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Evaluation of Work-Force Composition Adjustment KENT KUNZE Office of Productivity and Technology Bureau of Labor Statistics INTRODUCTION The purpose of this paper is to evaluate different methods of adjusting labor input in order to account for changes in labor productivity and multi-factor productivity. It attempts to define and identify those charac- teristics or dimensions of workers that contribute to productivity change. It also analyzes and compares the different methods of measuring and aggregating the changes in these characteristics and the associated productivity changes. Finally, it considers the most meaningful and workable procedure by which the Bureau of Labor Statistics (BES) could implement the measurement of the productivity effects of changes in work-force composition. Presently, BUS publishes labor productivity measures for several economic sectors (including the private business economy) and a number of 3 or 4-digit standard industrial classification (sac) codes for manu- facturing and nonmanufacturing industries. The measures for the economic sectors (private business, nonfarm business, manufacturing, and non-financial corporations) are published quarterly. The selected industry measures are published annually (see Bureau of Labor Statistics 1976, pp. 219-231~. Labor productivity is defined as the ratio of the quantity of output produced to the quantity of labor input used in producing it. The BES measures labor productivity as the ratio of constant-dollar output from a given sector to the total unweighted hours of labor input in that sector 334

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Evaluation of Work-Force Composition Adjustment 335 (Bureau of Labor Statistics 1976, pp. 219-2231. It is this measure of total hours that a work-force composition adjustment adjusts in order to correct for shifts in the demographic and organizational (industry and occupation) characteristics of the work force, as will be explained in the following sections. The following section outlines the basic model of work-force composi- tion adjustment and considers the demographic and organizational characteristics that may be important to this adjustment. The report then surveys the available data sources (surveys and censuses) that can be used to estimate an adjustment coefficient. The specific methods of composition adjustment used by a number of researchers are then reviewed and compared. The final section gives conclusions and recom- mendations. WORK-FORCE COMPOSITION C ONCEPT AND DEFINITION When measuring labor productivity, the commonly used measure of labor input is total hours of all workers measured as an unweighted sum over all types of labor: H = LiHi. (1) where H is total hours of labor input and Hi is the total hours of labor input for the ith category of labor. The assumption implicit in this method of aggregation is that all labor is homogeneous; each type of labor can be freely substituted for any other type, and such substitu- tions will have no effect on productivity growth. A consequence of this assumption is that productivity changes resulting from shifts in the composition of the work force are interpreted as changes in labor pro- ductivity. For example, if the proportion of experienced workers in- creases, there will be an increase in the total output (as measured in constant dollars) without any change in an unweighted labor input measure.' This would be registered as an increase in productivity, rather than as an increase in labor input. The basic assumption in measuring the productivity effects of changing work-force composition is that there are varying types of labor that cannot ~ These events could occur as a result of a change in consumer demand from a lower- to a higher-priced product.

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336 PAPERS be freely substituted for each other, or if they can be substituted, this cannot be done at a one-to-one ratio. Certain measurable characteristics, such as experience and schooling, make some workers more productive than others at some kinds of work. However, the effects of these different characteristics on productivity depend strongly on the nature of the occupation or industry. For example, a person with a bachelor's degree in farm management and 10 years experience in farming could undoubtedly run a more efficient and productive farm than a 16-year old illiterate. However, it is unlikely that if this person had the additional education provided by a Ph.D. in English, this would significantly improve the efficiency of the farm. Those characteristics that are measurable and relevant to productivity differences must be taken into account in relation to the type and quantity of labor demanded in the industry or occupation. The effects of any of the characteristics will also depend on the size of the increase in any one characteristic relative to the demand for it. For example, if everyone received an additional year of experience or an additional year of education, there would be an improvement in labor quality. But a large increase in the proportion of highly educated workers does not necessarily mean a large increase in labor quality. in fact, there is evidence that there may be some overeducation of the work force, which forces some people to work in jobs for which they are overtrained.2 This might lead to a decrease in productivity growth when people work in jobs that are not as challenging or exciting as those they were educated for. A MODEL OF WORK-FORCE COMPOSITION To construct a measure of aggregate labor input for use both in ac- counting for changes in unweighted labor productivity and in measuring multi-factor productivity, total input (hours) must be defined as a flow of labor services measured as total hours worked adjusted for labor composition. An adjusted measure of labor input (L ~ can be represented by a function (g) of the varying categories of labor input (Hi): L = g(H, , H2 , , H., ). (2) Assuming (g) is a linear homogeneous function, the percentage change in aggregate labor input is the derivative of the logarithm of (2) with respect to time: 2 See BES projections for 1985 of the number of college graduates and the number of positions requiring college degrees.

