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Aggregate production function and technological change in Canadian agriculture, 1935-65. 1970

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THE AGGREGATE PRODUCTION FUNCTION AND TECHNOLOGICAL CHANGE IN CANADIAN AGRICULTURE, 1935-65 by ALLISTER JOHN MCPHERSON B.Sc, University of Alberta, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of A g r i c u l t u r a l Economics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y , a v a i l a b l e fo r re ference and Study. I f u r t h e r agree that permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s fo r s c h o l a r l y purposes may be granted by the Head of my Department or by nils r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s fo r f i n a n c i a l gain s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department of A g r i c u l t u r a l Economics The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date June, 1970 ABSTRACT A study was undertaken to investigate the macro pro- duction relationships i n Canadian primary agriculture during the 1935-65 period. S p e c i f i c a l l y , the problem was to measure simultaneously the rate of disembodied techno- l o g i c a l change and technological change embodied i n machi- nery and implements, and material inputs. To estimate technological change, regression e s t i - mates were obtained for a l i n e a r homogeneous Cobb-Douglas production function, where r e a l gross a g r i c u l t u r a l output per person employed was the dependent variable, and a time index, weather index, and the annual flow of r e a l c a p i t a l services (including material inputs) per person employed were the independent variables. The data, which consisted of time series of thirty-one annual observations, was de- ri v e d mainly from publications of the Dominion Bureau of S t a t i s t i c s . The rate of disembodied technological change was estimated d i r e c t l y by specifying a term which allowed for s h i f t s i n the production function over time. To mea- sure the rate of embodied technological change, which was assumed to be capital-augmenting i n the vintage sense, several alternative values f o r the improvement i n the pro- ductive quality of machinery and implements, and material inputs were imposed on the o r i g i n a l data s e r i e s . Based on these al t e r n a t i v e s , a matrix of regression r e s u l t s was ob- tained, and the true value of the rate of embodied techno- l o g i c a l change was i n f e r r e d by choosing the "best" regres- sion. In addition, several alternative models were investigated. When disembodied and embodied technological change were s p e c i f i e d simultaneously, the "best" estimate of the annual rate of disembodied technological change was 1.76 per cent, while embodied technological change i n material inputs was estimated at 3*5 to 4»0 per cent annually. There was no evidence of a p o s i t i v e rate of embodied technological change i n machinery and implements i n any of the regressions. However, i t was concluded that t h i s a p r i o r i unexpected r e s u l t should be considered s u b s t a n t i a l l y biased. TABLE OF CONTENTS CHAPTER PAGE I. PRODUCTIVITY AND TECHNOLOGICAL CHANGE 1 Purpose of the Study 1 Productivity 2 Technological Change 5 Productivity Change in Canadian Agriculture . . 7 The Problem 12 II. CONCEPTUAL FRAMEWORK FOR MEASURING TECHNOLOGICAL CHANGE 17 Neutrality of Technological Change . . . . . . . 17 Disembodied Technological Change . . 21 Embodied Technological Change . . 23 Simultaneous Embodied and Disembodied Technological Change 33 Economies of Scale and Non-neutrality 36 Some Problems and Alternatives 40 III. METHOD OF ANALYSIS AND MEASUREMENT OF VARIABLES . 46 Model for Estimating Technological Change . . . 46 Measurement of the Variables 52 Gross agricultural output (Q) 52 Weather index (w) 54 Labor (L) 56 Flow of capital services (K) 59 V CHAPTER PAGE IV. EMPIRICAL RESULTS 73 Introduction 73 Results of the Basic Model 75 Model 1(a) 76 Model 1(b) BO Model 1(c) 61 Discussion of Model I S3 Model with a l l Technological Change Embodied . . 67 Model II 66 Model to Relax the Assumption of Constant Returns to Scale 69 Model III 90 Alternative Models to Relate Potential and Actual Output . . . . . . . 91 Model IV(a) 91 Model IV (b) 92 Testing the Production Function for Stability- Over Time 93 Model V 94 V. SUMMARY AND CONCLUSIONS 9$ Summary 96 Conclusions 101 Implications and Suggestions for Further Research 105 v i CHAPTER PAGE BIBLIOGRAPHY 10S APPENDIX 115 LIST OF TABLES TABLE PAGE I. Summary of Estimated Productivity Change in Canadian Agriculture 13 II. Gross Investment in Machinery and Implements, Canadian Agriculture, 1921-65 66 III. Coefficients of Multiple Determination and Standard Errors of the Estimate for Model 1(a) 7 7 IV. Dummy Variables Specified for Model V i : Q =Ae u te f +S^ 2 L 1- aK aeJ D,rS^=0.0, 1935-65 . . 95 V. Time Series Data for Canadian Agriculture, 1935-65 116 VI. Symbols Used in the Models 119 VII. Regression Estimates for Model 1(a): 2 Q=Ae u te f +S w f h w L 1 _ aK a,ft=-0.01 - . 0 3 , 1935-65 . . 120 VIII. Regression Estimates for Model 1(b): 2 Q=Ae u te f +S w f h w L X- aK a, iWo.O, 1935-65 129 IX. Regression Estimates for Model 1(c): 2 Q=Ae u te f +S^ h w L 1 _ aK a, fWo.O, 1935-65 130 X. Regression Estimates for Model II: 2 Q=Aef+S**hw L 1 _ aK a, ̂ =0.0, 1935-65 131 XI. Regression Estimates for Model III: 2 Q = A e u t e f + ^ h w L bK a, rWO.O, 1935-65 132 v i i i TABLE PAGE XII. Regression Estimates for Model IV(a): Q»=Ae u tL 1" aK a, /Uo.O, 1935-65 133 XIII. Regression Estimates for Model IV(b): Q=Ae u te f-s/ wL 1- aK a, rUo.O, 1935-65 134 XIV. Regression Estimates for Model V: Q ^ A e ^ e ^ ^ ^ L 1 - ^ ^ , A^=0.0, 1935-65 . . 135 XV. Regression Estimates for Model 1(a): Q ^ A e ^ e ^ S ^ ^ V - V ^ ^ O . O , for the 1935-49 and 1950-65 Subperiods 136 LIST OF FIGURES FIGURE PAGE 1. Factor Proportions and Productivity Change . . . . 4 2. Technically E f f i c i e n t Combinations of Labor and Ca p i t a l Inputs at Two Levels of Technology . . . 5 3. Productivity Changes over Time 13 ACKNOWLEDGEMENTS The author wishes to gratefully acknowledge the generous assistance and guidance provided by his thesis supervisor, Dr. G. R. Winter. In addition, the stimula- ting intellectual environment provided by the staff and fellow graduate students at the University of British Columbia was extremely motivating and enjoyable. Special appreciation i s extended to the author*s wife, Maria, whose assistance, patience and encourage- ment contributed immensely to the successful completion of this study. CHAPTER I PRODUCTIVITY AND TECHNOLOGICAL CHANGE I. PURPOSE OF THE STUDY The purpose of this study was to apply macroeconomic concepts to the measurement of technological change in Canadian agriculture during the period 1 9 3 5 - 6 5 * An attempt was made to measure three separate kinds of tech- nological change: technological change reflected in the improved productive quality of machinery and implements, technological change reflected in the improved productive quality of material inputs, and a l l other technological change derived as a residual. Technological change may be regarded as an advance in technology which i s : (l) knowledge used by productive units (firms or farms in this case) regarding the principles of physical, biological and social phenomena; (2) knowledge regarding the application of these principles to production such as the application of genetics to the development of better livestock or new varieties of crops; and (3) knowledge regarding the day-to-day operations of production such as management techniques. Technological change i s an important and perhaps the most important factor responsible for economic growth. 2 Economists have made s i g n i f i c a n t attempts since the mid 1950*s to measure the e f f e c t of the rate of technological change on a nation's rate of economic growth. Solow, f o r example, found that almost ninety per cent of the long-term increase i n output per unit of labor input i n the United States was attributable to technological change with the remaining ten per cent attributable to increases i n the quantity of c a p i t a l employed. 1 Although these r e s u l t s were extremely rough, more recent studies have confirmed that the e f f e c t of technological change on productivity over time has been substantial. I I . PRODUCTIVITY Since the eighteenth century, economists and policy makers have been interested i n productivity. E s s e n t i a l l y , productivity i s a measure which expresses the r e l a t i o n s h i p between output and the resources u t i l i z e d i n i t s production. More pr e c i s e l y , i t i s the r a t i o of output to a single input or to a composite of inputs. For example, the volume of output per man-year, and the number of bushels per acre are expressions of productivity. These r a t i o s are measures of performance r e l a t i n g the volume of output produced to XR.M. Solow, "Technical Change and the Aggregate Production Function," ''Review of Economics and S t a t i s t i c s , 39:312-20, August, 1957. 3 the volume of inputs used. Productivity, however, i s not synonymous with efficiency, since productivity simply expresses a physical relationship between output and input while efficiency implies an optimum level of performance in a productive situation in terms of the combination of inputs to produce a given level of output. As a description of a technical relationship between output and inputs, productivity i s a characteristic of the individual economic unit, and i t s changes, therefore, indicate that the productive resources within the unit have been reorganized so as to affect output. Alternatively, productivity changes may arise from a l l sourcesincluding shifts in production and employment of resources between units having different levels of productivity as well as productivity advances within individual units. This second concept i s more suitable for most economic and policy analysis at the macro level. For conceptual as well as practical reasons, labor productivity, that i s , output per unit of labor input, has been the most commonly studied measure of productivity, since labor usually represents a major proportion of value added i n production, labor input i s relatively easy to measure, and changes i n labor productivity are directly related to changes in real income per capita. In recent years, there has been an increasing volume of empirical 4 work on productivity at a l l levels of aggregation which has contributed to an extensive knowledge of the trends and magnitudes of productivity change* However, considerable scope remains for further investigation of the causes and sources of productivity increases. In a broad sense, changes in productivity may result from three sources: (1) the nature and rate of techno- logical change; (2) factor substitution in response to changes in relative input prices; and (3) economies of scale or increases in the u t i l i z a t i o n of existing productive capacity. The effects of changing factor proportions on pro- ductivity are easily shown in Figure 1 where Capital input .Q2 •Ql Labor input Figure 1 . Factor proportions and productivity change• Q]_ and Q2 represent the same level of output, but each i s produced by a different, although technically efficient, combination of capital and labor. In response to a change in relative input prices, a shift from Qi to Q2 would result in an increase in labor productivity as shown by an 5 increase i n the output-labor ratio and a decrease in capital productivity. Economies of scale exist when the percentage change in required inputs i s less than the percentage change i n the resultant output, when a l l inputs are increased in the same proportion. In this situation, i t i s obvious that pro- ductivity increases as output increases, since the output- input ratio increases. III. TECHNOLOGICAL CHANGE A production function shows, for a given level of technology, the maximum output level which can be obtained from given amounts of inputs. Technological change results in a shift i n the production function over time. In Figure 2, Labor input Figure 2. Technically efficient combinations of labor and capital inputs at two levels of technology. a shift i n the production indifference curve from position 1 to position 2 indicates that an increase in productivity 6 has occurred, since smaller amounts of capital and labor 2 are now required to produce the same level of output. In the usual case, and for purposes of this study, this increased productivity i s defined as the result of disembodied technological change. The increase in productivity shown in Figure 2 i s not the result of economies of scale, since the output level i s unchanged. Factor (input) substitution i s also eliminated as a possible source of increased productivity, since the level of output can always be produced at technology level 2 by a smaller combination of inputs employed in the same proportion, as shown by a ray (R) through the origin, than at technology level 1. An implicit assumption in Figure 2 i s that the pro- ductive quality of the inputs, labor and capital, does not improve over time. This homogeneity of inputs i s implied because the production indifference curves for two instances in time are drawn on the same indifference curve map. Consequently, a second type of technological change, namely embodied technological change, has been eliminated from Figure 2. Embodied technological change i s defined production indifference curve i s defined as a locus of technically efficient input combinations a l l of which are capable of producing the same level of output. 7 as a change in the productive quality of one or a l l of the inputs used in the production process. For example, tech- nological change may be embodied in labor as a result of improved health, higher educational attainments and training programs. Similarly, technological change may be embodied in capital in the form of improved designs.-^ Embodied technological change, therefore, gives rise to productivity increases as a result of increased output levels correspond- ing to inputs measured in "efficiency" units. Technological change may also be classified as neutral, labor-saving, or capital-saving. This topic i s discussed in Chapter II. Chapter II also outlines techniques whereby i t i s possible to estimate the rate of movement of the production function over time by a single number. This i s often used as a measure of disembodied technological change. IV. PRODUCTIVITY CHANGE IN CANADIAN AGRICULTURE Several empirical studies have attempted to measure productivity change in Canadian agriculture over the past three or four decades. In order to indicate the extent of productivity change in Canadian agriculture, and the attempts which have been made to identify the sources of -̂ H.A.J. Green, "Embodied Progress, Investment, and Growth," American Economic Review, 56:138-51, March, 1966. s productivity changes, a few of these studies are briefly discussed below. The f i r s t of the recent studies on productivity in Canadian agriculture was completed by Lok in the late 1950*s.^- Lok examined, for Canadian agriculture as a whole, the relationship between annual percentage changes in total productivity and real net return per farm over the years 1926-57• Lok concentrated on the estimation of a total productivity index. He aggregated constant dollar series for individual inputs into a single constant dollar index measuring total input, which was then divided into a constant dollar index of total output. Lok devoted considerable attention to the discre- pancies between productivity indexes when prices of different periods were used to weight the classes of out- puts and inputs in the construction of constant dollar series.^ As a result of this enquiry, he presented six indexes showing total productivity change in Canadian agriculture during 1 9 2 6 - 5 7 . ^ The estimates ranged from 4-Siepko H. Lok, An Enquiry into the Relationships Between Changes in Overall Productivity and Real Net Return per Farm, and Between Changes in Total Output and Real Gross Return, Canadian Agriculture, 1926-1957i Economics Division, Canada Department of Agriculture, Technical Publication 61/13 (Ottawa: 1961) . 5Ibid., pp. 1 0 - 1 1 . ^Lok, op_. c i t . , table 6 , p. 76. 9 a low increase of 19.8 per cent to a high of 59.1 per cent over the period 1926-57. Although Lok made no attempt to quantify the sources of these estimated productivity increases, he did o f f e r some general reasons such as re- search and education, economies of scale f o r i n d i v i d u a l firms, and greater adherence to the p r i n c i p l e of comparative 7 advantage. Furniss has used s i m i l a r methods and basic data sources as did Lok to estimate productivity change during the period 1935-60. He estimated that t o t a l a g r i c u l t u r a l productivity increased by 60 per cent over t h i s period, which i s equivalent to an annual growth rate of 1.9 per o cent. This compared with an annual growth rate of 2.2 10 per cent during the 1946-60 period. Furniss also investigated i n d i v i d u a l f a c t o r pro- d u c t i v i t i e s using the constant d o l l a r method and output- i n d i v i d u a l input r a t i o s . He found that labor productivity increased by 183 per cent during the 1935-60 period. S i m i l a r l y , the productivity of land and buildings increased ?Lok, op_. c i t . , pp. 20-21. g I. F. Furniss, "Productivity of Canadian Agriculture, 1935-1960: a Quarter Century of Change," Canadian Journal of A g r i c u l t u r a l Economics, 12, No. 2: 41-53, 1964. 9lbid.» p. 42. 1 G I b i d . , p. 51. by 42 per cent over the 1935-60 period, but only 14 per cent over the 1946-60 period. In I960, the ratio of total output to capital inputs ( a l l other inputs) was 36 per cent less than in 1935, and similar to the 1946 l e v e l . ^ Like Lok, Furniss made no attempt to quantitatively explain these estimated productivity changes in terms of techno- logical change, economies of scale, and factor substitution. He did, however, indicate the nature of the changes in input proportions over the time period studied, and suggested that substantially increased inputs of purchased feed, seed, f e r t i l i z e r s and pesticides had made an impor- tant contribution to increased total agricultural productivity. Mackenzie also has investigated productivity in 12 Canadian agriculture. Unlike Lok and Furniss, who investigated productivity change related to a gross measure of agricultural output, Mackenzie examined net labor x xIbid.,pp. 43-44. 12W. Mackenzie, "The Terms of Trade, Productivity and Income of Canadian Agriculture," Canadian Journal of Agricultural Economics, 9 , No. 2 : 1 -13 , 1961; W. Mackenzie, "The Impact of Technological Change on the Efficiency of Production in Canadian Agriculture," Canadian Journal of Agricultural Economics, 10, No. 1 : 41-53, 1962; and W. Mackenzie, "Regional Changes in Income, Terms of Trade and Productivity within Canadian Agriculture," Canadian Journal of Agricultural Economics. 11 , No. 2 : 41 -51 , 1963. productivity change in agriculture by deriving a measure of value added output in real terms (gross outputs less 13 material inputs) per unit of labor input. A comparison of Mackenzie's estimates with those of Furniss, indicates that net labor productivity has increased much less than gross labor productivity. This suggests that purchased inputs have contributed substantially to the phenomenal labor productivity increases estimated by Furniss. Mackenzie1^" extended his estimates of net labor product- i v i t y changes to a total net productivity index for Canadian agriculture by aggregating inputs into a measure 15 of total input in a manner suggested by Kendrick. On this basis, Mackenzie estimated that total net product- i v i t y for Canadian agriculture increased by 37«0 to 43*6 per cent from the 1944-46 period to the 1954-56 period. In a recent Doctoral dissertation, L i attempted to explain the increases in labor productivity in Canadian agriculture. This i s the only study which has specif- i c a l l y attempted to explain productivity changes in terms ^W. Mackenzie, 1961, p. 7. Mackenzie, 1962, p. 43. 15 J.W. Kendrick, "Productivity Trends in Agriculture and Industry," Journal of Farm Economics, 40:1554-64* December, 1956. "^Lew-king L i , "Technological Change in Canadian Agriculture" (unpublished Doctoral dissertation, University of Manitoba, Winnipeg, 1966). of technological change, economies of scale, and factor substitution. Using the Solow or geometric method, L i estimated the rate of disembodied technological change on the basis of both net value added and gross measures of output.^ He estimated that disembodied technological change has proceeded at an annual rate of 3.1 per cent in •i ti-the agricultural sector as a whole for the period 1946-65. Over the same period, net labor productivity increased by 176 per cent with 75.2 per cent of this increase attribu- table to technological change and the remainder, 24.8 per cent, attributable to increases in the capital-labor ratio. Productivity increases have been well demonstrated for Canadian agriculture. The main results of the above studies are summarized in Table I. However, interpretive analysis in the terms suggested in this chapter have only recently begun. Important aspects of the problem of under- standing productivity changes have not yet been investigated. V. THE PROBLEM In the real world, i t i s d i f f i c u l t to isolate the effects of technological change, factor substitution and economies of scale on changes in productivity. However, M. Solow, "Technical Change and the Aggregate Production Function," Review of Economics and Statistics, 39:312-20, August, 1957. l 8 L i , op_. c i t . , p. 112. 