The practical relevance of . relatively simple verSions of production functions is investigated in relation to more generalized production functions. Outputs,fixed capital assets, and wage rates are deflated by appropriate price indices. The rate of returns to capital is calculated by assuming that total value added is imputed to the factors of production, i.e., capital and labor. Further, using the stock value of human capital for each country,labor quality indices are con- structed by industry, allowing us to make adjustments for the international labor quality difference. The labor quality indices confirm differences in the quality of labor across countries, but significantly below the level suggested by Leontief. The determination of an appropriate Kye Woo Lee functional form is little affected by this adjustment, though the regression coefficients and other parameters (e.g., the elasticity of substitution) are affected. Although the estimation procedures are not flaw- less and the selection of one overall best specification of the production function is intractable, one or two specifications significantly outperform the others. Significant differences in the estimates of para- meters are obtained from the usual least squares estimates and the analysis of covariance estimates or the covariance transformation estimates. Country constants and time con- stants are obtained and interpreted. The generalized production functions often reduce to simpler forms such as CES, CD (Cobb-Douglas), and FC (Fixed Coefficients) in many industries. The elasticity of substitution of different industries is not far from unity and is not quite different from each other. Even in the variable elasticity of substitution cases, it approaches one or varies in a very narrow range around one. Averages of point estimates for VES support the conclusion that the elasticity of substitution is around one. Simultaneous estimation of the returns to scale with the elasticity of substitution is not quite success— ful, though fairly good estimates are obtained. The pro- duction function which allows for varying returns to scale with the output level is tested, but the added complication Kye Woo Lee is not quite fruitful either. This is, however, an area requiring further research. As to the value of the returns to scale, almost all industries exhibit the in- creasing returns to scale and the degree of homogeneity is different among industries. Having estimated the parameters of the production functions, their implications for international trade patterns are considered. The investigation of the validity of the factor—intensity hypothesis, one of the basic assump- tions of the modern trade theory based on the comparative advantage, reveals that reversals in factor intensity are quite possible in the empirically relevant ranges of factor-price ratios and factor—use ratios. Yet a somewhat different interpretation from the conventional View is made and the comparative advantage theory based on factor endowments is regarded as not seriously spoiled due to the reversals. A possible alternative hypothesis which does not require the factor—intensity assumption emerges from the observed differences in the economic efficiency of the countries. AN INTERNATIONAL STUDY OF MANUFACTURING PRODUCTION FUNCTIONS: ESTIMATION AND IMPLICATIONS BY Kye Woo Lee A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1972 To Yongwoo ii ACKNOWLEDGMENTS I am deeply indebted to many persons who have assisted me for the completion of this dissertation. I would particularly like to thank Professor Anthony Koo, who directed me to the topic of this dissertation, for his comments and encouragement at various stages of this study. I am also very thankful to Professor James Ramsey for his valuable suggestions on various aspects of this dissertation. Professor Paul Strassmann provided me with a very helpful opportunity to present the earlier version of this dissertation at the joint meeting of the Econometrics Workshop and the Economic Development Graduate Seminar of the Michigan State University. Under the pressing time constraint, Professor Robert Gustafson read the whole manuscript and gave me good advice. I also wish to thank Professor Jan Kmenta for his comments on the first draft of this dissertation, and Professor Leo Sveikauskas for his endeavor to point out various problems of production function study. Use of the Michigan State University Computing facilities was made possible through support, in part, from the National Science Foundation. iii Finally, I would like to thank many foreign government officials and my friends who helped me in gathering the data necessary for this study. iv llilll TABLE OF CONTENTS Chapter I. II. III. IV. INTRODUCTION 0 O O O O O O O O O O O O O O l. The Importance of Production Functions in Economic Analysis . . . . . . . . . 2. Statement of the Problem . . . . . . . 3 O overView O O O O O O O O O O O O O O 0 REVIEW OF EMPIRICAL STUDIES USING THE CES MODEL 0 I C O O O O O O O O O I O O O 1. Introduction . . . . . . . . . . . . . 2. Cross—Sectional Studies . . . . . . . 3. Time«Series Studies . . . . . . . . . SOME EXTENSIONS AND REFORMULATIONS OF THE CES MODEL . O I C O C O O O O O O C O O O 1. Composition of an Industry . . . . . . 2. Differences in Labor Quality . . . . . 3. Variations in Economic Efficiency . . 4. Returns to Scale . . . . . . . . . . . 5. Uses of the Capital Variable . . . . . 6. Model Discrimination and Pooling Observations O O O O O O O O O O O O O PRODUCTION FUNCTIONS AND THE COMPARATIVE ADVAN TAGE . O O O I C O O O O C C O O O O 1. Introduction . . . . . . . . . . . . . 2. Empirical Verifications of the Factor— Intensity Hypothesis . . . . . . . . . Page 10 11 ll 20 26 33 34 35 41 43 50 69 78 78 86 Chapter VI. SOME BIBLIOGRAP APPENDICES A. The Procedure of the Present Study . . . The Models and the Discrimination . . . Adjustments for the Economic Efficiency Difference . . . . . . . . . Adjustments for the Labor Quality Difference O O C O I O O O O O O O O O O The Data . . . . . . . . . . . . . . . . EMPIRICAL RESULTS . . . . . . . . . . . Production Functions for the Industries The Regression Results . . . . . . . . . Estimates of the Elasticity of substitution 0 O O O O O O I O O O O O 0 Estimates of Returns to Scale . . . . . Measurements of the Economic Efficiency IMPLICATIONS OF THE EMPIRICAL STUDY . . HY Q 0 O O O O O O O O O I O O O O O O 0 Empirical Studies Using CES Class of Production Functions . . . . . . . . . . Countries and Years Observed . . . . . . Sources of Statistical Data . . . . . . Labor Productivities, Wage Rates, and Input Ratios . . . . . . . . . . . . . . Ramsey's Specification Error Tests . . . vi Page 91 92 104 108 112 116 116 139 151 169 172 178 190 200 200 201 202 204 223 Table 1. 2. 10. 11. 12. 13. LIST OF TABLES Ranking of Countries by Labor Efficiency' (Present Study) . . . . . . . . . . . . . . . Ranking of Countries by Labor Efficiency (Gupta ' S StudY) O I O O O O O O O O O O O O 0 Summary of the Specification Error Tests (SET) 0 O O O I O I O O O O O O O O O I The Best Performing Models in the SET . . . . The Number of Significant Cases for coeffiCientS O O O I I O O O I O O O O O O 0 Regression Results of the Four Models . . . . Results of the Selected Regression Equations Estimates of the Elasticities of Slletitution o o o o o o o o o o o o o o o 0' Estimates of the Elasticity of Substitution by Previous Empirical Studies . . . . . . . . Estimates of the Returns to Scale from the K—MOde l o o o o o o q o o o o o o o o o o o 0 Economic Efficiency of Countries by Industry 0 O O O O O O O O O O O I O O O O O Ranking of Industries by Input Ratio and EffiCiency (UOSQAO) O O O O O O O O O O O O O Ranking of Industries by Input Ratio and Efficiency (Korea) . . . . . . . . . . . . . Vii Page 110 111 117 138 140 141 145 152 154 171 174 186 188 Factor—Price and Factor—Use Ratios and LIST OF FIGURES the Elasticity of Substitution . . . . The Fixed—Coefficients Case . . . . . Factor Substitutions and the Intersection of Isoquants . More than Two Intersections of the Isoquants The Substitution Curves in VES Production Functions . The Substitution Lines in the CES case Scatters of Factor Intensities (370) . Scatters of Factor Intensities (383) . Scatters of Factor Intensities (381) . The Relationship between the Elasticity and the Input Ratio (ISIC (ISIC (ISIC (ISIC (ISIC (ISIC (ISIC 205) 232) 251) 260) 311) 360) 391) Substitution Lines Figure 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 10ea lO—b lO—c lO-d lO—e lO-f 10-9 11. 12. Substitution Lines Crossovers (I) Crossovers (II) viii Page 52 79 82 84 87 89 157 158 159 162 163 164 165 166 167 168 179 181 CHAPTER I INTRODUCTION 1. The Importance of Production Functions In Economic AnaIQSIS Since World War II, main economic problems are centered around the issues of a. unemployment, b. low rate of economic growth, and c. inequitable distribution of income. Seriousness of the unemployment problem and investi- gation of possible remedies have received special attention in recent years. Many different lines of thought have been advanced by different economists. Primarily, labor is a factor of production, and the only way one can investigate the possibilities of increased employment of a foctor of production is by investigating the possibility of (a) increasing output and (b) substitut- ing this factor of production for other factors of produc— tion. Increase in output is an economic growth problem and production functions specify a long-run relationship between inputs and outputs. Therefore, production func- tions must play a crucial role in the theory of economic growth. Substitutability among the factors of production can not be studied outside a framework of production rela- tions. Whatever measure of substitutability one may choose, it would inevitably involve marginal product of that factor, and one can not say anything about it outside the production relations. To achieve a rapid economic growth, enormous efforts have been exerted to inquire into the causes of growth, investment behavior, monetary and fiscal policy, role of international trade, productivity and technological change, and investment in human capital. From the theoretical point of View, a discussion of growth rates inevitably introduces the concept of a pro— duction function. When we are analyzing growth rates, we are actually analyzing how changes in factors of production cause changes in output under different assumptions or how changes in one factor are compatible with changes in other factors to achieve a given rate of growth. The most impor- tant implication of any aggregate production function will be in the linking of changes in factor supplies and output,, in the aggregate, over time and thus to the understanding of economic growth (Nerlove, 1967, p. 56). The problem of income sharing is inextricably linked with the level of substitution possibilities, the way they are changing with the level of capital-labor ratio, and the change in factor-price ratio. It is quite clear that the main economic problems of recent years are interrelated, that all these relation— ships can only come out of the same estimated model of production, and that the substitution possibilities, returns to scale, and technological changes in the produc- tion relations are the key questions. For instance, many theoretically controversial issues would be resolved with the knowledge of the substitu- tion between factors at hand. (Although different measures of substitutability and their merits and demerits are dis- cussed in the literature (Singh, 1970), we will adhere in this thesis to the Hicksian definition of the elasticity of substitution concept.) In their landmark article, Arrow, Chenery, Minhas, and Solow (ACMS) suggested three important areas in which knowledge of the elasticity of substitution plays a crucial role: (1) The stability or instability of certain growth paths implied by some model, notably the Harrod—Domar model, depends in a critical way on the assumption of the value of the substitution between labor and capital (Solow, 1956, Swan, 1956, etc.). (2) The effects of varying factor endowments on international trade hinge on the shape of particular production functions. Either zero or unitary elasticity of substitution in all sectors of the economy leads to Samuelson's strong assump- tion about the invariability of the ranking of factor proportions. Variations in elasticity among sectors, however, imply reversals of factor intensities at different factor prices with quite different consequences for trade and factor returns. (3) ACMS reiterated the traditional importance of the elasticity of substitution for relative shares over time (Arrow, gt_al., 1961, p. 225; also see Morrissett, 1953). Certainly, these cases do not exhaust the list. For example, in development economics, the labor—intensive or capital-intensive argument is a classical controversy and is still discussed in the context of choice of techni- ques and investment criteria (resource allocation). But the validity of this argument depends on the magnitude of the elasticity of substitution of different industries. It has been argued recently that controversy between the manpower planning approach and human investment approach to educational planning stems mainly from the different assumptions about the substitutability between factors of production (between capital and labor, or between unskilled labor and skilled labor, etc.) (Blaug, 1967 and Bowles, 1970). However, there has been rare consensus about the estimates of the parameters in production functions and even the framework of its estimation. After an extensive survey of the empirical analysis of production functions, Nerlove described his major finding as "the diversity of results": Even slight variations in the period or concepts tend to produce drastically different estimates of the elasticity. While there seems little rhyme or reason for most of these differences, a number of possible sources of bias exist and may account for at least some of the discrepancies. (1967, p. 58.) Our knowledge about other parameters of the production relations is little better or is even in a worse state. 2. Statement of the Problem In the past decade in economic development, analysis of growth has relied mainly on Keynesian tools and has pro- duced proliferative aggregate growth models. In the field of resource allocation, controversy has centered around the validity of the classical principle of comparative advantage, which implies growth is promoted by specializa- tion following the comparative advantage. The classical approach derives this principle from international trade theory. Classical economists have sought to learn the sources of those comparative advantages within the context of their labor theory of value, but with the demise of the labor theory of value another explanation was needed. The modern version of the comparative advantage doctrine is essentially a simplified form of static general equilibrium theory. The optimum pattern of pro- duction and trade for a country is determined from a com- parison of the opportunity cost of producing a given commodity with the price at which the commodity can be imported or exported. In equilibrium, no commodity is produced which could be imported at lower cost, and exports are expanded until marginal revenue equals marginal cost. Under the perfectly competitive equilibrium, the opportunity cost of a commodity—~the value of the factors used to pro- duce in their best alternative use—-is equal to its market value. Market prices of factors and commodities can, therefore, be used to determine comparative advantage under competitive conditions. Long-term changes are not ignored, but they are assumed to be reflected in current market prices. The assumption of perfectly competitive equilibrium is quite critical in this argument, although it is con- tended that even in the case of imperfect markets or no market at all, the comparative advantage can be determined by the use of shadow prices. The Heckscher—Ohlin version of the comparative cost doctrine (Heckscher, 1950, Ohlin, 1933), which states that the major cause of comparative cost difference and inter- national trade patterns between countries is to be found in their relative factor endowments, has filled this gap, because it provides a measure of comparative advantage that does not depend on the existence of perfect competi- tion and initial equilibrium. In its simplest form, this theorem states that a country will benefit from trade (and ultimately promote economic growth) by producing commodities that use more of its relatively abundant factors of produc- tion. A country will export these commodities and import commodities which use more of its relatively scarce factors unless its pattern of domestic demand happens to be biased toward commodities using more of the country's abundant domestic factors of production.1 Because of the simplicity and plausibility of the theorem, it has considerable appeal and has received much attention in recent years. The critical assumptions in this theorem, however, are that: 1. Factors of production are comparable among countries, 2. Production functions are the same for each com- modity among countries exhibiting unitary returns to scale,2 lDistortion of the theorem due to the demand condi- tion happens when the relative abundance of a factor is defined in terms of the ratio of its quantity to the quan- tity of the other. If one defines the relative abundance of a factor in terms of its relative cheapness, the Heckscher- Ohlin proposition is valid. For more details, see Mookerjee (1958), Chapter three. To insure that abundance in terms of relative quantity and abundance in terms of relative price go together, the assumption of taste similarities has usually been made. 2Unit homogeneous production functions are not essen- tial in the theory of Heckscher-Ohlin. As Vanek (1962) indicated for the two—commodity, two—country case, if the degrees of homogeneity of the production functions for the two commodities are the same, the factor proportions theory remains intact. Distortions occur if the production func- tions for the commodities differ with respect to their degree of homogeneity. See Vanek, 1962, pp. 109—91. 3. The production functions are such that the commodities can be ranked unequivocally by factor- intensity (factor—use ratio). Presuming that these assumptions in fact hold, Heckscher concluded that differences in factor—prices are essential to differences in comparative costs (Heckscher, 1950, pp. 271—8). However, attempts which have recently been under— taken to verify empirically the Heckscher-Ohlin theorem have generally yielded results paradoxical to the theorem.3 Although some possible explanations have been made to attribute these contradictory results to statistical methods used, selection of unrepresentative data, and other difficulties in the empirical studies, the recent trend in studies has been to place the major blame on the Heckscher-Ohlin theorem itself, especially on its basic underlying assumptions. To date, the major study on the empirical relevance of several of these basic assumptions is the celebrated ACMS (1961) and Minhas (1963) papers. However, their investigation was not specifically designed for this pur- pose. Moreover, their study has several shortcomings as 3See, for example, Leontief (1953) for the U.S. case, Wahl (1961) for the Canadian case, Tatemoto and Ichimura (1959) for the Japanese case, Bharadwaj (1962) for the Indian case, and Roskamp (1954) for the West German case. will be discussed later in Cahpter II, Section 2. As a result, the study does not provide as definitive an approach to this problem as one would like. The purpose of the present study is to provide an additional contribution to the inquiry of empirical rele— vance of some of the fundamental assumptions of modern international trade theory, especially empirical signifi- cance of the factor intensity hypothesis. Without exception, all three critical assumptions mentioned above concern the production relations in each industry. There— fore, the efficient method of verifying the empirical validity of these assumptions would be to start with the correct specification of the production relations and the efficient estimation of the parameters. The unambiguous factor-intensity assumption also plays an important role in many other theorems of trade and development, as well as in the Heckscher—Ohlin theorem. In particular, the validity of the full factor-price equali- zation depends on the industries' comparative positions on the factor—intensity scale.4 In the traditional demonstra— tion of full factor—price equalization, it was assumed that one commodity would be deSignated as relatively 4The factor-price equalization theorem implies that free international trade in commodities tends to equalize factor prices between countries, thereby serving, to some extent, as a substitute for factor movements: 10 labor-intensive and the other commodity as relatively capital-intensive, irrespective of the relative price of the factors. This is called the strong Samuelsonian factor—intensity assumption, arising from an implicit assumption that the production function for the two com- modities, while different from each other, nevertheless both possess the property of having unitary elasticity of substitution between the factors. Samuelson (1951-52) thought that the reversal of factor intensities could occur theoretically, but that this was not apt to happen within the range of the factor-price ratio that might in practice be observed internationally (pp. 121-2). 3. Overview A brief review of past analyses of production functions is made in Chapter II. In Chapter III some extensions and reformulations of the previous models are attempted, in light of the shortcomings of past works. An empirical estimation of the production functions, incor- porating all the theoretical considerations discussed in the previous chapters, is presented in Chapter IV. The results are reported in Chapter V. Chapter VI concludes the study with a discussion of the implications of the estimates of the production parameters for the theory of international trade. CHAPTER II REVIEW OF EMPIRICAL STUDIES USING THE CES MODEL 1. Introduction Many economists seem to agree that the concept of production, efforts to discover the forces governing pro- duction, and attempts to express the relationship between inputs and outputs in functional forms have passed three main stages of thought. The first one began with the eighteenth century and can be characterized by a long period of fragmentary and unsystematic speculation, reach- ing its terminus in the 1880's. The second stage opened in the 1890's, with the explicit formulation of the mar- ginal productivity analysis, considered both as a theory 0f factor service pricing and as a theory of the functional distribution of income among factors of production. The final stage, starting early in the 1920's with the pioneer- ing work of Cobb-Douglas, is still in its course. Until the beginning of the 1960's, studies on pro— duction functions followed two main streams: the estima- tion of Cobb-Douglas (CD) production functions and the 11 12 input-output work begun by Leontief at about the same time as CD function. Since that time, the two approaches having been generalized, the most important discovery on the neo- classical side was the CES production function by Arrow, Chenery, Minhas, and Solow. A selective list of previous empirical studies which employed the CES production function as their frame- work to estimate the elasticities of substitution and other parameters is provided in Appendix A. The general impres— sion one gets from this review of empirical work can be summarized as follows: 1. As in many demand analyses, we find a tendency for cross-sectional estimates of parameters to be greater than time-series estimates (see Table 9). 