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Evaluation of Work-Force Com position A djustment L H L = Liv where i ~InH ~Levi = 1. 337 (3) Thus the growth rate of adjusted labor input is equal to a weighted average of the growth rate of its individual components.3 We can now further decompose the aggregate adjustment by trans- forming hours worked into labor services. We begin by adding and subtracting the growth rate of unweighted hours to the expression for the growth rate of labor services (3~: L rHl_ H- H L = Levi Hi H H . (4) The difference tHi/Hi-H/H1 is interpreted as the growth rate of the proportion (Hi/H) of total hours worked by the ith category of workers. Therefore, the growth rate of labor services can be expressed as the sum of the rates of change in work-force composition Qua and unweighted hours: L (L+ H L QL H where Q' AH, N .] (5) Taking the antilog of the integral of (5) over time results in the relation L = QrH. Reflecting changes in the composition of hours worked across the economy, the adjustment measure ~ Q~) transforms unweighted hours worked into a measure of aggregate labor services adjusted by work-force composition. 3So far, we have not discussed how to measure the weights (vi) of the aggregation func- tion. In (6) we assumed these weights to be equal to the wage rate. This has been the common, though restrictive, assumption because of the lack of possible alternatives for measuring these weights. We will try to clarify this point in the following section and later when we discuss the different methods of aggregation that have been used.

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338 PAPERS The next questions are, Which characteristics of labor need to be identified that relate to different quantities of labor service and what are the appropriate measures of these varying quantities? SPECIFIC CHARACTERISTICS OF WORK-FORCE COMPOSITION The previous section provided a framework in which it is legitimate to decompose the work force into specific types of labor inputs. This model allows one either to aggregate these specific inputs into one measure of labor input, adjusted by the composition of the work force, or to let the different types of labor stand alone as separate inputs in the production function. Which of these two options is followed depends on the separability properties of the inputs (Russell 1975, Blackorby and Russell 1976, Berndt and Christensen 19731. The questions still remain What constitutes different types of labor and how should we measure them? Traditionally, those who have done this type of analysis have either relied on some very restrictive assumptions or have arbitrarily determined what characteristics are important (Denison 1967, Jorgenson and Gri- liches 19671. These procedures were used for want of sufficient data or lack of technical knowledge of the production process. To be more specific, the neoclassical economic theory of production and distribution assumes that there is perfect competition and equilibrium in all the factor input markets. If thi, is so, each factor will be paid the value of its marginal product from the total value of output. In the labor market, each worker's wage is the value of the contribution to output made by that worker. A difference in wages then necessarily reflects a difference in productivity and a difference in labor inputs. However, there is some ambiguity even in a situation of perfect competition and equilibrium. Differences in wage rates can reflect differences in produc- tivity without there being a difference in the characteristics of labor. For example, there can be a wage differential for risky or hazardous work. Discrimination can also create wage differentials among workers with the same characteristics and productivity, although discrimination is not consistent with perfect competition. To facilitate the measurement of labor inputs, the neoclassical model has been used in two ways. One is to assume that perfect competition and equilibrium exist everywhere and therefore that any wage differential is also a differential in the quality and productivity of labor inputs (Jorgenson and Gollop 1977~. The second method is similar. Categories of labor with particular measurable characteristics are chosen a priori