13 TABLE I SUMMARY OF ESTIMATED PRODUCTIVITY CHANGE IN CANADIAN AGRICULTURE Average annual Author Productivity measure percentage growth rate Lok total productivity, 1926-57 0.5-1.5 Furniss total productivity, 1935-60 1.9 total productivity, 1946-60 2.2 labor productivity, 1935-60 4.1 land and buildings productivity, 1935-60 1.4 land and buildings productivity, 1946-60 0.9 capital productivity, 1935-60 -2.6 Capital productivity, 1946-60 0.0 Mackenzie total net productivity, 1944-58 2.3 Li disembodied technological change, 1946-65 3.1 net labor productivity, 1946-65 5*2 one way to gain a better understanding of productivity changes over time i s to separate the productivity changes into the broad source components of technological change, factor substitution, and economies of scale. This concept i s shown in Figure 3. In time period t, 50 units of output Capital input K2=20 Ki=15 ; _ _ t+l=100 1 1 1 t=50 1 1 1 L2=20 L1=25 Labor input Figure 3» Productivity changes over time. are produced with 25 units of labor and 15 units of capital. In the subsequent time period, t+1, 100 units of output are produced with 20 units of capital and 20 units of labor. Productivity has increased from time period t to time period t+1. Both capital and labor productivity ratios have increased and i t i s conceivable that total productivity has also increased, although this cannot be ascertained from the limited information. Furthermore, i t i s impossible to discuss why the increases in productivity have occurred. Input substitution has occurred and may have contributed to the increase in productivity, especially labor product- i v i t y . However, disembodied technological change and/or economies of scale may also have contributed to productivity change. The broad changes which have occurred in Canadian agriculture are similar to those portrayed in the simple example above. Productivity has increased in the agricul- tural sector. There has been a substitution of capital for labor, and agricultural labor productivity has increased more rapidly than in any other major sector of the Canadian economy during the post war period."^ When the real world ^The Dominion Bureau of Statistics has estimated that output per person employed in Canadian agriculture has increased by 5.5 per cent annually during 1946-67. This compares with 2.8 per cent for the commercial nonagricultural industries, and 4*3 per cent for the nonagricultural goods- producing industries. See: Canada, Dominion Bureau of Statistics. Aggregate.Productivity Trends, 1946-67. (Ottawa: Queen »s Printer, iy6ti). agricultural sector is considered, the changing quality of inputs over time further complicates efforts to attribute the sources of productivity gains. Embodied technological change must also be considered, and i t would seem reasonable on a pr i o r i grounds to postulate that the productive quality of many agricultural inputs has increased over time. This study was designed to investigate technological change in the aggregate primary agricultural sector in Canada during the time period 1935-65. Specifically, the problem was: (1) to measure the rate of disembodied tech- nological change; (2) to measure the rate of technological change which has been embodied i n agricultural machinery and implements; and (3) to measure the rate of techno- logical change which has been embodied in material inputs. This study did not investigate a l l of the possible sources of productivity change. An assumption was made regarding economies of scale, and the influences of the substitution of capital for labor were not estimated. Conceptually, the measurement of embodied technological change should have been extended to include a l l inputs. A p r i o r i , i t would be reasonable to expect that the quality of the labor force has improved over time. However, because of data limitations and the lack of suitable methods of analysis, this study was limited to the consideration of 16 embodied technological change in two inputs only. Chapter II i s a review of the theoretical framework and related empirical studies regarding the measurement of technological change. Various models are interpreted and evaluated in terms of their contribution to this study. The model and data, including methods of derivation, manipu- lation and assumptions, used in this study are outlined in Chapter III. The results of the analysis and related dis- cussion are presented in Chapter IV. Finally, Chapter V presents a summary and the main conclusions and implications of the study. CHAPTER II CONCEPTUAL FRAMEWORK FOR MEASURING TECHNOLOGICAL CHANGE This chapter reviews the theory which i s pertinent to the measurement of technological change, and the methods and approaches which have been developed. This review i s not exhaustive, but rather i t concentrates on those methods and studies which have provided the background for, and contributed most to, the methods used in this study, which are outlined in the following chapter. I. NEUTRALITY OF TECHNOLOGICAL CHANGE Economists distinguish among three types of techno- logical change as i t affects the shift i n the production function: neutral, labor-saving, and capital-saving technological change. In many methods, estimation of the rate of techno- logical change involves accurate specification both of the aggregate production function and of the form of technological change. In addition, i t has been customary, for reasons of theoretical and empirical convenience, to assume that technological change i s neutral. When consider- ing the question of neutrality, the usual procedure has been to make assumptions about the way in which technological change affects relationships between certain variables which are derived from the production function. Techno- logical change i s then neutral i f i t s effects do not alter the relationship between the chosen variables. Because there are several possible pairs of variables which may be chosen, alternative definitions or various forms of neutral technological change are possible. The most widely used and best known of these are the "Hicks" and "Harrod" definitions of neutral technological change. Technological change i s neutral i n the Hicks sense i f the ratio of the marginal product of labor to the marginal product of capital i s unchanged when the capital-labor ratio i s unchanged. When disembodied technological change i s assumed to be an exogenous function of time (t), the production function Y=f(K,L,t) i s implied. If technological change i s Hicks-neutral, the function becomes Y=A(t)f(K,L) where A(t) i s any function of time. This i s the general M.J. Beckmann and R. Sato, "Aggregate Production Functions and Types of Technical Progress: a S t a t i s t i c a l Analysis," American Economic Review. 59:88-101, March,1969. 2 A more complete treatment, from which the following discussion i s drawn, of the implications of the two definitions i s found i n : F. Halm and R. Matthews, "The Theory of Economic Growth: a Survey," Economic Journal, 74: 779-901, December, 1964. form of the production function used to estimate disembodied technological change in this study, and, therefore, Hicks- neutral technological change was implicitly assumed. The Harrod definition i s based on the comparison of points on the two production functions where the marginal product of capital is constant. Technological change i s neutral in the Harrod sense i f the capital-labor ratio which results in a constant marginal product of capital after technological change, also causes the capital-output ratio to remain constant. With two inputs, Harrod-neutral technological change i s shown algebraically as Y=f(K,A(t)L) which indicates that Harrod-neutral technological change may be described as "labor-augmenting", since the labor force i s measured in efficiency units, A(t)L. Technological change may also be "capital-augmenting". In this case the general form of the production function i s Y=f(A(t)K,L) which i s the mirror image of Harrod-neutral technological change with K and L reversed. Capital-augmenting techno- logical change i s a useful concept in the study of vintage- capital models, and i t i s equivalent to the concept of embodied technological change in capital which was used in this study. The concept i s more f u l l y discussed in Section III below. Technological change i s neutral in both the Hicks and Harrod sense when the el a s t i c i t y of substitution between labor and capital i s unity.^ The Cobb-Douglas production function, which was used in this study, possesses this property and, therefore, unequivocal neutrality was implicitly assumed. Recently, Beckmann and Sato have generalized the concept of technological neutrality by extending the principle that technological change i s neutral when the relationship between a specific pair of variables i s invariant through time, to relationships between variables other than those considered in the Hicks, Harrod, and Solow definitions.^ Under the Beckmann and Sato scheme technological change i s Hicks-neutral when the relation- ship between the marginal rate of substitution and the capital-labor ratio i s constant, Harrod-neutral when the relationship between the capital-output ratio and the interest rate does not change, and Solow-neutral when the relationship between output per worker and the wage rate i s invariant. After examining relationships between output-capital ratios, output-labor ratios, capital-labor ratios, interest rates, wage rates, marginal rates of 3Ibid., p. 6*29. ^Beckmann and Sato, op_. c i t . , p. 90. 21 substitution, and labor's share, Beckmann and Sato suggested a number of interesting new types of technological change. They applied regression analysis to time series data for the U.S., Japanese, and German private non-farm economies in order to empirically investigate the implications of their formulations of technological neutrality. They con- cluded that:^ (1) the traditional types of Hicks, Harrod, and Solow neutrality were for a l l countries at least as good as the unconvential types of neutrality; (2) Solow- neutral technological change performed particularly well; (3) general factor-augmenting technological change did not give a substantially improved explanation of observed data when compared with single-factor-augmenting technological change; and (4) irrespective of how technological change was specified, the estimated production function turned out to be close to a Cobb-Douglas or CES function.^ II. DISEMBODIED TECHNOLOGICAL CHANGE The rate of growth of total factor productivity i s conventionally defined as the difference between the rates 5Ibid., p. 9 5 . °The CES production function i s a more general function which allows the e l a s t i c i t y of substitution to be estimated. It also contains efficiency and distribution parameters so that both neutral and non-neutral technological change can be estimated. Although the CES function has been the most commonly applied in recent work, i t was not employed in this study because, being a more general form, i t i s more d i f f i c u l t to estimate than the Cobb-Douglas form. 22 of growth of real output and real input, where the rates of growth of real output and input are the weighted averages of the rates of growth of individual products and inputs. 7 Under various assumptions (including neutrality, perfect competition, and constant returns to scale) a change in total factor productivity may be identified with a shift in the production function, and changes in real output and input not accompanied by a change in total factor producti- vity may be associated with movements along the production function. Technological change i s also defined as a shift in the production function, and the terms technological change and total factor productivity have tended, i n practice, to be used interchangeably. Much of the empirical work during the 1950*s and early 1960 fs concentrated on this simple concept of deducting the contributions of increased capital and labor inputs to increased output, and attribut- ing the "residual" growth i n output to disembodied techno- logical change or total factor productivity. The arithmetic index and the Solow model were the most common methods employed. 7D.W. Jorgenson and Z. Griliches, "The Explanation of Productivity Change," The Review of Economic Studies, 34:249-83, July, 1967. This interchangeability depends upon the assumption of constant returns to scale, and i s valid only i f economies or diseconomies of scale do not exist. The arithmetic index which has been used by Abramovitz^ and Kendrick^ may be defined as C=Y/(wL+iK) where Y i s output, w i s the real wage rate in the base period, L i s labor input (in physical units) i n a given year, i i s the real return to capital in the base period, and K i s capital input (in physical units) in a given year.I-1- Perfectly competitive equilibrium i s implied since the weights, w and i , represent the marginal products of labor and capital, respectively. The c r i t i c a l assumption, however, i s that the marginal products of the inputs are changed only by technological change and always in the same proportion. Therefore, the marginal products are assumed to be independent of the ratio of the quantities of the inputs, which i s a very restrictive assumption and not reasonable over a longer period where substantial changes may be expected in the capital-labor ratio. The so called constant dollar method which, as noted in Chapter I, has been used by Lok, Furniss, and Mackenzie to estimate productivity change in Canadian agriculture 9 M. Abramovitz, "Resource and Output Trends in the United States Since 1870," American Economic Review, 4 6 : 5 - 2 3 , May, 1956. 10J.W. Kendrick, Productivity Trends i n the United States (Princeton: Princeton University Press, 1961TI 1:LE.D. Domar, "On Total Productivity and A l l That," Journal of P o l i t i c a l Economy. 70 :599, December, 1962, i s a special formulation of the arithmetic index. The constant dollar index may be defined as C=y/(l+k) where y, 1 and k are the values of output, labor and capital, respectively in base year prices. Aside from the normal assumption of competitive equilibrium, the basic d i f f i c u l t y with this method (as well as with the arithmetic index) i s that i t i s not suited to either a linear or an expo- 12 nential world. For example, i f the values of output and of inputs are linear with respect to time, C w i l l gradually approach a constant while i f they are exponential, the relative rate of growth of C w i l l approach the dif- ference between the rates of growth of output and of the fastest growing input. Solow derived, from very general assumptions, a somewhat better measure of the rate of movement of the production function than the arithmetic index.^ Solow1s derivation was based on a linear homogeneous production function of the general form Y=A(t)f(K,L). Solow defined technological change as "any kind of shi f t " 1 2 I b i d . , p. 607. ^R. M. Solow, "Technical Change and the Aggregate Production Function," Review of Economics and Statistics. 39:312-20, August,. 1957. i n the production function, and the term A(t) measures the cumulated effect of shifts over time. 1 4 With the further assumptions of perfect competition and neutral technological change, technological change between two periods i s given by A=Y-WkK-W1L where A, Y, L, and K are the percentage rates of change per unit of time of disembodied technological change, output, labor input and capital input, respectively. Ŵ  and W-L are the shares of capital and labor in output which, under the assumptions of this model, w i l l be equal to the elas- t i c i t y of output with respect to capital and labor, respectively. Given time series data on Y, L, K, Wj,. and Wi, A can be estimated. The interpretation of the Solow model i s straight- forward: disembodied technological change i s equal to the change in the output which i s not accounted for by the changes in capital and labor. Thus, as Domar has clearly pointed out, disembodied technological change i s estimated in this method as a residual, and for this reason he prefers to c a l l i t the "Residual." 1 5 1 4 I b i d . , p. 312. 1 5 E . D. Domar, "On the Measurement of Technological Change," Economic Journal, 71:712, December, 1961. Upon applying the above model to the U.S. non-farm economy for the period 1909-49, Solow concluded that: (1) technological change was neutral on average; (2) tech- nological change proceeded at an annual rate of about one per cent for the f i r s t half of the period and two per cent for the last half; and (3) 87.5 per cent of the increase in gross output per man-hour could be attributed to techno- logical change, and the remaining 12.5 per cent to increased use of capital." 1"^ Lave has applied the Solow model to U.S. agriculture and concluded that: (1) technological change in agriculture was twice as rapid as in the private non-farm sector; and (2) technological change accounted for 60 to 73 per cent of the increase in output per man-year during the 1850-1950 period with the remainder, 27 to 40 per cent, attributable to increases in c a p i t a l . A s noted in Chapter I, L i applied the Solow model to Canadian agriculture for the period 1946-65. Domar has provided a more general interpretation of the Solow model - the geometric index. Since any linear homogeneous production function with constant factor shares ^Solow, op_. c i t . , p. 320. *^L. B. Lave, Technological Change: Its Conception and Measurement. (New Jersey: Prentice-Hall, Inc., 1966), pp. 47-57. IS Domar, op_. c i t . w x w k i s of the Cobb-Douglas type, Domar identified L K as a weighted geometric mean. It follows that, i f both L and K w l wk are index numbers with a common base, then L K i s a geometric index of inputs, each weighted by i t s share in output in the base period. Disembodied technological change i s then the ratio between geometric indexes of out- 19 puts and inputs. 7 The geometric index i s simply a geometric index number with constant factor shares as weights, and therefore differs from the Solow model which uses current factor shares as weights. However, with this assumption, Domar has derived an index which circumvents the underlying assumption of an aggregate production function with the accompanying implications. Furthermore, since relative factor shares appear to have been quite stable over time and relative prices have not, the geometric index seems to better approximate reality than does the arithmetic index. The above models provide only indirect measures of disembodied technological change. They do not isolate the effects of "pure" technological change alone, but include in the measures a l l increases in output not accounted for by the growth of explicitly recognized inputs and, therefore, must be treated conceptually as residuals. Moreover, these 1 Q I b i d . , p. 713 methods are based on the notion that technological change i s disembodied, that i s , that a l l technological change consists of better methods and organization which improve the output performance of the inputs. The inputs are assumed to be homogeneous over time. However, many changes in technology must be embodied i n new capital i f they are to be u t i l i z e d . In the above methods, capital does not change in quality, form, or composition, nor does i t act as a vehicle for the introduction of technological change into the productive process. The methods reviewed in the next section were developed in an attempt to explicitly recognize this concept, and to provide estimates of the rate of embodied technological change. III. EMBODIED TECHNOLOGICAL CHANGE The concept of embodied technological change developed from the notion that capital investment and technological advance influence each other in such a way that their separation i s meaningless, i f not impossible. If techno- logical change cannot be implemented without introducing new kinds of capital, then capital investment may be regarded as the vehicle of technological advance, and capital, therefore, cannot be considered homogeneous. In this approach, technological change i s embodied in new capital, and may be regarded as a progressive reduction in 29 the cost of producing capital, or alternatively, as a 20 progressive improvement of the quality of capital. Therefore, capital goods embody the technology of their date of construction, and those built at different dates ("vintages") are qualitatively dissimilar. A separate pro- duction function i s required for each vintage and total 21 output i s the sum of output from a l l vintages in use. In 1959, Solow reconstructed his earlier disembodied 22 model to make allowance for embodied technological change. Solow began by interpreting his disembodied model as a linear homogeneous Cobb-Douglas function Q=Be?iLaK1~a where e^ i s an exogenous shift function which measures the rate of neutral disembodied technological change.2^ Solow's embodied method i s based on a vintage model of production. Output at time t, Q v(t), from the surviving capital equip- ment of vintage v i s given by a linear homogeneous Cobb- 2 0 F . Hahn and R. Matthews, "The Theory of Economic Growth: a Survey," Economic Journal, 74:343, December, 1964. 