2. The CES production function fits the inter— country cross-sectional data better than the U.S. regional or state data and the estimated standard errors are smaller than for cross—state data. This may be true becuase of the enormous range of the independent variable in the inter- national sample.1 3. However, there are numerous sources of possibly serious bias in the model of cross-sectional studies. In fact, it has been well pointed out that a serious defect lMinhas (1963, p. 15) articulated some advantages of an international study. See also Ball (1966), Bardhan (1967), and Robinson (1964) for criticism of this approach. 13 of cross—sectional studies is misspecification of the theoretical model. To see these possible sources of bias, let us specify the theoretical model used in most cross— sectional studies. The original ACMS model is as follows: .1/ (2.1.1) v = y[5x'p + (1—5)L'p] p where V: real value added. K,L: the factors of production, capital and labor, 7: the efficiency parameter, 6: the distributive parameter, p: the substitution parameter, where o = is the elasticity of substitution, 1: degree of homogeneity. Assuming unitary returns to scale, perfect competi— tion, and profit maximization, the marginal product of labor would be equal to the wage rate, i.e., BV/BL = W where W is the real wage rate. Then (2.1.2) %%-= y'pv1+p(1—5)L“Il+°) = w, or z o 1 — ————1 L 1 + p ln W. 1n y - ACMS provided a formal proof that if a homogeneous degree one production function is assumed, and if relation (2.1.3) holds, a corresponding production function (2.1.1) 14 is determined and T—é—E turns out to be the elasticity of substitution between labor and capital. For the purpose of statistical estimation of the parameters, they ran the regression: (2.1.4) 1n‘i’-= a+b an+e where a - 1 + 0 1n y 1 + p 1n (1 6) — 1 for each industry across countries. The production function (2.1.1) was transformed from the following equation: (2.1.5) v = (316'0 + aL‘p) ““9 1 where p = %~- 1, a = a fi and B is the constant of inte- gration, setting _ 8 (2.1.6) e—a+8 , (oc+B)--]’/p or 'Y-p=0t+8, (2.1.7) Y (2.1.8) A.= 1. p is called the substitution parameter, because it is derived from parameter b, the elasticity of substitu— tion. An attractive feature of this new class of produc- tion function is that the elasticity of substitution can 15 be any constant, not necessarily zero, unity, or infinity. If the equation (2.1.4) exists and if ACMS assump— tions are valid, then for any given value of p, the func- tional distribution of income is determined by 6, which is called the distributive parameter. We saw that the equilibrium condition implied by competitive labor and product market under constant returns to scale was: (2.1.2) y’pv1+p(1-aL'I1+p) = w. In a similar manner the equilibrium condition: (2.1.9) Y-pv1+pax_(l+p) where r is the real rate of return on capital, may be derived from the relation BV/BK = r, which is implied by the assumption that the capital market is also competitive. Dividing the first equilibrium condition (2.1.2) by the second (2.1.9) yields: __ 1+0 (2.1.10) i—g—£I%) =‘g Now, additional knowledge of K and r would allow one to estimate 6. The distribution of income can be calculated by Wage bill _ WL _ K (2'1‘11) Earnings of capital - FE" 3 If) ' 16 ACMS have estimated the values of 6 for four indus- tries in five countries. It was found that the magnitude of 6 is fairly stable in each industry across the countries. From the equation (2.1.6) one can derive the following relationship: (2.1.12) I‘g"€" mm The ratio of B and on was therefore considered fairly stable across the countries in each industry. A test was made by ACMS that indicated that for each industryEBis unlikely to be the same in different countries. A more symmetrical possi- bility was that international difference in efficiency may affect both inputs equally. This amounts to assuming in equation (2.1.5) that 8 and a probably vary across the different countries by approximately the same proportion. This equiproportional change in the labor and capital coef— ficients, a and B, was taken to represent the change in efficiency andwas considered as evidence of the "neutrality of technological change."2 Since from equation (2.1.7) the parameter Y will change in proportion to the proportional change in 8 and d, it is called the neutral efficiency parameter. From 6 and p we can use euqation (2.1.1) to estimate the efficiency parameter, y, in each country involved for each industry. 2Gupta (1968) criticized this method. 17 The data for four industries in the five countries analyzed indicated that,for each industry, the substitution and distributive parameters were fairly stable across coun- tries, while the efficiency parameter varied. Clearly this new class of production function opened up a new horizon for the study of production func— tions. In the CD function usually expressed as V = a Lb Kc, the exponents of the variables L and K carry the heavy burden of measuring all the magnitudes which are measured in the CES function by the separate parameters. The uni- tary elasticity of substitution in the CD functions often leads to conclusions that are unduly restrictive. To illuminate this point, let us consider some of the impli- cations of the CES function for factor reversals, resource allocation, income shares of factors, and technological change. For both fixed factor proportion and CD functions, it is possible to rank industries uniquely according to capital intensity. For the CES function, however, this is not generally true. Suppose there are two industries each with a CBS function and both hiring factors from the same competitive markets. The equilibrium capital-labor ratio for these two industries are: K _ _ l 1 W l 6 o c K _ 2 2 W 2 (17’2 ‘ x2 ‘ ‘1'??? ('5’ Taking ratios we obtain: l 2 X o d r 2 1 2 (l — 61) 62 o —0 (2.1.13) = constant - (g) l 2 When 01.? 02 factor reversals are possible. Minhas (1963) and more recently, Hodges (1966) have shown that such reverslas are possible within the empirically relevent range of factor prices. These results cast doubt on the strong factor intensity assumption in international trade theory (especially, Heckscher—Ohlin theory) and also on the classification of industries as capital-intensive and labor—intensive for the purpose of resource allocation. In general, the substitution possibilities differ among industries, and the assumption of uniform substitutability may not be valid. For the CES function, factor shares depend on the elasticity of substitution. From equation (2.1.9) the share of capital in output is: 19 -ovl+05K—(1+o)) 5 17:” v ___ Y-ovo 5 K-o = 6K-p[6K—p + (1 — (ML-p].-l 1 _ v 1 - 6 K p [l + ——g——(fi I For pi<0 (o > 1) the share of capital rises with an increase in the capital-labor ratio; for p > 0 (o < 1), it decreases; and for p = 0 (o = 1), it remains constant. The rate of growth of output can be expressed as a weighted average of the rates of growth of capital and labor measured in efficiency units, with weights being their respective elasticities a and 8: V=B (K+Ek)+d(L+EL) where EK and EL are efficiency correction factors of capital and labor inputs. a and B are constants if and only if c is one. They add up to one only when A is one. Substi- tuting the expressions for B and a: V Ni + éK) + 1(1 -6) (i. + EL) + 15w”) (EKK)"’-1)(f< + éK - i. - EL). 20 The first two terms correspond to the CD part, and the last term vanishes for either p = 0 (o = l) or K + E = L + EL. The rates of change in factor shares 8 andcfare: é=oV—0(K+EK) = OBIL + EL — K — EK] 0L=V-(L-EL) oB[L + EL — K - EK] Thus, the relative share depends both on the rates of growth of factors measured in efficiency units and also on the elasticity of substitution. The difference in the rates of increase in the marginal products of labor and capital, M and MK’ due to L technological progress is: (ML — MK I K = L) = . (E — E ). Therefore, types of technological change are also affected by the elasticity of substitution. When EL - EK < 0, the technological change is capital-using in the Hicksian sense, if c < l; but labor—using, if c > 1. 2. Cross—Sectional Studies Despite all these merits, the studies using the CES production function are subject to criticism, 21 especially because of the specification errors of their stochastic models. 2.1. The Constancy of Output Price Although equation (2.1.3) was formulated in real terms, ACMS measured both V and W in monetary units in their actual regression (2.1.4): the L is labor input in man-years, W is money-wage rate (total labor cost divided by L) in U.S. dollar per man-year, and V is value added in thousands of U.S. dollars. However, the use of monetary values in equation (2.1.4) can be justified if the price of product, P, is uncorrelated with wages. To see this, let us reformulate (2.1.4) to include price, P, explicitly: 1...: . “17 (2.2.1) ln PL a + b 1n P + e where V, W are now in monetary units. Then, a' + b' 1n W + 1nI>- b' 1n P + e V 17...: . ——9—— (2.2.3)1nL a+b an+1+plnP+e 'where a' = ln[Yl—O(1 - 6)-°] and now b' is the elasticity of substitution. Therefore, regression of (2.1.4) will result in an unbiased estimate of o, if P varies without systematic correlation with W (or if P is constant across '1- _ a ,. _____._ __. ._.__. .- . 22 countries as they assumed). Unless we admit that the entire world is a single, perfect market for manufacturing goods, we certainly can not proceed with the assumption of constant price.3 Gupta (1968) noted that estimates of the labor efficiency parameter derived from the fitted CES pro- duction functions lead to rankings of labor efficiency across countries which are contrary to common knowledge, especially for the five Latin American countries whose efficiency seems most overrated (see Table 2). Minsol gave the most plausible explanation of this phenomenon: Individual prices are systematically too high because of high rates of protection in Latin America, reflected in high value—added ratio. There is considerable evidence that this is the case. Without these countries, the rank— ing is not so implausible. This suggests that the assump- tion of the constancy of output prices is systematically in error in some cases. 2 2 The Constancy of the Efficiency Parameter The constant term in equation (2.1.4) contains the term 0 lnY, so that an estimate of equation (2.1.4) 1 + assumes that the efficiency parameter, y, is a constant across observations. ACMS (1961, III-B) offered evidence, however, that y does vary across countries, and that it is positively correlated with wages. Sveikauskas (1971) also v—‘v 3Empirical evidence of this argument is not ample, but see Kravis and Lipsey (1967). 23 showed the same results with U.S. interstate data. This provides another source of bias. Actually, ACMS attributed the difference in the production function among countries to this variation in y. If the efficiency parameter, y, varies from obser- vation to observation, equation (2.1.4) should be replaced by .Y. = ' ' ___—L.- (2.2.4) ln L a + b 1n W + 1 + p ln,y + e where a' = ln [(1 — o)-OP1—O]. If we estimate equation (2.1.4), instead of equation (2.2.4) which is the correct specification, we find that the mathe- matical expectation of the least squares coefficient esti- mators of the a and b are (see Kmenta, 1971, pp. 392-5): E(éi) = a' + 1 +00. d1 (2.2.5) E(8) = b' + p d l + D 2 where dl,d2 are the least squares coefficients of the equation (2.2.6) 1n~y= d + dzln W + residual. 1 I—g77 is different from zero, the least squares estimator of a (and b) based on equation (2.1.4) will be Given that biased unless <11 (and d2) equals zero: i.e., unless Y and 24 W are uncorrelated, b will be biased. If T—g—E and d2 are both of the same sign, the bias of b will be positive; otherwise it will be negative. 2.3. The Equiyalence of the Quality of Labor Labor services are ordinarily measured in terms of man-hours, man-years, or, when the data restrictions are unusually severe, in terms Of numbers of employes. In any case, estimation of equation (2.1.4) assumes that the quality of a unit of labor does not vary across countries. This will add an additional source of bias with the same production function, because, a priori (or based on the previous empirical works), we have sufficient grounds to assume that there are some differences in the quality of labor across countries and yet the theoretical concept of a production function refers to a relationship between homogeneous output and homogeneous inputs. Suppose that the "true," quality constant, labor input of one observation 1 is Li*, which we do not observe, and that the observed data for i (i=l,2,...,N) are: = 'k .- (2.2.7) 1n Li ln (Li ) G where G is an unknown nonstochastic variable, depending on i. Since Wi is defined as the wage bill divided by measured Li' then, 25 = * (2.2.8) 1n Wi 1n (Wi ) + G where Wi* is the true wage rate. Then equation (2.1.4) should read (2.2.9) 1n = a' + b' 1n W + (1*b')G + e fU< where a' = ln [(1 — 6)—6y1—0](see Lucas, 1969, p. 237). In this case, if, instead of equation (2.2.9), one esti- mates equation (2.1.4) 1n = a + b 1n W + e L"| 1 we are in the uneconomic region, i.e., the iso- quants have the wrong curvature. Finally, Kmenta's gain in generality involves not a trivial cost, since we require data on capital input, which was not necessary in the ACMS model, and whose mea— surement carries another series of problems. In the following discussion a more generalized production function, that is, the CES production function of degree 1(CES—1) shall be developed, without using the capital variable. If the production function in a particular industry if V = V(K,L) and assumed to be homogeneous of degree A, then 1 . V = L V(K/L, 1). If we define y = V/LA and x = K/L, 47 we can write the production function as y = f(x). The marginal products of capital and labor are respectively: av _ Ll-l g; ‘3? - dx av _1-1 __ it Let w' = W/L)”1 where W is the wage rate. If the labor and product markets are competitive, the equilibrium condition is _ 1—1 __ df W - L [1f(x) x a;- 7 — '- _di or W — Af(x) x dx Starting with the observed relationship 0:6 1n a + b 1n W + c 1n L + u V where the addition of the L term on the right side of the equation may be justified by the test of the statisti- cal significance of c. Setting (A - 1)(l — b) or, 0 II (3.4.5) >2 II we can write equation (3.4.4) as 6For a slightly different specification arriving at the same result see Paroush (1966). 48 v_ w (3.4.6) 1nT—lna+bln—X-:I+u. L L Because, if we write equation (3.4.4) in antilogarithmic form with the disturbance term left out for convenience, we get V _ c E-aWbL or jL-= a WbLCL’(A-1) L)‘ = a Wb(%0bIA-1). Returning to the natural logarithmic form, we get equation (3.4.6). The equation (3.4.6) may be rewritten as (3.4.7) ln y = 1n a + b 1n W'. By substituting the expression of W' into equation (3.4.7), we get the following differential equation: In y = 1n a + b 1n [1y — x gifl. 1 Solving this equation substituting Z = 5) we get -1 _IL y=(aSA'1-A'1x—(P-‘l-5Me)'1. where e is the constant of integration. This can be written as 1 _ (lo-1)) (3.4.8) y = x)‘ (a 51.1 x b - 1-19) . __L l-b 49 1 _ c-1/1—b _ b Since -3 — -E:B7S—- — - 1-b = b-1 , if we set o=M%-l). the equation (3.4.8) becomes 1 A/ -— - p y=xA(B+ bAlxp) = (Bx-lp + a)—A/p, where -1 a = a b 1‘1 , D=A(%'l), and B is a constant. Returning to the original variables, we get the following production function: V film-E)” + aJ'A/p (3.4.9) (BK—p + aL—p)—A/p. On the assumption of constant elasticity of substitution, we know that the degree of homogeneity is A, and the pro- duction function is CES-1. The estimator of the elasticity of substitution will be 6 6:1???5' 50 since we know in the CES function _ l o — 1+0 . 5. Uses of the Capital Variable 5.1. From CES to Generalized Production Functions T Clearly, the assumption of constancy of the elasti- city of substitution for different factor-use ratios along a given isoquant is too restrictive to be realistic. Intui- tively speaking, as one moves along a given isoquant, he would expect the elasticity of substitution to change, i.e., the elasticity of substitution to decrease (or increase) as the relevant input-use ratio goes to zero (or to infinity). If we denote the factors of production by K and L, the diagrammatic illustration would be easy to understand. Let us assume that the factor markets are perfectly competitive. Then, the marginal rate of technical substitution of L for K is equal to the factor-price ratio, w/r, where w and r are the price of L and K, respectively. The elasticity of substitution between L and K can be expressed as _ d(KgL) s _ d(K/L) . W/r (3°5-1’ 0- ds °i<71:"‘—7Td(w r m *where 3 stands for the marginal rate of technical substitution between the two factors of production. 51 Rays from the origin, a and B, in Figure 1, show different factor-use ratios at points A and B, respectively, and the slopes of kll1 and k212 show the factor-price ratio at points A and B. As the factor-price ratio, w/r, changes from the slope of k l to that of k l the equili- l l 2 2’ brium point in production moves from point A to point B along the isoquant I. We observe that the factor-price ratio, W/r, has declined at point B, since the line k212 becomes flatter than klll' At the same time, we note that the factor-use ratio, K/L, also decreases from a to 8 due to the decrease in factor-price ratio. This is equivalent to saying that, as the marginal rate of technical substitu- tion, 5, decreases, then K/L decreases, since 5 = (W/r) by the assumption of competitive equilibrium. Then, by the definition of the elasticity of substitution, the assump— tion of constancy of 0 implies that the rate of decrease in K/L is constant, as W/r decreases. However, one may think that this is only one special case of more general phenomena: the rate of decrease in K/L may be constant in some cases, but more generally it may change as W/r changes further. For instance, for the initial decrease of W/r, the rate of decrease in K/L may be increasing, since at the beginning there may be more room for the substitution of L for K. Therefore, 0 may cecrease at first. As the employment of labor increases faster than that of capital compared with the relative decrease in wage rates the rate of 52 \ \ \ :4 FIGURE 1 FACTOR-PRICE AND FACTOR-USE RATIOS AND THE ELASTICITY OF SUBSTITUTION 53 decrease in K/L may slow down, since less capital is now associated with each unit of labor. Therefore, the 0 may cease to decrease or the 0 may even increase. At this point one should note that the limitation of the ACMS assumption of constancy of o is.quite related to their other important assumption of ". . . the existence of a relationship between V/L (value added per unit of labor) and W (the wage rates), independent of the stock of capital," since they estimate 0 using this relationship. Hildebrand and Liu used K/L as an additional regressor in the ACMS regression equation (2.2.4) with the U.S. Census of Manu- facturers (1957) data and found that the coefficients of K/L were in the neighborhood of twice their respective standard errors or larger in 10 industries among the 17 industries tested. They further suggested that the elas- ticity of factor substitution must be somehow related to the varying capital—labor ratios: If one relies upon the goodness of fit of an empirical relationship as the initial basis for deriving a theoretical one as ACMS did, one probably would have to consider the three-variable relationship (V/L, W, and K/L) as better established than the two-variable one (V/L and W) (1965, p. 35). Thus we can think of at least two possible ways to generalize the CES production function. One approach would be to start from the relationship between a and K/L, relax— ing the assumption of the constancy of o, and to derive an explicit form of production function. Another approach 54 would be to start from an empirical relationship, adding K/L as one of the independent variables in the regression model. These are exactly the ways one finds in the liter— ature. Revankar, Sato and Hoffman followed the former and Lu and Fletcher, Yeung and Tsang follwed the latter. 5.2. The RSH Model The simplest formulation of the relationship between the elasticity and the capital—labor ratio may be a linear relationship. Thus using the linear relationship, Revankar (1967, 1971» Sato (1967) and Sato and Hoffman (1968), (RSH) derived an explicit form of production function which has the property of variable elasticity of factor substitution.7 Starting from the assumption 7VES functions derived independently by Revankar (1967), Sato (1967), and Sato and Hoffman (1968) are actually identical. Revankar compared his VES function with that of Sato and Hoffman, and found some differences (Revankar (1971), pp. 154-55, footnotes 2 and 3). But he compared his VES only with one version of the Sate-Hoffman VES function, for which capital share is a linear function of the capital-labor ratio. However, if we compare his VES and another version of the Sato-Hoffman functions, where elasticity of substitution is a linear function of the capital-labor ratio, the two functions are identical and have the same properties: The Sato-Hoffman VES func- tion reads as a a C v = BK1+C [(1 + c)L + bKlI+c a ac _ 1'+'c' b 1"+c or V - AK [L + 1+0 K] where the elasticity of substitution, 0, is a linear func- tion of the capital-labor ratio, that is, o = a + b X 55 (3.5.1) 0 = a + b (K/L), a = 1, they derived a production function of the following form: 1 c (3.5.2) v = AK1+C [L + (1%?) K]1+C where A and c > 0 are constants and the other variables are defined as before. It should be clear that the elas- ticity of substitution has variable values depending on the value of K/L. This is the reason it is called a vari- able elasticity of substitution production function (VES). This can be simplified as (3.5.3) v = AK“[L + BK]Y where a = l/(l+c), B = b/(l+c), and Y = c/(l+c). The marginal products of capital and labor, respectively, are: where X = K/L and a and c are constants (c > 0). Under the assumption of the unitary returns touscale, parameter a disappears. The Revankar VES function reads as v =yKa‘1‘5") [L + (p - 1)K]a‘5p where o = a + TEE-13X, y, 6, and p are parameters. If we put b = (p — 1)/(l - 6p), then, 1%" a” ‘ 50’ ___—b — — l+c " p 1 Therefore, the two production functions are identical. 