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Evc~luc~tio'' of Work-Force CO'}2~)sili()n A~j[ISIn7~l 339 as factors that may determine labor productivity, and the average wage of each category is used to measure the productivity differential (Denison 1967). While neither of these two methods has been proved to be a true representation of labor quality, the second seems more reasonable given what is known about how markets really work. The average wage of a large group of people with certain measurable characteristics age, sex, education, occupation seems to be a reasonable measure of its productivity, relative to another group's average wage. In this vein we consider the following characteristics that may determine or provide a measure of efficiency or ability: age, sex, education. industry, occupation, class of worker (employed versus self-employed). and experience. We also mention two factors that are hard to measure but still play a role in enhancing the ability of the labor force: effort and health. Industry The principal reason for disaggregating labor input by industry is that different industries often provide special training that increases the efficiency or ability of the laborer and that is not measured by the other characteristics. In general, however, industry wage differentials also reflect regional differentials, occupational differentials, union/non-union differentials, and disequilibrium differentials, rather than productivity differences arising from worker characteristics alone. Occupation The term occupation describes different types of labor that are in a position to handle different specific ranges of job functions. In effect occupation is a classification of skill; certain innate or acquired attributes are needed to do the work of certain occupations. Certain occupations also have unique relationships with particular capital goods. Conse- quently, disaggregation by occupation may capture quality dimensions of the work force that are not measured by the other characteristics described in this paper. Employment Class Employment class distinguishes wage and salary workers from self- employed workers. This distinction is necessary because the total com

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340 PAPERS pensation of self-employed workers as measured in the national income accounts includes returns on their capital investment as well as their labor services. The following dimensions of work-force composition relate to indi- vidual characteristics that may influence productivity: Age Young people entering the work force lack experience and training. On the other hand, older persons may be in poor health or may no longer be able to make the effort needed to be fully productive. These factors tend to raise or lower the productive capacity of the work force as the proportions of different age groups increase or decrease. Sex Traditionally, married women have not participated in the labor force continuously because they usually withdrew from it during their child- bearing years. This rather loose attachment to the work force caused married women to forgo the advantages of continuous experience in a profession, and they were therefore not able to achieve the higher earning levels that males achieved through continuous employment. Although there may have been substantial sex discrimination practiced in earlier periods, new laws forbid these practices. If these are effective, later data on earnings by sex will begin to reflect differences in productivity only, rather than including the effects of discrimination. Productivity differences should narrow as more women work and participate more continuously over the normal working years. Education A work force that has a higher level of education is better equipped to learn and utilize the newest and most efficient techniques of produc- tion; educated workers are usually more proficient at their occupations. It is unsatisfactory to count the work done by a college graduate, on average, as equal to the work performed by an elementary school graduate of the same age and sex.4 Going to school tends to prevent 4 How much more to attribute to college training is not easy to determine in practice. College graduates tend to be more able than high-school and elementary school graduates and would have earned somewhat more even without the additional schooling.