2 1 I b i d . , p. 837. op R. M. Solow, "Investment and Technical Progress," Mathematical Methods in the Social Sciences, K. J. Arrow, editor (Stanford: Stanford University Press, I960), pp. 89-104. 23 ^This i s a more restrictive formulation than the 1957 model, since disembodied technological change i s now assumed to advance at a constant rate,?\ , over time, and factor shares are also assumed constant over time. ) Douglas production function Q v ( t ) = B e A v L v ( t ) a K v ( t ) 1 - a . Embodied technological change, represented by 1\, i s assumed to be uniform, approximately exponential over time, and capital-augmenting (Solow-neutral). Therefore, a l l techno- logical progress appears as a steady improvement in the quality of capital goods at the rate ^ / ( 1 - a ) . 2 4 To emphasize obsolescence rather than depreciation, capital i s assumed to be subject to a constant force of mortality, m, and the average length of l i f e of capital i s l/m.2-> Labor i s homogeneous and the allocation of labor to capital of various vintages i s assumed to equate the marginal produc- t i v i t y of labor in a l l uses. As Solow demonstrated, i t i s then possible to derive a measure of "equivalent capital" at time t, J ( t ) , by summing the surviving capital goods of past vintages inclusive of time t, weighted according to their vintage. Output at time t, Q(t), i s then given by Q ( t ) = B e - m ( 1 - a ) L ( t ) a J ( t ) 1 - a . Using exogenous estimates of a (elasticity of output with respect to labor) and m, Solow estimated the value of 2 4Solow, op_. c i t . , p. 91 . 25 The theory does not require an explicit assumption about depreciation. However, the vintage composition of the stock of capital i s required, and since such information i s not usually available, i t must be derived by employing an assumption about depreciation. See: Ibid., p. 93• ft from time series of output, labor, and gross investment. For the U.S. private sector, 1919-53, he found that (S equalled about 0.025 which was substantially larger than the estimated value of 0.015 from his disembodied model. However, the difference i s in the expected direction, since in the embodied model only new capital benefits from technological advance rather than a l l capital goods as in the disembodied model. In 1962, Solow presented a slightly different method for estimating capital embodied technological change while drawing a distinction between actual and potential output. In this model a l l technological advance i s embodied in new capital goods, and the rate of embodied technological change i s , therefore, synonymous with the rate of improve- ment in the productivity of capital goods, ?\. Assuming that labor and capital of various vintages are allocated so that output i s maximized, that i s , the marginal produc- t i v i t y of labor i s equal in a l l uses, the equivalent stock of capital in year t, J(t ) , i s where I(v) i s gross investment in year v, and B(t-v) i s the amount surviving in a later year t. Potential output, P(t), i s then a function of the equivalent stock of capital <:DR. M. Solow, "Technical Progress, Capital Formation, and Economic Growth," American Economic Review, 52:76-S6, May, 1962. and the available labor supply, L(t), and i s given by P(t)=F(J(t), L( t ) ) . No explicit term representing technological change i s required because i t i s contained in J. 2? However, actual output, A(t), i s less than poten- t i a l output because of unemployment and idle capital. If ii(t) i s the unemployment rate, then A(t)=f(u)F(J(t), L ( t ) ) . To derive empirical estimates of and u, Solow used a linear homogeneous Cobb-Douglas production function A ^ ^ B I O ^ ^ ^ J ^ 1 " 3 . Solow f i t t e d the function using various estimates of the equivalent stock of capital which were derived by using various values for the improvement factor ?\. In an effort to determine whether the rate of productivity improvement differed between plant and equipment, differ- ent values of (\ were used for each component. The c r i t e r i a for determining the best estimate of r\ were the goodness of f i t and low standard errors of the regression coefficients. Solow* s above model provided two concepts used in this study: (1) the distinction between potential and 2 7 I b i d . . p. 77 33 actual output; and (2) the possibility that different kinds of capital may experience different rates of embodied technological change. IV. SIMULTANEOUS EMBODIED AND DISEMBODIED TECHNOLOGICAL CHANGE The methods outlined in Section II depend on the assumption that technological advance increases the produc- t i v i t y of old and new capital goods in the same way and in the same proportion. On the other hand, methods described in Section III are based on the opposite assumption that technological advance can be introduced into the produc- tion process only through new capital investment. In the real world, the truth most probably l i e s somewhere between these two extremes. The disembodied and embodied approaches to techno- logical change were synthesized by Phelps in a linear homogeneous Cobb-Douglas production function Q ( t ) = A e u t J ( t ) a L ( t ) 1 - a where u i s an estimate of neutral disembodied technological change, and J i s Solow*s equivalent capital stock (which 28 embodies technological change at rate n). In this model, 2 8 E . S. Phelps, "The New View of Investment: a Neo- classical Analysis," Quarterly Journal of Economics, 76:549-67, November, 1962. disembodied technological advance occurs i f u i O and 7̂ =0, while embodied technological advance occurs i f u=0 and ?\>0, When both u and A are p o s i t i v e , both kinds of technological change occur j o i n t l y . I n t r i l i g a t o r extended the Solow and Phelps models empirically i n two ways: (1) embodied and disembodied technological change were estimated j o i n t l y rather than separately as i n the Phelps method; and (2) technological change embodied i n improved quality of labor as well as improved quality of c a p i t a l was estimated. 2^ I n t r i l i g a t o r derived h i s model by adding Solow*s unemployment function r e l a t i n g actual and po t e n t i a l output to Phelps* embodied and disembodied model. His model, therefore, i s a l i n e a r homogeneous Cobb-Douglas production function r e l a t i n g actual output, Q(t), to equivalent c a p i t a l , J ( t ) , and equivalent labor, M(t), Q ( t ) = A e u t e b + c u + d u 2 J ( t ) a M ( t J 1 ' 3 where u i s a measure of neutral disembodied technological change, and J and M are c a p i t a l and labor inputs, respec- t i v e l y , weighted f o r quality change (embodied technological change). I n t r i l i g a t o r * s method of estimation was s i m i l a r to that used by Solow i n h i s 1962 model, that i s , the D. I n t r i l i g a t o r , "Embodied Technical Change and Productivity i n the United States, 1929-1958," Review of Economics and S t a t i s t i c s . 47:65-70, February, 1965. production function was estimated using the alternative capital and labor input series based on various assumed levels of embodied technological change. The results of the various regressions were then compared in order to choose among the assumed values for embodied technological change. Using data for the U.S. private sector, 1929-58, Intriligator concluded that embodied and disembodied tech- nological change must be treated simultaneously.^® Thus, Intriligator provided a method, which was used in this study, for determining the rates of disembodied and embodied technological change simultaneously. However, many of the stringent assumptions of the previous Solow models were necessarily retained: (l) disembodied techno- logical change i s Hicks-neutral and proceeds at a constant rate; (2) embodied technological change i s both capital and labor-augmenting; (3) the production function i s linear homogeneous and Gobb-Douglas; (4) the economy i s in a state of perfectly competitive equilibrium; and (5) the marginal productivity of labor i s equated over a l l vintages of capital. The validity and implications of two of these assumptions, neutrality and constant returns to scale, have been subject to considerable scepticism and inves- tigation. If they are not valid, biased estimates of Ibid., p. 6 9 . disembodied technological change w i l l result. This i s discussed in the following section. V. ECONOMIES OF SCALE AND NON-NEUTRALITY Walters has clearly pointed out that i f economies of scale are present i n the aggregate economy, i t s effects w i l l be included in the measure of disembodied techno- logical change.31 In view of i t s importance, Walters investigated the assumption of constant returns to scale by estimating the unrestricted Cobb-Douglas function Q=Ae u tK aL b with similar data to that used by Solow in his 1957 paper. Walters found that the sum of a+b was significantly greater than one, thus indicating economies of scale. According to his estimates, 27 to 35 per cent of the increase in output in the U.S. private non-farm sector could be attributed to economies of scale with a consequent reduc- tion i n the proportion attributable to disembodied technological change. However, as Walters indicated, the implications of economies of scale i n the aggregate produc- tion function are not clear, and his results cannot be regarded as overwhelming evidence against the hypothesis of constant returns to scale. 3lA. A. Walters, "A Note on Economies of Scale," Review of Economics and Statistics, 45:425-27, November, 1963. 37 Ferguson has expressed similar views.3 2 On the basis of a study of the U.S. manufacturing sector, 1929-63 he concluded that in aggregate studies covering long periods of time, a production function which i s homogeneous of degree one i s l i k e l y to provide economically more meaningful results even though these results may be s t a t i s t i c a l l y less significant than in the case of homo- geneity of degree greater than one (economies of scale).33 Ferguson also argued that homogeneity of degree greater than one should not be interpreted to mean that the aggre- gate economy i s subject to economies of scale. In addition, Kislev has suggested that many of the estimated aggregate production functions for U.S. agriculture, which have shown significant economies of scale, are biased in the direction of overestimating economies of scale.34 The neutrality assumption most commonly questioned has been that of Hicks-neutral disembodied technological change. For example, Resek questioned this assumption in Solow*s 1957 paper, as well as the method which Solow used 3 2 C .E. Ferguson, "Substitution, Technical Progress, and Returns to Scale," American Economic Review, 5 5 : 2 9 6 -305, May, 1 9 6 5 . 3 3 i b i d . , pp. 303 - 0 5 . 34-Y. Kislev, "Overestimates of Returns to Scale in Agriculture - A Case of Synchronized Aggregation," Journal of Farm Economics, 4 8 : 9 6 7 - 8 3 , November, 1966. to test the neutrality assumption.-^ Resek suggested that i f technological change i s non-neutral in the Hicks sense, then some of the observed increases in output per man could be attributed to the interaction of capital or labor and technological change. One method of relaxing the assumption of Hicks- neutrality i s to allow for other types of neutral techno- logical change such as Harrod-neutral and Solow-neutral which are non-neutral or biased in the Hicks sense. To the extent that other types of neutrality are also taken into account in a method or model, the assumption of neutrality would l i k e l y lead to less biased results than in the case where only Hicks-neutral disembodied technological progress i s a possibility. An example of this approach i s a study by David and van de Klundert of the private domestic sector of the U.S. economy, 1899-1960.3° They employed a homo- geneous of degree one CES production function incorporating both labor and capital-augmenting technological change. The relative rates of labor and capital augmentation can then be related to the usual concepts of neutral, labor- saving, and capital-saving technological advance in the 3 5 R . W . Resek, "Neutrality of Technical Progress," Review of Economics and Statistics, 45:55-63, February, 1963. 36p. A. David and T. van de Klundert, "Biased Efficiency Growth and Capital-Labor Substitution in the U.S., 1899-1960." American Economic Review. 55:357-94, June, 1965. Hicks sense. From the empirical application of their model, they concluded that technological progress in the private domestic sector of the U.S. economy was labor- saving in the Hicks sense. A quite different approach to non-neutrality and economies of scale has been suggested by Brown and Popkin.^^ They attempted to attribute changes in output over any discrete time period to the weighted change in inputs, economies of scale, and neutral and non-neutral techno- logical change. The method consisted of f i t t i n g a Cobb- Douglas production function to various time periods in order to identify time periods called "technological epochs," in which there was only neutral technological change. Within each epoch, the influences on output of neutral technological change, economies of scale and increased inputs were estimated, and the changes in the parameters of the estimated production function between epochs were then used to measure output change attributable to non-neutral technological change. However, the abrupt shift from one epoch to another i s an approximation, since i t i s l i k e l y that a shift in 3 7 I b i d . . pp. 3 6 2 - 6 3 . 38 J M. Brown and J. Popkin, "A Measure of Technological Change and Returns to Scale," Review of Economics and Statistics. 44:402-11, November, 1962. technology occurs gradually over time. Thus, the produc- tion function i s misspecified at the ends and beginnings of a l l periods. However, i f a study covers a long period of time, the misspecification as a result of using epochs compared with only one time period would be of a lesser degree. Using data for the U.S. private non-farm sector, 1890-1958, Brown and Popkin found evidence of economies of scale, and showed that the effects on output of non-neutral disembodied technological change were extremely small com- pared with the effects of neutral technological change. In summary, the evidence against the assumption of constant returns to scale i s not conclusive. The question of assuming neutrality i s really a question of specifying the correct type of technological change. However, i t must be noted that i f the assumptions are not valid, biased esti mates of technological change result. The use of these assumptions in this study i s discussed in the following chapter. VI. SOME PROBLEMS AND ALTERNATIVES The models and methods outlined in the previous sections provided the basic concepts which were used in this study. As an aid to interpreting the results of this study, i t i s useful to br i e f l y outline the basic problems and objections to these methods, and some alternative approaches. From an analytical point of view, disembodied techno- logical change has been treated as an exogenous variable which i s not explained by any economic phenomenon. It has been called the "Residual" and "a measure of our igno- rance. "^^ The embodiment hypothesis (technological change embodied in factor inputs) was an attempt to relate part of this residual to qualitative change in factor inputs. These attempts have also been c r i t i c i s e d . Although he conceded that the embodiment hypothesis i s a potentially f r u i t f u l method of analysis, Griliches has argued that in practice i t turns out to be a mere "relabelling of an already empty box." 4 0 Moreover, i f the assumption that technological change proceeds at constant exponential rates i s dropped, Jorgenson has shown that i t i s often impossible to distin- guish capital-embodied from disembodied technological change on the basis of available data. 4 1 However, David and van de Klundert have defended the embodiment approach. 4 2 ^^Domar, op,, c i t . , p. 709. ^ Z . Griliches, "Technological Change and Economic Theory: Discussion," American Economic Review, 55:344, May, 1965. 4 1D. W. Jorgenson, "The Embodiment Hypothesis," Journal of P o l i t i c a l Economy. 74 :1 -17 , February, i 9 6 0 . 42 David and van de Klundert, op_. c i t . , pp. 357-59. They argued that i t i s possible to infer the rate of factor augmentation from conventional measures of inputs and out- puts, and that this may be used to place prior restrictions on further attempts to empirically identify the sources of capital and labor augmentation. The conception and estimation of an aggregate produc- tion function raises numerous theoretical and practical problems, although these are less troublesome when the methods are applied to one sector such as agriculture rather than to the whole economy.43 There has been considerable discussion in the literature on the relevant concept of capital as i t relates to the production function. Harcourt has recently provided a useful review of the controversies.44 In addition to the above conceptual problems, there are numerous d i f f i c u l t i e s i n obtaining accurate measurements of inputs and outputs which are required for any empirical analysis. The measurement of aggregate capital i s particu- l a r l y d i f f i c u l t because: (l) i t i s usually purchased not hired; (2) i t i s durable; and (3) i t s cost i s ambiguous.45 43For a discussion see: L. B. Lave, Technological Change: Its Conception and Measurement (New Jersey: Prentice-Hall, Inc., 1966), pp. 13 -15; 37-38; 140-41. ^G. C. Harcourt, "Some Cambridge Controversies in the Theory of Capital," Journal of Economic Literature. 7:369-405, June, 1969. ^ E . D. Domar, "On Total Productivity and A l l That," Journal of P o l i t i c a l Economy, 70:602, December, 1962. Errors of measurement w i l l bias any estimate of technolo- gical change. Errors may arise from: (1) errors in the time series; (2) non-homogeneity of the series over time; and (3) errors stemming from the economy's not always being in long-run equilibrium. 4 0 The alternative approaches, largely inspired by Denison, Griliches and Jorgenson, are attempts to directly explain a large portion of the residual. Denison attempted to identify the important elements of quality change in labor inputs. 4 7 Increases in output not accounted for by increased amounts of inputs or quality changes of inputs were attributed to changes in total factor productivity. Growth in total factor productivity was then ascribed to particular sources that could be identified and quantified such as resource shifts, economies of scale and the effect of demand pressures. In this way Denison was able to explain a large part of the residual. The Economic Council has applied Denison*s methods to explain the growth of output in the Canadian economy.4^ Similar methods have ^ Lave, op_. ext., p. 63 . 4 7Edward F. Denison, The Sources of Economic Growth in the United States and the Alternatives Before Us, CED Supplementary Paper No. 13 (New York: Committee for Economic Development, 1962)• 4^Economic Council of Canada, The Challenge of Growth and Change, Fi f t h Annual Review (Ottawa: Queen's Printer, T9o"8), pp. 7 - 6 1 . been used by the Council to explain the growth in labor productivity in Canadian agriculture.^ Griliches attempted to explain productivity change in the U.S. agricultural sector by estimating a cross- sectional production function.-^ The Cobb-Douglas function estimated was homogeneous of degree greater than one with six independent variables: livestock expense, other current expense, machinery, land, buildings and man-years of labor. Griliches then adjusted the time series data on inputs for changes in quality, and combined these by using weights derived from the estimated production function. On this basis he was able to account for a l l of the observed increases in total agricultural productivity, 1940-60. More recently, Jorgenson and Griliches have examined the hypothesis that i f quantities of output and input are measured accurately, growth in total output i s largely explained by growth in total input.^ Within the frame- work of social accounting, the hypothesis becomes that i f real output and real input are accounted for accurately, 4 9 i b i d . , pp. 63-75. -^Z. Griliches, "The Sources of Measured Productivity Growth: United States Agriculture, 1940-60," Journal of P o l i t i c a l Economy, 71:331-46, August, 1963. W. Jorgenson, and Z. Griliches, "The Explanation of Productivity Change," Review of Economic Studies. 34:249- 83, July, 1967. the observed growth in total factor productivity i s negli- gible. In summary, these alternative approaches attempt to make the residual disappear by constructing new measures of the growth of the various inputs which w i l l , when taken together, f u l l y account for the observed growth of output. This approach i s questionable to the extent that i t i s tantamount to tampering with the data. Moreover, David and van de Klundert have questioned whether this represents an alternative approach. They suggest that i t would be more sensible to begin by trying to identify the form which factor augmentation has taken, and then proceed to tackle the intriguing, but quite distinct question of the sources of such augmentation. 