56 l c b 3V _ (1+c)V I+c (1+c)V (3.5.4) fi-T + b [L + (I;E)K] and c -—-‘V (3.5.5) 13%: “Ch [L + (T1EOKI so 1 av (""'1+c)V b av (3.5.6) fi=——fi—— + (FEW? Under the assumption of perfectly competitive factor mar— kets, we have SV/BK = r and BV/BL = W, where r and W are the wages of capital and labor, respectively. Thus, one may substitute and arrive at (3.5.7) r = (l/1+c)%+ (b/l+c)W. To estimate b and c we consider the equilibrium relation (3.5.8) r = (1/1+c)‘-I% + (b/1+c)W + e, where e is the usual disturbance term. However, we do not have any a priori reason for assuming a linear relationship between 0 and K/L. More possibly it may be a quadratic relationship as we have discussed. The possibility of deriving an explicit func- tional form of production (like the RSH form) depends on the feasibility of having an explicit integration of 57 (3.5.9) '1;(X)93,§=f 1 a 1n .xd ){+ A expjfi——3r—— where d(xn is the share of capital, )(is the capital—labor ratio, and A is a constant (Sato and Hoffman, 1968, p. 454). With several different quadratic relationships, one can hardly derive an explicit form of production function. If a profit-maximizing industry considers the fac- tor prices (W and r) as given, it will employ these two factors of production in such amounts as to equate the price ratio, g» to the marginal rate of substitution, 3, between capital and labor. Then the substitution functions, which relate the capital—labor ratio and the marginal rate of substitution, for the RSH model can be derived from the equation (3.5.4) and (3.5.5) as follows: (3.5.10) ‘1 cX r b 1+bc+mx If b = 0, the equation (3.5.10) reduces to the substitution function for CES or CD case depending on the value of "a" in equation (3.5.1). Yet in the RSH Model a = l by assump- tion. Therefore, the substitution function for the CD case results in: or 1n = 1n c + 1n X. 58 For the limiting Leontief type fixed coefficients (FC) case, § = X = (a constant) is the substitution function. In this case the K/L is independent of the W/r and the relevant substitution func- tion is given by a constant. One can compute the constant value by taking the simple arithmetic average of the ob- served sample K/L values for the industry concerned. One should recall that the production function (RSH) reduces to the CES or CD or FC case depending on the values of certain parameters. The empirical results of the regression analysis will allow the classical tests of significance at a certain level for the coefficients. This procedure, in addition to the specification error tests which will be discussed in the next section of this chapter, allows for the production function to vary its form across industries even within a generalized production function. Then, one may choose a substitution function corresponding to the production function chosen by the statistical tests of significance. 5.3. The K Model Kadiyala's (1971) study is interesting from this point of view. He started with a criticism of the RSH-VES production function. By its very nature, the assumption of the RSH function has the property that the elasticity 59 of substitution is either a monotone increasing function of the input ratio or a monotone decreasing function of the input ratio. This means that the elasticity of sub- stitution either reaches a maximum or a minimum as the input ratio increases. Kadiyala argued that this property is against one's intuition because the elasticity of sub- stitution of the labor for capital is the same as the elasticity of substitution of capital for labor. There— fore, he argues that one would expect the elasticity of substitution to increase as the input ratio increases, as well as decreasing when the input ratio gets sufficiently small. He proposed a production function of homogeneity of A in the inputs L and K, and with the variables related by a quadratic form: _ 20 o o 29 A/Zo (3.5.11) V - Y(wll L + 2w12 K L + w22 K ) where w w are assumed to be nonnegative and 11' “12' 22 wll + 2w12 + w22 = 1 under the assumption of unitary re- turns to scale, and Y stands for the efficiency parameter, which also absorbs the neutral technical change. Again, simplifying it: 29 (3.5.12) V = (0L + 28 Lo K3 4. OK2D)A/2p. It should be noted that this function takes care of all the previously discussed functional forms as special cases. 60 For example, if a = 0, it reduces to VES(RSH); if B = 0, it reduces to CES; if p + 0 (approaches to zero), it re- duces to CD; if p -> —|m, it reduces to the fixed coeffi— cients case; and if B = 0 and p = it reduces to a l '2‘! linear function. The elasticity of substitution, 0, is given by . . l - p + R where ‘0 (w w - w 2) (3.5.14) R = 3111122 12 -p o (wll X + w12)(w12 + w22X ) and X = K/L (Kadiyala, 1971, p. 6). A distinctive feature of this function is that the elasticity of factor substitu— tion increases (decreases), reaches a maximum (minimum), and finally decreases (increases) as the capital-labor ratio increases when p > 0 (p :< 0). Because, 2 _ 2 -p-l 2p _ —§ = p Iw11“22 “12 ) ”12 X (”22 x ”11) 2 ‘ 2 X"p + w + w Xp) (”11 12) (”12 22 assuming wllw22 - wlzz > 0, R increases (decreases) with X 1 1 as long as X a (wll/w22)75 (X < (wll/wZZIZE). Therefore, when p > 0, the maximum elasticity is attained at 1 X = (wll/m22)23 and the minimum is attained as X tends to its two end points. When p < 0 the maximum elasticity is 61 attained as x approaches 0 and w, and the minimum is attained 'l at X = (wll/w22)23 . In fact, the relationship between the elasticity of substitution and capital-labor ratio and the behavior of other parameters, as well as that of the elasticity of substitution, are the main questions we are investigating. Answers to these questions, therefore, must be obtained from the observed production relationship, not the other way around. To derive the substitution function for the K model, one can rewrite the equation (3.5.12) as follows: 2 = v2“A = 6L29 + 28 Lo Kp + 6sz. Then, a + B Xp Bxp'I+ axib'l HIS is the substitution function for the VESél and VES-1 case. If a = 0, the substitution function for the VES (RSH type) case results in: Bxp Exp-1 + 6x HIS 2p-l If 6 = 0, the substitution function for the VES (YT and LF type: see next subsection) can result in: = a + 8x0 Bx"-1 HIS 62 If 8 = 0, the substitution function for the CES case results in: _ a x-2p+l 8 ”IS If p + 0, the substitution function for the CD case results in: a + B _ a + B = — 'X. BX"1 + 6x”1 B + 6 ans If p + -m, the substitution function for the FC case results in: x = a constant. If B = 0 and p = , the substitution function for the N|l—' linear production function results in: = 2 B HIS 5.1. The YT Model Quite recently, Yeung and Tsang derived a more generalized production function based on the observed relationship. Their specification of the empirical relationship is a modified version of that of Hildebrand and Liu (1965): (3.5.15) 1n %-= 1n a + b 1n W + c 1n §.+ e. 63 Beginning with this relation and following the same proce- dure as we have done for the CESéA function, Lu and Fletcher (1968) derived the variable elasticity of substi- tution function: (3.5.16) v = [8169 + Omfp(.ILS.)'-<=(1+o)l-l/o 1 where p = B-- 1, _ l-b n _ l-b—c ' __1 a = a 5‘. If c = 0, this production function reduces to the CES-l function as a special case. It has been shown by Lu (1967) that this function has the properties of (a) positive mar- ginal products, (b) downward sloping marginal product curves over the relevant ranges of the inputs, (c) homo- geneity of degree of one (VES—l), and (d) variable elas- ticity of substitution as the capital-labor ratio varies. To show that the elasticity of substitution obtained from this function is variable, they use the relation: VK VL V VLK 0’: which holds when the production function is homogeneous of degree one (Allen, 1938, pp. 341-3). By substituting the derivatives of the new function (3.5.16) one gets b S l"C(l+-)?) 64 where s is the marginal rate of substitution of capital for labor and X is the capital—labor ratio. Since 3 is a func- tion of X, the elasticity of substitution varies with the capital-labor ratio. Using the perfectly competitive con- . . W dition, s = E7 _ b (3.5.17) 0 — 1. _eWE_ , l"C(1+-fi-) which can be used to estimate 0 empirically. Yeung and Tsang (1971) specified the empirical relationship in a slightly different manner: (3.5.18) 1n %-= 1n a + b In W + c 1n §.+ a 1n L + u. By setting d = (1 - 1)(1 — b), or 1 = 1 + T_§_E , they derived 1n i%~= 1n a + b 1n E§§T + c 1n %-+ u. Following the same procedure as we did for CES-A and VES-l, they obtained (3.5.19) v = [BK-p + anL‘°(§)'V]’A/0 where 1 73 . O‘IO 9’ n = 1 - b Au-b)-c' ;u%-- 1). D I 65 That this function is homogeneous of degree A can be readily verified. When A = 1, it reduces to the familiar VES—l function. Therefore, one may call it the VES-A production function. To show that the elasticity of substitution is variable, first it is necessary to find the marginal rate of substitution as follows (Yeung and Tsang, 1971, p. 5): (3.5.20) 3 = - ggl= V/ L = an(p - v)x""v+1 dL V K BO + anvxp’v From the definition of the elasticity of substitution, 0 = (dX/ds) (s/X) one can find that _ 1 (3.5.21) 0 — 1:. Bp(p _ v) . 8p + aanp-V Being a function of the capital-labor ratio, the elasticity of substitution is obviously not a constant. From equation (3.5.20), the substitution function for both VES-A and VES-l is: = an(p - v)xp-v+l (3.5.22) _ 80 + aanp V INS If v = O, which implies c = 0, equation (3.5.22) becomes 1+p W- 991 66 which is the substitution function for the CES case. In natural long form, 1n ”IS = 1n (%9) + (1 + p) 1n x. If both v = 0 and p = 0, then the substitution function for the CD case results in: In natural log = 1n (21) + In x. 1n 8 HIS As to the limiting FC case, the substitution function is X = a constant. If c = 0, which implies v = 0, the YT model reduces to the CES—l or the CES-A case depending on the significance of the coefficient d. In the CES case we know that as b + 0 (so p + w), the CES production function approaches the FC production function (see Arrow, gt_gl., 1961). Therefore, under the condition of c = 0 and b = 0, the YT model reduces to the FC case. In the case where b approaches zero while c ¥ 0, the equation (3.5.18) may be written as v = a(K/L)CLl+d (eliminating the error term for convenience for the moment) and the corresponding substitution function can be more easily computed by 67 fl__ 1 + d - c X. r c In natural 1n,. lnvl=1n(1+d'c)+lnx. r c One should note that associated with each production func— tion is a unique substitution function and that all substi- tution functions but one (equation (3.5.22)) give a linear relationship between 1n X and 1n ¥-. 5.5. The RZ Model A slightly different line of generalization was attempted by Revankar and Zellner (1969). That is, given a neoclassical production function with a given elasticity of substitution (constant or variable), they showed how this function can be transformed to yield a neoclassical general- ized production function with the same elasticity of sub— stitution, with the returns to scale variable, and satisfying a preassigned relationship to-the output level. All previous versions of generalization have allowed for any degree of homogeneity in the inputs. Yet the re- turns to scale are constant at any level of output. The RZ model attempts to generalize this point further. It was shown earlier by Soskice (1968) that if the returns to scale are themselves functionally related to output, a common procedure for estimating the elasticity of 68 substitution in CES will generally be inconsistent, even if it would not otherwise have been so. The salient feature of their study is that they showed that the elasticity of substitution associated with the generalized production function, V = g(f), is the same as that associated with the function f(L,K) and if f(L,K) is a neoclassical production function homogeneous of degree Af, a constant, then a production function, V = g(f), with preassigned returns to scale function, A(V), can be obtained by solving the following differential equation (1969, p. 242): dV = VA(V) (305023) ii. We With the following returns to scale function _ 8 and taking f in the Cobb-Douglas form, (3.5.25) f = 8 K8 L“ , with the returns to scale parameter 6 = a + B, one has the following production function of the Revankar and Zellner version: (3.5.26) VeYv = 6K8 L“ . 69 The elasticity of substitution of this function is one, the same as that of f (CD function) (see the same page (1967) for the proof). To derive the substitution function for the R2 model, let Z = VeYV. Then, the substitution function for the R2 model is H. = a r F X. which is also the substitution function for the CD case. In natural log, In E- = 1n‘g + 1n X. r B One should note that this model attempts to general- ize only the returns to scale and that it carries all the problems of the basic neoclassical production functions (f of equation (3.5.25)). Therefore, our task--to specify the basic production relations-—is not resolved by this model. 6. Model Discrimination and Pooling Observations Having seen, in the previous section, various ways (If generalization of production functions, we are now led 1x) a crossroad. A natural step one would take is to inves- tigate which model is "closer," in some sense, to the true 70 model than the others; we have to discriminate among the competing alternatives. Unfortunately, we do not have a very efficient compass at hand. Of course, some statistical procedures recently developed in the literature--for example, one approach in the article by Box and Cox (1964) and Box and Tidwell (1962) and that of Hill (l966)-—are useful for this purpose. One should realize, however, that although the procedures provide formal methods for discrimination between alternative models, they are no panacea for a lack of theory. Indeed, it has been noted that all the proce- dures are limited in the possible alternatives each proce- dure can handle. One is still left with the problem of reducing an infinite number of diverse possibilities to a manageable set. Thus, the subtleties of our economic models are pressing on the verge of our ability to discri- minate among them. We have to use at least two basic criteria of dis— crimination: economic criteria and statistical criteria. Basically, we have to investigate the properties of each functional form and the behavior of the parameters, and make sure they are consistent with our economic theories. In the previous sections, our main effort was exerted in this direction and we could make some tentative statement about each generalized production function, although there was no basis for making a definite choice. 71 Fortunately, however, in almost all cases each form of production function described above utilizes the linear least squares technique in actual estimation. Since this technique employs some explicit assumptions, we have another sound ground on which to discriminate among the alternatives: whether each stochactic specification of the functional forms satisfies the full ideal conditions of the linear least squares estimation technique. 6.1. The Specification Error TestsB 7 In our empirical study of this thesis we perform several specification error tests on the residuals from a single equation linear least squares regression analysis. The tests we are going to use were originally developed for the purpose of confirming the null hypothesis that the usual assumptions of the classical linear regression model as applied to a given population hold in a given situation. However, it is possible to use the tests for making a statis- tical choice among alternative models. The regression model employed in the tests is defined by equation: (3.6.1) Y = X8 + u where Y is an N X 1 vector of observations on the dependent variable, X is the N x K regressor matrix, B is a K x l 8This section follows Ramsey (1970) closely. 72 vector of regression coefficients, and u is the N x 1 vector of disturbance terms. The null hypothesis is that the full ideal condi- tions hold for model (3.6.1). The full ideal conditions (in addition to those listed above) are: (a) (b) Distribution of u is stochastically independent of the regressors. Distribution of u is N (0, ozIN); that is, u is normally distributed with zerO mean and constant variance. The alternative hypothesis is that the model (3.6.1) is misspecified in one or more of the following ways: (a) (b) (c) One calculating Group I error: omitted variables, incorrect functional form, simultaneous equation prob- lems; Group II error: heteroskedasticity; and Group III error: nonnormality of the distri- bution of the disturbance term. should first note that the residuals used in the test statistics are not the classical least squares residuals. This is because the classical least squares residuals, though distributed under the null hypothesis as normal with the null mean vector, have a population covariance matrix which is not proportional to the identity matrix. Therefore, the least squares estimates of the disturbance terms are neither statistically 73 independent, nor are they distributed with constant vari- ance for arbitrary regressor matrices. Consequently, one needs a set of (N-K) residuals which under the null hypothe- 2 sis are distributed as N(0, o I Such a set of resid— N-K) O uals was developed by Theil (1967) and further properties deduced in Theil (1968) and in Ramsey (1969). The Best Linear Unbiased Scalar Covariance Matrix Estimator (BLUS), denoted by u, is defined by G = A'Y where u is the (N — K) x 1 residual vector, A is an N x (N - K) matrix to be defined below, and Y is the N x 1 dependent variable vector defined in (3.6.1). The matrix A is defined in the following way: (3.6.2) A = (Kx(N“K)) x = (KxK) (Nx(N-K)) Al (NxK) xl (N-K)x(N-K) ((N-K)xK) _ _ -1 . _ 1/2 . A1 — PD P , where D = P'MllP, P is the (N — K) x (N - K) matrix of eigenvectors of the matrix M11, and M11 is obtained from 1 X'] by partitioning M conformably to M = [I - X(x'X)' N the partition of X, i.e., 74 M00 M01 M = M10 M11 D is a diagonal matrix whose nonzero elements are the eigenvalues of M11. Under the null hypothesis 8 is distributed as 2 N(0,o I The distributions under the alternative N-K)’ hypotheses fall into three groups corresponding to the three groups of error: Group I : 8 ~ N(c, 0), Group II : 8 ~ N(0, 02). Group III : nonnormally distributed, where u is an (N — K) x 1 vector C being the mean of u conditional on X, 0 is an (N - K) x (N - K) covariance matrix, and 02 is a diagonal covariance matrix whose non- zero elements are unequal. The five specification error tests developed by Ramsey (1969 and 1970) are RESET, RASET, BAMSET, WSET, and KOMSET. A brief summary of these five test statistics is provided in Appendix E.9 The five tests are, to a con- siderable extent, complementary to rather than substitutes for each other. The first two tests (RESET and RASET) are 9See also Ramsey, J. B., "Program DATGENTH: A Computer Program to Calculate the Regression Specification Error Tests: RESET, WSET, RASET, BAMSET, and KOMSET." (Revised) Econometrics Workshop Papers No. 6704, Michigan State University, July, 1970. 75 tests for the group I misspecifications, the BAMSET is a test for group II errors, and WSET is a test for group III errors. KOMSET is a test for both group I and group III errors. RESET is able to detect cases where RASET's power is reduced, i.e., where the relationship between Di and q1i is nonmonotonic (see Appendix E). The power of RESET, however, presumably depends on the full ideal con- ditions holding for the regression of u on the regressors qj, j = 1,2,...,k. KOMSET's power, on the other hand, is not reduced by nonmonotonicity and does not need the assumptions required by RESET. KOMSET's drawback is that for small sample sizes its power is low. 6.2. PoolingObservations One of the most detrimental factors of the inter— national comparison approach in economic analysis is lack of adequate data for analyses.10 Frequently, necessary data are not obtainable for crucial variables. Even available data are limited for one or a short period of time. This leads one to draw a very dubious conclusion from one's empirical study. 10Minhas (1963) explained some statistical reasons for conducting an international study of production func- tions. In addition to this, we have a vested interest in conducting a study using intercountry cross-sectional data. For example, we will find implications of the estimated garameters of the production functions for international rade patterns, which needs a cross-country comparison. 76 Therefore, one approach to many economic problems is to conduct cross-country analyses including both developing and developed countries. Another approach is to use the historical data of advanced countries, spend— ing the time-series data penuriously. Since each approach presents serious econometric difficulties, they will pro- vide a better basis for testing economic models if they can be used in combination, broadening the base of our data. The value of a cross-country analysis for generat- ing some preliminary hunches can not be denied. But unless technological changes can somehow properly be taken into account in the use of cross-sectional data, use of results may lead to erroneous inferences concerning the relation- ships between inputs and outputs and among inputs. And the same applies to application of cross-sectional analy— sis to projections into the future. The specification errors discussed in the previous chapter, and biases due to such errors are far more serious in the estimates of a cross-country analysis. The time-series analysis also suffers from similar specification errors (see Chapter II, Section 3). As in any time-series analysis, a second problem is that of serially correlated residuals. Another problem arises if the wage rate should be treated as an endogenous variable. Further, problems arise if wage rates are 77 correlated with the error term in the regression, whether because of an upward sloping labor supply curve to the industry or because of an upward sloping aggregate labor supply curve together with nonindependence across industries of error terms in labor demand curves. Unfortunately, even using what appear to be highly restrictive assumptions, one finds it impossible to deter— mine even the sign of the asymptotic bias. Therefore, these biases are not a promising route to reconciliation of the time-series results with the cross-sectional results. The third possibility which appears to be more pro- mising is, therefore, to bring together sources of the time-series and cross-sectional approaches to have a better, more balanced picture of the production relationships and their implications. Consequently, one needs to apply more formal econometric methods to estimate the production function. CHAPTER IV PRODUCTION FUNCTIONS AND THE COMPARATIVE ADVANTAGE 1. Introduction As we have seen (Chapter I, Section 2), the development policy and the modern trade theories based on the comparative advantage hypothesis depend so critically on the assumption concerning the production relations that once the validity of those assumptions is rejected, the theorems themselves have little practical value. Those assumptions, particularly the factor-intensity hypothesis, are in turn related invariably to different forms of pro- duction functions. If the amounts of capital and of labor employed per unit of their respective outputs were technologically fixed so that no substitution between factors is possible in response to any change in their prices (Leontief type fixed coefficients case), the ranking of different indus- tries in accordance with the relative magnitude of the two input coefficients would certainly be valid. (The techni- cal capital—intensity and the factor-intensity coincide in this case.) This can be shown in Figure 2, on which the 78 79 FIGURE 2 THE FIXED-COEFFICIENTS CASE 80 product contours for the two commodities have been drawn. In this case, to produce some anount of commodities A and B, say a1 and bl’ factors of production, capital (K) and labor (L), must be combined in the ratio given by Ca and 1 Ob respectively. The relative prices of the two factors 1! have no effect on factor proportions, and as long as Oala2 is to the right of Ob b one can say that, irrespective 1 2' of factor prices, commodity A is always labor—intensive relative to commodity B. It still would be meaningful even if, in response to a given change in relative prices of the two inputs, capital were substituted for labor or vice versa, provided the downward or the upward shifts of the capital-labor input ratio were so uniform as not to disturb to any significant extent the relative position of the individual industries on the capital-labor intensity scale. One example is the case in which the production functions for the individual industries possess the unitary elasticity of substitution between the two factors as in the CD production function.1 (The technical capital—intensity and the factor-intensity still match each other.) However, it is not necessary that Oa must always 132 be to the right of the Ob1b2‘ The same conclusion holds, 1For the proof, see Ferguson (1969), p. 383, foot-L note. 81 even if one relaxes the assumption of the linear limita— tionality and the two isoquants intersect each other within the range where factor substitution is possible. At the relative factor price indicated by qq (or a) in Figure 3, commodity A is relatively labor-intensive as indicated by 00 for B and 00' for A. Although the relative factor-price changes from qq to pp(=p'p'), the factor—use ratio for B is OT and the factor-use ratio for A is OT'. Still commodity A is labor-intensive. As long as one curve intersects the other only once, an unequivocal defini- tion of factor intensity is possible. If the elasticities are the same as in the CD production function among the individual industries, one isoquant intersects the other only once.2 But what if the two isoquants intersect more than once, say twice? This is the case, if the elasticities of substitution of the individual industries are different from each other. If some industries responded to a given change in the relative price of the two factors by a much larger shift in their relative inputs than others, then their comparative position on the capital-labor intensity scale would often be reversed (in this case the technical capital—intensity does not coincide with the factor-intensity). 