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Evaluation of Work-Force Composition Adjustment 341 younger persons from taking full-time work, thus reducing employment and increasing the average age of entry into full-time employment. Under these circumstances, highly educated new entrants have less experience at older ages (Denison 1967~. E. xpertence The average number of years a worker has been employed in a position affects his ability to do the work of that position. People starting new jobs need an orientation period before their knowledge of the process permits high performance. As new and more complex technologies are introduced, the experienced worker can often make the transition with less effort or retraining (although sometimes, it is the older, more experienced workers who resist change). Measurement of experience might make it unnecessary to measure age, since it is a better measure of skill than age is. However, productivity may peak at a certain age in some occupations because of the loss of physical vigor. In measuring the effects of work-force composition, we should always keep in mind that two or more of the preceding dimensions may interact. Individual factors when isolated from one another may not always prove to be important, but when they interact, there may be a significant joint effect on the level of labor input. Consider, for example, age and sex; age is a better measure of experience for the male work force than for the female work force because many women raise families and re-enter the work force later in life. The interaction of age and sex may thus distinguish different types of labor better than either classification alone. This may also be true of occupation and industry, since certain occupa- tions are present only in one industry. This completes the list of the principal dimensions that are measur- able. There are other dimensions of labor that influence productivity, but because of their subjective nature they are rather hard to measure. For example, the health of the work force is important when some workers cannot perform up to their capacity because of illness. On the other hand, if they cannot work at all, this will be reflected in the number of hours worked (Denison 1967~. Other considerations include the effort applied to perform the task and the amount of innate ability a worker has. These are rather vague and subjective attributes that are usually impossible to measure. It is also doubtful that there has been a large change over time in the pro- portion of workers having more or less of these characteristics (Denison 1967).

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342 DATA SOURCES FOR WORK-FORCE COMPOSITION PAPERS A major concern in deriving an adjustment for work-force composition is the availability of detailed and accurate data. Given the model of work-force composition in the previous section and the specific dimen- sions of labor composition (sex, age, education, occupation, industry, worker-class, and the interaction effects), the data necessary for calcu- lating the full adjustment coefficient are (1) hours worked per year, cross-classif~ed by each dimension and (2) average compensation per hour worked for at least 1 year, cross-classif~ed by each dimension. Average compensation per hour worked is the only measure of a worker's marginal product that can be used given the available data. If we had direct measures of the marginal product cross-classif~ed by all the dimensions, it would not be necessary to assume that the wage rate is equal to marginal product. Given the lack of data and the lack of a model to construct such data, we must continue to use this restrictive assumption. Hours worked are necessary because they are a measure of the produc- tive time of the workers. Sick time, leave time, and holidays are not productive time, and therefore should not be included in the hours measure. Cross-classif~cation of the data is needed to capture the interaction effects and to specify each type of labor. By type of labor we mean not whether or not a worker has more or less education, or is male or female. A type of labor means a worker of a given age, sex, level of education, working in a given industry, and so on. Cross-classification means that each dimension must be disaggregated according to the levels that differentiate the productivity of the individuals. For some dimensions this is straightforward -sex can only be male or female. For other dimensions, disaggregation entails making a judgment to how much detail is necessary; age can be divided by single years or 10-year intervals, industry can be divided into the nine 1-digit sac codes or the thousands of 7-digit sac codes. Each additional level of disaggregation of a dimension of the work force adds not one additional type of labor but the sum of an the dis- aggregations of all the other dimensions. For example, given two sexes, three age groups, five industries, four occupations, two education levels, and two worker classes, the total number of labor types that would be the product of all these is 480. If we disaggregate age by four age groups instead of three, there would be 640 types of labor. Consequently, cross-classif~cation and disaggregation of each dimension puts great demands on the availability and handling of data. More precisely,

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Evaluation of Work-Force Composition Adjustment 343 data availability will restrict the number of dimensions and disaggrega- tion levels. AVAILABLE DATA A number of data sources are available for compiling the information needed, but none of these sources meets all the requirements outlined above. We will review each data source in terms of the information it offers and the information lacking. The two most important data sources available on a monthly basis are the current employment statis- tics (790 survey) and the current population survey (cPs). The appendix (Table A1) summarizes the surveys described in this section. CURRENT POPULATION SURVEY The cPs is conducted every month for the week that includes the twelfth of the month. Approximately 57,000 households are interviewed. The March survey collects annual earnings, and the May survey collects weekly and hourly earnings. This is the only survey and only data source other than the decennial census that collects information on all the dimensions of the labor force we have outlined. This information includes employment and hours worked for 48 industries (51 after 1975), 21 occupations, sex, five worker classes, age by S-year intervals, and education by number of years up to 16 (open class above 16 years). In order to use the cPs it would be necessary to limit the number of dimensions to an order that would allow enough degrees of freedom to make estimates from the sample statistically significant. If all the dimen- sions were used at the level of disaggregation collected by the survey, there would be more than 2,000,000 types of labor from a sample of just over 100,000 people, which is hardly appropriate. The number of dimensions or the level of disaggregation could be limited in order to achieve enough degrees of freedom in two different ways. One is to assign a zero value a priori to some types of labor. For instance, one would not expect to find teenagers with advanced graduate degrees working as laborers in the mining industry. The second method of limiting the level of disaggregation is to test aggregated dimensions against disaggregatecl dimensions for reliability and significance. s The earnings measures from cPs are not as reliable as the measures of other surveys. The annual earnings information collected refers to sThis problem is discussed further in the concluding remarks.