5 2 The methods used in this study, which are presented in the following chapter, draw heavily on the models for measuring embodied and disembodied technological change which are reviewed in earlier sections of this chapter. 5 2David and van de Klundert, op_. c i t . , p. 358" CHAPTER III METHOD OF ANALYSIS AND MEASUREMENT OF VARIABLES This chapter presents the basic model and data series which were used to estimate the rates of disembodied and embodied technological change in Canadian agriculture, 1935- 65. The chapter consists of two parts: (1) Section I con- tains a description of the basic model, a definition of variables, and a discussion of the implied assumptions; and (2) Section II outlines the sources, methods, and assump- tions which were employed in the derivation of the time series data used to estimate the model's coefficients. The actual data series are presented in Table V which i s found in the Appendix. Also found in the Appendix i s Table VI which presents a l i s t of symbols representing variables, para- meters, and coefficients used in this study. I. MODEL FOR ESTIMATING TECHNOLOGICAL CHANGE The basic model for this study was a linear homoge- neous Cobb-Douglas production function of the form < & « A e u t e f + £ w + h w 2 L 1 _ aK a where: (1) Q represents annual gross output in the primary agricultural sector at base period prices; (2) L represents the number of persons employed in the agricultural sector; 47 (3) K represents the equivalent annual flow of mate- r i a l inputs (intermediate goods purchased from other sectors of the economy), and capital services including livestock, land, buildings, and machinery and implements a l l measured at base period prices; (4) t i s a time index, 1 , 2 , . . . , 3 1 , representing the years during the time period 1935-65; (5) w i s an annual index which was designed as a proxy for environmental influences on output, and measures the observed deviations from the expected long-term trend of a weighted composite of crop y i e l d s ; 1 (6) e u t i s a shift function designed to measure the annual percentage rate (lOOu) of neutral disembodied tech- nological change; and 2 (7) e* , + s w +* l w i s a function designed to relate actual output, Q, to potential output through the weather index, w. The model was estimated in natural log-linear form using the least-squares regression technique, which provided estimates of: the constant, InA*(=lnA+f); the disembodied 2 technological change coefficient, u; the w and w coefficients, g and h; and the elast i c i t y of output with respect to capital,a. The time series data consisted of 31 obser- vations on Q, t, w, L, and K for each of the years 1935-65. For purposes of this study, the index i s called a "weather index". The rates of technological change embodied in mate- r i a l inputs, and in machinery and implements,^ and ft respectively, were measured indirectly in the manner sug- gested by Solow and Intriligator, and reviewed above i n Chapter II. A p r i o r i values of ^ and /\ were used to 2 construct alternative series for the K variable. Thus, there was an alternative time series for K for each possible combination of £ and 7\ . The model was estimated using each of these alternatives which resulted in a matrix of regres- sion equations where each regression was.computed on the basis of a different time series for K. The real world values of (3 and rl were then inferred by choosing the "best" regression equation using the c r i t e r i a of goodness of f i t , significance levels of the estimated coefficients and low standard errors. The above model i s based on several important assump- tions: (1) disembodied technological change was Hicks-neutral and proceeded at a constant rate; (2) embodied technological change proceeded at a constant rate and was capital-augmenting in the vintage sense, which implies that the productive quality of mate- r i a l inputs improved at an annual rate of 100^ per cent, and that the productive quality of new gross investment in ^Section II below outlines the detailed method used to derive the alternative K series. machinery and implements improved at an annual rate of 100 A per cent, that i s , machinery and implements purchased in any year were 100?\ per cent more productive than those purchased in the preceding year; (3) the Cobb-Douglas production function was linear homogeneous in labor and capital, which implies constant returns to scale and unitary elas t i c i t y of substitution between capital and labor; (4) the agricultural sector was in a state of per- fectly competitive equilibrium; (5) inputs of labor, livestock, land, and buildings were homogeneous over time; (6) capital inputs were utili z e d at a constant rate; and (7) labor was allocated so that i t s marginal product was equated over a l l vintages of machinery and implements. The construction of the data series, particularly for the flow of capital services and the derivation of the equivalent stock of machinery and implements, required seve- r a l additional assumptions. However, these are more conveniently discussed in the following section, which deals specifically with measurement of the variables. The above assumptions are highly restrictive. Assumptions (4) , (6) and (7) were necessary because of the lack of feasible operational alternatives. Assumption (5) i s a serious deficiency in this study, since i t i s unreason- able to expect that the productive quality of these inputs has remained unchanged over the entire period, 1935-65. However, within the context of the general approach of this study, alternative methods for measuring embodied techno- logical change in more than two factors are not available. In a recent study of technological change in Canadian agri- culture, 1946-65, L i empirically investigated the hypotheses of constant returns to scale, Hicks-neutral disembodied technological change and unitary elas t i c i t y of substitution between capital and labor. He did not find any s t a t i s t i c a l evidence which would reject any of these hypotheses.3 How- ever, since this study covered a longer time period, a dummy variable was devised to investigate whether the para- meters and coefficients of the production function changed significantly during the time period under study. The assumption of constant returns to scale was also relaxed in an alternative model. Therefore, i t was possible to compare the regression results under assumptions of economies or diseconomies of scale and constant returns to scale. The specific models used to investigate the s t a b i l i t y of the production function, and to relax the assumption of constant •'Lew-king L i , "Technological Change in Canadian Agriculture," (unpublished Doctoral dissertation, University of Manitoba, Winnipeg, 1968), pp. 76-87. returns to scale are outlined in detail in the following chapter. It i s well known that variations in climatic condi- tions account for substantial year-to-year variation in agricultural production, especially in the output of f i e l d crops. One method of allowing for this involves directly adjusting the f i e l d crops component of output with a weather index. This method has been used by L i , and was used as an alternative in this study. 4 Such a procedure, however, results in the use of an independently calculated measure of potential output to estimate the production function. However, since the production function i t s e l f i s intended to provide an estimate of potential output, i t i s somewhat circular to impose an independently calculated measure of potential output at the outset. 5 Therefore, i t i s more logical to introduce an expression into the production function which would relate potential and actual (observed) output. The basic functional form, ê +SW+hw , used in this study i s similar to the one used by Solow to relate poten- t i a l and actual output in the U.S. economy through the unemployment rate. This particular expression may ^Ibid., pp. 43-44. 5R. M. Solow, "Technical Progress, Capital Formation, and Economic Growth," American Economic Review, 52:77, May, 1962. 6Ibid., p. 78. duplicate the l e f t half of the normal curve, and, a p r i o r i , i t seemed to possess the right general shape. However, as outlined in the following chapter an alternative functional form was also investigated. II . MEASUREMENT OF THE VARIABLES Annual time series data for gross output (Q), weather index (w), labor input (L), and the flow of capital services (K), were required to estimate technological!change. The main data sources were publications of the Dominion Bureau of Statistics. The series on output and capital services were measured at 1935-39 constant prices. This base period was chosen of necessity, since the o f f i c i a l price indexes, which were used as deflators, are constructed on the base period, 1935-39=100. A description of the derivation of the required time series follows. Gross Agricultural Output (Q) Since the specification of the production function included material inputs, the relevant concept of output was gross output rather than a measure of value-added pro- duction. Gross agricultural output consists of three components: (1) cash receipts from the sale of farm products (excluding inter-farm transfers); (2) income in kind; and (3) changes in farm-held inventories of f i e l d crops and 5 3 livestock. Each of these components was further subdivided into f i e l d crops, livestock and livestock products, and forest and maple products which, in turn, were deflated by the appropriate price index. 7 Total cash receipts, income in kind, and inventory changes for livestock and products, and for f i e l d crops were'deflated by the animal products and f i e l d products components, respectively, of the Canadian farm products price index.^ Total cash receipts and income in kind from forest and maple products were deflated by the lumber and timber component of the general wholesale price Q index. Gross output at base period prices was obtained as the sum of the deflated livestock, f i e l d crops, and forest and maple products output. One variant of the gross output series was constructed by dividing the total f i e l d crop component by the weather index described in the following section. This had the effect of increasing gross output in years of unfavorable 'For a detailed description of the various income components see data source: Canada, Dominion Bureau of Statistics, Handbook of Agricultural Statistics. Fart I I : Farm Income - 1 9 2 6 - o " 5 [Ottawa: Queen's Printer, 1 9 6 7 ) • Price index data source: Canada, Dominion Bureau of Statistics, Prices and Price Indexes (Ottawa: Queen's Printer, various issues)• 9 Price index data source: Ibid. This was not the most ideal price deflator, but a more suitable alternative was not available. weather conditions, and reducing the measure of output in years of better than average weather conditions. The data series for gross output, Q, and gross output adjusted for weather influences, Q*, are shown in Table V, columns 2 and 3 , respectively. Weather Index (w) This variable was designed as a proxy for environ- mental effects on agricultural output. The concept used to construct the index was suggested by S t a l l i n g s . ^ He employed time series of crop yields from experimental plots where as many variables as possible were held constant. A trend was estimated to account for changing s o i l f e r t i l i t y and seed quality over time. The crop yield variation about the estimated trend provided an indication of the year-to- year influence of weather on yields. Such a method in- volves two basic assumptions: (1) a l l variations in yield due to non-weather influences not correlated with weather are randomly and normally distributed with an expected value of zero; and (2) the trend of yields i s linear, and can be removed by the simple regression of yield on time. In view of the nonavailability of suitable experi- mental plot data, and the very aggregated nature of this x uJames L. Stallings, "A Measure of the Influence of Weather on Crop Production," Journal of Farm Economics. 43:1153-59, December, 1961. study, average annual yields were used to construct the index. In this respect the method used in this study differed from that suggested by Stallings. The significance of this depar- ture in method i s that additional factors influencing yields, which could be held constant in an experimental plot, were included in the weather index developed for this study. For example, the weather index may include such non-weather influences as annual variations in seed and f e r t i l i z e r application, cultural practices, and crop damage by pests. The weather index, therefore, i s an "ex post" measure of a l l influences on crop yields after removal of the long-term linear trend. However, this did not seem to be a serious limitation for purposes of this study. F i r s t l y , i t i s reasonable to assume that the effects of technological change on crop yields were accounted for by the estimated linear trend. Therefore, the weather index would not remove the effect of technological change which this study attempted to measure. Secondly, since the correlation of the weather index with the other explanatory variables, labor and capital, was negligible, the weather index would not ex- plain any of the annual variation in output properly attributable to changes in labor and capital inputs. The actual weather index was constructed by estim- ating a simple regression of the form y=a+bx for each crop considered, where y i s the average yield in bushels per acre, and x i s a time index representing each of the years 1935-65. The crops considered were a l l wheat, oats, and barley in each of the Prairie Provinces. A weather index was computed for each crop in each province from the regression results by dividing the observed yield by the predicted yield value. The nine individual indexes were then combined into a single aggregate weather index by weighting each according to their value of production as a proportion of the total value of production of a l l three crops in the three provinces. x ± The aggregate weather index i s shown in Table V, column 4. Labor (L) Labor input was measured in man-years on the basis of the number of persons employed annually in agriculture as reported in the Labour Force Survey. The number of persons employed includes those paid and unpaid, fourteen years of age and over. Regular quarterly and monthly surveys were not initiated u n t i l 1945• Prior to this, only annual estimates based on the number of persons employed at the beginning of June are available. There- x xSource of yield and value of production data for 1935 -62 : Canada, Dominion Bureau of Statistics, Handbook of Agricultural Statistics. Part I: Field Crops (Ottawa: Queen's Printer, 1964J; and for 1963-65: Canada, Dominion Bureau of Statistics. Quarterly Bulletin of Agricultural Statistics (Ottawa: Queen's Printer, various issues). fore, to ensure a continuous and comparable series, the employment estimates used for the years 1946-65 were the June estimates rather than the annual averages. This i s a potential weakness in that agriculture i s characterized by considerable seasonal fluctuation in employment. How- ever, a comparison of the June estimates with the annual averages for the 1946-65 period shows that, while the June estimates were slightly larger in magnitude, the trend and year-to-year changes diverged very l i t t l e . Although the Labour Force Survey i s the only source of employment data for agriculture, the estimates have severe limitations which must be recognized. In addition to the sampling error of the survey i t s e l f , a simple measure such as the number of persons employed f a i l s to take into account the changing structure and quality of the agricultural labor force. The average hours of work per week have been declining. Therefore, other things being equal, a simple measure of labor input such as the number of persons employed would be biased upwards in the later years. The age and sex composition of the labor force i s ignored. The proportion of people employed who are in the younger age groups has been declining. Therefore, to the extent that older workers are.less productive because of their age, the labor input estimates could be biased upwards in the later years. On the other hand, however, i t i s very l i k e l y that the quality of the labor force has improved over time, as a result of improved health stand- ards and s k i l l s . This would result in a downward bias in the measurement of labor input. Ideally, the labor input series should have been adjusted for these influences. How- ever, for purposes of this study such adjustments were not attempted because of the lack of appropriate information on the relationships involved. Any adjustments made would tend to be highly arbitrary and, therefore, questionable. Moreover, the effects of the various influences are to some extent offsetting. L i attempted to account for the changing age and sex composition of the agricultural labor force by developing 12 the concept of a man-equivalent. However, a comparison of L i f s data with the labor input series used in this study for the 1946-65 period reveals that the average annual per- centage decline in the two series differed by less than 0.2 per cent, and the year-to-year movements were very similar. This was interpreted as an additional indication that, given data limitations, detailed adjustments to the labor input series were not j u s t i f i e d . Thus, the measure of annual labor input used in this study was the June estimate of the total number of L i , op_. c i t . . p. 45* 59 persons employed i n a g r i c u l t u r e . 1 ^ The series i s shown i n Table V, column 5« Flow of C a p i t a l Services (K) Measurement of the flow of c a p i t a l services required s i x data seri e s at base period p r i c e s : (1) quantity of l i v e - stock and poultry on farms; (2) stock of buildings; (3) amount of b u i l d i n g depreciation; (4) amount of land input; (5) stock of machinery and implements; and (6) quantity of material inputs. The method of aggregating these compo- nents into a single measure of the flow of c a p i t a l services i s outlined following a b r i e f discussion of the derivation of each s e r i e s . Livestock and poultry. The value of l i v e s t o c k and poultry on farms at base period prices was derived by d i v i - ding the current value of l i v e s t o c k and poultry by the animal products component of the Canadian farm products price i n d e x . 1 4 The r e s u l t i n g series i s shown i n Table V, •^Source fo r 1935-45: Canada, Dominion Bureau of S t a t i s t i c s , Canadian Labour Force Estimates, 1931-45. Refe- rence Paper No. 23 (Ottawa: Queen's Printer, 1958); and f o r 1946-65: Canada, Dominion Bureau of S t a t i s t i c s . The Labour Force (Ottawa: Queen's Printer, various i s s u e s ) . 1 4 C u r r e n t values of l i v e s t o c k and poultry data source: Canada, Dominion Bureau of S t a t i s t i c s , Quarterly B u l l e t i n of A g r i c u l t u r a l S t a t i s t i c s (Ottawa: Queen's Printer, various i s s u e s ) . Price index data source: Canada, Dominion Bureau of S t a t i s t i c s , Prices and Price Indexes (Ottawa: Queen's Printer, various i s s u e s ) . 60 column 7. Buildings. Since the value of the stock of buildings was not available separately, i t was necessary to derive the series from published estimates of gross investment and the depreciation figures which are discussed below.x^ The net stock of agricultural buildings at base period prices in year t+1 was defined as the net stock in year t plus gross investment in year t+1 minus depreciation in year t, a l l at base period prices. Gross investment at base period prices was obtained by deflating the current dollar e s t i - mates of gross investment by the building materials component of the price index numbers of commodities and services used by farmers. The real net stock of buildings series i s shown in Table V, column 8. Building depreciation. The published estimates for building depreciation do not include depreciation on ''New construction (gross investment) data source for 1935-48: 0. J. Firestone, Private and Public Invest- ment in Canada 1926-1951, Department of Trade and Commerce Tottawa: King's Printer, 1951), p. 154; and for 1949-65: Canada, Dominion Bureau of Statistics, Private and Public Investment in Canada. Outlook and Regional Estimates (Ottawa!! Queen's Printer, various issues). 1 6 Price index data source: Canada, Dominion Bureau of Statistics, Price Index Numbers of Commodities and Services Used by Farmers (Ottawa: Queen's Printer, various issues). buildings located on rented farms. Therefore, i t was necessary to adjust the published estimates. 1 7 It was assumed that the average value of buildings on rented farms, part owner - part tenant farms, and owner operated farms was equal. It was further assumed that one-half of the farms classified as part owner - part tenant had buildings on the rented portion. Using these assumptions, an annual adjustment factor was defined as the percentage of total farms operated by tenants plus one-half of the percentage of part owner - part tenant farms. The adjustment factor was calculated from Census of Agriculture data for census years, and values for intercensal years were interpolated. The building depreciation series adjusted to include rented farms was then obtained by dividing the published series by one minus the adjustment factor, which increased the published estimates by approximately fifteen per cent. To arrive at depreciation at base period prices, the adjusted series was deflated by the building materials -I d price index. The f i n a l series i s shown in Table V, column 9. 