2But it is not necessary that the elasticity of substitution is equal to unity. 82 I Q' B i (’ P I 1.. I I ' I, [I] ~ I l I II I I I I I ’l I l/ 1' I’ A (l/ a 0 q p. FIGURE 3 FACTOR SUBSTITUTIONS AND THE INTERSECTION OF ISOQUANTS 83 The distinction between capital— and labor—intensive indus- tries must lose in such a case much of its analytical use- fulness. Neither in explanation of the pattern of international trade nor in the study of development policy would it be permissible to utilize it as technological datum. Figure 4 shows the case where isoquants of the two commodites intersect twice. At the relative factor- price qq, the factor-use ratios are 00 for commodity A and 00' for B, indicating B is a capital—intensive indus— try. Yet at the relative factor—price pp, the factor-use ratios of OT' for A and OT for B indicating A is now a capital-intensive inudstry. Thus, if A is capital-intensive at one factor-price ratio and B is capital—intensive at another factor—price ratio, there must be a relative factor-price at which the two commodities use the two factors at the same rate. This must be the case where the two isoquants have the same slope at that relative factor- price. This point can be located on the diagram. Since ala1 cuts bb at S and S', the slope of the two curves must be the same. Let this be M on bb and N on a a l 1' the slopes at N and M are the same, there must be a member Since of the family of A-curves which is tangential to bb at M. Let a be this curve. It can now be easily seen that 2&2 as long as relative factor-prices (price of labor to price of capital) are higher than the slope at M, A will use factor quantities in a ratio steeper than the slope O M, 84 al - 5- .- uvuv', a2 :1 FIGURE 4 MORE THAN TWO INTERSECTIONS OF THE ISOQUANTS 85 and in all such cases, on previous reasoning, A will be relatively labor-intensive. Similarly, if relative factor prices are lower than the slope at M, B will be relatively labor—intensive. We can no longer say that A is labor—intensive unequivocally at all relative factor— prices. As we have seen, the CD production function pre- cludes the possibility of factor-intensity reverslas, since all industries have the unitary elasticities of substitution. All industries respond to changes in the relative factor- price at the same rate, so that relative factor—intensities of the industries are not disturbed. A distinctive feature of the CES production func— tion is that it allows for interindustry differences in the relative ease or difficulty with which factor inputs can substitute for each other with a given change in the relative price of the two factors. The elasticity of sub— stitution of an industry is nonetheless a constant for all factor-intensity ratios. Thus, the factor-intensity ratio changes at a constant rate (equal to the elasticity of substitution) as the factor—price ratio changes. Therefore, if the elasticities of the two industries are not equal, the relative magnitude of the two factor-intensity ratios may be reversed between the two industries as the factor price ratio changes.3 3But not necessarily at the rate of one as in the CD case. 86 Since the VES is a more generalized case of the CES, it also allows for the possibility of interindustry differences in the elasticities of substitution and the possibility of the factor-intensity reversals. The only difference is that the factor—intensity ratio does not change at a constant rate, but at a variable rate as the factor-price ratio changes. This may lead one to suspect that the VES production functions not only allow the possibility of the factor—intensity reversals, but also increase the chances of the possibility as exaggerated on Figure 5. 2. Empirical Verifications of the Factor—Intengity Hypothesis Minhas (1963), with the CES, set out to demonstrate empirically that a meaningful distinction can be made between capital— and labor—intensive industries. He found that reversal is a rather common phenomenon. Using equations (2.1.2) and (2.1.9), and a perfect equilibrium condition, he found a log—linear relation for each indus- try between the factor-intensity ratio, K/L, and the factor- price ratio, W/r: _ 1 g 1 El 1 + p 1n 8 + 1 + p 1n r , (4.2.1) 1n ms: where a = (l - 5)Y-p. B = 5Y_po 87 FIGURE 5 THE SUBSTITUTION CURVES IN VES PRODUCTION FUNCTIONS IS: 88 and all other parameters (a, B, and p) are defined as in equation (2.1.1) and estimated from the least squares fit of equation (2.1.4). This relation between K/L and W/r may be represented by a straight line as shown in Figure 6, if we plot the function on a double logarithmic scale. If the slopes of the two lines--that is, the elasticity of substitution between capital and labor in the two indus- tries--happen to be exactly the same with the levels of both lines to be also equal, the factor-intensity of both industries will be identical throughout any factor-price ratio. If the slopes of the two lines are the same yet the levels of the two lines are different, they will be paral- lel, which means that the capital-labor ratio in one of the two industries will be higher throughout than the other. In case the elasticities are unequa1-—that is, the slopes of the two lines differ-—they are bound to intersect somewhere. The crossover points might, however, be located to the right or to the left of the usual or even possible range of observed capital-labor or price ratios. In this case, one industry can still be, for all practical pur- poses, unequivocally characterized as using more capital per unit of labor than the other. Minhas found that in fact crossovers occur within the practically relevant range so often that the strong factor-intensity assumption, the conventional distinction between capital- and labor- intensive industries, is of limited practical validity. tflfifi 89 FIGURE 6 THE SUBSTITUTION LINES IN THE CES CASE '1'2 90 Leontief (1964), in his review of Minhas' work (1963), performed some supplemental computations to have the estimates of the intercept of equation (4.2.1) for more industries than Minhas presented, and claimed that Minhas' own empirical evidence justified the opposite conclusion.4 Of the theoretically possibel 210 crossover points (i.e., C31) between the lines of 21 industries, only 17 are found to be located within the empirically relevant range of factor-price ratios, spanned on the one end by those observed in the United States and on the other by those in India. Moreover, most of these crossovers occur between industries whose curves run so close together throughout the entire range that for all practical purposes their capital—labor intensities would be considered identical. Hence, the modern theory of international trade appears to have been vindicated. One may notice several weaknesses in the Minhas and Leontief studies. First, the labor input was not adjusted for the difference in the efficiency of a man-year 4Minhas presented estimates of 8 only for six indus- tries and no word is given in explanation of the origin of the estimates. Leontief computed for 21 industries alto- gether using Minhas' results of the statistical analysis of the return on capital invested in the same industries in different countries. Equation (4.2.1) is written in the following form to get the estimates of the intercepts: in (g) = 1n (‘31-) -, (o + 1) In ‘11:”- 91 of labor among countries. This point was questioned by Leontief, but he did not make any adjustments in his com— putations. Second, due to the lack of international data on the capital variable, certain indirect procedures had been used to estimate the equation (4.2.1). The use of indirect estimation procedures is likely to cause biasedness in the estimates. Third, their analyses assumed that all indus— tries' production relations were subject to constant, unitary returns to scale and did not allow for nonunitary returns to scale or for variable returns to scale. Fourth, the assumption that all industries have the same (CES) form of production function is too bold to accept. Fifth, they did not allow for differences in the economic efficiency among various parts of the world. 3. The Procedure of the Present Study In our estimation of the industry-production func- tions and verification of the factor-intensity reversals, all the theoretical considerations discussed in Chapter III will be incorporated. First, a more finely defined industry classification is used. Second, labor inputs are adjusted for differences in the quality of labor among countries. Second, data on fixed capital assets for dif- ferent industries in each country are used to estimate production parameters directly. Third, nonunitary returns 92 to scale and varying returns to scale are allowed in our models. This will eliminate the bias in the estimates of the elasticity of substitution because of the incorrect specification of the returns to scale (see Maddala and Kadane, 1967). Fourth, by pooling cross-sectional and time-series observations a larger and better sample is used. Fifth, differences in economic efficiency among countries are adjusted by using the error components model. Sixth, the industry production function is not restricted to a certain form because we use more general- ized production functions which may reduce to the CES or CD or FC form based on empirical results. Moreover, a discri— mination is made among the competing, generalized produc- tion functions based on the specification error tests. Having estimated the parameters of production functions, the substitution function for each industry is to be derived from the production parameters. The possi- bility of factor-intensity reversals is investigated directly from the comparison of the substitution functions for each industry. Each industry's substitution function is drawn on the double natural log axes and the incidences of crossovers among the substitution functions are counted. 4. The Models and the Discrimination The four alternative production functions proposed for consideration in this empirical analysis are as follows: 93 l. Yeung, Tsang (4.4.1) v = [BK'p+anL“p(%)'V]‘A/D (YT) 2. Revankar, Sato, = l/(1+p) p/(l+p) Hoffman (RSH) (4° 4'2) V BK IL+ 0-1—1..me 3. Revankar, Zellner (4.4.3) VeYv = 6K8La (R2) 4. Kadiyala (4,4,4) v = [aL20+ BLpr+ KZle/ZO where V is output, K is the capital input, L is the labor input, and the Greek symbols represent parameters. The parameters defined in each function are specific to that function only. The stochastic formulation of the four models are as follows: 1. YT (445) 1n (37-) =a+ban. +cln (5) +dlnL. +2 . . L it It L it It lit 2. RSH = .11. (4.4.6) 1n Vit a + b 1n Kit + c 1n [L + 1+p K]it + €2it 3. RZ (4.4.7) ln Vit + YVit = a + b 1n Kit + C 1n Lit + €3it 4. K 20/A_ 20 o 0 2p (4.4.8) Vit — b Lit + c(2) Lit Kit +‘d Kit + €4it (i = 1,2,000'N; j = 1,2,3'4; t = 1'2'3’ooo'T)’ where Vit is the value added of each manufacturing industry at the three-digit level in i-th country in year t, K is the value of the fixed capital assets, L is the labor input 94 in terms of man-year, and W is the wage rates which is the ratio of the wage-bill for the man—years and the labor in- put. All these four variables are adjusted as explained in the next section. That is, V and K are adjusted value by the wholesale price indices, W is adjusted by the con- sumer price indices, and L is adjusted for the labor quality differences by the effective human capital stock indices. 6.. are the disturbance terms as defined in the error com- ijt ponents model, that is, e. = u. + v + w. . In other words, it 1 t it all four models are estimated within the context of error components model.5 When one pools cross—sectional and time-series observations, the question of the appropriate restrictions on the variance-covariance matrix is of special interest and significance. This is because the relationship of the disturbances over the cross—sectional units, i.e., country in this study, is likely to be different from that of the disturbances of a given cross—sectional unit over time. Suppose we have N = 12 numbers of individual countries over T = 3 periods of time. As an example, let us con- sider the YT model. Then the statistical model we consider now is: v _ 1< (4.4.9) 1n (i- - a + b 1n wi + c 1n (L)it + a 1n Li t t )it + 8 o 0 1t 5 . . . A covariance model 18 also run for each industry. 95 Where i = 1,2,000112; t = 1'2'30 In matrix notation the regression equation can be written as (4.4.10) Y = XB + e where Y is an (36 x 1) vector of observations on the dependent variable, ln (%) X is an (36 x 4) matrix of regressors, it' B is a (4 X 1) vector, and e a (36 x 1) vector of error terms. Without loss of generality we assume that each column vector of the X matrix sums to zero, i.e., each Xit is now trans- formed into deviation from its column mean. Therefore, the constant term, a, disappears and X isra (36 x 3) matrix and B is a (3 X 1) vector. Clearly, various kinds of prior specifications with respect to the disturbances will lead to various kinds of restrictions on the variance-covariance matrix. One approach to the specification of the behavior of the disturbances with pooled cross—sectional and time- series data is to combine the assumptions that we frequently make about cross-sectional observations with those that are usually made when dealing with time-series data. As for the cross—sectional observations, it is frequently assumed that the regression disturbances are mutually independent but heteroskedastic. Concerning time-series data, one usually suspects that the disturbances are autoregressive, though not necessarily heteroskedastic. When dealing with pooled 96 cross-sectional and time—series observations, one may com— bine these assumptions and adopt a cross-sectionally hetero— skedastic and timewise autoregressive model.6 Or by dropping the assumption of mutual independence, one may have a cross-sectionally correlated and timewise auto- regressive model.7 A simpler model of the specification of the behavior of the disturbances when combining cross-sectional and time— series data has been adopted by the proponents of the so- called error components model. The basic assumption here 6The characterization of this model is as follows: 2 2 E(eit ) = oi (heteroskedasticity) ) = 0 (i ¢ j) (cross—sectional independence) it 0 2 where u. = N(0, o 2), e. ~N(0. -E$-§Or 1t Hi 10 - , 1 E(s ) = 0 for all i,j. i,t-1“jt 7The specification of the behavior of the disturbances of this model is as follows: E(€. t2) = Git (heteroskedasticity) E(eits jt) = Oij (mutual correlation) Sit = p. lei t-l + “it (autoregre351on) Where Uit~N(or “i i), EH61 l_lr:1jt)t_ = OI ipj=1'2'oo o [no For the method of estimation of the regression coefficients using these two models, see Kmenta (1971), pp. 510-14. 97 is that the regression disturbance is composed of three independent components. That is, (4.4.11) Sit = ui + vt + wit Where ui represents the time invariant, individual effects, vt represents the period specific, individual invariant effects, and wit represents the remaining effects which are assumed to vary over both individuals and time periods. The model assumes that E(ui) = E(vt) = E(wit) = O for all 1 and t, °i ' i=i' ' _ A E(uiui) ‘ o ifii' I (4.4.12) 2 l _ = I E(vtvt) — V‘: t t o, tft' E(w w ') = 02 i=i' t=t' it it w ’ 0, otherwise. This implies that sit is homoskedastic with its variance— covariance matrix written . = 2 2 2 (4.4.13) E(ee ) 0wI(3x12) + Cu A-+ 0v B where I(3x12) is an (36 x 36) identity matrix, and A and B are (36 x 36) matrices defined as: .99- § 98 111 111, o,...,o 111 111 o, 111,..,o A= 111 o, o, ,o 5 E 1i1 o, 0,..111 111 L _. where there are 12 rows and columns of the (3 x 3) block matrices; 100 100 100 010,010,...,01o 001 001 001 100 100 100 010,010,...,01o B: 001 001 001 100 100 100 010,010,...,01o 001 001 001 L. __ where there are 12 rows and columns of the (3 x 3) block matrices. The coefficients of correlation between sit and ejt (i#j) is 2 Cov(€. , 8. ) 0 1t' 3t = T———2" 2 (iaéj) (4.4.14) /’ Var(eit) Var(ejt) u v w (cross-sectional correlation) The coefficient of correlation between sit and eis(t#s) is 99 Cov(eit,s. ) 0: (4.4.15) 1. 13 = 2 2 2 (tfs) JVar(eit) Var(ei;) Cu + 0v + ow (timewise correlation) Finally, the coefficient of correlation between eit and E. is is Cov(e. ) ,6. (4.4.16) 1t 35 = o (i#j, t#s) JVar(€it) Var(€js) (cross-sectional and timewise independence) By using these results, one can find elements of the variance—covariance matrix, and obtain estimates of the regression coefficients that have the same properties as Aitken's generalized least squares estimator.8 Wallace and Hussain (1969) expounded an alternative way to estimate the regression coefficients in the error ~components model, which may be called the covariance transformation method. They suggested a covariance trans— formation matrix, Q, such that 8For alternative procedures for estimation of the variance-covariance matrix, see Wallace and Hussain (1969), Nerlove (1971a), Henderson (1971), Searle (1968), and Amemya (1971). Mundlark (1963) presented a good rationaliza- tion for the error components model in the context of estimat- ing production functions. Barlestra and Nerlove (1966) considered a model that is at the same time much more com- plicated and yet more simplified than the model considered here, in that they analyzed the compound problem of lagged dependent variables and error components, but with error components only arising in a single direction. Maddala (1971) considered both models. Hildreth considered the error components model in a simultaneous equation context. 100 (4.4.17) Q — I(3x12) 3 A 12 B + ___—7(3le J(3x12)' where J(3x12) is (36 x 36) matrix with ones everywhere and all other matrices have been previously defined. It may be noted that Q is idempotent: , 2 Q = Q and Q = Q Thus the transformation matrix is singular with rank (12 - l)x (3 - 1). Applying the transformation matrix to the model described in the equation (4.4.10), the ui,v components t of sit are swept out and we get (4.4.18) QY = QXB + Qw where w is the (36 x 1) vector representation of wit' Using the ordinary least squares procedure to the transformed model, we have the following normal equation: (X'Q'QX)§ = (X'Q'QY). Since Q is idempotent, g = (X'Q'QX)“1 X'Q'QY = (X'QX)-l X'QY. Consequently, we can show that § is unbiased and consistent,9 and the variance-covariance matrix is If an ordinary least squares procedure is applied to the equations (3. 6. 4) and (3.6.5) treating ui ,vt as con- stant coefficients of dummy variables in the 1covariance model, the estimates (say, 8 ) will be still unbiased but the variance of 80 will not Be the best one: 101 E<§é') = Et<§ — B)(§ — e)'1 = 0: (WHY1 _ 2 l _1 _ _]_._ _L '1 .. OW [X (INT 'T- A N B + NT JNT)X] In this context where there are only exogenous variables present (no lagged dependent variables present), Wallace and Hussain compared the generalized least squares (Aitken) estimators with estimators produced by a covari- ance transformation technique. They showed the following conclusions which are very much meaningful for the practi- cal researcher (1969, p. 66): first, covariance estimators have the merit of computational easiness--no iterative estimation is required in the usual case of unknown E<§o) = E[(x'X)'1 X'Y] l = E [(X'X)' x'(x80+ 5)] = E {E}, + (X'X)"l X's] + (x'X)'1x'E(e) II on El A u» o on o V II E [(éo — e)(éo — B)'] E rmvx)'1 1 1 x'e(x'X)' x‘:‘] 1 x'E(ee')X(x'X)' 1 1 (x'X)' _ 2 I - I I '- - INTOE (X X) X X(X X) _ 2 I - — INT°e(x X) 2 2 2 _ =' . I I (cu + 0v + ow) (x X) NT .. 