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344 PAPERS the earnings of the previous year. Misreporting often occurs because the interviewed person is not the wage earner, but another member of the household. The earnings data do not include employers' contributions to social security and other benefits, although adjustments can often be made for these discrepancies. Current Employment Statistics (790) Survey The 790 survey, also called the establishment survey, is another monthly survey referring to the week that includes the twelfth day of the month. This survey covers approximately 165,000 nonagricultural establish- ments. It provides data on employment, hours, and earnings for produc- tion and nonsupervisory workers. For nonproduction and supervisory workers, it only provides employment. The extensive coverage of the 790 survey and its comparability to the National Income and Product Accounts (NIPA) prepared by the Bureau of Economic Analysis (BEA) make it one of the most significant sources of information for productivity analysis. It provides reliable data for 3- and 4-digit sac levels for the manufacturing industries. The output information for productivity measures is derived from the NIPA, and it is highly desirable to have comparable input data. The 790 survey provides this comparability. A number of problems arise in using the establishment survey. The survey does not collect hours and earnings data for nonproduction and supervisory workers. It does not survey the agricultural sector. It collects data on hours paid for, rather than on hours worked. Finally, and this is most important to the composition measure, it does not disaggregate data by demographic classifications other than sex. These deficiencies, however, do not altogether preclude the use of the 790 survey. By using the survey's aggregate measures as control totals for the cPs disaggregated measures by individual industries, data com- parable to the NIPA disaggregated by industry and demographic classifi- cation could be prepared on an annual basis. 6 At present, the BUS iS studying the possibility of modifying the 790 survey with a periodic supplement to collect some or all of the following information: (1) hours worked by production workers and nonproduc- tion workers and (2) hours paid and earnings of nonproduction workers. Reporting would be on an annual or quarterly basis. This modification 6 Using the cPs and the 790 survey would entail developing a control adjustment model of some type.

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352 PAPERS Method of Aggregation The aggregation function used to compute adjusted labor input is a transcendental logarithmic (translog) function (Christensen et al. 19731. Although it is assumed to be homogeneous of degree one, it does not assume constant elasticities of substitution between different types of labor. This is a less restrictive assumption that is made by either the Cobb-Douglas or the constant elasticity of substitution (CES) model. The aggregation of this type of function is estimated using a discrete approxi- mation to a Divisia index (Diewert 1976~. Data To compute Divisia indexes, one needs not only total hours worked but also compensation per hour by each dimension. The following four matrices of 81,600 cells were computed for each year: an employment matrix, an hours matrix, a weeks worked per year matrix, and an average compensation per hour matrix. To construct these matrices for all four components of labor input for each year (1947-1973), the authors used a multi-proportional matrix model generalizing the RAS method (Stone and Brown 1962~. The statis- tical principles underlying this model are an extension of those that underlie the bi-proportional matrix model of Bacharach (19651. The data used in this model were derived from Census Bureau reports and special labor force reports (which are estimated from the cPs) for all the demographic and industry classification. These data were than controlled to establishment-based (790) survey data so that they would correspond to the output data of the NIPA. Results Rates of change of labor input for the period 1947-1973 and two sub- periods as estimated by Gollop and Jorgenson are shown in Table 5, along with annual index numbers for the same period. WALD ORE Waldorfs study is very limited in its coverage of the economy and annual information. Composition adjustment coefficients were calculated for several 2-, 3-, and 4-digit sac level industries for discrete and unrelated time intervals. These industries are some of those for which BUS pub