'Source of published estimates: Canada, Dominion Bureau of Statistics, Handbook of Agricultural Statistics. Part II: Farm Income - 1926-63 TTJttawa: Queen's Printer, 1967). 18 Source of price index data: Canada, Dominion Bureau of Statistics, Price Index Numbers of Commodities and Services Used by Farmers (Ottawa: Queen's Printer, various issues). Land. The total agricultural land area in Canada increased by less than seven per cent from 1931 to 1966. In fact, between 1941 and 1961 there was a slight decline in total agricultural land area. However, the ratio of improved to unimproved land increased substantially from 1.1 in 1935 to 1.6 in 1965. The land input series devel- oped for this study was an attempt to account for the shift towards improved land. For census years the acreage of improved and unimproved land was taken from the Census of Agriculture, and estimates for the intercensal years were interpolated. The value of total agricultural land at base period prices was then calculated by multiplying the acreage estimates by the average value per acre during 1935-39 for improved and unimproved land, respectively.^ The resulting series i s shown in Table V, column 10. Material inputs. Material inputs refers to the goods and services other than durable capital which are purchased from the non-agricultural sector of the economy and con- sumed in the process of production. The material input The average values per acre for 1935-39 were deri- ved by dividing the total value of land and buildings (less the value of buildings as derived for this study) by the number of acres. This resulted in a 1935-39 average value per acre of $19.60 and $4«20 for improved and unimproved land, respectively. Source of total value of land and buildings data: Canada, Dominion Bureau of Statistics, Quarterly Bulletin of Agricultural Statistics (Ottawa: Queen's Printer, various issues)• series at base period prices was derived as the sum of six items: (1) total machinery expenses (excluding machinery repairs) deflated by the price index for gasoline, o i l , and grease; (2) f e r t i l i z e r and lime expense deflated by the price index for f e r t i l i z e r ; (3) feed expense deflated by the price index for feed; (4) machinery repair expense deflated by the price index for farm machinery; (5) build- ing repairs (adjusted to include buildings on rented farms by employing the adjustment factor outlined above under building depreciation) deflated by the building materials price index; and (6) the sum of other crop and livestock expense, el e c t r i c i t y , telephone, and miscellaneous expense 70 deflated by the price index for hardware items. This procedure resulted in a time series of material inputs at base period prices which embodies no technological change, that is,j2>s=0. The basic series i s shown in Table V, column 12. Material input series embodying technological change at various rates were derived from the basic series by assuming that technological change occurred at a constant * uSource of a l l material inputs expense data: Canada Dominion Bureau of Statistics, Handbook of Agricultural Statistics, Part II, Farm Income - 1926-55 (Ottawa: Queen's Printer, 1967). Source of a l l price index data: Canada, Dominion Bureau of Statistics, Price Index Numbers of Com- modities and Services Used by Farmers (Ottawa: Queen's Printer, various issues)• 64 annual rate of 100^ per cent. Augmented material input series for various values of ^ were easily derived by multiplying each observation, n, of the$=0 series by (1+P)n where n=0, 1 , 2, 30. To test the sensitivity of the method of estimating embodied technological change to the price indexes used to deflate the current dollar value of material inputs, an alternative series for material inputs was derived by deflating the total current dollar value of material input expenses by the general wholesale price index. 2 1 The general wholesale price index was chosen for this purpose because i t i s often used as a reference level against which to com- pare the movements of other price indexes. The alternative series for material inputs when^=0 i s shown in Table V, column 13 . Based on this alternative, material input series embodying varying rates of technological change were constructed in the manner outlined in the preceding para- graph. Stock of machinery and implements. Since a purpose of this study was to measure embodied technological change in machinery and implements, knowledge about the vintage composition of the stock of machinery and implements was ^Source of wholesale price index data: Canada, Dominion Bureau of Statistics, Prices and Price Indexes (Ottawa: Queen's Printer, various issues)• required. Assuming that machinery and implements purchased in any year were 100 A per cent more productive than those purchased in the preceding year, the equivalent stock of machinery and implements was defined as the sum of the surviving machinery and implements of different vintages, after weighting each vintage by the appropriate rate of embodied technological change. Thus, derivation of the equivalent stock of machinery and implements embodying tech- nological change at an annual rate of 100 ?\ per cent required a time series of past gross investment measured in volume terms, that i s , at base period prices, and knowledge about the service l i f e of machines and implements, that i s , know- ledge about the rate of replacement of old investment goods. Gross investment at base period prices was obtained by deflating the current dollar estimates of gross investment (Table II, column 2) by the farm machinery component of the price index numbers of commodities and services used by farmers. 2 2 The resulting series of annual gross investment at base period prices for 1921-65 i s shown in Table II, column 3. To test the sensitivity of the method of measur- ing embodied technological change to the price deflator, gross investment was also deflated by the USDA index of Source of price index data: Canada, Dominion Bureau of Statistics, Price Index Numbers of Commodities and Services Used by Farmers (Ottawa: Queen's Printer, various issues). 66 TABLE II GROSS INVESTMENT IN MACHINERY AND IMPLEMENTS, CANADIAN AGRICULTURE, 1921-65 Gross investment at 1935-39 prices Year Gross Deflated by- Deflated by investment DBS price adj. USDA (1) at current prices* index price index (2) (3) (4) thousands of dollars 1921 59140 53088 50590 1922 28725 31952 30237 1923 42240 45468 42753 1924 32105 31353 31261 1925 33790 34515 33489 1926 69200 70902 68719 1927 88600 90872 87377 1928 116200 119057 115164 1929 100300 102872 99405 1930 72000 74227 72217 1931 26700 28135 26202 1932 23400 24867 22180 1933 15500 16830 15784 1934 30600 32347 32797 1935 34400 36021 35318 1936 44200 45194 44964 1937 62900 64712 62837 1938 67400 64745 648O8 1939 63000 60811 59886 1940 82600 78072 74684 1941 88400 81027 78929 1942 71500 62500 60287 1943 39400 33646 32059 1944 72900 61675 57949 1945 90400 78540 71069 1946 131600 110774 104527 1947 214900 170150 159067 1948 280500 198093 178095 1949 352395 222612 193517 1950 389640 236002 197286 1951 423065 226480 206072 1952 449805 230197 227980 1953 440255 223821 220017 1954 296050 149596 148993 1955 323745 162850 160747 1956 371495 177409 176902 67 TABLE II (continued) Ye ax investment Gross Gross investment at 193?-39 prices Deflated by Deflated by DBS price adj. USDA (1) index price index (3) (Jtl thousands o f d o l l a r s 1957 1958 1959 I960 1961 1962 1963 1964 1965 325655 347620 410650 423065 389640 456490 547215 612155 683780 145512 151750 146861 153339 165318 175867 166430 174532 149059 150266 170269 163969 200519 191401 216940 209499 240007 227547 *Source of gross investment data for 1921-25: Kenneth Buckley, Capital Formation i n Canada 1696-1930 (Toronto: University of Toronto Press, 1955), pp. 131-32; for 1926-46: 0. J. Firestone, Private and Public Invest- ment in Canada 1925-1951. Department of Trade and Commerce, TOttawa: King's Printer, 1951); and for 1949-65: Canada, Dominion Bureau of Statistics, Private and Public Invest- ment in Canada, Outlook and Regional Estimates (Ottawa: Queen's Printer, various issues). farm machinery prices paid by farmers. * Before the USDA index was used as a deflator, however, i t was converted to a 1935-39 base and adjusted for the changing value of the U.S. dollar in Canadian funds. Gross investment as deflated by the adjusted USDA index i s shown i n Table II, column 4. •'Source of price index data: United States Depart- ment of Agriculture, Agricultural Statistics (Washington: Government Printing Office, various issues). Four basic hypotheses about the rate of replacement of old investment goods have been used in total factor pro- ductivity studies: (1) accounting depreciation i s set equal to replacement; (2) gross investment in some earlier period i s set equal to replacement; (3) a weighted average of past investment with weights derived from studies of the survival curves of individual pieces of equipment i s set equal to replacement; and (4) each investment generates a series of replacement investments over time. 2 4 For purposes of this study, hypothesis (2) was adopted, and the stock of machin- ery and implements was measured by assuming a service l i f e of thirteen years after which the machine or implement i s discarded. 2 5 Thus, the stock of machinery and implements i s a thirteen-year moving sum of past gross investment at base period prices. This method i s analogous to that used by Hood and Sc o t t , 2 o and to that suggested by G r i l i c h e s . 2 7 2 4D. W. Jorgenson and Z. Griliches, "The Explanation of Productivity Change," Review of Economic Studies.34:255, July, 1967. 25 service l i f e of thirteen years was also used by Hood and Scott. See: W.C. Hood and A. Scott, Output, Labour, and Capital in the Canadian Economy, Royal Commission on Canada's Economic Prospects (Ottawa: Queen*s Printer, 1957), p. 473. 2 6 I b i d . , pp. 234-37. 27 'Z. Griliches, "Measuring Inputs in Agriculture: a C r i t i c a l Survey," Journal of Farm Economics. 42:1417, December, I960. The equivalent stock of machinery and implements in year t, J(t ) , was defined as J ( t ) - £ (i+^)vKv) v«=t-12 where I(v) i s the amount of gross investment in year v at base period prices, For<\*=0, the stocks of machinery and implements derived from gross investment deflated by the DBS price index and the USDA price index are shown in Table V, columns 14 and 15, respectively. Two assumptions are inherent in this method: (1) a l l machinery and implements have the same service l i f e which is constant over time; and (2) gross investment at base period prices i s an unbiased measure of the quantity of machinery and implements actually brought into production during any given year. Implicit in the assumption of a common service l i f e for a l l machinery and implements i s the condition that the service lives of various kinds of machinery and implements can be averaged into one repre- sentative figure, and that annual gross investments consist of constant proportions of the various kinds of machinery and implements.* In order to provide an indi- cation of the sensitivity of the method to assumptions about the service l i f e of machinery and implements, stocks of machinery and implements based on alternative assump- Hood and Scott, 0 £ . c i t . , p. 239. tions about the service l i f e and rate of replacement of old investment goods were also constructed. 29 However, the results of the regression models using the various alter- natives followed the same pattern, and differed only slightly in degree from the results using the basic assump- tion of a thirteen-year service l i f e . 3 0 The issue involved in the assumption that gross investment at base period p r i - ces i s a measure of the quantity of machinery and imple- ments actually brought into production, i s the separation of the values of transactions in new investment goods into a price and a quantity component. This i s commonly achieved by deflating current dollar estimates by an appropriate price index. However, an error in this separ- ation w i l l affect the magnitude of the flow of capital services, and result in a biased measure of technological 29The alternatives were: (1) a fifteen-year sum of past gross investment; (2) a service l i f e function, l-(t/15), which allows for a constant proportion of gross investment in any year to be discarded in each successive year; (3) a service l i f e function, l - ( t 2 / l 5 2 ) , which allows for an increasing proportion of gross investment in any year to be discarded; and (4) a service l i f e function, 1-(>/t/ -715), which allows for a decreasing proportion of gross investment i n any year to be discarded in each successive year. 30 ^ Since the regression results based on the alter- native assumptions did not affect the conclusions of the study, they are not reported. change. Aggregation of the flow of capital services. The annual flow of capital services from the stocks of land, buildings and livestock was derived by assuming an annual rate of return on the investment at base period prices. The flow of capital services from the stock of buildings also included depreciation. A method suggested by Griliches was employed to convert the stock of machinery and implements into an annual flow of services. 3 2 Assum- ing that there i s no deterioration with age, and that the flow of services i s constant over the l i f e span of a l l machines and implements, then the annual flow of services equals an annuity for the service l i f e at the rate of return. As Griliches points out, under these assumptions the annuity equals the sum of annual interest and depre- ciation charges, with the interest charges f a l l i n g and the depreciation charges ri s i n g as the machine ages.33 Thus, the annual flow of capital services derived for this study was the arithmetic sum of four components: (l) the value of livestock, plus the value of land, plus 31 This also applies to a l l of the other variables in this study where the quantity has been measured by deflating current dollar estimates by a price index. 3 2Griliches, loc. c i t . Ibid. the stock of buildings, a l l at base period prices multiplied by the rate of r e t u r n ; 3 4 (2) building depreciation at base period prices; (3) material inputs at base period prices; and (4) the stock of machinery and implements at base period prices multiplied by the factor for a thirteen-year annuity at the rate of ret urn. 3 5 3 4 An annual rate of return of six per cent was assumed. An alternative rate of eight per cent was also employed. However, the regression results using an eight per cent return followed the same pattern and differed only slightly in degree from the results using the basic assumption of a six per cent return. The alternative results are not reported. •^The factor for a thirteen-year, six per cent annuity i s 0.11296. CHAPTER IV EMPIRICAL RESULTS I. INTRODUCTION The multiple regression routine of the "UBC TRIP" computer program was used to provide least-squares regres- sion estimates. x The basic model was estimated in natural logarithmic form ln(Q/L)=lnA+ut+f+gw+hw2+ln(K/L)a+ln ^ whereJJU i s a disturbance term about which the usual assump- tions were made.2 The output of the program included: (1) the estimated regression coefficients; (2) the standard errors of each regression coefficient; (3) the F-ratio and associated probability for each regression coefficient;3 (4) the standard error of the estimate, S; (5) the coeffi- X J . H. Bjerring and R. H. Hall, UBC TRIP (Triangular Regression Package) (Vancouver: University of Br i t i s h Columbia, Computing Centre, 1968). The usual assumptions are that/t-are random vari- ables with zero expectation and constant variance, and are pairwise uncorrelated. See: J. Johnston, Econometric Methods (New York: McGraw-Hill Book Company, Inc., 1963), p. 107. 3 ^The F-ratio i s equivalent to the more common " t " test. See: Ibid.. pp. 123-25. The associated probability i s the level at which the estimated regression coefficient i s significantly different from zero. See: Bjerring, op. c i t . . pp. 48-49. 74 cient of multiple determination, R2; and (6) the Durbin- Watson d s t a t i s t i c , "d". There are two constant terms, InA and f, in the above model. However, only one constant term, lnA*=lnA+f, was estimated. If desired, InA* can be separated by noting that actual output must equal potential output when the weather index equals one, which implies f+g(1.0)+h(1.0)2=0. Therefore, f can be determined, and thus InA, from estim- ates of g and h. A l l other coefficients are presented i n the same units as they appeared in the models. The method of determining the rate of embodied tech- nological change in machinery and implements and in material inputs, ^ and respectively, involved estimating a regres- sion equation for each possible combination of the a pr i o r i imposed values for r\ andfb. Thus, i f m alternative values of A and n alternative values of ^ were imposed, there would be mxn possible combinations resulting i n a mxn matrix of estimated equations. The regression estimates for the various models are presented i n the Appendix.4 For each equation the estimated regression coefficients, R2, standard error of the estimate (S), and the Durbin-Watson 4Each page of a particular table shows the regression estimates for several values of £ given a specific value of (\, that i s , the regression estimates for an individual equation are shown as a column with the column heading and table heading identifying the ^ and J\ value, respectively. d s t a t i s t i c ("d") are shown. A l l values of "d" are insigni- ficant (that i s , reject the hypothesis of serial correlation) at the one per cent level of significance unless otherwise noted. The standard errors of the regression coefficients are shown in parentheses. The associated probability of the F-ratio for each coefficient i s shown immediately below each standard error. The results of five models are reported. Model I was the basic model outlined in Chapter III. In Model II a l l technological change was assumed to be embodied in machinery and implements and material inputs. The assumption of constant returns to scale was relaxed in Model III, while two alternatives for relating actual and potential output were investigated in Model IV. Finally, Model V was an attempt to assess the s t a b i l i t y of the production function over time by introducing a dummy variable into the model. II. RESULTS OF THE BASIC MODEL Three alternative sets of data were employed to e s t i - mate the production function Q=Ae u te f +S w + h w 2L 1- aK a. The regression results for each alternative set of data are discussed individually as Models 1(a), (b) and (c). Model 1(a) (Table VII) In this model, material inputs were deflated by the individual price indexes (Table V, column 12), and gross investment in machinery and implements by the DBS price index (Table V, column 14). ^ values ranging from 0.0 to 0.08 were used to construct alternative series for material inputs, thus embodying technological change at the corres- ponding annual rates of 0.0 to 8,0 per cent. Likewise, r\ values ranging from -0.01 to 0.03 were used to construct a l - ternative stocks of machinery and implements embodying technological change at the corresponding annual rates of -1.0 to 3.0 per cent. The matrix of regression results for the various combinations of j\ and ^ are shown in Table VII. (Table VII consists of nine pages; each page reports the results for a specific Rvalue). 2 Table III shows the R values and standard errors of the estimate for a l l combinations of ft and ^. It i s evident from Table III that: (1) for a given A value, R 2 . increased and then decreased as the value of ^ increased; and (2) for a given ^ value, R 2 decreased as the value of A increased. Some individual numerical results are summarized as follows: ft value ^ value Highest R 2 (.9877) -0.01 0 . 0 3 5 Lowest R 2 (.9847) 0 . 0 3 0.0 77 PQ EH CO o Q 9 19 CO # Q OS S .-3 o w M n EH O M « § O EH Q D-. CO H fx) EH S EH o o CO EH B o M Cm o -co o o o o L A o o L A C A o CA o L A cv o cv o O O to L A O O H CA OvO t o ^ O O O A t o O L A tO L A o o L A - * O L A t o L A ONO C—.—I O L A •00 L A vO CV OLA - 0 0 » A O O o I tO O N •CO L A ONO to o vO O tO L A ON O o c v MD O -CO L A ONO O C A o o tO L A ONO VO L A voo •00 L A ONO -3-tO o O tO L A ONO O L A 0\0 -CO L A ONO o o OvO tO L A ONO ON ON tO L A ONO O H vO O tO L A ONO vO vO O "CO L A ONO -4 -C0 vO O tO L A ONO cv cv OvO tO L A ONO cH -4 -OvO tO L A ONO O O v O v O tO L A ONO t o H sOO tO L A ONO vO L A vO O tO L A ONO <r\0 t o tO L A ONO 0AO 0\0 tO L A ONO H C A OvO tO L A ONO O O vOvO tO L A ONO O CV vO O t o L A ONO L A tO vO O tO L A ONQ CVoA vQtO tO L A ONO xO CV OAO OLA OvO tO LA tO LA ONO ONQ ovo tO LA ONO t o o vO O tO L A ONO vOvO vO o tO L A ONO OACV \ o - c a t O L A ONQ ON ON L A tO tO L A ONQ OAO OvO tO L A ONQ OM3 O^O tO L A ONO OCV vO O tO L A ONO - 4 - t o vO O tO L A O O H L A \OtO tO L A ONQ t o H LAOS tO L A ONO CM CV 0\0 - 0 0 L A ONQ ON tO \OvO tO L A ONQ MD O t O L A ONQ C A rH vOtO tO L A ONQ o t o vOtO t o L A ONO vO CV rH -3- O - L A OvQ tO L A tO L A ONO o o t o o vO O tO L A O O LAO vO O tO L A ONO CV -4 -vOtO tO L A ONO O O L A ON tO L A ONO L A - 4 " OvO O L A 0 0 tO L A tO L A ONO ONQ OCA vO O tO L A O O - 4 - 0 vOtO t o L A O O OvO vOtO tO L A O O O C A L A ON tO L A ONO -*o L A O tOvO O O H OA LAO C-vO vO O tO L A tO L A O O ONO H O H -4" vOtO t o L A ONO L A O o t o iH L A ON tO L A ONO o L A O L A ON tO L A ONO L A rH O C V O A L A O t0\O ONO CV o o t o LAO tOvO O O L A CV o OA i - l vOtO •CO LA o o CV CV N O t o tO L A o o H I A vOtO tO L A o o o o U N O tO L A o o vO L A L A O tO L A o o L A tO L A O tO L A o o O-d- CAr—I LAO LAO tO LA tOvO o o O O L A O CV C A L A O L A O tO L A tO vO o o o o H\0 l A O tOvO O O O C A - 4 H tOvO o o CA o CO d) •p a u ft a •H o X! co a> % <D CO s •H •p CO a> a> £1 -p o CO u o u u - a u T5 p CO 78 ft value ^ value Lowest S (.0551) - 0 . 