2 which is greater than EKBB') 0W(X'QX)"l l l 2 . _ 1‘ _‘1 -l ow[X (INT T A N B)X] 102 variances. Second, covariance estimators are unbiased, no matter whether we do or do not have access to prior information about the variance components. Third, al- though the Aitken estimators are more efficient for known variance components for finite samples, covariance esti- mators are asymptotically equivalent to Aitken (known variances) and the iterative Aitken (unknown variances) estimators, in the case of weakly nonstochastic X's (do not repeat but are bonded). Only for strongly nonsto— chastic X's (repeat in repeated samples) the Aitken (known variances) and the iterative Aitken-Zellner type (unknown variances) have smaller asymptotic variances than covariance estimators. This large sample property is virtually a repetition of small sample property. Fourth, covariance analysis serves to "clean up" speci- fication error no matter whether the error is the non- stochastic constant coefficient of the dummy variable variety or is made up of additive stochastic components (random variables). Taking advantage of these properties, we apply this covariance transformation technique to the four stochastic equations ((4.4.5) - (4.4.8)) for each of the 19 industries. For the YT model, two parameters of equa- tion (4.4.1), B and a, which can not be obtained from the stochastic regression equation (4.4.5) are estimated by the two-step least squares technique. Considering the 103 estimates of the coefficients of the firstvstep least squares equation (4.4.5) as prior information and holding them fixed in the production function, YT (4.4.1), we can set up the second-step least squares equation. That is, equation (4.4.1) can be rearranged using coefficients of the first—step regression equation (4.4.5) as: (4.4.19) v""/A = 816‘" + ou’w‘L“p (Elf-V0 01' (4.4.20) v* = BK* + aL* where v* = V'p/X, K* = K'p, and L* = nL'p(§)'9. B and a can now be estimated by following the same proce- dure as we did to estimate equation (4.4.5). In RSH model I—%—E-was estimated from the equili— brium relation 1 = (___... V it 1 + p (4.4.21) r )(E) ( ) W. + s it + T—%—3 1t it' Then, this is also transformed by the 0 matrix as is equation (4.4.18). The remaining two models (R2 and K) are estimated by what may be called two-step maximum likelihood method, using the covariance transformation technique. For example, in K model, if one knows the value of p and A, one can estimate the rest of the parameters by the usual least 2p/A t 20 t squares method with Vi as the dependent variable and 20 D p t 1t as regressors. However, as p and 104 A are unknown, these parameters are estimated by the maximum likelihood approach. Thus, the likelihood func- tion is written as a function of the data and of the unknown parameters p and A by substituting for the other parameters, b,c,d, their maximum likelihood estimators as the functions of the variables of Vi 29/0, L 29, 2Litp' Kitp' Kitzp’ maximize the second—step likelihood function: i.e., pick t it The values of 6 and X are chosen to that value of 6 and X such that the corre5ponding ordinary least squares regression minimizes the sum of squared errors. A confidence region for 6 and_X can be obtained by using the fact that —21n l, where l is the likelihood ratio, is asymptotically distributed as Chi-Square. The tests on specification errors are applied to each of the four stochastic regression equations, which are transformed from the equations (4.4.5) - (4.4.8) by the covariance transformation matrix, Q, as we did in equation (4.4.18), in each of the 19 industries at the three-digit ISIC level. The tests have been explained earlier (Chapter II, Section 6 and Appendix E) and the results will be discussed in the next chapter. 5. Adjustments for Economic Efficiency Differences When dealing with cross-sectional and time-series observations in this study, we decided to use the error 105 components model (Section 4 of this chapter). We also have chosen the covariance transformation technique, instead of estimating the components of the errors, to estimate the regression equations (4.4.5) — (4.4.8). The virute of this model is that this gives one unbiased estimate of regression coefficients by allowing for vari- ations in the disturbance 'term in both time and cross- country directions. Therefore, the differences in economic efficiency across countries are readily accounted for. An alternative way to allow for the difference in efficiency across countries is to use the analysis of co- variance model. This is a generalized regression model utilizing aspects of the analysis of variance. This model supposes that each cross—sectional unit and each time period is characterized by its own special intercept. This feature is incorporated into the regression equation by the introduction of binary dummy variables which are supposed to account for constant effects associated with both the time direction and cross-sectional units but not readily attributable to identifiable causal variables.10 Again, using the same number of observations, say N = 12 and T = 3, the regression equation for the YT model becomes: 10For the use of this model in the study of produc- tion functions, see Koch (1962). (4.5.1) ln ( where zit = 1 = 0 Uit = l = 0 l. u 106 K a + b 1n Wi + C 1n (f)it + d In Li t t +52 + 0), then country two is more efficient than country one by Y2 in using the same amount of inputs in that industry. The equation (4.5.1) contains 4 + (12 - 1) + (3 - l) regression coefficients to be estimated from 12 x 3 observa- tions. The use of the covariance model to deal with pooled cross—sectional and time-series observations stems from concern about possible bias in the estimates of the causal parameters. That is, the disturbance e. 1t satisfy the usual assumptions of the classical linear regres- is supposed to sion model by using the dummy variables. If the model is correctly specified and the classical assumptions are satisfied, the ordinary least squares estimates of the regression coefficients will be unbiased and efficient. One of the practical reasons for choosing the error components model is that the commonly used covari- ance model can not be used together with the specification error tests on the stochastic models. As it was shown before (equation (3.6.2)), the specification error tests to be used in this study need the X matrix (regressors) partitioned into x0 matrix (K x K) and x matrix [(NT-K)x K]. 1 As the inverse of X0 is required for the subsequent tests, it is necessary that the X0 chosen should be nonsingular. 108 The analysis of covariance model sometimes yields X0 mat- rix which is singular depending on the number of observa- tions and of dummy variables. However, to estimate the actual magnitude of the differences in economic efficiency across countries, we use the covariance model separately on equation (4.4.5)-(4.4.8) for each industry. 6. Adjustments for the Labor Quality_Difference In order to measure the labor input in efficiency units two methods discussed earlier were applied. The method using wage differentials, however, resulted in irregular cases more often than the method measuring effec- tive human capital stock. This may be attributable to the measurement errors in wage rates and labor inputs or dis- tortions in factor markets and in exchange rates. There- fore, the human capital stock approach was used in the actual calculation of labor inputs in each industry. The labor efficiency index for each country was calculated as a weighted sum of the proportions of three skill categories in the industry's labor force. The problem is to give an appropriate weight to each skill category. The weights should represent the productive capacity of each category. Although the three skill categories were more comprehen- sive than the labor force grouped by the level of schooling, one may roughly identify each skill category by the level 109 of schooling. The relationship between earnings differen- tials and school-years of the workers is well established in the U.S. Attributing the earnings differentials to the difference in the productive capacity, weights for three skill categories were obtained from Denison's study (1969). The professional and technical persons were weighted 200% more than the manual workers, and administrative and clerical persons, 100% more. The effective human capital stock indices and ranks are given in Table l. The labor efficiency of the U.S. is used as a standard: the labor efficiency of other countries is expressed as a ratio to that of the U.S. in all industries except for the leather finishing and 11 In many industries the high- tannering industry (291). est rankings are given to Canada and the U.S., and the lowest to Korea (R) and Costa Rica. The mean of all-industry average indices is 0.9085 and it varies from 0.8179 to 1.0133 across countries. This implies that the U.S. labor force is 25% more efficient than the least efficient labor force in the sample and is 10% more efficient than the average other countries' labor force, which is significantly different from the three times suggested by Leontief. Obviously, different industries in the same country have different ranks. One may also notice that irregular cases 11For this industry, the Australian index is used as a denominator. 110 Nmmo. GOHuMH>mo pumpamum mmom. com: oooo.H N.N H H N N N N m N H I H H m ¢ v m .¢.m.D MNNm. N.v m m I I m m m w I I I I I I I I .M.D ommm. m.m I I m m NH I I m v I 0H m OH I OH HH MHmmUOSm MMNm. m.m m 0H m w m b I m m m w v m m m b >63HOZ vhmm. 5.5 m m h m HH m m HH N m m m w w m m OOHxOE thm. m.m OH m m CH m HH m NH HH v m HH HH m m CH MOHOM thm. v.¢ v v m m CH m ¢ m m N m N m N N w GMQMh Nme. m.m m h v m h m m m m m w m h H h m MHUGH mHmm. v.m I HH OH m w v I OH 0H m 5 OH m 0H HH H MUHm mumOU Hmmm. m.m h m m h m m h m m H m h H h m N MGHQU MMHo.H H.N N N H H H H N H m I N m N m H m mwmcmo omom. m.m m m I I v OH H m h b HH m m m m m mHHmuumsd mMsmm xmvcH xcmm mmm mHm NMN .muw>¢ .mH0>¢ Hmm Hmm ohm 0mm va wmm HNm HHmUmmm HmN omN th owN HmN HMNIMDM kHuGSOU AMODBm BZMmHMmV =NUZMHUHmmm m0m¢H= Mm mmHMBZDOU m0 UZHMde H mHmdfi 111 as we observed in the rankings of countries by labor efficiency in the ACMS study are not observed. The rank- ings in Gupta's study using the results of the ACMS study are given in Table 2 for comparison. As pointed out by Gupta (1968), five Latin American countries demonstrate unusually the highest ranks, which does not occur in the present study. TABLE 2 RANKING OF COUNTRIES BY "LABOR EFFICIENCY" (GUPTA'S STUDY) 2:64:24. “423:“ United States 5.8 5 Canada 7.0 9 New Zealand 7.2 10 Australia 10.6 16 Denmark 10.0 14 Norway 9.9 13 United Kingdom 10.5 15 Ireland 13.3 19 Puerto Rico 6.9 8 Columbia 5.7 4 Brazil 2.4 1 Mexico 3.5 2 Argentina 6.3 6 El Salvador 6.7 7 South Rhodesia 12.2 18 Iraq 5.0 3 Ceylon 7.9 11 Japan 8.1 12 India 11.6 17 112 7. The Data The numerical values of the data used in this empiri- cal study are tabulated by country and year and are included in Appendix B and D. Only a brief description of the sample and variables follows: 7.1 Countries in the Sample The countries which comprise the sample were selected to include both developed and less developed countries as well as intermediate countries as long as the required variables were available in a reasonable number of indus- tries. The original sample involved 27 countries, yet it was reduced to 12 countries mainly due to the avialability of observations on capital variable. The characteristics of the countries are as follows: Africa Asia Europe America Australia 29 - Japan U.K. Canada Australia Norway U.S.A. LDC Rhodesia China - Mexico - (Southern) (Rep.) India Costa Rica Korea (Rep.) 7.2. Industry Data have been collected for manufacturing indus- tries at the three-digit level of the International 113 Standard Industry Classification (ISIC) compiled by the U.N. Each two-digit industry is represented by at least one three-digit industry, which involves a sufficient number of countries sampled. For instance, the textile industry (23) at the two-digit level is represented by spinning and weaving (231) and knitting mills (232) at the three-digit level of the ISIC. Official reports of the industrial census of each country were the main source of data (see Appendix C). Many of these reports do not follow the ISIC system, in which case appropriate adjustments were made. 7.3. Time Period The data pertain to different years between 1954 and 1968 and each country was observed three times in sequence. Significant efforts were made to have the same starting years and termini for all countries involved, although these efforts were not quite successful. A third observation for each country usually fell on the mid-point of the time span, which varied from five to ten years depending on the data availability of the country. Years of irregular character, such as years of an unusually high rate of unemployment, were avoided, if possible, in order to reduce the effects of business cycle and capacity utilization (Lovell, 1968). 114 7.4. Value AddedLLabor Input, and Wage Rates V, L, and W are defined basically the same as in the ACMS study, but L and W are measured in efficiency units as specified in the previous discussion. All money values were converted into U.S. dollars at the official exchange rates or at free market rates where multiple exchange rates prevailed. Time-series of wage rates of each country were deflated by the cost of living index expressed by the consumer price index, and capital and value added by the wholesale price index. 7.5. Capital Input and Rental Price Due to the restricted availability of the data on capital and rental price of the capital, a limited number of countries and industries were selected for the study. The capital input collected in the study is the net value of the tangible fixed assets such as land, buildings, and other structures, machines and other equipment, etc. The returns to capital were calculated from the factor—income share function: r = (V — LW) / K. 115 1:6. Skillyggmposition of the Labor Force For the proportions of the occupational groups in the labor force for each industry, statistics were obtained mainly from an extensive study on manpower planning made by Horowitz, gt_al. (1966). Their industry classification is a more aggregated two—digit level. Yet this may not seriously hurt the validity of this study, if one assumes that the three—digit industry selected in our sample represents the majority of the two-digit industry, respec- tively. CHAPTER V THE EMPI RI CAL RESULTS 1. Production Functions for the Industries The regression results will be discussed in detail in the next section. Only the results of the specification error tests on the four models are summarized in Table 3. For the purpose of comparison, classical criteria of model discrimination, the explanatory power of the regression R2 and F values are also reported in the last column of each table. R2 obtained from the covariance model ranges from 0.819 to 0.999.1 If one were to regard the coefficient of determination as a sole criterion of model discrimination, one must conclude that each and every model is a "good fit." It can not be a useful means to discriminate models. However, it is apparent from the tables that in a number of cases each model is misspecified. Since the specifi- cation error tests are performed on the stochastic models which are formulated using the error components model, R2 values reported in the tables are those of the regression 1Except for R2 model and three cases in K model. 116 117 ON.o mH.o 30.0 .<.z .«.z .¢.z .<.z s.<.z Lame “my Ammo «am.mv Amacofi.o Amzvmm.o Amxvmm.wfi Am NM mmm BM Hmeos “mafia: nausea mom OHmH mamme mommm oneaonHommm mma mo smazzam m mflmdfi 118 Table 3 Footnotes 1The relevant degrees of freedom are given in brackets below each statistic quoted. 2"R " indicates null hypothesis (no specification error) rejected at 10% level as well as at 5% level, "Al" indicates nonrejected at 5% level but rejected at 10% level, "A2" indicates non—rejected at both 5% and 10% level. 3"I.D." stands for "Kolmogorov Statistic being ill defined for this problem." 4Properties of the residuals (BLUS) conditional on 5. 5Properties of the residuals (BLUS) conditional on f. 6Properties of the residuals (BLUS) conditional on 7"N.A." implies that no nonsingular matrix was discovered. As the inverse of X0 is required for the tests, it is necessary that the X0 chosen should be nonsingular. Therefore, the specification error tests are not available for this model. 119 “om.mc 05.0H Asm.mv mm.o Lam.mv Ho.mfi Aom.mv mfi.m .w .Nm.o no.0 Hm.o m:.o Nm m.m.H Ammvmm.o Amavmfi.o Am Nm mmm 9N Hmdoa Ammaapwme mnanmdcdm s wcdnmoz .wcdccdnmv Hmm onH memme mommm 20He¢0HmHommm was so Nmazzsm Ammscflucoov m mamas 120 Hom.mv Afim.mv m:.o Afim.mc NA.HN Aom.mc om.Hm _m om.o mm.o 65.0 o¢oz. AN¢V:H.O “mavsfi.o AN x Nm mmm RN Hmdoz Amaaaz mcdppdcsv «mm onH Ammsawucoov m mummy mBmmB mommm ZOHB Nm mmm EN H0002 Amaadz Boos nmnpo s mcscmflm .Haas samv fimm onH AGOSGHHCOUV m mnmda memma mommm ZOHB¢UHmHUmmm mma mo Hm¢zzbm 122 Aom.mv 00.0 _m 00.0 55.0 03.0 Am¢0m~.o Amacmfi.o Amavnfi.o Bmmzom EmmB O¢Oz Amv Ammvfim.mfi ANV Ammvmm.mfi Amy .Afi Nm mmm 8% H0002 Ammnspndm a wonspacnsmv com onH Acoscfiucoov m mamas mamma mommm onadonHommm mma m0 Nm¢zznm 123 15m.m0 Amm.mv mN.0 A5N.mv 5m.mm ml No.0 mm.o 35.0 .¢.z .¢.z .¢.z . .¢.z Ammv Amy Homv A~ Nm mmm EH Hoses Aunmopnmamm a gamma .masmv H5m onH memma mommm onaaonHommm mme mo wmaszam Avmncwucoov m mamma 124 “Hm. NV 50. H mm (\N [\.v Aom. n0 :0. N: Rd HH.0 30.0 Hm.0 “mavsfi.o .0.H AN<05H.0 Emmzom BWmS .¢.z N Am “~4055 m MN “N A Am<00 Am 0 Amdvmm.fi 8mmz¢m Ammv AH Nm mmm 8% H0002 Ammanmfiapsm s wudpnfinmv owm onH H00SQHHGOUV m mqmdfi mBmMB momma ZOHH¢0HmHUmmm Mme mo HMdSEDm 125 «Hm.mv .¢.z .<.z .<.z . .¢.z .¢;z a Amm.m0 Ammo 1N0 Acme 505.50 no.5 mm.o Am AmnHSmHGHm 80:000H 0 00HH0CG09V HmN UHmH memma mommm ZOHB¢UHmHUmmm mmB m0 Hmmmzbm Acmsaflunooc m mamma 126 ~0m.mv Afim.m0 NH.o “Hm.mv m5.m Rom.m0 02.55 fid H0.o 5m.0 00.0 .<.z .¢.z .¢.z .4.z .¢.z Lam. Ame Ammv A5N.m0 A~40m~.o Ammcmo.o Ammvom.fifi Ammcmm.fl Am H0002 “0&0500Hm H0npfimv 00m UHmH mamme mommm ZOHBmUHmHommm mmB m0 Hmmmmbm Awmsnwucoov m mamms 127 Amm.m0 Asm.mv 00.: 1:5.m0 03.05 Ann.m0 ss.s5 .m mm.0 Nw.0 5m.0 Am¢0mH.o Am¢vmm.o Am mmm BM H0002 AmHmoHamso Hwfinpms0sH onmmv Ham UHmH Acmscflucoov m mummy mammfi mommm ZOHB¢onHUmmm mmB mo Hm¢22Dm 128 Hom.m0 HHm.NV 00.0 Afim.mc mm.n~ Aom.m0 mm.mm _m 30.0 no.0 05.0 Am tad Nm mmm BM H0002 Amposuonm Hmoaamno nmspov mam onH mamas mommm ona mmm 8% H0002 Acmsnfluqoov m 00009 Amodnocfiwmm asoflonpmmv Hmm onH 05009 00000 one¢oH5Hommm mma mo 5002200 130 53~.50 50~.~. .Nm.n Amm.mv 50.3 “00.50 50.0w Rd 05.0 0N.0 55.0 Am¢05fi.0 A~¢0HN.0 AN¢VNN.0 9mm202 .¢.z 2000 “04005.0 Ammo “00005.0 1300 550055.0 8mm3 .¢.z “NV 2N4vm0.N 200 104000.H . Amy 200005.0 8mm2mm 8mm¢m .<.z AHN.50 250050.0 500.50 AN Nm mmm BM H0002 chHHSpomuscmz 9203000 300 0HOH Ammficanoov m mqmdfi mawmfi mommm ZOHB¢UHmHommm mmB mo Hm¢22Dm 131 505.50 “am.mv 50.0 Asm.~0 Hm.fi “05.50 H5.0 4m No.0 50.0 50.0 .¢.z Am¢vmfi.0 AN Nm mmm BM H0002 Aaompm 0 conH0 ~05 onH A009GHUGOUV m MHmms .memma mommm ZOHB Nm mmm EH H0002 500000000: 9000000000Icozv 005 0909 09009 00000 2099009000000 009 00 0000000 50000000000 0 00009 133 000.00 000.00 00.00 000.00 00.0 hd 30.0 m0.o mm.o Amdvmfi.o 000000.0 000me.0 BmmEOM .¢.z 0000 600000. N 000 000000 0 0000 000000. 9mm: .¢.z 000 000000.00 8mmz¢m 0000 000000.0 8mm Nm mmm BM H0602 0000000000 00000000000 000 0909 09009 00000 2099009000000 009 00 0000000 0000:00co00 0 00009 134 Aum.mv AmN.NV owé Amm.mv oH.oH Anm.mv Hm.: fid mm.o am.o :m.o Am Nm mmm 9% H0002 chfinammom e wsduadspmfinmv Hmm onH mamma mommm onadonHommw mma mo Hm¢szam Acmscwunmov m wands 135 Aom.mv mm.o .fim.mv mo.o afim.mv :N.H Aom.mv mo.m tial mo.o Noo.o no.0 mm.o Aomv Amy .m.H Am Nm mmm EN H0002 Ammaodso> nopozv mmn onH mamma mommm ona¢onHommm mus mo mm¢zzam Acmscwucoov m mqm¢a 136 A3N.mv Am~.mv :m.o Amm.mv u~.u~ A:~.mv am.o¢ hd 50.0 00.0 :m.o .¢.z .<.z .<.z .¢.z .¢.z Ammv ANV Ammv Afim.mv Ammvmm.o Ammvflm.o Afi Nm mmm 8H Houoz Amcdnspomhscmz magmasnpmch Hon onH mamma mommm one¢onHommm mma mo mm¢zzsm Acmscflucouv m wands 137 equations of the error components model. Here, we observe significant variation in the values of R2 among models for each industry. Despite this fact, one has to be careful in applying the R2 criterion, because for some industries the model with the largest R2 is strongly rejected by the tests for specification errors.2 One may conclude from these results that the coefficient of determination some- times does not seem to be a useful means for discriminating alternative models and is not a correct instrument, either. The model which consistently performs better than any other model is the YT model, as shown in Table 4. In 16 out of 19 industries, the YT model performs better than other models. The next competing one is the RSH model. In seven industries (grain mills, knitting mills, pulp and paper, printing and publishing, electrical machinery, ship building and repairing, and scientific instruments) the RSH model leads the other models. Yet in six out of those seven industries, the YT model competes so keenly with the RSH model that the two models can not be discriminated by the specification error tests at the 10% significance level. Only in grain mills does it perform better than the YT model absolutely. The RZ model performs relatively well in five industries (leather tannering and finishing, cement, basic iron and steedq Shipbuilding and repairing, and 2For example, see ISIC 260, 321. 138 TABLE 4 THE BEST PERFORMING MODELS IN THE SET ISIC Bestl Sggggd 205 Grain Mills RSH YT 231 Textile (Spinning and Weaving) YT 232 Knitting YT,RSH 251 Wood Mills YT 260 Furnitures & Fixtures YT 271 Pulp, Paper Board, & Paper YT, RSH RSH 280 Printing & Publishing YT=RSH 291 Tanneries & Leather Finishing YT=RZ, RSH 300 Rubber Products YT 311 Basic Industrial Chemicals YT 319 Other Chemical Products YT 321 Petroleum Refining YT 334 Cement Manufacturing YT=RZ 341 Basic Iron & Steel RZ, YT=RSH 360 Non—Electrical Machinery YT 370 Electrical Machinery YT, RSH 381 Shipbuilding & Repairing RSH, YT=RZ 383 Motor Vehicles RZ=K YT=RSH 391 Scientific Instruments RSH, YT l The model listed first in the column is judged best at 5% level. When two models are related by equal Sign, they are equally good at the same Significance level. When they are separated by comma, the second model is judged equally good as the first model at 10% level. If one model is reported, the test is at 5% level. 139 motor vehicles), but it competes with other models, espe— cially with the YT, in all of these five industries. In many industries the Specification error tests are not available for the K model, mainly because no nonsingular matrix of X0 is obtainable in the tests. Therefore, it is hard to judge about this model based on the tests. How- ever, the regression results obtained from the stochastic model are so poor in terms of the Student—t tests in almost all of the industries (except for five industries) that one may be justified in ruling out this model from competition. The K model competes with other models only in the motor vehicle industry as far as the specification error tests are concerned. 2.. Regression Results Now a discussion of the regression results obtained from each of the four models for each industry is in order. A survey of the results suggests that the relations fit the data rather well, and small, reliable standard errors have been observed in a number of industries. The only excep- tion is the K model. Table 5 summarizes the number of Significant cases for each regression coefficient. Since each regression equation has been run for 19 different industries, the maximum number of significant cases is 19. When ordinary least squares are applied to the regression 140 TABLE 5 THE NUMBER OF SIGNIFICANT CASE FOR COEFFICIENTS Number of Significant Case1 Model A ‘ b e 82 l. YT 15 9 l4 2. RSH 15 12 - 3 0 R2 7 6 "’ 4. K 5 5 5 1"t" tests are at the 10% Significance level. 