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Evaluation of Work-Force Composition Adjustment 353 fishes separate productivity measures. The dimensions considered were occupation and sex for production workers only, although there is a long discussion of education levels and some empirical evidence on them. Occupation was disaggregated by 26 different types. Waldorf uses a discrete approximation to the Divisia index for aggregation, thus as- suming a translog production function. The data were constructed from the BES Industry Wage Surveys. They included average earnings and employment cross classified by sex and occupation. Table 4 gives the results of this study. As can be seen, adjustment for sex and occupation produced little change in estimated labor input for most of the selected industries and time intervals. SUMMARY A comparison of the results of adjusted labor input as derived by Denison, Gollop and Jorgenson, and Kendrick, along with an hours worked series developed by BUS are shown in Table 5. These results are not strictly a comparison of the different dimensions used, but also a comparison of different methods of aggregation and adjustment. Consequently, one must make comparisons with care. For the period 1947-1966 (the time span of Kendrick's study), labor input was calculated as increasing by 1.32 percent annually by Gollop and Jorgenson, 1.12 percent annually by Denison, and 0.92 percent annually by Kendrick. These increases are all substantially larger than the annual increase in unweighted hours calculated by BES, 0.63 percent annually. Denison found that the adjustment for labor composition contributed to labor input at an annual rate of 0.65 percent from 1947 to 1969. Gollop and Jorgenson found that labor composition contributed to labor input at an annual rate of 0.74 percent for the same period. However, they also found that changes in composition increased labor input by 0.66 percent from 1947 to 1973. Thus there was an annual increase of only 0.25 percent from 1969 to 1973, a substantial decline in the rate of growth. Some of the difference in the composition adjustment between Denison and Gollop and Jorgenson is undoubtedly attributable to the fact that Denison did not adjust his labor input for changes in its industry com- position. That this dimension adds to the size of the composition adjust- ment is seen by comparing Kendrick's measure with the BUS measure of hours. Kendrick, who adjusts for industry alone, found that labor input grew by almost 50 percent more than BES, 0.92 percent compared with 0.63 percent.

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356 TABLE 5 Growth Rates of Labor Input: Private Domestic Business Economy PAPERS Hours Worked Composition Adjustment Labor Input Gollop Gollop Gollop Deni- and Deni- and Deni- and Ken BES son Jorgenson son Jorgenson son Jorgenson duck 1947-1966 0.63 0.46 0.56 0.66 0.79 1.12 1.32 0.92 1947-1969 0.79 0.57 0.68 0.65 0.74 1.22 1.42 - 1947-1973 0.90 - 0.76 - 0.66 - 1.43 - Body of table shows annual percentage growth rates. CONCLUSIONS AND RECOMMENDATIONS It is evident that these studies consider work-force composition adjust- ment in one form or another to be important in the measurement and explanation of productivity change. Denison adjusts only for the demo- graphic characteristics (the supply characteristics), while Kendrick adjusts using only industry mix (possibly a quality effect, but also possibly a measure of increased capital used with labor and/or improved resource allocation among industries). Gollop and Jorgenson combine all the Denison-Kendrick dimensions along with occupation. All these studies show that there is a significant difference between the unadjusted hours used by the BLS and adjusted hours. The questions that need to be answered are, What are the important dimensions of adjustment and can they be measured for all time intervals? What is the appropriate aggregation method? Can reasonably accurate data be derived for the dimensions considered and the method of aggregation used? DIMENSION The studies reviewed suggest that (1) education and experience are important dimensions for measuring labor quality, (2) class of worker is important because the earnings estimates for the self-employed include returns to capital as well as to labor, (3) age and sex need to be examined in relation to individual industries over specific time periods, (4) occupa- tion seems to be the least important dimension, and (5) industry may be a useful dimension for measuring labor input quality change; work must