0 1 0.035 Most significant u (0.35 per cent level) 0.03 0.0 Most significant a (0.39 per cent level) - 0 . 0 1 0.035 Thus, the "best" regression based on the c r i t e r i a of goodness of f i t , low standard error of the estimate and significance level of the a coefficient occurred when ^= -0 .01 and ^=0.035. This implies embodied technological change in material inputs at the annual rate of 3»5 per cent, and in machinery and implements at the negative annual rate of 1.0 per cent. A negative rate of embodied technological change in machinery and implements was disturbing and contradicted a p r i o r i expectations. This should most l i k e l y be considered as a spurious result for various reasons which are discussed below. The specific negative value of ft which would have given the "best f i t " was not determined, since in view of the questionable nature of the results, the model was not extended to include higher negative values for A . When fWO.O, that i s , no embodied technological change in machinery and implements, the "best" regression (R =.9873) occurred when $=0,035 to 0 . 0 5 . The a coefficient was most significant when $=0.04. This indicates embodied technolo- gical change i n material inputs at an annual rate of approximately 4*0 per cent. These results, therefore, sup- port the hypothesis of substantial embodied technological 79 change in material inputs. The u coefficient, which i s an estimate of the annual rate of disembodied technological change (100 u), had the largest value and was most highly significant in the regres- sions when (*<=0.0. This was the expected result, because i f embodied technological change i s not specified, then a l l increases in total factor productivity would show up as disembodied technological change. However, with respect to ft, the a p r i o r i expected results did not occur. The largest and most significant value of u occurred when =0.03, rather than when ft=0.0. This was further evidence which suggested that the results with respect to embodied technological change in machinery and implements must be regarded as sus- pect. When no embodied technological change was specified (ft=^=0.0), u=0.0246 indicating an annual rate of disembodied technological change of 2.46 per cent. In this regression the u coefficient was highly significant (1.44 per cent level) with a relatively low standard error (.0095)* When (\=0.0 andf-0.04 (the "best" regression), u=0.0113. Thus, when embodied technological change was specified the e s t i - mate of disembodied technological change declined, which supports the conclusion that a large portion of increases in total factor productivity can be attributed to techno- logical change embodied in material inputs. It should be 80 noted, however, that the estimates of u became increasingly imprecise and less significant as >̂ increased. The a coefficient was highly significant with low standard errors in a l l regressions. The g coefficient was not significant, while h was significant at about the 15 per cent le v e l . Although the test for serial correlation was inconclusive for some regressions, seria l correlation was not considered to be a problem. Model Kb) (Table VIII) This model was an attempt to assess the sensitivity of the method to an alternative deflator for gross invest- ment in machinery and implements. In this model gross investment was deflated by the adjusted USDA price index (Table V, column 1 5 ) . A matrix of regression results was obtained for the various combinations of ^ and $ as outlined for Model 1(a) above. The regression results when/l̂ O.O are reported in Table VIII. The regression results of Model 1(b) exhibited the same trends as Model 1(a) with only slight differences i n the magnitudes of the estimated coefficients and statis- t i c a l measures. The main results are summarized as follows: Highest R 2 ( .9879) Lowest R 2 ( . 9 8 5 3 ) Lowest S ( .0547) value ^ value - 0 . 0 1 0 . 0 2 5 - 0 . 0 3 5 0 . 0 3 0 . 0 - 0 . 0 1 0 . 0 2 - 0 . 0 3 5 ft value Rvalue Most significant u (0.73 per cent level) 0.03 0.0 Most significant a (0.31 per cent level) - 0 . 0 1 0.025-Q.035 Thus, the general conclusions drawn from the results of Model 1(a) are applicable to this model as well. In view of the questionable nature of the results with respect to ft, only the regressions in which ft=0.0 are reported. When ft=0.0, the "best" regression occurred when $=0,035, although R2=.9875 in a l l regressions when =0.025 to 0 . 0 5 . Thus, this model gave a slightly lower estimate of embodied technological change in material inputs (approx- imately 3«5 per cent compared with 4.0 per cent in Model 1(a)). The u coefficient was almost identical in Models 1(a) and (b), (0.0113 compared with 0 .0115) . The a coefficient, however, was slightly more significant in Model 1(b) than in Model 1(a). Thus, the results of Model 1(b) were very similar to those of Model 1(a), which suggests that the method was not particularly sensitive to the alternative deflator for gross investment in machinery and implements. It was con- cluded that the results of Model Kb) did not provide significant evidence for preferring the adjusted USDA price index over the DBS index for purposes of this study. Model 1(c) (Table IX) This model was an attempt to assess the sensitivity 82 of the method to an alternative deflator for material inputs. In this model, a l l material inputs were deflated by the general wholesale price index (Table V, column 13) . A matrix of regression results was obtained for the various combinations of ({ and ^ as outlined for Model 1(a) above. The results when ^=0.0 are reported in Table IX. Like Model K b ) , the results of Model 1(c) showed the same trends as Model 1(a). The main results are summarized as follows: ft value $ value Highest R 2 (.9880) - 0 . 0 1 0.025-0.035 Lowest R 2 (.9856) 0.02 0.0 Lowest S (.0543) - 0 . 0 1 0.025-0.03 Most significant u (0.13 per cent level) 0.02 0.0 Most significant a (0.26 per cent level) - 0 . 0 1 0.025-0.03 In view of the questionable nature of the results with res- pect to ̂ , only the regressions i n which ^=0.0 are reported. When ^=0.0, the "best" regression occurred when f=0.035 to 0 .04 , although R2=.9877 in a l l regressions when $=0.03 to 0 . 0 4 . Thus, this model gave an estimate of embodied technological change in material inputs which was identical to the estimates of Models 1(a) and (b). However, the u coefficient was considerably greater in magnitude (0.0168 compared with 0 .0113) , and was more highly signif- icant (10.72 per cent level compared with 38 .53) , than in Model 1(a). 83 Thus, the results of Model 1(c) were similar to those of Models 1(a) and (b). The most notable difference was the larger and more highly significant estimate for disembodied technological change in Model 1(c). The results with res- pect to however, suggest that the method was not sensi- tive to the alternative deflator for material inputs, and i t was concluded that the results of Model 1(c) did not provide significant evidence for preferring the general wholesale price index over the individual price indexes as a deflator for material inputs for purposes of this study. Discussion of Model I On a p r i o r i grounds, the results of Models 1(a), (b) and (c) with respect to A were unexpected. There are two possible interpretations: (1) the rate of embodied tech- nological change in machinery and implements was, in fact, negative during the 1935-65 period; or (2) the method failed to provide an unbiased estimate of embodied techno- logical change. Although this study did not provide sufficient evidence upon which to base a choice between these explanations, there are strong reasons to suspect that the latter interpretation i s the more plausible and r e a l i s t i c . Several factors could account for a biased or spur- ious estimate of embodied technological change. Fi r s t , there was the problem of obtaining an accurate measurement of the real stock of machinery and implements. Since individual machines and implements are extremely heteroge- neous, they must be aggregated in value terms, and then deflated by a price index to arrive at a measure of the stock in volume terms. 5 There are two alternatives for measuring capital goods in value terms. Capital goods may be valued in terms of input costs, or in terms of their a b i l i t y :to produce (either on the basis of output or capa- c i t y ) . The f i r s t alternative attributes a l l increases in output to changes in the productivity of capital i t s e l f , while the second alternative attributes a l l increases in output to changes in productivity in the production of capital goods. Clearly, neither of these extreme altern- atives was satisfactory for purposes of this study. The crucial problem was the separation of such elements as design improvements and serviceability (embodied technolo- gical change) from changes in the cost of production; that i s , the separation of increases in the productivity of machines and implements from increases in productivity in the production of machines and implements. On the assumptions that the suppliers of machinery and implements 5The following discussion i s largely drawn from: Richard Ruggles and Nancy Ruggles, "Concepts of Real Capital Stocks and Services," Output. Input and Productivity Measure- ment (Studies in Income and Wealth, Vol. 25• Princeton: Princeton University Press, 1961), pp. 3#7-411. do not have monopolistic power in the product markets, and that the price charged continues to reflect chiefly the price of inputs available to the agricultural machinery and implement industry, the appropriate measure of machin- ery and implements for purposes of this study was gross investment deflated by a price index in which quality changes of the machines and implements have been accounted for. However, for purposes of constructing a price index, quality changes are d i f f i c u l t to define and measure. Therefore, i t i s important to note that the price deflator used in this study may have been biased. If so, there would be errors of measurement in the data series for real gross investment in machinery and implements. However, even i f the estimate of real gross invest- ment was unbiased, the d i f f i c u l t problem of measuring the stock of machinery and implements and the flow of services would remain. In this regard, the restrictive nature of the assumptions used in this study i s outlined in Chapter III. Secondly, there was the potential problem of chang- ing rates of u t i l i z a t i o n of machinery and implements over time. This may have been particularly important in this study i n view of the cycles in gross investment. As shown in Table II, column 3, real gross investment reached peaks in 1928, 1952 and 1965, and troughs in 1933 and 1958. This resulted in a stock of machinery and implements which actually declined during 1936-43 and 1959-65, but exper- ienced very rapid growth during 1944-56. Since agricul- tural output maintained a steady upward trend throughout the entire period, either other inputs were substituted for machinery and implements in the short run, and/or the u t i l i z a t i o n rate of machinery and implements was a variable. The model could accommodate substitutability between capital and labor, but i t did not take into account varying u t i l - ization rates for capital. Thirdly, the assumption of a constant rate of embodied technological change in machinery and implements may have been inappropriate. It i s possible that a cyclical pattern during the 1935-65 period may have obscured a long- term trend in the rate of embodied technological change. Fourthly, i t was possible that the time series data did not provide enough independent variation to allocate simultaneously with a high degree of confidence the i n - creases in total factor productivity to three sources - disembodied technological change, and embodied technolo- gical change i n both material inputs and machinery and implements. In fact, there was reason to suspect that the model could not choose between alternative combinations of embodied technological change in machinery and implements and material inputs. For example, the correlation coef- f i c i e n t between the flow of capital services, K, when ft=0.0 and ^=0.02 on the one hand, and when ̂ =0.02 and P=0.0 on the other, was 0.9938 indicating an almost exact linear relationship between the two measures of K. There- fore, the regression results for the two alternatives would not provide sufficient information to use as a basis for choice between the alternative combinations of the rates of embodied technological change. This suggests that the model may not have been capable of estimating the rate of embo- died technological change in more than one factor. Thus, there were many reasons to suggest that the estimate of a negative rate of embodied technological change in machinery and implements may have been substantially biased. III. MODEL WITH ALL TECHNOLOGICAL CHANGE EMBODIED In Model I technological change, both embodied and disembodied, was specified to occur simultaneously. Of course, when /l=̂ =0.0 a l l technological change was assumed to be disembodied. For Model II a function of the form was estimated. Therefore, a l l technological change was assumed to be embodied in material inputs and machinery and implements. A matrix of regression results for the various combinations of ft and (3 was estimated using the as same data series as in Model 1(a). The results whenft=0.0 are reported in Table X. Model II (Table X) The regression results of Model II indicate a posi- tive rate of embodied technological change i n material inputs. However, like Model I, the "best" regression occurred when ft=-0.01, indicating a negative rate of embodied technologi- cal change in machinery and implements. Therefore, for reasons outlined above, only the results when ft=0.0 are reported. A comparison of the results of the "best" regression when ft=0.0 in Model II, with those of Model 1(a), reveals the following points: (1) Model II provided a higher estimate for the rate of embodied technological change in material inputs (5.0 per cent compared with 4.0 per cent); (2) R 2 was higher in Model 1(a) (.9673 compared with .9870); and (3) the elasti c i t y of output with respect to capital, a, was larger (0.5416 compared with 0.4572), more highly significant, and had a substantially lower standard error in Model II than in Model 1(a). The higher estimate of embodied technological change in Model II was expected, since any technological change which was estimated in Model 1(a) as disembodied technolo- gical change was measured as embodied technological change in Model II. The higher R 2 in Model 1(a) provided some evidence that embodied and disembodied technological change should be treated simultaneously, although this evidence must be considered weak i n view of the lower standard error of the estimate, more highly significant a, and lower standard error of a in Model II. As expected, the el a s t i - city of output with respect to capital was higher in Model II, because a l l increases in total factor productivity were assumed to be the result of embodied technological change. The relatively greater indeterminancy of the est i - mates of a in Model 1(a) compared with Model II was most li k e l y due to the very high correlation between the flow of capital services, K, and time, t, in Model 1(a). Of course, this problem of multicollinearity did not arise in Model II, since t was not a specified variable. IV. MODEL TO RELAX THE ASSUMPTION OF CONSTANT RETURNS TO SCALE A l l other models reported in this study were homo- geneous of degree one, that i s , constant returns to scale was assumed. Model III was an attempt to assess the sensitivity of the method to the assumption of constant returns to scale. In Model III labor was introduced into the production function as an independent variable. The function Q = A e u te f + g w f h w 2L bK a was estimated using the same data series as i n Model 1(a). Estimates were obtained for ft=0.0 only, and are reported in Table XI. Model III (Table XI) A comparison of the results of Model III with the results of Model 1(a) (whenA=0.0) shows the following: (1) Model III provided a slightly higher estimate of the rate of embodied technological change in material inputs (4.0 to 5«0 per cent compared with 4»0); (2) Model III gave a substantially higher estimate of the rate of disembodied technological change (1.99 to 1.88 per cent compared with 1.13); (3) the R 2 values were lower i n a l l regressions in Model III (.9513 compared with .9873 in the "best" regres- sion); (4) the a coefficient was substantially lower in Model III, had large standard errors and was not s i g n i f i - cant; and (5) serial correlation in the disturbance terra may have been a problem in Model III, since the Durbin- Watson test was inconclusive in a l l regressions. The relatively large standard errors for u and a were evidence of the problem of multicollinearity i n Model III. The three explanatory variables, t, K and L, were highly correlated. For example, the correlation coefficient between labor and the flow of capital services when A=(*=0.0 was -0.9506. This increased to -0.9676 whenft=0.0 and (5=0.08. In Model III the sum of a+b was less than one, which indicated the possibility of diseconomies of scale in the aggregate production function. However, this result was not s t a t i s t i c a l l y significant.