26, 6, and 6 refer to the equation (4.4.5)-(4.4.8). equations, in addition to the error components and the covariance models, Significantly different estimates are obtained. The complete results of the four regression models are tabulated in Table 6. The regression results of the best-performing model only are reproduced in Table 7. By performing the statis- tical tests of Significance on the estimated parameters, the null hypothesis being that each coefficient is equal to zero, one observes the instances in which the general- ized production functions are reduced to simpler produc- tion functions for each industry. These reduced versions of production functions are indicated in the last column of Table 7. In many industries the generalized production 141 TABLE 6 REGRESSION RESULTS OF THE FOUR MODELS Model ISIC No. (Sample Size) Coeffi- cients 205(33) 231(33) 232(33) 251(30) ' 260(33) YT a 6.043 3.810 3.276 —0.996 -0.658 6 0.231 0.807 1.313 0.593 0.376 (0.127) (0.397) (0.174) (0.226) (0.225) 8 0.207 -0.150 -0.307 0.489 0.229 (0.114) (0.312) (0.117) (0.196) (0.120) a -0.560 -0.315 —0.553 0.221 0.498 2 (0.204) (0.196) (0.082) (0.125) (0.207) R 0.637 0.448 0.761 0.822 0.405 RSH 3 6.418 3.332 4.341 -0.013 1.993 8 0.271 0.388 0.304 0.909 0.301 (0.108) (0.125) (0.141) -(0.118) (0.092) 8 0.215 0.502 0.475 0.262 0.937 2 (0.161) (0.180) (0.146) (0.102) (0.149) R 0.193 0.509 0.592 0.742 0.767 )7(1+0) 0.0021 0.0041 0.0013 0.0016 0.0049 RZ 3 10.346 -456.696 -3.476 -241.852 -22.830 8 1.048 -33.014 0.254 -15.568 0.370 (0.377) (32.12) (0.327) (8.927) (1.357) 8 0.271 23.425 0.735 14.384 —0.043 2 (0.524) (45.90) (0.339) (7.562) (2.211) R 0.200 0.034 0.300 0.164 0.003 Y 0.0005 -0.016 -0.0005 -0.010 -0.001 K 8 -644.460 -12.345 116.370 -13.752 -144.927 (285.847) (18.986) (230.679) (179.821) (83.095) 8 645.649 12.848 -116.028 14.057 145.946 (286.125) (19.244) (231.157) (180.858) (83.261) 8 -645.838 -12.731 116.028 -13.358 -145.967 2 (286.404) (19.548) (231.637) (180.338) (83.428) R 0.990 0.518 0.992 0.0005 0.998 p -0.0003 -0.005 -0.0005 -0.001 -0.001 A 1.3 1.4 1.3 1.6 1.4 “_43— .4. 142 TABLE 6. Continued Model ISIC No. (Sample Size) Coeffi- cients 271(30) 280(33) 291(24) 300(33) 311(36) YT a 3.195 0.328 2.235 3.236 1.101 6 0.978 0.085 0.031 0.751 0.658 (0.154) (0.166) (0.259) (0.127) (0.103) 8 -0.104 0.729 0.604 0.051 0.304 . (0.148) (0.111) (0.251) (0.077) (0.107) d -0.324 0.227 -0.453 -0.287 0.012 2 (0.164) (0.126) (0.215) (0.010) (0.085) R 0.740 0.809 0.533 0.661 0.871 RSH 3 3.215 0.472 2.279 6.312 0.272 6 0.560 0.733 0.620 0.220 0.910 (0.163) (0.073) (0.210) (0.106) (0.085) 8 0.227 0.515 0.070 0.374 0.273 2 (0.265) (0.106) (0.214) (0.169) (0.112) R 0.381 0.839 0.327 0.368 0.822 Y/(1+p) 0.00005 0.0027 —0.00002 -0.00082 —0.0014 Rz a -28.441 18.453 2.995 -27.040 46.177 8 -0.480 5.249 0.376 —0.065 5.543 (1.845) (2.659 (0.179) (1.982) (1.764) 6 2.255 -1.232 0.003 -1.278 -4.900 2 (2.992) (3.700) (0.183) (3.131) (2.579) R 0.020 0.112 0.217 0.008 0.227 Y -0.001 0.0005 -0.002 -0.001 0.001 K 8 -107.679 -77.667 161.547 —184.889 «19.289 (126.519) (222.238) (159.234) (47.502) (111.303) 8 108.722 78.215 -161.831 186.101 19.800 (127.053) (222.441) (159.811) (47.609) (112.257) 8 -108.723 -77.762 163.119 -186.319 -l9.308 2 (127.589) (222.644) (160.391) (47.718) (111.779) R 0.996 0.999 0.976 0.996 0.993 p -0.001 -0.0003 —0.0005 -0.001 -0.001 A 1.6 1.8 1.3 1.1 1.6 143 TABLE 6. Continued Model ISIC No. (Sample Size) Coeffi- cients 319(33) 321(27) 334(27) 341(33) 360(30) YT 8 5.809 0.082 2.881 5.509 7.254 8 0.743 1.222 0.714 0.734 0.616 (0.158) (0.288) (0.230) (0.232) (0.208) 8 -0.102 -0.043 0.068 -0.112 0.384 (0.145) (0.140) (0.133) (0.156) (0.195) a -0.515 0.085 -0.270 -0.494 -0.903 2 (0.099) (0.113) (0.173) (0.223) (0.214) R 0.702 0.485 0.768 0.372 0.724 RSH 8 5.321 4.260 5.167 8.864 7.581 8 0.498 -0.146 0.325 0.197 0.858 (0.083) (0.215) (0.116) (0.140) (0.180) 8 0.089 1.382 0.148 0.058 -0.663 2 (0.113) (0.323) (0.145) (0.206) (0.282) R 0.628 0.779 0.276 0.075 0.447 Y/(1+p) —0.00320 0.00069 0.00085 -0.00204 0.01286 R2 8 -48.857 25.853 -1.564 -128.347 760.590 8 3.244 7.794 0.228 1.446 17.002 (2.713) (1.739) (0.102) (4.104) (55.102) 8 -3.581 -9.798 0.212 3.026 -30.858 2 (4.525) (2.668) (0.116) (7.803) (74.264) R 0.044 0.446 0.246 0.018 0.001 V -0.0005 0.002 —0.001 -0.001 0.004 K 8 -147.921 —330.491 3.329 -216.459 400.082 (135.652) (121.922) (127.524) (104.752) (259.236) 8 149.114 333.324 -2.860 218.174 -398.794 (136.114) (122.585) (128.110) (105.130) (258.589) 8 -149.311 —335.169 3.392 -218.891 398.504 2 (136.577) (123.252) (128.697) (105.509) (257.943) R 0.993 0.963 0.964 0.988 0.993 p -0.001 -0.001 -0.001 -0.001 0.001 A 1.3 1.7 1.7 1.9 1.6 TABLE 6. Continued Model ISIC No. (Sample Size) Coeffi- cients 370(30) 381(30) 383(33) 391(27) YT a 3.907 —0.519 13.501 3.050 6 0.302 1.011 0.049 0.651 (0.239) (0.332) (0.275) (0.124) 8 0.191 -0.101 —0.545 0.495 (0.152) (0.197) (0.358) (0.151) a -O.156 0.203 -1.018 -0.504 2 (0.068) (0.181) (0.423) (0.009) R 0.325 0.339 0.235 0.836 RSH - a 4.437 3.114 13.683 3.397 6 0.438 0.344 -0.602 0.991 (0.099) (0.160) (0.393) (0.164) 8 0.405 0.728 0.600 -0.353 2 (0.119) (0.255) (0.669) (0.199) R 0.853 0.535 0.076 0.694 Y/(l+p) -0.00937 -0.00238 0.00434 -0.00086 RZ 8 324.916 -4.746 -l697.586 16.600 6 -11.963 0.459 19.863 0.993 (14.914) (0.295) (85.080) (0.752) a 18.146 0.899 -28.245 -0.892 2 (18.122) (0.470) (139.514) (0.917) R 0.035 0.382 0.002 0.840 Y -0.001 -0.0005 -0.014 0.0005 K 8 -341.576 54.099 3.522 -139.731 (256.797) (415.215) (44.645) (206.024) 8 343.177 —53.294 —3.264 140.337 (257.441) (415.826) (45.498) (206.638) 8 343.785 53.490 2.945 -139.943 2 (258.087) (415.521) (46.266) (207.253) R 0.998 0.987 0.089 0.993 p -0.001 -0.0003 -0.005 -0.001 A 1.1 1.8 1.43 1.3 RESULTS OF THE SELECTED REGRESSION EQUATIONS 145 TABLE 7 x—r’ Regression Results ’ ISIC1 Chosen \ Reduced No. Form a B 6 a Form 205(33) RSH 6.418 0.271 0.215 VES-1 (0.108) (0.161) 82 N2 231(33) YT 3.810 0.807 -0.150 —0.315 CES-l (0.397) (0.312) (0.196) S1 N2 N2 232(33) YT 3.276 1.313 —0.307 —0.553 VES+A (0.174) (0.117) (0.082) S2 82 82 RSH 4.341 0.304 0.475 VES-l (0.141) (0.146) (y#0) 82 $2 251(30) YT -0.996 0.593 0.489 0.221 VES-A (0.226) (0.196) (0.125) 82 82 S1 260(30) YT -0.658 0.376 0.229 0.498 VES-A (0.225) (0.120) (0.207) S1 S1 82 271(30) YT 3.195 0.978 -0.104 -0.324 CES-A (0.154) (0.148) (0.164) 82 N2 Sl RSH 3.215 0.560 0.227 (VES-1, (0.163) (0.265) y=0) 82 N2 280(33) YT 0.328 0.085 0.729 0.227 CD (0.166) (0.111) (0.126) N2 82 S1 RSH 0.472 0.733 0.515 (VES-1, (0.073) (0.106) y=0) 82 82 146 TABLE 7 (Continued) Regression Results ISIC Chosen - Reduced No. Form 8 B 8 8 Form 291(24) YT 2.235 0.031 0.604 —0.453 00 (0.259) (0.251) (0.215) N2 82 S2 Rz 2.995 0.376 0.003 CD (0.179) (0.183) (Y=0) S2 N2 300(33) YT 3.236 0.751 0.051 —0.287 CES-A (0.127) (0.077) (0.010) S2 N2 S2 311(36) YT 1.106 0.658 0.304 0.012 VES-l (0.103) (0.107) (0.085) S2 S2 N2 319(33) YT 5.809 0.743 -0.102 -0.515 CES-1 (0.158) (0.145) (0.099) 82 N2 S2 321(27) YT 0.082 1.222 -0.043 0.085 CES-l (0.288) (0.140) (0.113) S2 N2 N2 344(27) YT 2.881 0.714 0.068 -0.270 CES-1 (0.230) (0.133) (0.173) S2 N2 N2 RZ -1.564 0.228 0.212 CD (0.102) (0.116) (y=0) s S 1 2 341(33) Rz —128.3 1.446 3.026 (4.104) (7.803) N2 N2 YT* 5.509 0.734 -0.112 -0.494 CES-1 (0.232) (0.156) (0.223) 82 N2 S2 147 TABLE 7 (Continued) I L Regression Results t i ISIC Chosen Reduced No. Form a b e a Form RSH 8.864 0.197 0.058 (0.140) (0.206) N2 N2 360(30) YT 7.254 0.616 0.384 —0.903 VES—1 (0.208) (0.195) (0.214) S2 S1 S2 370(30) YT 3.907 0.302 0.191 -0.156 F.C. (0.239) (0.152) (0.068) N2 N2 S2 RSH* 4.437 0.438 0.405 (VES-1, (0.099) (0.119) y=0) S2 S2 381(30) RSH 3.114 0.344 0.728 (VES-1, (0.160) (0.255) y=0) S2 S2 YT* -0.519 1.011 -0.101 0.203 CES-l (0.332) (0.197) (0.181) s2 N2 N2 Rz -4.746 0.459 0.899 CD, (0.295) (0.470) (Y=0) N2 S1 383(33) RZ -l697.6 19.863 —28.2452 (85.080) (139.514) N2 N2 K 3.522 -3.264 2.945 (44.645) (45.498) (46.266) N2 N2 N2 YT* 13.501 0.049 —0.545 —1.018 F.C. (0.275) (0.358) (0.423) N2 N2 S2 “If ‘7 "1 r a ‘ _ --M ! HL»~‘-:n_——Vn “mv*1-- 148 TABLE 7 (Continued) Regression Results ISIC Chosen Reduced No. ‘ Form 3 B 6 a Form (0.164) (0.199) y=0) S2 S1 YT* 3.050 0.651 0.495 —0.504 VES-A (0.124) (0.151) (0.009) S2 S2 S2 1 The numbers in the parentheses are the sample size. 281 indicates the estimate is Significant (not equal to zero) by the t-test at the 10% level and 52 indicates the estimate is significant both at the 5 and 10% levels. N2 indicates the null hypothesis that the coefficient is equal to zero cannot be rejected both at the S and 10% levels. 3Numbers in parentheses are the standard errors. functions reduce to simpler functional forms. Only seven out of 19 industries Show that the generalized production functions maintain their functional forms (ISIC No. 205, 232, 251, 260, 311, 360, 391). In the previous section, we encountered a problem of choosing the best model among alternatives, even after the tests for specification errors were performed. The YT and the RSH models compete in six industries. These are knitting mills (232), pulp and paper board (271), printing and publishing (280), electrical machinery (370), Ship- building and repairing (381), and scientific instruments 149 (391). The statistical tests of Significance solve this problem in many instances. In the pulp and paper board, printing and publishing, electrical machinery, and Ship building and repairing industries, the RSH model reduces to the CES and other Simpler models, since Y in the l + 0 equation (4.4.21) is not significant at the 10% level.3 This implies that the elasticity of substitution is a constant or zero, because the RSH model was built on the assumption that 0 = a + Y x, a = 1. In the same industries, the YT model which is competing with the RSH also reduces to the CES or the CD or the FC cases. Therefore, one may conclude that the choice between the two models does not make any significant difference for those four industries, if the two functional forms reduce to the same model. The problem of choosing one functional form still remains in two industries, both the YT and the RSH models demonstrating the VES functional form (232, 391). Similarly, we have seen that the RZ model com— petes with other models in the five industries (291, 334, 341, 381, and 383). In the leather industry (291) the competing YT model reduces to the CD. The point estimate of Y in the R2 model is 0.0005 and its 90% confidence interval is (0.0539, —0.0529). On the basis of these 3The values of y/(l + p) were estimated by equa- tion (4.4.21). 150 results one would not reject the null hypothesis that Y = 0, in which case the R2 model reduces to the CD. In the basic iron and steel industry (341) and in the motor vehicle industry (383) none of the coefficients of the R2 model are Significant, while other competing models have Significant coefficients. In the Ship building and repairing industry (381), based on the same argument as in the leather industry, Y can not be said to be different from zero, and the RZ model reduces to the CD case. The competing YT model reduces to the CES case, yet its estimate of the elasticity of substitution, 8, is equal to 1.011, with a standard error of 0.332. At the 5% significance level one may not reject the null hypothesis that b = l, in which case the CES reduces to the CD. Therefore, the YT model and the R2 model do not conflict with each other for this industry, either. Taking account of all these results on the statis- tical tests of Significance, one may choose the most appro- priate model for each industry given the data. This chosen form is marked by a star in Table 7. Corresponding to the chosen functional form, we get the reduced func- tional form for each industry based on the "t" tests. This will be discussed in the next section. 151 3. Estimates of the Elasticity of Substitution Let us now consider the first-order parameter, the elasticity of substitution in the industry production func- tion. The estimates of the elasticity of substitution for the 19 industries Shown in Table 8 are computed with the parameters of the production functions chosen in the pre— vious section. The estimates are compared with other cross-sectional and time-series estimates for U.S. manu- facturing. Table 9 presents a summary of this comparison. First, as is claimed at the outset, the time-series esti- mates of elasticities of substitution between capital and labor in U.S. manufacturing are consistently well below the cross-sectional estimates. Interestingly enough, the estimates of this study based on the pooled cross-sectional and time-series observations are more in line with the cross-sectional results than with those of the time-series studies. Second, among the 19 industries seven industries represent the VES case (205, 232, 251, 260, 311, 360, 391), eight represent the CES case (231, 271, 300, 319, 321, 334, 341, 381), two represent the CD case (280, 291), and two represent the FC case (370, 383).4 Therefore, the conven- tional assumption as in the ACMS study that all industries 4Four more industries also exhibit the VES case and one more industry exhibits the CD case, if one considers the second best performing models. 152 TABLE 8 ESTIMATES OF THE ELASTICITY OF SUBSTITUTION ISIC Reduced Estimates Standard t—test3 by No.1 Form of 0 Deviation H0:0=l CES-A 205 VES-l 1.185 .234 .307 (33) 231 CES-l .807 .397 A2 .630 (33) 232 VES—A:YT 1.075 .031 .992 (33) RSH 1.034 .003 251 VES-l 1.059 .017 1.067 (30) 260 VES-A .998 2.685 1.010 (33) 271 CES-1:YT .978 .154 A2 .901 (30) RSH 1.007 .007 280 CDzYT 1.000 n.o. .913 (33) RSH 1.162 .161 291 CD 1.000 n.o. .354 (24) 300 CES—l .751 .127 R1 .776 (33) 311 VES-l 1.363 .182 .894 (36) 319 CES-1 .743 .158 A2 .649 (33) 321 CES-l 1.222 .288 A2 1.189 (27) 334 CES-1:YT .714 .230 A2 .227 (27) CDzRZ 1.000 n.o. 341 CES-l .734 .232 A2 .652 (33) 153 TABLE 8 (Continued) ISIC Reduced Estimates Standard t—test 0 by No. Form of 0 Deviation H0:0=1 CES-l 360 VES—A 1.887 13.732 .883 (30) 370 CES:RSH .730 .215 .506 (30) 381 CES—1:YT 1.011 .332 A2 .896 (30) 383 — n.o. n.o. —.108 (33) 391 VES—leT 1.704 6.210 .908 (27) RSH .960 .035 all-industry 1.042 average 1Numbers in the parentheses indicate the sample size. 2n.o. stands for not obtained. 3A2 indicates that one can not reject the null hypothesis both at the 5 and 10% levels, while R1 indicates one can reject the null hypothesis at the 10% level but not at the 5% level. are subject to the same production function has very limited validity. Estimates of 0 based on this assumption may also be biased, too. The last column of Table 8 reports the estimates of the elasticity of substitution from the CES— model developed in this paper (Section 5 of Chapter III): even with the allowance for nonunitary returns to scale, one can notice a significant bias in the estimates. 154 Amm.v .o.: .o.c ¢N.H mm.H .o.a .o.: o>.H .o.c .o.s mm. wH.| mm. mucmfisuumcH wo.H .o.c Ho.m mo. vo.~ .o.: www.w .o.s Hmn. «N. mm. mH. COHumuuommsmna . .o.c mu. hm. m~.H hm. mu. ow. . we. om. mw. .5002 .HuomHm hm .o.c om. «m. Hm. .o.c mm.H .o.c mv wo.H om. ms. .HuomHmsoz .o.s hm. mm. hm.H mm. Hm. m5. .o.: «m. om.H Hm. mo. mHmumz OHmmm mo.H mm. mm.H Nm. mm. mm. Hm. mH.H Hm.) mm. mm. NH.H| .oum .TGODm .o.s . .0.s m¢.H ¢m.l .o.c NN.H .o.s mm. om.H Hm. .o.: Edeonumm om. mm mm.H «H. .o.s mm. vmm.H ~H.HI mm. m~.H mm. HH.HI mHMOHETBU mm. up. m¢.H mv.H mm. .o.c mh. .o.c mm. on. mm. mm. Hmnnsm mm. on. an. mm. mm. mm. AooHHV mm. Hv. pm. 56. mm. Hmnummq H~.H mm. .o.c No.H .o.s hm. Awwnmv wa. ow. mH.H mH. ww. mcHucHum wH.H ms. wo.H hh.H om.H mm. MM.H vm. Hm. No.H mm. mo. Hmmmm .o.c . mm. NH.H mo.H om. mm.o Hm. . NH.H mm.H No.H musuHsusm em. mm mm. mm. ea. mm. wo.H mm. mv Hm. ow. om. anfisn .o.s om. mv.H Ho.H .o.c mp. mmo.H v¢.H mH. mo.H mo. mm. Hmnmmm< om. mm. mm.H >~.H mm.H Hm. om. we. mH. 0H.H mm. wH. mHHuxme mm. mu. mH.m mm. mm. Hm. mH.H .o.s ov. «N. mm. hm. poem Pm .0 .6 90 .6646 m m 0 6 u a m auumsocH GOHDOTmImmouU ucmmmum mmHHmmnwfiHB )AIV ! II" I)" mMHQDBm mDOH>mmm Mm ZOHBDBHBmmDm m0 MBHUHBmfim mmB m0 mflBflSHEmm m mqmdm. 155 TABLE 9 Footnotes a = McKinnon (1962) b = Kendrick (1964) c = Ferguson (1965) d = Lucas (1963 or 1969) e = McKinnon (1963) a' = Minhas (1963) b' = Minasian (1961) c' = Solow (1964) d' = Lin-Hildebrand (1965) e' = Murata—Arrow (1965) f' = Arrow, §E_§1. (1961) l = Automobiles only. 2 = Industries in this study are finer three-digit classifications. 3 = Knitting (232) 4 = Industrial Chemicals (311) 5 = Shipbuilding and Repairing (381) 6 = Based on a comparison of the U.S. and Japan only. Third, in many industries the elasticity of substi- tution is not very different from unity and from each other.5 5This is quite similar to Minhas' estimates. Hutcheson (1969) performed a test to determine whether one can reject the null hypothesis that the estimates of the elasticities of different industries are the same. He found with Minhas' estimates that one can not reject the null hypothesis even at the 25% significance level. Similar tests may be performed on the present estimates. 156 One can not reject at the 10% level the null hypothesis that the elasticity of substitution is equal to one in seven industries of the CES cases where the standard errors and the distribution of the elasticity can be obtained. From these, one may conclude that the CD pro— duction function performs relatively well and gives a good approximation in many industries. As to the industries where the production func- tions approach the fixed coefficients case, one can conduct an indirect test whether the production relations of the industries, in fact, reduce to the FC case. The scatters of V/L and V/K of the two industries (370 and 383) are plotted on Figure 7 and Figure 8, respectively. The picture emerging from these Scatters does not support the argument that the production relations in these indus- tries are really the fixed coefficients case. The capital— intensity corresponding to the Similar labor intensity varies quite significantly from country to country, and in many instances one can observe a negative relationship between the capital and the labor intensities. Further, as one compares these two scatter diagrams with that of other industry, say, Shipbuilding and repairing (381) which is not the FC case (Figure 9), one can hardly observe any distinct differences. The reason that several models are competing in these two industries may not be the fact that each model fits the industry well, but may V/L 157° 110—- O 8‘ 060-“- H 3 O O a D O 10__ O 69 O O . .o 31 0 1 n 5 1.0 2.0 ($1,000) ‘575 F‘IGURE 7 SCATTERS OF FACTOR INTENSITIES (370) V/L 180_J \D 0 ($1,000) 10 158 SCATTERS OF FACTOR INTENSITIES (383) r... o 0 o F o o O o O O O O O o O O o O _ 2 0. 5 9 0 ° 2 0.2 1.2 ($1,000) 2.6 FIGURE 8 V/L 100.. 50 ($1,000) 10~ SCATTERS OF FACTOR INTENSITIES (181) 159 O O 0 O O 0 O O °o O 0 0 ° 0 O o ‘3 O O 0 Z l L. 1K 1.0 3.5 7.50 ($1,000) FIGURE 9 160 be the fact that each model is too poor to exclude other models. For Six industries, where the two models compete so keenly that one can not make a choice between them, estimates of the elasticity of substitution of both models are reported (232, 271, 280, 334, 381, and 391). Although the numerical values are different, one may not deny that the two estimates are practically the same, if one con- siders the standard deviations of the estimates. Fourth, although the distribution does not work out well for the VES cases because of the small sample Size, the variations in the elasticity of substitution are relatively very small. The mean estimates of the elasticities and their standard errors are given in Table 8. In contrast to the CES cases, the elasticities are generally greater than one and the standard errors are very small. The relative insensitivity of their elas- ticities to the capital-labor ratio can be more easily observed from Figure 10 (a through 9). The elasticity is drawn on the vertical axis with variations in the capital-labor ratio on the horizontal axis. In most VES cases the elasticity varies in a very narrow range around one, except for a few erratic cases. One may be able to say that although the VES cases are Significant statistically in several industries, the mean of the estimates of the variable elasticities gives a very good 161 estimation, and the practical relevance of the VES in pre- ference to the CES or the CD is reduced. This argument is also supported by our other findings, especially the fact that the generalized production functions reduce to the cases Simpler than the VES in two-thirds of the 19 industries. Fifth, since one could not determine the Sign of the first and the second derivatives of equation (3.5.21) with respect to the input ratio, so the plot of the elas- ticities of substitution and capital—labor ratios does not support a ubiquitous functional form. In the grain mill industry, the chosen functional form is the RSH, which is derived from the assumption of the linearity between the elasticity and the input ratio. In the other Six industries exhibiting variable elasticity of substi- tution (205, 232, 260, 311, 360, 391), no a priori rela- tion is assumed in deriving the production function (YT form) and the production function iS derived from the empirical relationship. One finds no uniform relation between the elasticity of substitution and the input ratio. Therefore, it is hard to establish, a priori, any definite relationship between the elasticity and capital-labor ratio, and the approach to the estimation of production functions based on a priori assumptions as to the relationship is cast into doubt. In particular, none of the six industries shows a linear relationship C '.‘J . 1:50 .40 '— C) (‘0 .1. . O 1'.) .30 ’1! g. in) Piflj 162 "12 p'". 137.5 '53. Jeni ...L‘H. FIGURE 10-a 329-4 . THE RELATIONSHIP BETWEEN THE ELASTICITY .1 AND THE INPUT RATIO (ISIC 205) IV! C . .15.”! . OF.) 1 .1. (173.3 thf: CI.L.O U . 3.2. . 01') O 163 FIGURE 10-b (ISIC 232) .1. N Hf) _ {:0 .1. . 1+0 .05) .1. .357 5‘ r ‘.-’ . 1:0 ") ‘ fl (3 LI) J .. )J f. O . *JL.‘ 164 1.1 L “1'—"'3 r 'I i 1 - - _‘I'T‘H' +-h*1“+—r+ ‘H‘j‘i‘ + + 1)— ++++ FIGURE 10"C bk: (ISIC 251) . 3 t‘lN ,_. ...— fl.-(tlfi .604! ‘Wfllq “n.o., ..Ii- 165 .11 (:L 0 E: ___. .1 3 c: 1.0 __ _-. 0) F .._ j“ '1 F .. C: _- +++~fi :1 . C3 .— WE” “)7 ‘H‘ 'H' + ,1 .. + 0 +4- +_ (‘J T .J ’ m I ) [x 0) "— L- i". -i- or“ (to 330 '95 280 323 FIGURE 10-d (ISIC 260) ...L- 150 C... an .4 g ..QO ,q 0 9‘! o . (if) L: . 7:7 0 11 fl 166 P“ I n 1' 1- E“ r y— i i 1 i i j 1 i 1 i i i 1 i i I 1 l in ' ’ 0 r5 3 :09 :32. :35 T;q. 253 3"2. 32“ :32. FIGURE 10-e LL L-) 'J 1 (ISIC 311) .L L) m 1.0. C v ! 167 i l l 1 J I I l i J- L J,_L_i i i l T l 7 .0) . T) '3 '1 2.. 7 152‘ I: 10." 141-3.; 5.: v _n 2 '72. 6“ FIGURE lO—f . L (13.3 r._._..r_.... ’4, -. ' I" l/ \x 1 i\\ I (ISIC /P_— 0 m H ”"3 .10. '1‘." L-.. L» . (:0 167 —d-—- 21".) H 2%..) 352‘ l: 1.3.") 148.9 5.: .3 FIGURE 10-f 81. (ISlC 360) r-"" r— // [.1 v--- ""7 1 5' I" “ \z "7 --’ ' fl 1 l \\_ 2 l (- '1. p.- p- ' I'sj I; . 1' 1+.L:1( ‘ (.5 i”. (u .1. FT) 1:3 1 .L 1 1 1 1L1J_“L_th) 168 '1‘“) o I»: 'g— 7‘ “7}? ”£5.14 .5.'.) FIGURE (ISIC - ...) ——l. A (”‘7 1 169 between the elasticity and the capital-labor ratio. If one finds any linear relation, the Slope of the linear function is close to zero. 4. Estimates of Returns to Scale v One of the special features of this study is to estimate the returns to scale and the elasticity of factor substitution simultaneously. If the chosen functional form for each industry is RSH, the returns to scale is unity by assumption. Yet in all other functional forms, the returns to scale are to be estimated by the parameters of the production functions. In the K production function, it is estimated by the two—step maximum likelihood esti- mation procedure. In the YT model, the returns to scale are assumed to have a special functional form: _ d A_1+1TSO Therefore, it can be estimated by the knowledge of a and 3 of equation (4.4.5). Being different from other produc— tion functions, the R2 form demonstrates variable returns to scale; i.e., the returns to scale vary as the output (or value added) changes. The RZ production function is no longer homogeneous, though it is homothetic. The vari- able returns to scale can be estimated by the specific form of the returns to scale function. The returns to scale function incorporated in the R2 form are: 170 where 0 = a + B. a and B are estimable from equation (4.4.7). In the K model, although the estimates of para— meters from the second-step maximum likelihood estima- tion are very poor (as are the estimates of the elasticity of substitution), the estimates of returns to scale seem relatively sound; the sums of the squared errors of the regression (4.4.8) for different industries are very small for the selected values of 6 and f. The numerical values of I, the estimates of the returns to scale, of 19 industries are given in Table 10 with the sums of squared errors. From this table, one may say that the hypothesis of unitary returns to scale are hard to accept and a majority of industries exhibit increasing returns. In some industries they approached a value of two. Also Significant is the fact that returns to scale are differ- ent among industries. From the YT model the estimation of the returns to scale are very inconclusive. In the 18 industries, for which the YT function has been chosen or is a close competitor, d, the estimate of the coefficient of log L iS insignificant in five industries and I, the estimate of returns to scale, has negative values in nine industries. 171 TABLE 10 ESTIMATES OF THE RETURNS TO SCALE FROM THE K MODEL ISIC No. X SSEl 205 1.8 0.00000014 231 1.4 0.00002960 232 1.3 0.00000408 251 1.6 0.00000156 260 1.4 0.00000087 271 1.6 0.00000118 280 1.8 0.00000004 291 1.6 0.00000252 300 1.1 0.00000284 311 1.6 0.00000258 319 1.3 0.00000218 321 1.7 0.00000537 334 1.7 0.00000279 341 1.9 0.00000251 360 1.6 0.00000368 370 1.1 0.00000160 381 1.8 0.00000020 383 1.4 0.00192974 391 1.3 0.00000345 lSums of Squared Errors. 172 One may want to say from this that the hypothesis of unitary returns to scale can not be rejected (Yeung and Tsang, 1971). Yet, a more plausible explanation would be that the returns to scale function are specified in- correctly in the YT model or that the multicollinearity problem is serious among the regressors. In the five industries for which the R2 model is significant (291, 334, 341, 370, 383), only one industry (334) has significant coefficients for both the capital and the labor variables. The sum of the two is less than zero, indicating decreasing returns to scale. Further, the estimate of Y is not Significantly different from zero in this industry. One may say that the added com- plication of a variable return to scale has not been overwhelmingly justified. Yet this is probably not because of an incorrect theory of variable returns to scale; rather, it is because of an incorrect specifica- tion of the returns to scale function and/or the produc- tion function. 5. A Measurement of Economic Efficiency Although the error components model accounts for the difference in the economic efficiency among countries, one does not know the magnitude of the difference, unless one estimates the components of the variance. Instead of estimating the error components, a covariance technique 173 is used on the best—performing production function for each industry to get the difference in the economic efficiency among countries. The economic efficiency of the countries in each industry is measured from the country constants (in the same time periods) of the best-performing regression equations applying the covariance model on equations (4.4.5) - (4.4.8), as we did in equation (4.4.18) for the YT model. In all the 19 industries except in the leather tannering (291), the a of equation (4.5.1) measures the U.S. constant in the first time period (for 291, the a represents the Australian constant). The results are reported in Table 11. The reported numbers are deviations of other countries' con- stants from the U.S. constants in the first time period. Therefore, the economic efficiency of the U.S. indus- tries can be measured by taking an arithmetic average of the deviations of all other countries' constants from the U.S. constants in every industry. For example, to compare the efficiency in Spinning and weaving (231) and planing, sawing, and other wood mills (251) indus- tries of the U.S. in the first time period, get the arithmetic average of deviationscflfother countries' constants from the U.S. constants; —l.02 for 231 and .55 for 251. One can say that industry 231 is more efficient than 251, because in industry 231 average other countries 174 =.umouum= 0:» mo 05Hm>um 0:90 .uc0oHMHS0Hm 0HH0oHumHu0um 000 mua0umcoo mfiHu 0:0 mhucaoo 0:» noan 90 =.ummulm= may no 00:00HMH00H0 mo H0>0H 0:9 m .0H0>Huowmmmu .monnmm mfiHu OHHSU 000 000000 0:» mo mun0umqoo meHe 0.H 0H0.0 000.0 000.0 000.00H 00H.0mH M00.H 000.0 000.0 0H0.00 00 000.0 H00.0 0000.0 0000.0 0000.0 0Hm.0 000.0 000.0 0000.0 m0 00.0 mH.01 00.0 00.0 00.0 00.01 mH.0 MH.0 00.0 009 00.01 00.01 00.0 00.0 00.0 00.01 00.0 H0.0 0H.0 H09 00.0 1 00.0 00.0 00.0 00.0 00.0 00.0 00.0 .0.0.0 1 1 1 1 1 1 1 1 1 .M.0 1 1 00.0 H0.01 00.0 1 00.01 m0.01 00.HH1 0Hmovonm 00.H1 H0.H1 00.01 00.01 00.0 00.0 m0.H1 00.H1 00.0H1 003Hoz 00.01 00.0 0H.01 00.01 00.0 00.0 00.01 00.01 00.HH1 OOwaz 00.H1 00.01 H0.01 00.01 00.0 00.0 MH.01 00.01 00.0H1 00HOM 00.01 00.0 00.01 HH.0 00.01 0m.0 0m.H H0.01 0m.0H1 c0000 HH.H1 00.01 00.01 00.01 00.0 00.0 00.01 00.01 00.HH1 0H00H 00.01 00.H1 00.0 1 00.0 00.0 00.01 00.H1 0m.HH1 00Hm 0umou 00.H1 0H.01 H0.01 00.01 00.0 00.0 00.01 00.01 H0.HH1 00Hsu H0.01 1 00.0 00.01 0H.H 0H.0 00.01 00.01 0H.0H1 000000 MH.H1 00.0 00.01 00.01 00.H 00.0 H0.01 00.01 00.0H1 0HH0H0090 000 H00 000 H00 000 H00 0m0 H00 000 0090500 UHmH H00 1:08 H0. 000 H .0.00 mmBmDQZH wm mmHMBZDOU m0 NUZflHUHmmm UHZOZOON HH mflm<8 175 cmacHucou .HH mandfi 0m0.0 0m0.0 000.0 00m.0 H00.0 000.0 000.0 H00.0 000.0H H00.0 0 000.0 000.0 000.0 H00.0 000.0 000.0 H00.0 H00.0 0000.0 0000.0 0 0H.0 00.0 00.0 H0.0 00.0 00.0 0H.0 00.0 00.0 0H.0 09 00.01 00.0 00.0 00.0 00.0 00.0 mH.0 no.0 0H.0 0H.0 09 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 .¢.0.D 1 00.01 00.01 1 1 0m.H1 1 0H.H H0.01 00.01 .M.D 1 1 1 m0.m1 00.01 00.01 1 1 1 00.01 0Hmmwonm 00.m1 00.01 m0.0 mm.H1 00.01 00.01 0m.H1 1 mm.01 00.01 003hoz 00.01 mH.m1 00.H mm.H1 00.01 00.H1 00.01 mm.0 00.H1 00.01 00mez 00.01 00.01 H0.H 0H.01 00.01 00.01 mm.01 m0.H 00.01 0m.01 00H0M 0m.01 m0.01 00.0 H0.01 00.01 H0.01 00.01 m0.H 00.01 00.01 00000 0H.01 H0.01 00.0 no.0) 00.01 00.H1 00.H1 00.0 0H.01_ 00.01 0H00H 1 Hm.0H1 1 m0.01 H0.01 1 1 1 00.01 m0.0 00Hm 0umoo 00.01 00.01 HH.H 00.01 00.0: Hm.01 0H.01 00.0 00.01 00.01 0GH£U m0.H1 00.H1 00.0 00.01 00.m1 00.01 00.01 00.0 00.H1 0H.01 000000 H0.m1 00.01 00.0 1 1 0m.H1 00.01 00.H 00.H1 00.01 0HH0HumS< H0m m0m H0m, 00m 00m H00 000 H0m 0Hm HHm muunsou UHmH _.:,:‘-_'-:_ 1. .: _ . 176 produce less than the U.S. by 1.02 (thousand U.S. dol— 1ars) per man-year with the same amount of inputs, while in industry 251 average other countries produce more than the U.S. by .55 (thousand U. S. dollars) per man- year. In the second time period, with the same amount of inputs, all countries (including the U.S.) produce .01 (thousand U.S. dollars) more per man-year in industry 231 than in the first time period, while .04 (thousand U.S. dollars) less in industry 251. The efficiency of industry 251 is dropping as time goes on, while that of industry 231 is increasing slightly. The efficiency of other countries can be mea— sured by taking the difference of each country's constant from the mean of the deviations of all countries' con- stants from the U.S. constant. For instance, to compare the economic efficiency of industry 231 and 251 in Korea, take the difference of Korean constants in those two industries from the average deviation of other coun- tries' constants from the U.S. One gets .13 in industry 231 and .30 in industry 251, because the average devia- tions of other countries from the U.S. constants is -l.02 in industry 231 and .55 in industry 251, while the Korean constants are -.89 in 231 and .85 in 251. Therefore, in Korea industry 251 is more efficient than industry 231 compared with average other countries. 177 In all the 19 industries the country- and the time-dummy variables as a whole are quite significant (not equal to zero) at the usual significance levels, though each country or time constant alone is sometimes not significant in several industries. One may notice that the assumption of constant economic efficiency across countries can result in a serious bias. One should note, however, that the term "economic efficiency" is a very inclusive concept. It contains almost every- thing specific to the country which can not be attribut- able to other causal variables such as factor prices and factor-use ratios: the difference in entrepreneur— ship, research and technology, market structure, factor mobility, demand condition, transportation and other external economies, and even weather may be included. Further implications of the difference in economic efficiency between countries and between industries will be discussed in detail with a two-country (U.S.A. and Korea) example in Chapter VI. CHAPTER VI SOME IMPLICATIONS OF THE EMPIRICAL RESULTS Now it is time to consider the incidence of factor- intensity reversals. Corresponding to the reduced form of the chosen production function of each industry, the "substitution function" is plotted for the 19 industries. The lower left-hand ends of the substitution curves corre- spond to the country whose relative factor price (W/r) is the lowest in the industry in question. Similarly, the upper right-hand ends of the substitution curves indicate the country's relative factor price which is the highest in the industry. These two points define the empirically relevant range for the relative factor prices. The number of factor intensity crossover points located within the relevant range for W/r was then counted. The relevant range for W/r (also for K/L) is two times larger than that used in the Minhas-Leontief analysis. Restricting our attention for convenience to the 12 industries which show natural log-linear substitution functions, we get Figure 11. First, the picture emerging from the 12 substitution lines drawn by the computer con- firms the hypothesis that factor-intensity reversals are quite possible in the real world. We observe ten crossovers 178 3.55 7 l b. ~O.5 0.0 g I J 1r -o.8 0.0 3.20 7.20 FIGURE 11 SUBSTITUTION LINES AND CROSSOVERS (I) 180 in 12 industries. Even if we delete the motor vehicle in- dustry (383), the possible fixed coefficients case whose substitution function is a horizontal line at the mean capital-labor ratio, we still observe three crossover points. Since it is now justifiable to regard the substi- tution functions of the 12 industries as linear functions, one may want to apply the least squares technique directly to the stochastic version of the substitution function. For instance, for the YT model, which reduces to the CES, eit (6.1.1) 1n (gm: 1n (9%) + (1 + p) ln (gm; + or __1__ l + p 1n (9%) it + eit (6.1.2) In @it -ln (9%) + This is so, particularly because the estimates of the intercept term of the substitution functions (3, a and §) are not significant very often. The results of this tech- nique are shown on Figure 12. As noted, the intercepts are changed in many industries, but the slopes of the lines and the overall picture remain unchanged. The number of crossovers increased to 14 from ten. Without the motor vehicle industry, there are still seven crossovers. It may be hypothesized that, since the rest of the seven industries involve the existence of curvature in the non- linear "substitution curves" of the VES case, they might produce more crossover points than in Figure 12. Leontief emphasized the relative frequency or prabability of 1n (K/L) O \D_- h 1!) Ln ._ «3 Ln . w o *—‘ hi; I 1 l I 1 -0.|8 0.0 3.20 7.120 FIGURE 12 SUBSTITUTION LINES AND CROSSOVERS (II) 182 crossovers. For instance, he said, ". . . of the theo- retically possible 210 crossover points between the 21 lines entered on the graph, only 17 are found to be located within the wide range of factor-price ratios." Yet it should be relaized that the total number of possible cross- overs is quite deceptive, because a certain number of them may occur in the negative quadrants for which there is no economic relevance, and which should therefore be discarded. The absolute number of crossovers is of more economic significance than the relative number, and is the appropriate figure to consider. Second, however, most of the crossovers occur between industries whose substitution lines run close together throughout the entire range. This is a direct result of our findings that the elasticity is different in different industries, but that the difference is quite small in many industries. Even if the lines intersect, they do so at a very acute angle. Consequently, the difference in factor intensity among those industries will be minimal. Further, reversals occur mainly among the industries whose capital-labor ratios are relatively low. Moreover, reversals are observed at the left-hand ends of the substitution lines; i.e., the reversals occur at lower factor-price ratios and the range of factor price ratios relevant to the reversals is small compared with the whole range of the relevant factor-price ratios. For 183 example, the substitution lines of industry 280 and 381 crossover at the left-hand ends of the lines. Yet only four out of 30 observations in industry 381 and six out of 33 observations in industry 280 fall in the range of factor- price ratio below the crossover point and those observations are relatively underdeveloped countries. Therefore, one may say that for all practical purposes the capital-labor intensities of the industries whose substitution lines crossover may be considered identical and reversals are possible between the underdeveloped and the developed countries, but not among the developed or among the under- developed. Considering the evidence of factor-intensity reversals in this sense, the incidence of the reversals is not as significant or as pervasive as it would otherwise be. The operational and predictive value of the compara- tive advantage theoren based on the conventional dis- tinction between capital and labor-intensity assumptions is not spoiled by the possibility of the factor-intensity reversals. Yet this does not exclude the possible criticism of the theoren on other grounds. For instance, the chief criticism based on modern growth theory is that compara- tive advantage is essentially a static, long-run equi- librium concept which ignores a variety of dynamic elements such as structural disequilibrium in factor markets, 184 variation in the quantity and quality of factors over time, economies of scale, complementarity among industries, etc. (see Chenery, 1961). At the intuitive level, the theorem based on the factor endowments seem to contain some elements of truth. Yet we have seen in the previous chapter that each country demonstrates a different degree of economic efficiency independent of the factor prices. A survey of the country constants (Table 11) suggests that not only the factor endowments, but also the economic efficiency of each country in the use of the factors of production may ex- plain many trade flows among countries. The observation is that the capital (labor) abundant, high (low) wage— rate countries would tend to hold a comparative advantage in those industries which show higher economic efficiency, even though those very industries happen to be relatively labor- (capital)-intensive at the prevailing relative cost of labor and capital. This implies that the economic efficiency might compensate or overcome the disadvantage of the country's factor endowments. (As in the simple version of the Heckscher-Ohlin theorem, the transpor- tation costs are not considered). As an example, let us examine two countries: the U.S., a relatively capital abundant‘Country; and Korea, a relatively labor abundant country. Table 12 compares the capital-labor ratio and the economic efficiency of each industry for the U.S. ' mi (‘1' g. . ‘.J’ 1' 185 The numbers in the economic efficiency colums are the mean deviations of other countries from the U.S. constants in the covariance model. Therefore, the smaller the values, the more efficient are the U.S. industries. Generally, the capital intensive industries of the U.S. show higher economic efficiency than the average other countries [pulp and paperboard (271), other chemical products (319), basic iron and steel (341), and motor vehicles (383)I. Yet several industries which are rela- tively capital intensive demonstrate negative efficiency [petroleum refining (321)] or positive but very small advantage in terms of economic efficiency than other countries [industrial chemicals (311) and cement manu- facturing (334)]. On the other hand, quite a few labor intensive industries exhibit relatively higher economic efficiency than the average other countries [textile (231), knitting C232), rubber products (300), nonelectrical machinery (360), electrical machinery (370), and scientific instru- ments (391)]. Therefore, if the data and the methods of the Leontief study (1953) are correct, the so called "Leontief paradox," that the U.S., in fact, concentrates on the production and export of relatively labor intensive industries, may possibly be explained by the comparative advantage in the economic efficiency of the U.S. industries. 186 TABLE 12 RANKING OF INDUSTRIES BY INPUT RATIO AND EFFICIENCY (U.S.A.) £§£§_ K/L Efficiency 205 42.70 -11.37 231 33.73 -1.02 232 16.39 -1.02 251 22.80 .55 260 15.40 .79 271 145.52 -0.82 280 22.15 -0.28 291 - -0.97 300 22.45 -1.42 311 52.55 -0.40 319 129.15 -2.06 321 249.61 1.25 334 199.34 -0.62 341 83.06 -l.89 360 32.94 -4.37 370 18.69 -1.72 381 15.78 .68 383 56.90 -6.12 391 16.81 -2.74 187 As to the Korean industry, one observes the reverse facts. The numbers in the economic efficiency column of Table 13 are the deviations of the Korean constants from the mean deviation of all countries from the U.S. constants. Therefore, the greater the values, the higher are the efficiencies of the Korean industries. In general, labor intensive industries have higher ef- ficiency [knitting (232) and textiles (231)] and capital- intensive industries have lower efficiency [grain mills (205), other chemicals (319), petroleum refining (321), and motor vehicles (383)]. However, some labor intensive industries demonstrate lower efficiency [furnitures and fixtures (260), leather (291), basic iron and steel (341), and electrical machinery (370)] and several capital in- tensive industries show higher efficiency than the average other countries [wood mills (251), industrial chemicals (311), and cement manufacturing (334)]. Our tentative findings on the relative profit— ability of different industries among countries seem to receive support from other empirical studies. For in— stance, Hufbauer (1970) recently tested several con- temporary hypotheses which attempt to explain the trade patterns among countries. The result is that each of the hypotheses explains the existing trade pattern so well that one can not select a single hypothesis. The tested hypotheses emphasize many possible sources of the 188 TABLE 13 RANKING OF INDUSTRIES BY INPUT RATIO AND EFFICIENCY (KOREA) ISIC KgL Efficiency 205 36.65 -.93 231 13.32 .13 232 8.99 .89 251 20.91 .30 260 6.37 -.23 271 18.44 .10 280 15.34 -.53 291 11.26 -l.05 300 .67 .00 311 51.77 .04 319 20.56 -.48 321 146.30 .00 334 285.73 .09 341 9.91 -.70 360 11.77 .18 370 12.24 -.47 381 13.27 .53 383 15.24 -2.27 391 5.76 -.31 189 differences in economic efficiency: human skills, economics of scale, stages of production, technological gaps, product cycle, and preference similarities. But not a single device lends an exclusive explanation. In order to check the validity of this observation and to explain the sources of the observed difference in the economic efficiency, much further research and careful investigation are needed. 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APPENDIX A EMPIRICAL STUDIES USING CES CLASS PRODUCTION FUNCTIONS Cross-Sectional Studies (1) Cross-Country Author Sample ACMS (1961) 19(2) Minhas (1963) 19 Murata-Arrow (1965) 23 Fuchs (1963) 19 Bardhan (1967) 19 Clague (1969 2 (2) Cross-Region Minasian (1961) Solow (1964) Lin-Hildebrand (1965) Dhrymes (1965) Zarembka (1970) Time-Series Studies Author McKinnon (1962) McKinnon (1963) Kendrick (1964) Ferguson (1965) Maddala (1965) Lucas (1963) Hodges (1966) Author Philpot (1970) Year Industry Method 1950-5 1950-5 1953-9 1950-5 1950-5 1957-8 S. state 1957 S. region 1956 S. state 1957 U.S. state 1957 U.S. state 1957-8 Year Country Industry 1947-58 U.S. 2-digit 1899-57 U.S. 2 1947-58 U.S. 2 1931-58 U.S. 2 1949-61 Canada 2 1948-58 India 2 Data 8-country 1953, 58, 63 Cross-Sectional and Time-Series Studies 3-digit V/L vs. W 3 V/L vs. W 2 V/L vs. W 3 V/L vs. W With a shift variable 3 V/L vs. W 3 K/L vs. W/r 2 V/L vs. w 2 V/L vs. W 2 V/L vs. W and K/L 2 V/L VS. W V/L vs. r 2 V/L vs. W with Kmenta method (1967) Method with lagged dependent variable with an estimator of technological changes with total capitals ad- justed for capacity utilization with lagged adjustment processes with non-competitive equilibrium conditions Method V/L vs. W allowing for labor quality difference and non-unitary returns to 200 scale APPENDIX B COUNTRIES AND YEARS OBSERVED Time Country 1 2 3 Developed 1. Australia 1961 1965 1968 2. Canada 61 65 67 3. Japan 62 64 66 4. Norway 57 59 63 5. U.K. 54 58 63 6. U.S.A. 58 63 67 Less Developed 1. China (R) 1961 1965 1968 2. Costa Rica 58 63 68 3. India 59 61 64 4. Korea (R) 63 66 68 5. Mexico 55 60 65 6. Rhodesia (S) 62 63 64 *For the following countries (year), the value of the fixed capital assets is obtained from the estimate made by the present author:‘ U.K. (1963), Korea (1963). 201 APPENDIX C SOURCES OF STATISTICAL DATA Australia: "Secondary Industry, 1960-61," "Manufacturing Industry, 1964-65 (1967-68)," Bureau of Census and Statistics, Canberra. Canada: "Taxation Statistics (1963)," Taxation Division, Department of National Revenue, "Corporation Financial Statistics (1965)," "Corporation Statis- tics (1967L" Corporation and Labor Unions Returns Division, Dominion Bureau of Statistics, The Queen's Printer, Ottawa, 1963 (1968, 1970L."Genera1 Review of the Manufacturing Industries of Canada, 1961," "Manufacturing Industries of Canada, Section A, Summary for Canada, 1965 (1967)," Bureau of Statis- tics, Ottawa, 1965 (1968, 1970). China (R): "The Report on Industrial and Commercial Sur- veys. No.1“ (No.2)," Ministry of Economic Affairs, Taiwan, 1968 (1969), "General Report on the 2nd (3rd) Industrial and Commercial Census of Taiwan, Vol. III," Commission of ICCT, 1963 (1968). India: "Annual Survey of Industry, 1959 (1961, 1964)," Central Statistical Organization, Department of Statistics, Calcutta, 1963 (1965, 1968). Japan: "Census of Manufacturers: Report by Industry, 1962 (1964, 1966)," Research and Statistics Division, Ministry of International Trade and Industry, Tokyo, 1964 (1966, 1968). Korea (R): "Report on Mining and Manufacturing Census, 1963 (1966, 1968)," Korea Development Bank, Seoul, 1965 (1968, 1970). Mexico: "Censo Industrial, 1956 (1961, 1966), Datos de 1955 (1960, 1965)," Direccion General de Estadistica, Mexico, 1959 (1965, 1967). Norway: "Industri Statistikk, 1963," "Norges Industri Produksjonsstatistikk, 1957 (1959)," Statistisk Sentralbyra, Oslo. Rhodesia (S): "The Census of Production in 1964: Mining, Manufacture, Construction, Electricity and Water Supply," Central Statistical Office, Salisbury, 1966. 