^ In summary, the assumption of constant returns to scale resulted in a slightly lower estimate of the rate of embodied technological change in material inputs, and a more substantially lower estimate of the rate of disembodied technological change. However, there was insufficient s t a t i s t i c a l evidence to reject the hypothesis of constant returns to scale, V. ALTERNATIVE MODELS TO RELATE POTENTIAL AND ACTUAL OUTPUT The principal means for relating potential and actual output used in this study i s outlined in Chapter III. Two alternative forms were investigated. These are discussed individually as Models IV(a) and (b). Model IV(a) (Table XII) As described in Chapter III, a measure of potential output, Q* (Table V, column 3), was derived from actual The F - tes t employed i s out l ined i n : Gerhard Tintner , Econometrics (New York: John Wiley and Sons, I n c . , 1952), pp. 89 -91 . Subs t i tu t ion of the appropriate values gave F=1.27. For one and twenty - f ive degrees of freedom, the c r i t i c a l values of the F d i s t r i b u t i o n are 7.77 and 4.24 fo r the one and f i v e per cent l e v e l s of s i g n i f i c a n c e , r e s p e c t i v e l y . gross output by dividing the f i e l d crops component by the weather index. The production function Q,=AeutL"L"'aKa was then estimated using the same data series as in Model 1(a) for t, L and K. The results whenA=0.0 are reported in Table XII. A comparison of the results of Model IV(a) with the results of Model 1(a) (whenA«=0.0) reveals the following: (1) the "best" regression in Model IV(a) occurred when f=0.G5 compared with f «=0.04 in Model 1(a); (2) the R 2 values were lower and the standard errors of the estimate were larger in Model IV(a) (.9784 and .0675 compared with .9873 and .0560 in the "best" regressions); and (3) the u and a coefficients were very similar in magnitude and level of significance. Thus, directly adjusting gross output for weather influences resulted in a slightly larger estimate of the rate of embodied technological change in material inputs. However, on the basis of goodness of f i t and low standard errors of the estimate, this method was inferior to the method of relating actual and potential output which was used in Models I, II, III and V. Model IV(b) (Table XIII) Another function which seemed a pr i o r i to have the right general shape for relating potential and actual out- 93 put i s the logarithmic, reciprocal function y = e f - s / w . Therefore, the production function Q ^ A e ^ e ^ V " ^ was estimated using the same data series as in Model 1(a). The results when ft=0.0 are reported in Table XIII. Compared with Model 1(a), Model IV(b): (1) gave a lower estimate of the rate of embodied technological change in material inputs (3»0 per cent compared with 4*0); and (2) provided estimates of the rate of disembodied techno- logical change which were lower and also less significant. The Rfc values were lower and the standard errors of the estimate were higher in Model IV(b) compared with Model 1(a). Thus, the use of the logarithmic, reciprocal function to relate potential and actual output resulted in a lower estimate of both the rate of disembodied and embodied tech- nological change. However, on the basis of goodness of f i t and low standard errors of the estimate, this method was also inferior to the principal means for relating actual and potential output used in Models I, II, III and V. VI. TESTING THE PRODUCTION FUNCTION FOR STABILITY OVER TIME Since agriculture experienced significant structural changes during the time period under study, i t was desirable to test whether the estimated regression relationships were stable over time. The technique involved the use of a dummy variable, D, which took the value zero for a l l years prior, to a given date, and the value one for a l l subsequent years. On the basis of the most highly significant dummy variable, the 1 9 3 5 - 6 5 period was divided into two sub-periods, and the hypothesis that the estimated regression coefficients were equal in a l l three time periods was tested by computing the appropriate F-ratio. Model V (Table XIV) Using the same data as in Model 1(a) whenA=(*=0.0, the production function Q = A e u V + ^ h w V - a K V D was estimated for each of the dummy variables, D]_ to D13, l i s t e d in Table IV. A significant regression coefficient for D implies that a significant change occurred in at least part of the relationship from one period to the other. The results of ten regressions with the most highly significant D coefficients, j , are reported in Table XIV. In three regressions, D2, D5 and DD, j was significant at the five per cent level, and in one regression, DD, j was s i g n i f i - cant at the one per cent level. In addition, the regression with the dummy variable D D also had the highest R , lowest standard error of the estimate, and the most highly signi- ficant value for a. Therefore, i t was concluded that a significant shift occurred between the 1 9 3 5 - 4 9 and 1 9 5 0 - 6 5 TABLE IV DUMMY VARIABLES SPECIFIED FOR MODEL Vi : Q = A e u te f + g w f h w 2 L 1- aK ae j D, M-O.O, 1935-65 Dummy variable Zeros Ones D l 1935-40 1941-65 D 2 1935-45 1946-65 D3 1935-46 1947-65 D4 1935-47 1948-65 D 5 1935-48 1949-65 D6 1935-49 1950-65 D 7 1935-50 1951-65 1935-51 1952-65 D 9 1935-52 1953-65 DlO 1935-53 1954-65 D l l 1935-54 1955-65 D 1 2 1935-55 1956-65 D 1 3 1935-60 1961-65 periods in at least part of the true production function relationship. In view of this result, the period under study was divided into two subperiods, 1935-49 and 1 9 5 0 - 6 5 . Model 1(a) was estimated for each subperiod when f\=^=0.0. The 96 regression results are shown in Table XV. The regression results indicate that disembodied tech- nological change occurred at a more rapid annual rate during the 1950-65 period than during the 1935-49 period (3.08 per cent compared with 1.09). In addition, the u coefficient was highly significant in the regression for the 1950 -65 period, but not significant in the regression for the 1935- 49 period. However, the elast i c i t y of output with respect to capital was more highly significant in the regression for the 1935-49 period than in the regression for the 1950- 65 period. To test the hypothesis that the true set of regres- sion coefficients was equal in a l l three time periods, 1935-49, 1950-65 and 1935-65, that i s , that the observations for the 1950-65 period belonged to the same relationship as those for the 1935-49 period, an F-test was performed.7 On the basis of the test, there was insufficient s t a t i s t i - cal evidence to reject the hypothesis of equal sets of regression coefficients for a l l three time periods. This result should not be interpreted as a contra- diction of the results obtained when the dummy variable was 'The test i s outlined i n : Johnston, op_. c i t . , pp. 136-37. Substitution of the appropriate values gave F=2.47. For five and twenty-one degrees of freedom, the c r i t i c a l values of the F distribution are 4.04 and 2.68 for the one and five per cent levels of significance, respectively. introduced into the model. The F-ratio tested the hypo- thesis that the entire set of regression coefficients was equal i n a l l three time periods, that i s , that the regres- sion relationship taken as a whole did not shift s i g n i f i - cantly over time, while the significant regression coefficient for D indicated that a l l or part of the regres- sion relationship shifted between the two subperiods. Therefore, interpreting the results of the F-test and dummy variable together, i t was concluded that only part of the relationship shifted significantly over time. Although i t was not possible to determine precisely which part(s) of the relationship underwent a significant shift, the regres- sion results for the two subperiods (Table XV) suggest that the constant term and u were significantly different in the two periods. Both the constant term and u were almost three times as large with relatively much lower standard errors in the 1950-65 period than in the 1935-49 period. CHAPTER V SUMMARY AND CONCLUSIONS I. SUMMARY Productivity increases in Canadian agriculture over the past three or four decades have been well demonstrated. However, only recently have research efforts been directed specifically towards identifying the sources of the obser- ved productivity gains. The purpose of this study was to identify the kinds and magnitudes of technological change which have contributed to total productivity gains in Canadian agriculture. More specifically, the problem was to measure: (1) the rate of disembodied technological change; (2) the rate of technological change embodied in agricultural machinery and implements; and (3) the rate of technological change embodied in material inputs. Disembodied technological change was defined as a shift in the production function. Therefore, the rate of disembodied technological change was a measure of the effect on output of new technology which can be implemented with reliance on the existing resources or inputs. General improvements in farm management and decision-making, adoption of soil-testing practices, and more efficient feed rations for livestock are obvious examples of disembodied technological change. Embodied technological change, on the other hand, i s a measure of the effect on output of improved technology which must be implemented in conjunction with improved or new kinds of inputs. In particular, an attempt was made to measure the rate of embodied technological change corresponding to changes in the productive quality of machines and implements. Intuitive examples of technology which might be expected to be embodied in machines and implements include: power options, electric starters, improved tillage implements, and entirely new machines such as hay conditioners, side delivery rakes and machinery for handling specialty crops. Similarly, the rate of embodied technological change in material inputs was an attempt to measure the corresponding changes in the productive quality of material inputs. Intuitive examples are improved seed varieties, more effec- tive herbicides and pesticides, improved f e r t i l i z e r s , and feed additives. The analysis was carried out for the aggregate p r i - mary agricultural sector in Canada for the 1935-65 period. To measure technological change, regression estimates were obtained for a linear homogeneous Cobb-Douglas production function where gross agricultural output per person employed was the dependent variable, and a time index, weather index, and the annual flow of capital services (including 100 material inputs) per person employed were the independent variables. The data required was derived from publica- tions of the Dominion Bureau of Statistics, and consisted of time series of thirty-one annual observations. Gross output and the flow of capital services were measured at 1935-39 base period prices, while labor input was measured as the number of persons employed in agriculture. Hicks-neutral disembodied technological change was measured by specifying an exponential shift function, e u t , in the production function. Embodied technological change was assumed to be capital-augmenting in the vintage sense so that machines and implements, and material inputs pur- chased in any year were 100 ^ and 100$ per cent, respec- tively, more productive than those purchased in the prece- ding year. Alternative data series for the flow of capital services were constructed by imposing several values for A and $. A matrix of regression results was obtained, and the true values of r\ and were inferred by choosing the "best" regression on the c r i t e r i a of goodness of f i t , signi- ficance levels of the estimated coefficients and low standard errors. Several alternative models were investigated: (1) a l l technological change was assumed to be embodied; (2) the assumption of constant returns to scale was relaxed; (3) alternative means of relating actual and potential output 101 were investigated; and (4) the st a b i l i t y of the production function relationship over time was tested. Two hypotheses were s t a t i s t i c a l l y tested: (1) constant returns to scale; and (2) equal sets of regression coefficients in the three time periods, 1935-65, 1935-49 and 1950-65. II. CONCLUSIONS Many rigorous assumptions are implied in the models as well as in the derivation of the data series, especially the stock of machinery and implements and the flow of capital services. Therefore, the results obtained must be interpreted carefully. It i s particularly important to note that the rate of technological change was assumed to be constant over time. Therefore, the estimated values must be interpreted as long-term trends. Also, there were undoubtedly errors of measurement in the variables. These errors could arise from several sources: (1) the flow of capital services and gross output were measured i n constant dollars, and there- fore, i t was implicitly (and heroically) assumed that the agricultural sector was in long-run equilibrium; (2) prob- lems of aggregation of economic units (farms) may have been present, since the analysis and data employed were for the aggregate agricultural sector taken as a whole; (3) the ut i l i z a t i o n rate of inputs was assumed constant 102 over time; and (4) aggregation oyer products and inputs may have introduced errors, since non-homogeneous products and inputs were aggregated into single measures of gross output, labor input, and capital input. In addition, there was the problem of aggregating capital inputs which have imputed rates of return and those which have market returns. If errors of measurement were present, and i t seems l i k e l y that they were, a dependence between the disturbance term and the observed values of the explanatory variables would exist, which invalidates one of the basic assumptions of the linear regression model. Thus, the regression estimates would be biased. x In addition, the results of the study are based on the assumption that the model was correctly specified, that i s , that the model was a true expression of the production relationships which actually existed in the aggregate agri- cultural sector. Of course, this i s an untestable assump- tion which can only be qualitatively judged on the basis of conformity with existing knowledge and theory, and whether the results are reasonable on such a pr i o r i grounds. With the preceding qualifications kept in mind, the following main conclusions appear to be j u s t i f i e d . J. Johnston, Econometric Methods (New York: McGraw-Hill Book Company, Inc., 1963), pp. 148-50. 103 1 . This study confirmed that there has been sub- stantial technological change in Canadian agriculture. When a l l technological change was assumed to be disembodied, i t was estimated to be about 2.46 to 2.70 per cent per year. On the c r i t e r i a of goodness of f i t and most significant u coefficient, the "best" estimate of disembodied technolo- gical advance (when embodied technological change was not specified) was 2.70 per cent annually during the period 1935-65. 2. When a l l technological change was assumed to be embodied, the "best" estimate was an annual rate of embodied technological change in material inputs of about 5.0 to 6.0 per cent. 3 . When both disembodied and embodied technological change were specified simultaneously, the estimates of the annual rate of disembodied technological change ranged from 1.13 to 1.76 per cent, while embodied technological change in material inputs was estimated at 3*5 to 4.0 per cent annually. On the c r i t e r i a of goodness of f i t and most significant u coefficient, the "best" estimate of the annual rate of disembodied technological change was 1.76 per cent. It should be noted, however, that in these "best" regressions the disembodied technological change coefficient, u, had a relatively large standard error and was not signi- ficant . 4. There was no evidence of a positive rate of embodied technological change in machinery and implements in any regressions. It was concluded that the disturbing and a p r i o r i unexpected result of a negative rate of embodied technological change in machinery and implements should probably be considered substantially biased, for several reasons discussed. 5. The results of this study suggest that both dis- embodied and embodied technological change should be treated simultaneously. This i s particularly evident when a com- parison i s made between regressions in which a l l technolo- gical change was specified as disembodied (that i s , A«=p»0.0), and those where both disembodied and embodied technological change were specified. In a l l cases the R 2 values were higher and standard errors of the estimate lower when dis- embodied and embodied technological change were specified simultaneously. The evidence i s not as conclusive, however, when the simultaneous case i s compared with the one in which a l l technological change was specified as embodied. Although the R 2 values were slightly higher in the simul- taneous specification, the standard error of the estimate was slightly lower when a l l technological change was assumed to be embodied. 6. There was insufficient s t a t i s t i c a l evidence to reject the hypothesis of constant returns to scale. How- ever, the assumption of constant returns to scale did result in slightly smaller estimates of the rate of both disembo- died and embodied technological change. 7. A comparison of the regression results for the 1935-49 and 1950-65 subperiods strongly suggests that dis- embodied technological change occurred at a more rapid rate during the 1950-65 period. Moreover, this result i s generally compatible with the results of previous studies. However, there was no s t a t i s t i c a l evidence which would re- sult in the rejection of the hypothesis that the regression relationship, when taken as a whole, was equal in the two subperiods. Therefore, the production function as a whole appeared to be stable during the period under study. III. IMPLICATIONS AND SUGGESTIONS FOR FURTHER RESEARCH The macro theory of production i s closely related to the theory of economic growth. This study was an investi- gation of the macro production relationships in Canadian agriculture, and as such i t was an attempt to contribute to the existing knowledge about the sources of increased agri- cultural output per unit of input. In this regard, the single most interesting result was the large role attribu- ted to technological change embodied in material inputs by a l l models investigated. However, there are no direct policy implications which can be drawn from this study. The following suggestions are offered for further research. F i r s t l y , the present models could be refined in several ways. Alternative functional forms such as the CES production function could be investigated. The measurement of capital was particularly troublesome, and much additional research i s needed to improve these estimates. Similarly, the measurement of labor input was extremely rough and the possibility of technological change embodied in labor should be investigated. In addition, i t would be desirable to recognize the regional differences in Canadian agriculture by disaggregating the analysis on a geographical basis. Secondly, i t would be extremely interesting to bring such variables as the rate of research and development into the analysis. L i t t l e i s known about the determinants of tech- nological change, and any meaningful policy variables must focus on the factors which influence the rate of technolo- gical change. Nelson has made some useful observations on this t o p i c . 2 Thirdly, there i s the cost of technologi- cal change relative to i t s benefits. In other words, there i s the d i f f i c u l t and important question of the allocation of resources to technology generating activities from a ''See: M. Brown, The Theory and Empirical Analysis of Production (Studies in Income and Wealth, Vol. 31 . New York: Columbia University Press, 1967), pp. 479-99. 107 social point of view. 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A P P E N D I X 116 TABLE V TIME SERIES DATA FOR CANADIAN AGRICULTURE, 1935-65 Gross output at 1935-39 prices Year (1) Unadjusted (2) Adj. for weather influence (3) Weather index (4) Number of persons employed (5) - thousands of dollars - thousands 1935 714692 742933 0.918 1298 1936 678246 817834 .631 1319 1937 649721 857883 .522 1339 1938 750782 766809 .954 1359 1939 951000 853852 1.251 1379 1940 953247 880568 1.191 1344 1941 858138 897597 .893 1224 1942 1254375 982733 1.705 1139 1943 960235 930630 1.100 1118 1944 1181643 1109792 1.153 1136 1945 954172 1023127 .838 1144 1946 1047257 1052000 .990 1271 1947 1065622 1182537 .804 1172 1948 1118516 1162991 .930 1186 1949 1087086 1277670 .740 1114 1950 1005254 1016709 .975 1066 1951 1164233 1079363 1.210 991 1952 1351133 1092384 1.418 927 1953 1419083 1240862 1.235 911 1954 1117393 1342909 .719 906 1955 1208839 1152614 1.107 880 1956 1355879 1202803 1.258 808 1957 1150249 1224544 .868 777 1958 1182370 1238907 .872 746 1959 1219542 1282430 .892 739 I960 1289950 1294529 .993 690 1961 1137814 1509708 .535 712 1962 1413262 1404225 1.013 694 1963 1537396 1432982 1.149 695 1964 1509317 1563203 .934 679 1965 I6O8468 1523514 1.107 649 117 TABLE V (continued) Livestock Net stock Deprecia- Year and of tion on Land (6) poultry buildings buildings (10) (7) (8) (9) thousands of dollars at 1935-39 prices 1935 574396 1351600 54064 2061640 1936 612201 1306375 47224 2077880 1937 569502 1268903 42502 2093700 1938 560188 1236533 45045 2107140 1939 634961 1202311 42166 2119180 1940 639664 1171179 37722 2131920 1941 499441 1145869 34060 2140159 1942 519636 1122314 31813 2151800 1943 678595 1101082 33240 2159920 1944 651395 1080617 31482 2168040 1945 612398 1064009 31559 2176160 1946 593449 1053283 33795 2184280 1947 573854 1041234 34325 2191980 1948 472120 1029240 31180 2199680 1949 516500 1029270 30034 2207380 1950 521528 1028256 28844 2214240 1951 595575 1027105 25888 2222534 1952 645360 1031220 26184 2232720 1953 590031 1036225 28551 2244900 1954 555845 1036990 28993 2255400 1955 596788 1036171 31040 2266180 1956 576233 IO364OI 32197 2275497 1957 586229 1033005 32858 2284940 1958 677764 1031552 35982 2290960 1959 720340 1029046 37854 2296980 I960 711098 1023844 39172 2306920 1961 737124 1036476 41700 2317120 1962 718105 1051472 42958 2331980 1963 769765 1065067 44270 2346680 1964 810433 1075297 45253 2361380 1965 726664 1086587 4836I 2376500 TABLE V (eoncluded) Material inputs, p«=0 Deflated Deflated Gross stock of machinery and implements,A=0 Year by ind i v i - dual price indexes (11) (12) by whole- sale price index (13) Deflated by DBS price index (14) Deflated by adj. USDA price index (15) thousands of dollars at 1935-39 prices 1935 1694H 174961 707466 682666 1936 174075 177669 707192 684877 1937 182278 182353 740551 716453 1938 199377 196460 770781 747772 1939 225497 213413 760690 738939 1940 225657 208516 747890 726246 1941 246423 223922 709860 690011 1942 286355 272181 669488 650893 1943 336784 314609 628907 .610735 1944 346837 324221 662447 642482 1945 367195 340750 716120 691371 1946 413327 371739 8IOO64 780114 1947 454735 372595 947867 906384 1948 411694 354737 1109939 1049161 1949 413859 358113 1287357 1197714 1950 411939 349656 1458647 1332163 1951 434667 342790 I620382 1473427 1952 434064 376873 1789768 1641521 1953 445886 387063 1935517 1786854 1954 457683 400376 2004086 1856918 1955 471260 411014 2104436 1957378 1956 513563 439299 2248199 2102221 1957 497298 429753 2332036 2196022 1958 536079 462155 2400357 2278292 1959 556777 488396 2454901 2349632 I960 559201 495272 2451181 2365097 1961 56OOO4 499659 2402147 2344468 1962 577179 519177 2349804 2314920 1963 612057 546143 2314521 2309035 1964 644093 578178 2306781 2312462 1965 664227 595164 2316591 2312029 TABLE VI SYMBOLS USED IN THE MODELS Symbol Derivation Meaning Q Variable Gross agricultural output (thousands of 1935-39 dollars) Q* Variable Gross agricultural output adjusted for weather influences (thousands of 1935-39 dollars) L Variable Number of persons employed (thousands) w Variable Weather index (potential=actual output when w=1.0) K Variable Annual flow of capital services (thousands of 1935-39 dollars) t Variable Time index (1, 2, 31) D Variable Dummy variable (ones and zeros for specified time periods) j\ Parameter Rate of embodied technological change in machinery and implements ft Parameter Rate of embodied technological change in material inputs e Number e = n i ^ o [ l + ( l / n ) ] n and lne=l A Estimated Constant coefficient (lnA*=lnA+f) f Estimated Constant coefficient u Estimated Regression coefficient for rate of disembodied technological change g Estimated Regression coefficient for w h Estimated Regression coefficient for w2 j Estimated Regression coefficient for D a Estimated Elasticity of output with respect to capital b Estimated Elasticity of output with respect to labor TABLE VII 2 REGRESSION ESTIMATES FOR MODEL 1(a): Q=Ae u te f + s w + h w L 1 _ aK a, ft—O.Ol, 1935-65 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 lnA» 3.0982 (1.0002) 3.0320 (.9602) 3 .0549, (.9441) 3.1006 (.9256) 3.1616 (.9408) 3.2387 (.8837) 3.4251, (.8383) 3.6303, (.7929) 3 .8337, (.7495) 4.0210 (.7096) u .0225 (.0092) .0208 .0153, (.0109) .1695 .0137 (.0113) .2336 .0124 (.0117) .2975 .0113 (.0120) .3577 .0104 (.0123) .4092 .0094 (.0129) .4794 .0090 (.0133) .5125 .0091 (.0137) .5206 .0094 (.0141) .5171 g .1055 (.2299) .6540 .1052 (.2260) .6494 .1054 (.2254) .6481 .1053 (.2251) .6479 .1052 (.2250) .648O .1049 (.2252) .6494 .1038 (.2262) .6539 .1024 (.2277) .6602 .1006 (.2296) .6677 .0988 (.2315) .6754 h .1522 (.1078) .1665 .1541 (.1059) .1539 .1547 (.1056) .1513 .1554 (.1054) .1489 .1561 (.1053) .1468 .1569 (.1054) .1449 .1587 (.1058) .1422 .1604 (.1065) .1404 .1621 (.1073) .1394 .1637 (.1082) .1387 a .4925 (.1682) .0069 .5062 (.1622) .0044 .5030 (.1597) .0041 .4959 (.1567) .0040 .4861 (.1533) .0039 .4736 (.1499) .0040 .4430 (.1424) .0045 .4089 (.1350) .0054 .3750 (.1277) .0068 .3437 (.1211) .0085 R 2 .9871 .9875 .9876 .9876 .9877 .9876 .9875 .9873 .9871 .9869 S .0563 .0554 .0552 .0552 .0551 .0552 .0554 .0558 .0563 .0567 "d" 1.540 1.589 1.601 1.611 1.619 1.624 1.630 1.629 1.626 1.621 TABLE VII (continued),A=-0.005 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 InA* 3 .2746, (.9933) (.9638) 3 .1479, (.9493) 3.1736, (.9322) 3.2166 (.9134) 3.2778, (.8927) 3.4382 (.8486) 3.6290 (.8033) 3.8249 (.7591) 4.0105, (.7179) u .0236 (.0093) .0173 .0162 (.0111) .1511 .0146 (.0115) .2135 .0131 (.0118) .2789 .0118 (.0122) .3430 .0108 (.0125) .3992 .0095 (.0131) .4809 .0089 (.0135) .5213 .0089 (.0139) .5331 .0092 (.0142) .5295 g .1044 (.2331) .6614 .1048 (.2286) .6543 .1051 (.2277) .6522 .1052 (.2272) .6511 .1053 (.2268) .6503 .1051 (.2268) .6510 .1043 (.2273) .6539 .1029 (.2285) .6598 .1012 (.2301) .6668 .0993 (.2319) .6745 h .1528 (.1039) .1704 .1543 (.1071) .1582 .1547 (.1067) .1554 .1553, (.1064) .1528 .1559 (.1062) .1506 .1566 (.1061) .1484 .1582 (.1063) .1452 .1600 (.1069) .1428 .1617 (.1076) .1412 .1633 (.1084) .1402 a .4621 (.1668) .0099 .4864 (.1626) .0059 .4865 (.1603) .0054 .4828 (.1576) .0050 .4761 (.1546) .0048 .4663 (.1512) .0048 .4402 (.1440) .0051 .4086 (.1366) .0059 .3760 (.1292) .0072 .3450 (.1224) .0088 R 2 .9868 .9873 .9874 .9874 .9875 .9875 .9874 .9873 .9871 .9869 S .0571 .0560 .0558 .0557 .0556 .0556 .0557 .0560 .0564 .0568 "d" 1.515 1.563 1.575 1.587 1.597 1.605 1.615 1.618 1.618 1.615 TABLE VII (continued), A=0.0 0 . 