202 203 United Kingdom: "Census of Production, 1954 (1958, 1963)," Bureau of Trade, HMSO, London, 1958 (1969), "Capital, Output and Employment 1948-1960." A Programme for Growth 4, The Department of Applied Economics, University of Cambridge, London, Chap- man and Hall, 1964. United States: "Census of Manufacturers, 1958 (1963, 1967), General Summary," "Supplementary Employee Costs, Cost of Maintenance and Repair, Insurance, Rent, Taxes and Depreciation and Book Value of Depreciable Assets: 1957," Bureau of Census, Department of Commerce, Washington, 1961 (1964, 1968), "Statistics of Income: Corporation Income Tax Returns, 1962- 1963 (1965, 1967)," U.S. Internal Revenue Service, Washington, 1966 (1968, 1971). 204 APPENDIX D LABOR PRODUCTIVITIES, WAGE RATES, AND INPUT RATIOS 205 Country t V/L W K/L Australia 1 61.7033 23.0563 61.9411 2 159.3380 50.2602 148.4645 3 85.0011 26.9267 81.5814 Canada 1 94.7265 35.0808 40.6842 2 150.9918 34.3391 93.7258 3 162.8201 36.6540 100.3878 China (R) 1 7.9754 .7324 8.3495 2 7.2908 1.7509 6.3696 3 5.3600 1.7028 6.6258 India 1 4.9209 1.0523 2.3479 2 3.8769 .9704 4.0354 3 4.4542 1.6415 3.5795 Japan 1 53.2017 7.3953 29.2066 2 54.3149 8.9732 34.9856 3 62.4704 10.2032 42.5636 Korea (R) 1 7.8607 1.9026‘ 36.6517 2 5.2673 1.2051 9.4810 3 4.4933 1.0376 7.9548 Mexico 1 12.7955 3.2980 6.6488 2 15.3441 5.3585 17.5860 3 15.3439 3.3594 12.0693 Norway 1 30.5437 19.8548 216.4569 2 41.1594 19.8861 235.3245 3 63.1028 23.3150 301.2946 Rhodesia (s) 1 2.6491 1.7166 6.3351 2 3.3951 1.6816 6.7681 3 3.0445 1.7305 5.6716 Costa Rica 1 12.1064 2.2141 23.1347 2 41.5438 7.3541 55.1740 3 29.9046 5.7513 54.1610 U.S.A. 1 155.9615 49.8875 42.7003 2 196.8580 54.4702 94.0390 3 240.9091 58.9717 106.8033 (Mean) 54.7494 15.2647 57.6698 H 205 231 L V/L W K/L L 7.5597 37.6438 21.2211 39.4531 27.1318 7.3648 73.6890 40.2867 78.9826 32.1679 7.2705 42.1675 21.7915 41.6981 29.2586 12.7559 53.2326 28.4170 86.7440 55.2425 8.3771 69.9234 27.3603 68.7199 53.4169 9.0029 70.4019 28.8122 108.7181 42.7701 20.4954 5.7088 1.1454 9.3458 62.0852 26.4391 5.0744 1.9223 15.3436 73.8014‘ 32.4866 6.7441 2.0285 16.4325 89.9233 26.3756 4.9887 3.3198 4.2770 1048.1482 31.3265 5.7356 3.6243 4.3209 1078.7451 35.0434 5.8461 3.8666 5.2430 1136.2405 31.0508 14.8997 5.3326 18.9719 355.7686 33.2379 16.0354 5.6221 19.1431 360.0507 36.2572 17.4844 6.3417 18.9310 338.1220 8.4226 4.9225 1.8513 13.3171 87.4675 13.9859 4.4478 1.4102 9.2263 96.3944 13.0674 4.0209 1.2999 8.0191 128.7856 57.4052 12.0079 5.1589 7.7499 189.8549 30.1024 15.4588 8.3178 20.1385 103.1355 50.9774 30.8606 11.1488 23.9938 107.3238 1.7791 23.9808 14.0024 73.7901 11.2268 1.8013 28.8871 14.3287 81.3832 10.1726 1.6210 33.4791 16.3527 90.8175 10.2439 1.5996 3.8056 1.7454 3.7532 4.8739 1.5358 4.1472 1.8556 3.7677 4.8321 1.5760 4.5902 1.9532 4.0334 4.7647 5.2103 23.3426 5.6044 44.4907 .5815 2.2367 17.7506 7.8874 59.8764 .6274 2.7846 24.9519 8.4679 63.2196 1.0609 118.9840 52.5806 33.5611 33.7304 619.4080 113.1030 70.7177 37.6552 38.1637 576.8550 111.8000 82.5964 41.6953 54.9579 616.0000 26.1626 26.4280 12.5876 35.4774 222.9237 am... 1. V____—_..__ 1_______ , , 206 APPENDIX D (Continued) 232 Country t V/ L W K/L Australia 1 38.5419 19.4571 26.2462 1 2 78.6198 37.1313 51.8782 3 41.0149 19.7926 29.4842 Canada 1 39.4714 22.3205 12.4719 2 48.8273 21.4188 17.5584 3 52.2505 22.7548 17.4615 China (R) 1 4.2639 1.2318 3.9310 2 3.3972 1.6607 2.9308 3 7.4384 3.3078 3.7348 3 India 1 3.5455 1.8841 2.4311 4 2 4.5482 2.1223 2.4325 3 5.1971 2.6672 3.7156 Japan 1 11.4902 4.3224 8.2989 2 15.0407 5.0688 10.8979 3 17.4429 5.5908 11.6982 Korea (R) 1 3.3958 1.2641 8.9886 2 3.7870 1.0268 3.2770 V 3 2.2473 .9951 2.7647 ‘ Mexico 1 10.2277 5.7847 5.4140 2 16.5861 8.0365 14.3352 3 27.4604 10.2426 12.1555 Norway 1 21.4777 12.3970 41.0145 ( 2 23.5079 12.8784 49.8671 1 3 28.4240 14.4279 59.2112 1 1 Rhodesia (s) 1 1.9690 .8123 1.2135 3 2 1.9705 .8343 1.0840 l 3 1.9698 .8986 .9339 ( Costa Rica 1 33.0023 5.5816 18.2947 ' 2 18.3569 7.7149 31.7636 . 3 18.8704 7.5413 25.2382 5 U.S.A. 1 51.6238 31.3982 16.3871 1 2 62.0741 34.0156 14.7812 3 74.3507 37.4354 17.8035 (Mean) 23.4060 11.0308 16.0515 G—IAM LR 207 251 L V/L W K/L L 20.2355 40.3957 22.4612 27.7584 45.0335 24.1494 87.4576 45.1496 58.8433 47.2622 23.9692 10.3577 5.2963 7.2878 209.0447 23.9712 50.6904 32.7385 63.9418 71.5901 23.3199 70.6167 34.7789 51.8005 70.3916 21.9917 75.8886 41.7143 61.0112 69.3534 4.4877 5.6056 1.4968 7.7939 13.3282 10.9053 7.9040 2.5569 17.0000 25.8259 20.2136 12.3720 2.6047 15.1711 35.1518 2.3936 3.7495 1.6289 4.2481 15.7440 1.7001 3.1308 1.7518 4.1913 12.7685 4.0398 2.8142 1.8297 4.1131 20.1353 474.0869 13.2340 4.9937 9.4237 306.4316 465.9104 16.6815 5.8024 12.1815 308.6964 457.5488 20.7855 6.4052 14.0666 317.1147 12.6595 10.0061 2.6276 20.9199 8.7029 23.7421 7.7017 1.8829 11.9509 15.2935 44.3121 6.5377 1.5613 8.7418 24.6013 26.5192 8.9125 3.6666 5.9122 39.5448 10.3158 15.5802 7.1382 18.8204 12.3466 15.1780 17.1445 4.7456 11.8782 33.1988 5.2943 23.6684 16.2430 47.3378 9.9760 4.9315 27.9914 16.5062 56.1807 8.4418 5.4450 33.6622 17.8209 64.1752 9.1985 1.7544 21.4114 6.8766 39.7749 .7586 1.8072 17.6588 8.0814 29.3539 1.0551 1.9474 11.3995 8.3976 31.9518 .6614 .2552 .8394 .9251 213.3460 54.7843 35.4404 22.8017 484.3660 220.4860 70.5669 39.9897 32.9976 463.0310 240.6000 84.6133 44.7171 51.5694 440.1000 73.0086 27.7775 14.2302 27.1067 103.9716 208 APPENDIX D (Continued) 260 Country t V/L W K/L Australia 1 39.7473 21.8450 26.3892 2 83.6372 42.8824 61.0575 3 44.4457 21.9160 32.4652 Canada 1 51.7872 31.1541 15.1638 2 68.1093 30.4258 11.6179 3 74.0447 31.7569 21.1660 China (R) 1 3.8441 .6004 2.5715 2 2.7524 1.7990 2.4190 3 3.2132 1.5307 6.5491 India 1 4.6505 2.9292 4.4970 2 4.3952 3.2020 5.0117 3 5.3490 3.7154 5.5096 Japan 1 15.7575 6.0146 10.4208 2 19.6239 6.9872 13.1798 3 22.6616 7.5957 15.5448 Korea (R) 1 4.6770 2.0817 6.3653 2 4.0502 1.5525 4.7724 3 3.7053 1.3870 3.9673 Mexico 1 6.4542 3.9174 2.6532 2 11.4847 6.8033 8.8483 3 13.8753 6.1508 4.7473 Norway 1 27.3825 16.9960 28.0341 2 31.9247 17.7771 32.8619 3 35.6989 17.4484 35.9952 Rhodesia (S) 1 1.8441 1.0262 .8896 2 1.9539 1.0819 .9042 3 2.1196 1.0650 .7584 Costa Rica 1 16.0370 5.6431 13.0251 2 19.2890 7.1921 14.2357 3 20.1237 8.9368 17.1132 U.S.A. 1 67.5919 41.1858 15.4033 2 79.8868 43.6726 14.7949 3 91.6230 46.1730 17.5504 (Mean) 26.7881 13.4681 13.8329 209 =.____________________________________________ 271 L V/L W K/L L 18.3692 77.2423 32.1089 132.2625 7.7390 21.0087 173.6963 67.3224 267.6774 8.9244 22.9789 95.9181 36.2584 179.0267 8.9551 33.6038 129.7060 54.1541 172.1674 61.0614 33.1366 161.9073 ' 52.6837 297.4396 53.8926 36.3233 148.7316 56.9747 349.5193 56.8279 12.1773 9.8169 2.1131 19.7340 7.2229 4.6825 8.5407 2.3548 17.6695 12.3243 6.2468 10.6012 3.5532 39.2729 12.3723 13.2601 10.0591 3.9308 20.5307 32.8848 16.3185 7.7036 4.0918 22.2207 34.3504 17.1835 8.1951 4.2188 30.8024 39.8870 100.7998 30.5503 9.4739 62.9015 120.5392 107.6566 42.3599 10.6630 65.5959 124.2084 122.1854 44.9271 11.6218 72.1876 123.4580 4.2904 13.7877 3.0949 18.4416 6.1703 7.2214 13.0262 2.4823 16.0077 7.9039 9.3862 9.3324 2.1421 13.7904 9.9472 16.9239 27.4444 8.2214 56.2730 12.7712 18.7008 43.1829 14.2260 82.0632 8.9646 18.3066 64.6726 18.3688 95.5085 13.8964 13.1127 43.4487 20.4583 161.1944 19.0053 12.3013 43.4080 20.7217 185.5144 19.0855 14.3790 42.1665 22.8740 244.4143 19.3986 2.3276 3.7416 1.5225 5.9997 1.3565 2.1516 3.8143 1.5737 5.3803 1.4856 2.1888 4.7918 1.7684 5.5310 1.3733 .7379 .9478 .7761 347.5990 127.2465 57.8866 145.5213 202.5910 376.5480 157.1133 64.6434 187.0502 208.3690 425.3000 176.6822 69.6826 223.6842 222.1000 55.7313 57.7939 22.0397 106.5128 48.6356 V' 210 APPENDIX D (Continued) A 280 Country t V/L W K/L Australia 1 58.6352 31.7775 53.6854 2 126.7830 66.0158 128.9738 3 74.0647 36.2366 70.3370 Canada 1 78.5560 43.1551 31.8296 2 142.6197 44.3573 54.9918 3 158.7716 48.3729 58.1063 China (R) 1 4.5051 1.7284 6.0265 2 4.5358 2.5473 5.6125 3 5.1712 3.0784 5.7114 India 1 7.1639 4.6036 6.8373 2 7.1762 4.9057 6.7684 3 6.4732 4.7275 6.5765 Japan 1 26.5998 9.2863 10.4322 2 34.5447 11.5261 15.1053 3 42.7185 12.5395 21.1320 Korea (R) 1 8.5548 3.3966 15.3377 2 6.8745 2.5412 9.2554 3 6.1215 2.7489 6.8695 Mexico 1 12.8106 7.0813 7.1801 2 24.4863 11.5301 18.2859 3 39.2028 13.4429 22.4082 Norway 1 .32.2493 20.3389 68.8891 2 37.3872 21.2537 76.9477 3 42.8735 19.4451 51.7778 Rhodesia 1 4.8798 3.0513 2.7711 2 5.3237 3.2728 2.9674 3 5.2432 3.2944 2.9474 Costa Rica 1 21.0184 9.9118 20.1686 2 23.1801 12.9663 20.9502 3 17.2900 10.8957 18.5284 U.S.A. 1 91.6902 53.4454 22.1524 2 112.4674 57.5115 28.6043 3 130.1259 60.3171 37.8896 (Mean) 42.4272 19.4335 27.7593 u 211. Y L V/L w K/L L 31.3758 36.1086 23.3472 30.2223 4.0390 36.4787 82.1565 49.9661 66.5786 4.2000 37.4932 43.8121 23.9147 37.8557 3.6530 70.7411 43.4931 43.6204 10.3044 5.9728 1.1162 7.7912 .9756 10.2006 2.9793 1.8403 8.3619 .9436 13.9238 2.6632 2.0264 9.2560 1.2615 57.7150 5.1691 2.7235 2.3104 10.1198 64.2251 4.1023 2.7687 2.2827 10.6265 77.3597 4.1013 2.4495 2.5838 9.4408 270.5632 21.1613 7.9414 13.6196 11.6186 311.7104 26.0816 9.1883 16.0577 11.6805 326.8119 26.7572 9.5058 17.0654 12.3678 13.3542 4.4667 2.2395 11.2592 1.3744 18.5572 3.3955 1.3425 6.4013 2.3065 21.4258 4.2459 1.3308 16.6910 1.2502 35.3876 7.5947 6.1199 6.4564 13.0037 24.1027 19.6654 9.2182 10.1768 3.7685 38.9224 24.2939 9.3651 10.4074 5.5813 11.1050 21.8876 16.5738 77.8669 1.1016 10.8993 24.3824 15.2481 89.9153 .9897 23.3685 38.6080 17.6579 206.7961 .9333 2.2702 2.1147 2.1720 .7635 16.6193 5.4021 17.9244 .2174 1.0939 14.2435 7.6670 16.5901 .2571 1.1107 17.8835 6.2485 25.2117 .1655 864.1010 913.2430 1031.0000 133.9700 19.0980 9.8001 29.5701 4.6615 212 APPENDIX D (Continued) _fi' 300 Country t V/L W K/L Australia 1 54.8387 29.6268 47Jn56 2 106.3981 59.6317 111.7421 3 60.9227 30.2572 66.6365 Canada 1 96.5658 45.7003 4737218 2 129.6802 44.6352 58.1239 3 151. 3850 46.9303 89.1158 China (R) 1 10.26247 1.6002 7k0878 2 6.3700 2.2092 11.0187 3 6.8310 2.4337 8.5448 India 1 15.1661 6.0873 7.8337 2 17.5479 6.6573 8.6207 3 12.2104 6.7589 12.3772 Japan 1 24.2656 7.4736 15.4289 2 29.5557 8.5384 19.5290 3 34.0219 9.5623 21.3282 Korea (R) 1 5.8753 2.5689 .6667 2 4.3327 1.8878 5.1713 3 4.7298 1.7511 4.4574 Mexico 1 18.1104 6.1946 7.5077 2 37.7082 12.8362 28.9958 3 52.3829 14.2711 18.8471 Norway 1 34.8027 19.9011 73.9356 2 39.2039 20.5984 82.1231 3 44.0573 23.0264 100.2690 Rhodesia (S) 1 4.8990 2.1891 5.1880 2 4.9891 2.3093 5.0773 3 5.7996 2.2585 5.2052 Costa Rica 1 29.5818 7.3118 39.3592 2 29.1685 10.0164 47.5230 3 78.3140 22.9309 198.9289 U.S.A. 1 94.1982 51.0747 22.4487 2 109.9554 54.2574 39.0698 3 122.9857 53.3091 45.5500 (Mean) 44.7611 18.7514 38.2564 =r 213 334 L V/L W K/L L 13.7650 93.0345 28.1943 179.3016 2.8017 17.4239 235.5488 66.1902 514.0990 2.9529 17.9756 125.8617 34.75832 346.5912 2.9803 16.7061 224.0165 48.4682 973.7682 3.2073 16.1082 322.5659 49.7083 895.9735 2.7241 16.2898 295.2558 49.9816 1039.4595 2.7344 5.2209 26.7247 2.6868 21.4604 7.8555 7.8734 34.3396 5.1093 77.6980 6.5208 12.7852 59.6278 5.1124 128.7637 4.8539 24.8705 9.6999 3.8183 38.1527 25.4750 28.0651 9.8823 4.4050 34.6545 28.1024 36.0237 9.3761 4.6008 29.9977 31.3889 127.5279 81.5798 14.8665 142.8730 19.8782 130.9464 74.5735 15.0247 177.8465 20.6022 129.6251 91.5152 16.0105 201.4429 18.8452 14.1677 49.2776 7.1060 285.7339 1.1150 17.5745 42.9640 4.1982 89.0302 2.2104 19.4024 24.6535 2.8214 145.0468 5.3033 47.5351 17.1743 6.7289 30.9042 9.6317 9.6420 74.8158 13.6202 119.5749 5.2865 16.5728 84.2287 21.5614 160.1130 5.7253 2.6690 60.5744 21.9765 169.8445 1.1539 2.7919 67.3484 24.7191 243.7867 1.1178 3.3395 94.0820 28.9946 413.1523 1.1225 1.1404 1.0410 .9847 .1609 .1973 .2816 347.8420 176.2275 53.6986 199.3379 41.1270 414.9590 220.9435 62.1166 392.9383 34.8630 516.7000 232.8708 66.0709 528.9192 32.6000 61.1579 105.1394 24.5388 280.7209 11.9326 214 APPENDIX D (Continued) E -..E Vii 319 Country t w V/L w K/ L J I» Australia 1 104.8100 26.3179 66.0665 1 2 239.4667 90.7001 152.4671 3 126.9355 30.0776 80.0668 Canada 1 133.0101 36.2827 185.5722 2 226.6934 32.8529 146.0172 3 253.2573 32.4313 199.1584 China 1 13.8511 1.9018 12.0795 2 15.3681 3.7000 36.1226 3 18.4194 4.1777 40.6627 U.K. 1 4.7011 2.0315 8.1261 2 6.0302 2.3408 12.7219 3 8.1488 2.7171 15.7923 India 1 15.5032 4.4441 9.0595 2 15.4926 5.0160 11.6071 3 14.6896 5.6687 12.5313 Japan 1 51.5173 9.5590 23.3515 2 66.3250 10.7176 28.1696 3 80.5597 12.5069 33.7656 Korea 1 17.1416 3.7380 20.5606 2 15.7302 2.9049 13.8406 3 15.0275 3.9275 17.0816 1 Mexico 1 25.5805 9.4399 11.5953 4 2 43.7552 15.3951 26.4963 , 3 71.5713 20.2989 36.5904 a Norway 1 56.0558 21.1632 137.5936 3 2 61.1036 22.0116 156.1068 , 3 77.4240 25.6295 206.5206 ' '1 Costa Rica 1 27.6828 7.9308 23.1833 L 2 48.2208 14.4803 33.8126 5 3 65.6694 16.2294 47.4702 U.S.A. 1 186.7074 56.8095 129.1458 I 2 239.1285 62.2392 177.3288 ' 3 261.9307 64.6925 148.8494 I (Mean) 79.0155 20.0102 68.4702 215 321 L V/L W K/L L 13.6205 182.0457 27.6139 431.4798 5.3241 15.0005 392.7090 63.4226 943.4790 5.6312 16.2204 276.2014 34.5550 491.9717 6.1817 23.6655 41.6732 21.1176 997.3544 13.4176 20.9303 323.5966 53.3580 1378.4024 6.3686 16.3571 343.7597 60.7967 1376.5104 6.3119 15.5101 49.2448 2.7866 26.4925 2.1177 36.3654 51.9123 5.5066 72.4780 7.2691 42.4047 62.6583 5.7688 111.4424 8.8728 132.5874 10.2399 2.5757 35.7013 14.4188 140.2697 7.2696 3.0445 45.5644 17.2323 146.7771 11.7399 3.7735 47.7683 18.0236 53.5188 111.0797 17.8247 218.6681 4.1993 56.6935 110.2331 18.1254 201.2450 3.6891 70.1926 74.5508 15.4310 ' 201.5227 4.0547 145.5430 99.0974 15.5440 488.9478 14.2143 162.8038 120.9860 15.9722 339.2603 15.4934 163.3238 153.2140 17.3704 418.3683 15.9638 8.8550 24.6970 3.6734 146.2987 .1872 11.6020 108.3756 7.4047 89.4255 1.0309 20.0296 96.8961 3.7628 80.5531 1.7554 65.0774 146.6570 21.3194 291.8968 4.9508 38.0775 68.4726 15.0812 48.6350 .4616 57.9296 134.4352 23.2264 66.2887 .5576 8.6924 8.2285 8.1926 .5909 .8846 .8123 363.6130 145.1396 64.6692 249.6063 146.0250 396.6610 243.5256 69.6512 1396.4297 119.2970 481.3000 383.3661 76.0004 1982.4836 106.7000 83.1009 139.7695 24.7917 451.0472 20.3611 I i ‘- *II’II' I. —{—'= 7‘ 216 APPENDIX D (Continued) 341 Country t V/ L W K/ L Australia 1 60.4987 26.7163 100.4227 2 133.9474 57.6349 242.8927 3 68.2649 29.6344 151.6086 Canada 1 122.2496 46.5493 169.3438 2 127.9411 45.6570 236.4608 3 120.4242 46.7173 275.2344 China (R) 1 17.7863 2.3884 11.0199 2 7.6566 3.2003 33.5781 3 10.1903 3.9001 43.3067 U.K. 1 3.9774 2.1859 4.8117 2 4.5248 2.4772 6.9542 3 4.8267 2.6875 10.3524 India 1 9.0767 5.0245 27.6889 2 8.1889 4.9544 24.3339 3 8.9648 4.8563 45.8530 Japan 1 27.9891 12.1043 58.0662 2 41.5843 13.2082 74.7697 3 46.0248 14.3853 87.6555 Korea (R) 1 8.9574 2.8932 9.9085 2 10.3493 2.5203 9.4151 3 6.2664 1.8833 7.6579 Mexico 1 34.6309 19.9281 30.3302 2 30.1520 12.1100 90.3212 3 76.0260 20.2529 103.6232 Norway 1 48.6773 20.4265 106.5467 2 43.8011 20.6212 131.0643 3 60.3712 30.2185 236.2839 Rhodesia (S) 1 4.5019 2.2158 15.7092 2 3.7130 2.2245 15.0737 3 4.8085 2.3340 14.0184 U.S.A. 1 103.2555 60.4389 83.0562 2 133.5255 66.8720 102.5374 3 140.6124 68.2482 125.3156 (Mean) 46.4778 19.9234 81,3702 ___../ 217 381 L V/L W K/L L 40.5032 32.9712 25.7077 24.2801 12.9200 47.1639 73.5936 58.2527 47.1698 12.6268 48.4304 44.9014 35.1864 25.3083 16.5837 80.2671 57.5305 41.8038 33.2577 15.2368 79.5299 90.3616 45.4384 44.9108 15.7154 81.4533 84.8694 46.8185 28.2084 16.0625 6.8314 8.1133 3.6871 7.5679 3.4906 12.5281 6.0234 3.2854 10.2670 6.1457 12.7928 10.1462 3.9288 13.2044 8.7891 414.6571 2.8283 2.1913 1.1718 251.9826 433.6618 3.1929 2.4432 1.7301 253.7344 430.2154 3.8042 2.6769 3.0136 186.7972 147.0100 6.0358 5.6172 5.4975 26.3109 163.4173 5.2668 5.3020 5.7163 20.7140 278.4598 5.0098 4.8607 4.4266 23.7869 408.7842 25.2229 12.7265 18.6203 124.3916 429.7877 38.4887 13.6239 21.3249 129.1045 419.4533 45.8302 14.4478 28.9625 131.8479 9.1873 6.8365 3.5940 13.2730 4.1871 12.5744 9.7040 2.3240 9.6287 7.1331 22.4097 4.3210 2.3533 9.0229 8.3848 25.5771 14.2660 6.1761 10.3845 .4654 22.4971 9.5905 8.0689 27.8352 .5598 32.2971 23.9055 9.2337 30.1095 1.3524 11.7507 31.0570 21.8169 46.0589 23.4069 11.7806 36.7388 23.3237 59.7840 21.6653 10.0743 41.9749 25.7037 69.7428 24.1657 3.1824 3.4086 3.3718 796.6740 74.1471 55.2020 15.7800 144.4420 819.7570 83.8192 61.3329 12.6752 139.5100 915.8000 94.1258 63.6380 13.3398 169.3000 188.9477 32.4893 20.3589 21.4098 60.0271 -. ”___, _..... 218 APPENDIX D (Continued) 360 Country t V/L W K/L Canada 1 69.1354 40.6110 53.7965 2 116.4462 40.9163 60.5534 3 116.9570 42.2572 79.0633 China 1 5.8093 1.6013 4.6338 2 5.1965 2.2792 5.5182 3 5.4211 3.4406 6.2983 India 1 6.1136 3.8081 6.1877 2 6.4731 4.8817 7.5574 3 7.5522 4.4241 11.7672 Japan 1 27.4099 9.1748 17.9731 2 31.1516 10.2792 21.0062 3 35.2813 11.2819 22.7445 Korea 1 5.5044 2.5786 11.7671 2 5.5844 1.9125 8.2277 3 4.8559 1.8212 6.7968 Mexico 1 10.1575 4.8927 8.7294 2 4.9380 2.6617 4.9311 3 33.2907 12.0152 22.6975 Norway 1 33.6966 20.1988 48.6055 2 39.9077 20.8093 60.7959 3 47.6944 23.0592 53.6378 Rhodesia 1 3.4960 1.8126 2.0361 2 3.3801 1.8103 1.5074 3 3.0371 1.9026 1.4489 Costa Rica 1 25.7859 7.0150 37.3420 2 20.3814 11.5864 15.7902 3 5.9579 9.7415 22.4078 U.S.A. 1 91.9060 55.8470 32.9355 2 116.2905 62.4596 37.0724 3 139.5297 66.3486 44.7431 (Mean) 34.2781 16.1143 23.9524 ____—/ H 219 370 L V/L W K/L L 44.8436 71.1016 40.3139 28.9457 81.6544 45.8282 113.2085 36.4384 37.7614 71.7899 50.6223 107.2834 36.7531 44.3152 80.3998 14.5682 7.7381 1.4699 5.6945 11.3934 27.1942 7.0710 2.2898 12.3593 28.4186 35.7224 8.2205 2.5764 8.9123 57.5056 75.0089 9.0610 4.4665 8.6624 61.3295 107.7967 9.1774 4.4812 8.6803 75.5541 154.4485 8.2813 4.6150 13.1858 119.6004 747.1904 30.7523 7.9075 17.6072 691.4634 767.5857 30.7575 8.5551 18.4450 753.3555 766.5869 34.3796 9.1302 17.7471 780.7134 ; 11.6043 8.1894 2.5010 12.2430 8.1790 ; 17.6678 8.9390 1.9537 8.6186 14.6171 19.4157 6.6076 2.0406 7.1649 22.1192 27.6033 16.6814 6.4041 8.2115 51.8748 38.3305 26.8864 11.9740 16.7226 26.5246 29.7870 37.8758 11.4863 15.0851 62.0401 u. 11.7188 37.2771 20.1383 47.0994 11.3308 : 11.2210 40.9277 21.2219 55.1250 11.5311 ' 14.7831 53.0602 23.7228 62.4132 15.6092 .9832 2.9923 1.4335 2.1133 1.4689 .9589 3.0649 1.5794 2.0198 1.6435 .9265 3.5738 1.7210 1.7075 1.7506 .1460 32.7417 7.3086 20.5582 .0617 .3651 26.7633 8.0053 19.5821 .1830 .5251 41.5638 12.2139 38.2843 .6735 . 1348.2450 92.6269 51.4964 18.6949 1122.2840 f 1459.3770 110.3116 58.4869 21.4713 1511.8190 5‘ 1864.5000 122.0615 60.1446 29.4685 1874.9000 5 256.5183 36.9726 15.4277 20.2967 251.7263 “1 220 APPENDIX D (Continued) 383 Country t ‘7 V/L W K/L Australia 1 50.4457 30.7570 45.3305 2 93.5083 57.1881 112.7721 3 55.9502 29.3762 59.5513 Canada 1 101.1472 51.6897 52.6662 2 151.5120 56.0408 62.6359 3 288.8630 89.5603 167.5164 China 1 7.7566 2.2286 6.8162 2 7.6236 2.6955 17.0373 3 17.1018 3.1015 22.9362 U.K. 1 4.1235 2.3989 3.3748 2 4.6598 2.8140 5.3466 3 6.0773 3.1929 4.9766 India 1 14.1976 5.5537 13.4563 2 11.3358 5.7515 14.2265 3 13.5871 6.0616 16.8698 Japan 1 34.0136 8.9533 28.3240 2 43.6188 9.8295 28.1574 3 44.7911 10.7166 33.1303 Korea 1 6.6924 2.3994 15.2429 2 8.9851 .1831 8.1452 3 11.2125 2.3422 9.5965 Mexico 1 56.1776 9.6664 33.8304 2 49.2815 13.9178 19.3795 3 60.2196 19.1675 39.9399 Norway 1 29.1498 20.7968 30.7424 2 36.2952 21.6745 57.5957 _ 3 50.6422 23.9381 59.7679 Costa Rica 1 19.0089 10.4157 13.6088 2 19.3060 9.0176 9.7683 3 .5319 12.0738 34.1137 U.S.A. 1 116.9579 59.2654 56.8988 2 180.5938 71.2884 70.2266 3 172.7354 69.7381 104.6785 (Mean) 81.1244 21.9332 38.1412 391 L V/L W K/L L 37.9923 48.7042 26.6041 35.0103 1.4997 55.9313 97.2857 54.1163 81.1109 1.6717 58.1663 34.7621 17.8150 48.5511 2.1214 42.7892 77.1440 45.7959 27.5771 8.5229 55.8288 131.2903 39.3987 48.1742 7.3780 34.0834 131.9667 43.1487 67.6333 8.6592 5.6772 5.8016 1.3685 3.5662 .2364 10.3077 3.4059 2.1047 4.4802 .4872 12.1549 4.6092 2.9945 6.2842 1.6821 357.0906 314.4020 399.2260 29.8506 8.4308 4.5498 7.4588 3.9369 45.3288 9.1068 5.1118 7.6284 3.9207 62.3315 8.6996 4.7052 12.0234 10.4347 286.2823 23.4243 8.8788 9.9094 43.8598 344.6216 29.3746 10.0705 12.2348 43.2361 392.8910 32.6236 11.2515 11.1132 52.2669 3.5833 6.7574 3.0490 5.7630 .9477 7.8850 6.6558 2.2029 7.6166 .9754 11.2752 5.9107 2.1428 8.7545 1.0931 15.9525 7.8891 4.9314 5.3024 1.3676 11.8975 26.5350 12.4087 13.0230 .8942 29.4580 41.2398 16.1542 21.3222 1.2161 1.0986 34.4287 21.7617 25.7044 .1755 .9944 36.4389 22.9900 28.0138 .1530 1.4667 48.5143 25.0553 27.9760 .2625 .1197 .1327 .4535 577.1880 88.0966 54.7545 16.8122 184.7320 693.8210 115.5829 57.6392 36.9699 178.5420 739.4000 127.6443 59.3284 30.9331 217.8000 40.5965 43.8638 20.7531 22.6277 28.8176 222 APPENDIX D (Continued) 311 Country t ‘ v/L w K/L 1, Australia 1 80.6469 33.0616 132.5493 14.3085 2 248.8187 82.7535 396.5261 14.8227 3 129.8542 38.5976 213.1340 18.0268 Canada 1 130.1543 50.2199 83.1318 25.3552 2 21.5975 4.3030 20.4549 14.8338 3 162.2220 34.9179 166.9922 28.4704 China 1 14.7305 3.3944 51.7904 6.9555 2 23.7855 4.7592 62.5805 10.9164 3 28.3274 5.7383 68.8643 12.7108 U.K. 1 5.5140 2.5754 10.5204 119.4624 2 6.9859 2.7904 12.2893 162.0513 3 9.6510 3.2177 16.3472 162.0513 India 1 13.8205 4.1754 28.4814 30.9783 2 27.6073 5.2508 55.5734 47.7245 3 14.6080 5.5690 55.0902 54.8684 Japan 1 43.7757 11.8919 76.9995 243.0139 2 60.4602 12.5830 98.7714 249.6041 3 65.0365 14.0505 115.9382 236.3874 Korea 1 23.4754 6.4445 51.7692 2.9014 2 22.9107 6.3352 36.2883 3.7317 3 34.9408 6.0105 105.0162 5.7181 Mexico 1 23.4220 8.6178 41.6400 20.1214 2 43.4696 14.2852 81.9908 8.5095 3 98.3012 22.0353 123.7603 15.4910 Norway 1 49.0431 23.8091 245.7559 9.9698 2 60.2911 23.4846 277.6800 10.1341 3 69.4800 26.4044 355.8770 10.6224 Rhodesia 1 6.0526 1.9939 12.4985 .9345 2 7.7587 2.1523 24.3691 .9772 3 5.8822 2.4047 24.2519 1.1935 Costa Rica 1 107.7267 13.1582 43.1900 .0460 2 173.4372 21.6937 101.9515 .0314 3 113.6723 15.3413 47.1981 .2455 U.S.A. 1 168.8886 60.3436 52.5542 276.7830 2 238.5140 67.3472 87.6005 279.4860 3 277.9070 70.1064 241.9344 294.0000 (Mean) 72.5770 19.7728 100.5934 66.4850 APPENDIX E RAMSEY'S SPECIFICATION ERROR TESTS "RESET" stands for regression specification error test. It is shown in Ramsey (1969) that under fairly general conditions the distribution of the (N - K) X 1 vector under the alternative hypothesis of Group I speci— fication errors is multivariate normal with covariance matrix given by 321 and a mean vector C which is non- N-K 2 null. [That is, 6 ~ N (c.5'I Further, it is shown N-K) '1 that the mean vector can be approximated by a linear sum of vectors qj, j = l, 2, ... , K. The (N - K) x 1 vectors qj are defined by qj = A'?(j+l), where A' is an (N - K))