0 .02 .025 .03 .035 .04 .05 .06 .07 .08 lnA» 3.4487 (.9824) 3.2685 (.9634) 3.2562 (.9514) 3.2602 (.9368) 3.2838 (.9198) 3.3282 (.9006) 3.4587 (.8583) 3.6305 ( . a i 3 5 ) 3.8163 (.7689) 3.9983 (.7270) u .0246 (.0095) .0144 .0173 (.0112) .1323 .0155 (.0116) .1910 .0139 (.0120) .2574 .0125 (.0124) .3246 .0113 (.0127) .3853 .0096 (.0133) .4798 .0089 (.0137) .5302 .0087 (.0141) .5476 .0090 (.0144) .5456 g .1030 (.2360) .6688 .1039 (.2312) .6603 .1043 (.2302) .6577 .1048 (.2294) .6554 .1050 (.2288) .6538 .1050 (.2285) .6536 .1045 (.2285) .6548 .1033 (.2294) .6596 .1017 (.2307) .6661 .0998 (.2323) .6736 h .1536 (.1106) .1735 .1546 (.1083) .1619 .1549 (.1078) .1592 .1553 (.1074) .1566 .1558 (.1071) .1542 .1565 (.1069) .1519 .1579 (.1069) .1481- .1596 (.1073) .1452 .1613 (.1079) .1431 .1630 (.1086) .1417 a .4322 (.1647) .0138 .4649 (.1623) .0080 .4676 (.1605) .0071 .4675 (.1582) .0065 .4642 (.1555) .0060 .4572 (.1524) .0058 .4362 (.1456) .0059 .4079 (.1382) .0065 .3771 (.1308) .0076 .3468 (.1238) .0092 R 2 .9865 .9870 .9871 .9872 .9873 .9873 .9873 .9872 .9870 .9868 S .0577- .0566 .0564 .0562 .0560 .0560 .0560 .0562 .0565 .0569 "d" 1.495* 1.538 1.551 1.564 1.575 1.585 1.600 1.607 1.609 1.608 #Test for serial correlation i s inconclusive at one per cent level of significance. TABLE VII (continued), A=0.005 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 InA' 3.6323 (.9675) 3.4140 (.9598) 3.3827 (.9509) 3.3691" (.9389) 3.3722 (.9239) 3.3968 (.9070) 3.4924 (.8671) 3.6415 (.8237) 3.8120 (.7791) 3.9862 (.7366) u .0258 (.0096) .0117 .0185 (.0113) .1114 .0166 (.0118) .1658 .0149 (.0122) .2295 .0133 (.0125) .2979 .0120 (.0129) .3632 .0100 (.0135) .4713 .0089 (.0139) .5333 .0086 (.0143) .5598 .0087 (.0146) .5624 g .1013 (.2389) .6772 .1025 (.2340) .6677 .1032 (.2328) .6643 .1038 (.2318) .6613 .1044 (.2310) .6587 .1045 (.2304) .6574 .1045 (.2299) .6568 .1035 (.2304) .6604 .1020 (.2314) .6659 .1003 (.2327) .6726 h .1546 (.1120) .1761 .1551 (.1096) .1655 .1553 (.1090) .1629 .1556 (.1086) .1603 .1559 (.1081) .1579 .1564 (.1079) .1554 .1577 (.1076) .1512 .1593 (.1078) .1478 .1609 (.1082) .1453 .1626 (.1088) .1435 a .4009 (.1620) .0193 .4398 (.1615) .0110 .4456 (.1602) .0096 .4485 (.1584) .0086 .4486 (.1560) .0078 .4450 (.1533) .0073 .4299 (.1469) .0069 .4056 (.1398) .0073 .3774 (.1324) .0082 .3485 (.1254) .0097 R2 .9861 .9867 .9868 .9869 .9870 .9871 .9871 .9871 .9870 .9868 S .0584 .0573 .0570 .0568 .0566 .0564 .0563 .0564 .0567 .0570 »!d" 1.478* 1.515 1.527 1.540 1.552 1.564 1.582 1.593 1.599 1.601 *Test for serial correlation i s inconclusive at one per cent level of significance. TABLE VII (continued), A=0.01 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 lnA» 3.8218 (.9485) 3.5755 (.9523) 3.5298 (.9467) 3.4979 (.9371) 3.4821 (.9255) 3.4868 (.9109) 3.5456 (.8750) 3.6624~ (.8337) 3.8161" (.7901) 3.9781 (.7470) u .0270 (.0096) .0093 .0199 (.0115) .0905 .0180 (.0119) .1384 .0162 (.0123) .1974 .0145 (.0127) .2644 .0130 (.0130) .3318 .0106 (.0137) .4517 .0092 (.0141) .5304 .0086 (.0145) .5665 .0085 (.0148) .5769 g .0992 (.2417) .6863 .1008 (.2369) .6763 .1016 (.2356) .6724 .1024 (.2344) .6688 .1032 (.2334) .6652 .1036 (.2327) .6630 .1040 (.2317) .6605 .1036 (.2316) .6618 .1023 (.2323) .6664 .1007 (.2333) .6723 h .1557 (.1133) .1780 .1559 (.1110) .1686 .1559 (.1104) .1662 .1561 (.1098) .1637 .1562 (.1093) .1614 .1566 (.1089) .1589 .1576 (.1084) .1545 .1590 (.1083) .1507 .1606 (.1086) .1477 .1622 (.1091) .1454 a .3686 (.1586) .0269 .4120 (.1600) .0154 .4202 (.1593) .0134 .4262 (.1579) .0116 .4294 (.1561) .0103 .4292 (.1538) .0094 .4204 (.1481) .0085 .4015 (.1413) .0084 .3763 (.1342) .0091 .3495 (.1270) .0103 R 2 .9858 .9864 .9365 .9866 .9867 .9868 .9869 .9869 .9869 .9867 S .0591 .0580 .0577 .0574 .0572 .0570 .0567 .0567 .0569 .0572 "d" 1.465* 1.494* 1.505* 1.517 1.530 1.542 1.563 1.578 1.587 1.592 ;.*Test for serial correlation is inconclusive at one per cent level of significance. TABLE VII (continued), 7\ =0.015 p 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 lnA» 4.0086 (.9257) 3.7466 (.9403) 3.6915 (.9377) 3.6453 (.9322) 3.6147 (.9235) 3.5980 (.9120) 3.6172 (.8809) 3.7002 (.8427) 3.8273 (.8007) 3.9767 (.7581) u .0282 (.0097) .0074 .0214 (.0115) .0715 .0195 (.0120) .1114 .0176 (.0124) .1637 .0159 (.0128) .2253 .0142 (.0132) .2930 .0114 (.0139) .4222 .0096 (.0144) .5167 .0087 (.0148) .5686 .0084 (.0151) .5868 g .0969 (.2444) .6957 .0987 (.2397) .6856 .0996 (.2385) .6816 .1005 (.2372) .6774 .1014 (.2361) .6736 .1022 (.2351) .6701 .1032 (.2337) .6656 .1032 (.2332) .6649 .1023 (.2334) .6677 .1009 (.2341) .6727 h .1569 (.1145) .1793 .1568 (.1123) .1711 .1568 (.1117) .1690 .1568 (.1111) .1668 .1568 (.1106) .1645 .1570 (.1101) .1623 .1577 (.1093) .1578 .1588 (.1091) .1537 .1603 (.1091) .1502 .1619 (.1094) .1475 a .3368 (.1546) .0367 .3826 (.1578) .0215 .3924 (.1576) .0186 .4007 (.1568) .0161 .4065 (.1556) .0142 .4098 (.1538) .0126 .4077 (.1489) .0106 .3946 (.1427) .0100 .3739 (.1358) .0103 .3493 (.1238) .0113 R 2 .9855 .9860 .9862 .9863 .9864 .9866 .9867 .9868 .9867 .9867 S .0597 .0586 .0584 .0581 .0578 .0576 .0572 .0571 .0571 .0573 "d" 1.455* 1.476* 1.485* 1.496* 1.508* 1.520 1.542 1.561 1.574 1.581 *Test for se r i a l correlation i s inconclusive at one per cent level of significance. TABLE VII (continued), f\=0.02 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 InA* 4.1926 (.8998) 3.9287 (.9247) 3.8655 (.9252) 3.8109 (.9227) 3.7637 (.9176) 3.7311, (.9094) 3.7108 (.8844) 3.7544 (.8505) 3.8506 (.8106) 3.9805 (.7692) u .0294 (.0098) .0058 .0231, (.0116) .0546 .0212 (.0121) .0866 .0194 (.0125) .1298 .0175 (.0129) .1851 .0157 (.0133) .2483 .0126 (.0140) .3820 .0103 (.0146) .4930 .0090 (.0150) .5625 .0084 (.0153) .5933 g .0943 (.2470) .7055 .0960 (.2426) .6962 .0970 (.2414) .6920 .0980 (.2402) .6879 .0992 (.2389) .6833 .1001 (.2378) .6793 .1017 (.2360) .6725 .1024 (.2349) .6694 .1020 (.2346) .6700 .1009 (.2350) .6736 h .1583 (.1157) .1799 .1580 (.1137) .1729 .1579 (.1131) .1712 .1578 (.1125) .1692 .1577 (.1119) .1673 .1577 (.1113) .1651 .1580 (.1104) .1609 .1589 (.1099) .1567 .1601 (.1097) .1529 .1616 (.1098) .1498 a .3056 (.1500) .0495 .3514 (.1550) .0303 .3626 (.1552) .0262 .3723 (.1550) .0227 .3807 (.1544) .0196 .3868 (.1532) .0173 .3913 (.1493) .0139 .3849 (.1439) .0123 .3695 (.1374) .0119 .3482 (.1306) .0125 R 2 .9852 .9857 .9859 .9860 .9861 .9863 .9865 .9866 .9866 .9866 S .0603 .0593 .0590 .0588 .0585 .0582 .0578 .0575 .0574 .0575 »d" 1.448* 1.462* 1.469* 1.478* I .488* 1.499* 1.522 1.542 1.558 1.569 *Test for serial correlation i s inconclusive at one per cent level of significance. TABLE VII (continued),A=0.025 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 lnA« 4.3683 (.8715) 4.1121 (.9046) 4.0460 (.9081) 3.9829 (.9090) 3.9277 (.9073) 3.8808 (.9027) 3.8241 (.8846) 3.8303 (.8562) 3.8912 (.8199) 3.9962 (.7803) u .0306 (.0099) .0045 .0248 (.0116) .0407 .0230 (.0121) .0653 .0211 (.0125) .1002 .0193 (.0130) .1460 .0174 (.0134) .2025 .0140 (.0142) .3341 .0113 (.0148) .4563 .0095 (.0153) .5446 .0086 (.0156) .5911 g .0915 (.2493) .7153 .0931 (.2454) .7070 .0941 (.2442) .7031 .0952 (.2430) .6987 .0964 (.2418) .6941 .0976 (.2406) .6896 .0998 (.2385) .6810 .1010 (.2370) .6758 .1013 (.2362) .6739 .1006 (.2362) .6759 h .1597 (.1168) .1800 .1594 (.1149) .1741 .1592 (.1144) .1726 .1589 (.1138) .1710 .1588 (.1132) .1694 .1586 (.1126) .1675 .1586 (.1116) .1637 .1591 (.1109) .1597 .1601 (.1104) .1558 .1614 (.1104) .1523 a .2753 (.1451) .0655 .3201 (.1514) .0420 .3317 (.1521) .0366 .3428 (.1525) .0317 .3525 (.1524) .0275 .3609 (.1518) .0239 .3715 (.1491) .0186 .3715 (.1447) .0156 .3621 (.1388) .0143 .3451 (.1323) .0143 R 2 .9850 .9854 .9855 .9857 .9858 .9859 .9862 .9863 .9864 .9864 S .0608 .0600 .0597 .0594 .0591 .0589 .0583 .O58O .0578 .0578 "d» 1.444* 1.451* 1.456* 1.463* 1.471* 1.480* 1.501* 1.523 1.541 1.556 *Test for se r i a l correlation i s inconclusive at one per cent level of significance. TABLE VII (continued), ft=0.03 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 lnA» 4.5352 (.8410) 4.2911 (.8812) 4.2243 (.8873) 4.1612 (.8914) 4.0988" (.8928) 4.0434 (.8918) 3.9587 (.8807) 3.9253~ (.8587) 3.9496 (.8273) 4.0238 (.7904) u .0318 (.0099) .0035 .0264 (.0116) .0300 .0248 (.0121) .0483 .0231 (.0126) .0748 .0212 (.0130) .1117 .0194 (.0135) .1589 .0158 (.0143) .2796 .0127 (.0150) .4095 .0104 (.0155) .5149 .0090 (.0159) .5803 g .0884 (.2514) .7254 .0900 (.2479) .7180 .0909 (.2469) .7144 .0920 (.2458) .7105 .0932 (.2446) .7058 .0945 (.2434) .7012 .0971 (.2411) .6915 .0991 (.2393) .6840 .1001 (.2380) .6796 .1000 (.2375) .6792 h .1613 (.1178) .1795 .1608 (.1161) .1747 .1606 (.1156) .1734 .1603 (.1151) .1722 .1600 (.1145) .1708 .1598 (.1139) .1693 .1595 (.1128) .1660 .1596 (.1119) .1623 .1602 (.1113) .1585 .1613 (.1110) .1549 a .2475 (.1398) .0849 .2897 (.1472) .0571 .3012 (.1484) .0503 .3123 (.1493) .0442 .3232 (.1497) .0383 .3330 (.1497) .0334 .3482 (.1483) .0254 .3548 (.1449) .0204 .3517 (.1399) .0177 .3399 (.1339) .0167 R2 .9847 .9851 .9852 .9853 .9855 .9856 .9859 .9861 .9862 .9863 S .0613 .0606 .0603 .0601 .0598 .0595 .0590 .0585 .0582 .0581 "d" 1.442* 1.443* 1.446* 1.451* 1.457* I .464* 1.483* 1.503* 1.524 1.540 *Test for serial correlation i s inconclusive at one per cent level of significance. TABLE VIII REGRESSION ESTIMATES FOR MODEL 1(b): Q=Ae u te f + g w + h w L 1 _ aK a, A=0.0, 1935-65 ? 0.0 .02 .025 .03 .035 .04 .05 7O6 .07 " .08 lnA» 3 .2307, (.9738) 3.1352 x (.9485) 3.1461 (.9355) 3.1745, (.9209) 3.2165, (.9034) 3.2761 (.8846) 3.4326 (.8431) 3.6204 (.8005) 3.8125 (.7580) 3.9952 (.7181) u .0226 (.0094) .0221 .0157 (.0110) .1630 .0142 (.0114) .2242 .0128 (.0118) .2894 .0115 (.0122) .3536 .0105 (.0125) .4103 .0092 (.0130) .4927 .0087 (.0135) .5338 .0086 (.0139) .5477 .0089 (.0143) . 5461 g .1123 (.2314) .6363 .1100 (.2276) .6375 .1097 (.2270) .6376 .1093 (.2266) .6368 .1089 (.2263) .6389 .1083 (.2263) .6406 .1069 (.2268) .6457 .1049 (.2282) .6534 .1028 (.2298) .6617 .1007 (.2315) .6700 h .1498 (.1084) .1758 .1522 (.1066) .1619 .1529 (.1063) .1588 .1537 (.1061) .1559 .1544 (.1059) .1534 .1553 (.1059). .1510 .1571 (.1061) .1471 .1591 (.1067) .1444 .1610 (.1074) .1424 .1627 (.1082) .1412 a .4693 (.1634) .0078 .4878 (.1599) .0052 .4866 (.1579) .0048 .4824 (.1556) .OO46 .4759 (.1528) .0044 .4664 (.1498) .0045 .4409 (.1430) .0048 .4099 (.1360) .0056 .3780 (.1290) .0068 .3475 (.1224) .0084 R 2 .9870 .9874 .9875 .9875 .9875 .9875 .9875 .9873 .9871 .9869 S .0566 .0557 .0556 .0555 .0554 .0554 .0556 .0559 .0563 .0567 »d" 1.532 1.572 1.583 1.593 1.602 1.609 1.618 1.620 1.619 1.617 H TABLE IX 2 REGRESSION ESTIMATES FOR MODEL 1(c): Q=Ae u te f +S w + h w L 1 - aK a,A=0.0, 1935-65 0.0 .025 .03 .035 .04 .05 .06 .07 .08 lnA» 3.6030 (.8502) 3.4889 (.8064) 3.5180 (.7905) 3.5627 (.7732) 3.6228 (.7548) 3.8032 (.7071) 3.9636 (.6695) 4.1233 (.6347) 4.2673 (.6028) u .0270 (.0079) .0021 .0197 (.0094) .0450 .0185 (.0097) .0645 .0176 (.0100) .0860 .0168 (.0102) .1072 .0165 (.0104) .1232 .0159 (.0108) .1482 .0158 (.0111) .1657 .0157 (.0115) .1805 S .1399 (.2333) .5607 .1440 (.2271) .5384 .1436 (.2265) .5385 .1429 (.2261) .5400 .1416 (.2261) .5434 .1441 (.2270) .5381 .1401 (.2281) .5512 .1357 (.2295) .5662 .1318 (.2310) .5798 h .1379 (.1092) .2158 .1377 (.1063) .2040 .1383 (.1060) .2006 .1392 (.1058) .1971 .1402 (.1057) .1936 .1408 (.1061) .1932 .1433 (.1065) .1872 .1460 (.1072) .1817 .1484 (.1078) .1774 a .4037 (.1414) .0081 .4244 (.1345) .0040 .4199 (.1320) .0038 .4128 (.1291) .0037 .4030 (.1261) .0037 .3722 (.1178) .0040 .3458 (.1116) .OO46 .3194 (.1058) .OO56 .2956 (.1005) .0067 R2 .9870 .9876 .9877 .9877 .9877 .9376 .9875 .9373 .9872 S .0567 .0552 .0551 .0550 .0550 .0552 .0555 .0559 .0562 "d» 1.612 1.690 1.703 1.712 1.720 1.721 1.722 1.719 1.715 TABLE X 2 REGRESSION ESTIMATES FOR MODEL II: Q=Ae f +S w + h w L 1 - aK a, A =0.0, 1935-65 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 InA' .9269 (.1821) 1.8038 (.1519) 2.0018 (.1468) 2.1899 (.1425) 2.3679 (.1389) 2.5358 (.1360) 2.8419 (.1316) 3.1109 (.1289) 3.3464 (.1271) 3.5523 (.1260) g .2024 (.2566) .4426 .1571 (.2343) .5150 .1498 (.2309) .5291 .1436 (.2283) .5416 .1385 (.2264) .5527 .1343 (.2252) .5628 .1283 (.2242) .5785 .1244 (.2246) .5907 .1218 (.2258) .6000 .1201 (.2274) .6075 h .1073 (.1203) .3842 .1305 (.1098) .2437 .1346 (.1083) .2224 .1382 (.1070)~ .2051 .1412 (.1061) .1916 .1439 (.1056) .1811 .1481 (.1051) .1671 .1511 (.1053) .1593 .1534 (.1059) .1553 .1552 (.1066) .1534 a .8578 (.0229) .0000 .7133 (.0173) .0000 .6806 (.0163) .0000 .6495 (.0154) .0000 .6201 (.0145) .0000 .5923 (.0138) .0000 .5416 (.0126) .0000 .4970 (.0116) .0000 .4579 (.0107) .0000 .4237 (.0100) .0000 R2 .9829 .9858 .9862 .9865 .9863 .9869 .9370 .9870 .9868 .9867 S .0636 .0580 .0572 .0565 .0561 .0558 .0555 .0556 .0559 .0563 »d» 1.175* 1.383* 1.425 1.461 1.491 1.514 1.545 1.559 1.563 1.562 *Test for serial correlation i s inconclusive at one per cent level of significance. TABLE XI 2 REGRESSION ESTIMATES FOR MODEL I I I : Q = A e u t e f + S w + h w L bK a, A=0.0, 1935-65 p 0 .0 .02 .025 .03 .035 .04 .05 .06 .07 .08 InA* 7.9914 6.6457 6.3933 6.1298 6.0039 5.9191 6.0063 6.3616 6.7965 7.2655 (3.9918)(4.0902)(4.1326)(4.1479)(4.1294)(4.1144)(4.0219)(3.8433)(3.6421)(3.4580) u .0320 .0257 .0240 .0223 .0210 .0199 .0188 .0194 .0207 .0226 (.0113) (.0150) (.0160) (.0170) (.0177) (.0185) (.0196) (.0200) (.0202) (.0202) .0088 .0961 .1428 .1984 .2460 .2929 .3476 .3452 .3157 .2730 g .1759 .1600 .1566 .1530 .1509 .1492 .1492 .1527 .1575 .1630 (.2427) (.2418) (.2416) (.2414) (.2412) (.2411) (.2411) (.2413) (.2414) (.2417) .4816 .5209 .5297 .5388 .5442 .5484 .5484 .5394 .5270 .5132 h .1171 .1258 .1279 .1302 .1316 .1329 .1337 .1324 .1301 .1272 (.1143) (.1141) (.1141) (.1141) (.1141) (.1142) (.1144) (.1146) (.1147) (.1149) .3165 .2804 .2726 .2641 .2587 .2543 .2524 .2579 .2670 .2787 b .5014 .4892 .4895 .4910 .4947 .4996 .5119 .5241 .5333 .5388 (.1734) (.1721) (.1712) (.1701) (.1690) (.1681) (.1672) (.1677) (.1692) (.1715) .0077 .0086 .0082 .0077 .0070 .0064 .0052 .0045 .0042 .0043 a .1212 .2313 .2506 .2701 .2779 .2820 .2688 .2351, .1968 .1578 (.3113) (.3197) (.3217) (.3208) (.3166) (.3123) (.2977) (.2759) (.2527) (.2316) .7005 .4826 .4490 .4124 .3924 .3787 .3787 .4067 . .4492 .5088 R2 .9500 .9507 .9509 .9511 .9512 .9513 .9513 .9511 .9509 .9506 S .0574 .0570 .0569 .0568 .0567 .0567 .0567 .0568 .0569 .0570 1.565 1.564 1.567 1.572 1.577 1.582 1.590 1.593 1 .594 1.593 *Test f o r s e r i a l c o rrelation i s inconclusive at one per cent l e v e l of significance i n a l l regressions. TABLE XII REGRESSION ESTIMATES FOR MODEL IV(a): Q»=Ae u tL 1~ aK a, A =0.0, 1935-65 r 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 lnA» 3.6856 3.5147 3.4985 3.5009 3.5218 3.5626 (1.1384)(1.1258)(1.1125)(1.0955)(1.0749)(1.0510) 3.6865 (.9966) 3.8550 (.9380) 4.0384 (.8794) 4.2188 (.8240) u .0232 (.0112) .0449 .0158 (.0134) .2460 .0140 (.0139) .3245 .0123 (.0143) .4040 .0108 (.0148) .4787 .0095 (.0151) .5427 .0076 (.0158) .6373 .0067 (.0163) .6856 .0064 (.0166) .7039 .0065 (.0169) .7048 a .4416 (.1953) .0302 .4730 (.1940) .0204 .4766 (.1920) .0185 .4769 (.1894) .0170 .4742 (.1862) .0160 .4680 (.1824) .0153 .4483 (.1737) .0147 .4208 (.1642) .0154 .3905 (.1546) .0167 .3605 (.1456) .0187 R2 .9774 .9780 .9781 .9782 .9783 .9784 .9784 .9784 .9783 .9781 S .0691 .0682 .0680 .0678 .0677 .0676 .0675 .0676 .0678 .0681 »d" 1.850 1.890 1.901 1.912 1.921 1.928 1.936 1.937 1.934 1.928 TABLE XIII REGRESSION ESTIMATES FOR MODEL IV(b): Q»Ae u te f- e/ wL 1~ aK a, A=0.0, 1935-65 r 0.0 .02 .025 .03 .035 .04 .05 .06 .07 .08 .nA« 3.5670 3.4818 3.4973 3.5303 3.5829 3.6566 3.8430 4.0645 4.2930 (1.2718)(1.2668)(1.2551)(1.2392)(1.2193)(1.1956)(1.1403)(1.0786)(1.0155) 4.5091 (.9550) u .0186 (.0125) .1449 .0112 (.0151) .4685 .0096 (.0157) .5542 .0081 (.0163) .6301 .0068 (.0168) .6921 .0058 (.0173) .7379 .0045 (.0182) .7935 .0041 (.0188) .8106 .OO44 (.0194) .8046 .0051 (.0198) .7869 g -.3176 (.0469) .0000 -.3197 (.0467) .0000 -.3206 (.0467) .0000 -.3216 (.0467) .0000 -.3226 (.0468) .0000 -.3237 (.0468) .0000 -.3257, (.0471) .0000 - . 3 2 7 4 (.0475) .0000 - .3288 (.0479) .0000 - .3299 (.0483) .0000 a .5188 (.2186) .0238 .5363 (.2188) .0201 .5346 (.2172) .0196 .5300 (.2149) .0194 .5220 (.2119) .0195 .5104 (.2083) .0201 .4804 (.1997) .0222 .4441 (.1899) .0257 .4063 (.1799) .0306 .3702 (.1702) .0366 R 2 .9749 .9752 .9752 .9752 .9752 .9752 .9750 .9748 .9745 .9742 S .0772 .0767 .0767 .0767 .0767 .0767 .0770 .0774 .0778 .0783 "d" 1.874 1.898 1.905 1.910 1.914 1.917 1.918 1.913 1.906 1.899 TABLE XIV 2 REGRESSION ESTIMATES FOR MODEL V: Q=Aeutef+SW+kw L 1~ aK ae j D,A =P=0.0, 1935-65 D Dl D 2 D3 D 4 D 5 D6 D 7 DlO D 1 X D13 InA* 3.0299 (1.0150) 3.8387 (.9437) 3.4769 (.9667) 3.4531 (.9444) 3.0012 (.9461) 2.1505 2.5205 3.0274 2.8945 3.0478 (.9038)(1.1566)(1.1064)(1.1095)(1.0178) u .0192 (.0101) .0673 .0320 (.0096) .0027 .0274 (.0095) .0079 .0280 (.0093) .0057 .0244 (.0089) .0105 .0190 (.0081) .0257 .0193 (.0100) .0610 .0224 (.0099) .0309 .0215 (.0099) .0377 .0196 (.0101) .0615 g .0813 (.2328) .7272 .1258 (.2225) .5833 .0983 (.2322) .6779 .0990 (.2269) .6690 .0814 (.2218) .7152 .1349 (.1979) .5084 .0973 (.2312) .6796 .0851 (.2382) .7218 -.1100 (.2355) .6485 .1752 (.2395) .4775 h .1549 (.1089) .1638 .1376 (.1044) .1969 .1525 (.1088) .1702 .1545 (.1063) .1552 .1642 (.1040) .1230 .1478 (.0927) .1195 .1619 (.1085) .1446 .1564 (.1113) .1689 .1469 (.1105) .1930 .1229 (.1117) .2816 a .5037 (.1705) .0066 .3635 (.1536) .0291 .4273 (.1621) .0137 .4299 (.1584) .0114 .5076 (.1587) .0038 .6443 (.1509) .0003 .5874 (.1938) .0055 .5047 (.1865) .0117 .5246 (.1858) .0089 .4967 (.1700) .0071 j .0541 (.0399) .1840 -.0787 (.0378) .0455 -.0538% (.0394) .1811 - .0690 (.0390) .0854 -.0844 (.0398) .0418 - .1335 (.0384) .0020 - .0719 (.0497) .1569 - .0399 (.0472) .4104 -.0471 (.0442) .2976 .0501 (.0386) .2038 R2 .9874 .9885 .9874 .9880 .9885 .9909 .9875 .9868 .9871 .9873 S .0568 .0544 .0568 .0555 .0542 .0484 .0566 .0581 .0576 .0570 "d» 1.551* 1.683 1.736 1.717 1.895 1.659 1.692 1.444* 1.434* 1.620 *Test for serial correlation i s inconclusive at one per cent level of significance. 136 TABLE XV REGRESSION ESTIMATES FOR MODEL 1(a): Q=Aeut e f + e w + h w 2 L 1 " aK a, ^ = 0 . 0 , FOR THE 1935-49 AND 1950-65 SUBPERIODS 1935-49 subperiod 1950-65 subperiod InA' 1.1424 (1.4696) 3.2510 (1.3939) u .0109 (.0120) .3872 .0308 (.0134) .0406 g .1925 (.2775) .5096 -.0701 (.3930) .3395 h .1179 (.1243) .3680 .2608 (.2023) .2222 a .8119 (.2449) .0077 .4461 (.2340) .0804 R2 .9706 .9738 S .0539 .0481 »d" 2.122 1.513* *Test for serial correlation i s inconclusive at one per cent level of significance.

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