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Evaluatio,' of Work-Force COM] positio'' A djustme'' i 357 be done on the inter-industry wage structure to see how much reflects nonquality factors. A study of age and sex by industry could be under- taken using the Gollop and Jorgenson data. However, it would first be advisable to test the significance and sensitivity of the multi-propor- tional matrix model. AGGREGATION Several econometric studies at both the industry level and the aggregate economy level have estimated the proper functional form of the aggre- gation function (Berndt and Wood 1975, Berndt and Christensen 1973, Mohr 1975~. All of these studies (both cross-sectional and longitudinal) suggest that the Cobb-Douglas function is not the proper form. Although some of these studies also suggest that a homogeneous function might not be proper, the Divisia index method (translog function) is probably the least restrictive specification. DATA If the number of dimensions could be sufficiently limited and still pro- vide accurate measures of work-force composition, then the cPs com- position data could be controlled to establishment-based industry totals, which correspond to the NIPA estimates of output. If these restrictions are not possible, then there should be further study using the multi- proportional matrix model. REFERENCES Bacharach, Michael (1965) Estimating non-negative matrices from marginal data. International Economic Review 6(September):294-310. Berndt, Ernst R., and Christensen, Lauritz R. (1973) The translog function and the substitution of equipment structures, and labor in U.S. manufacturing, 1929-1968. Journal of Econometrics 1(1):81-114. Berndt, Ernst R., and Christensen, Lauritz, R. (1973) The internal structure of func- tional relationships: separability, substitution, and aggregation. The Review of Eco- nomic Studies 40(July):403-410. Berndt, Ernst R. and Wood, David O. (1975) Technology, prices, and the derived demand for energy. Review of Economics and Statistics 57(August):259-268. Blackorby, Charles, and Russell, Robert R. (1976) Functional structure and the Allen partial elasticities of substitution: an application of duality theory. The Review of Economic Studies 43(June):285-292. Bureau of Labor Statistics (1976) BLS Handbook of Methods for Surveys and Studies. Bulletin 1910:219-231. Washington, D.C.: U.S. Department of Commerce.

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358 PAPERS Christensen. L. R.. J`'rgens`'n, D. W. and Lau. L. J. (1973) Transcendental Logarith- n, ic let odu cti`,n fr`'n tiers. Row ol Ec`~n'i~ s `~! Sadistic s 55 ( Feb ~ ~` al y): 28-45. Denison, Edward F. (1967) Why Growth Rates Differ. Washington, D.C.: Elrookings Institution. Denison, Edward F. ( 1 974) Accou,'~i,~g Jor U. S. Economic Growth 1929-1969. Washing- ton, D.C.: Brookings Institution. Diewert, W. E., (1976) Exact and superlative index numbers. Journal of Econometrics 4:115- 145. Gollop, Frank, and Jorgenson, Dale (1980) U.S. productivity growth by industry 1947-1973. In John W. Kendrick and Beatrice N. Vaccara, eds., New Develop~ne''ts i'' Productivity Measurement and Analysis. NBER Studies in Income and Wealth, Vol. 44. Chicago: University of Chicago Press. Jorgenson, D. W., and Gollop, F. (1977) U.S. productivity growth by industry: 1947- 1973. Social Systems Research Institute, University of Wisconsin-Madison. SSRI Workshop Series 7712(September): 17-77. Jorgenson, D. W., and Griliches, Z. (1967) The explanation of productivity change. The Review of Economic Studies 34(July:249-282. Kendrick, John (1973) Postwar Productivity Trends i'' the United States. 1948-1973. Prepared for the National Bureau of Economic Research. New York: Columbia Uni- versity Press. Mohr, M. F. (1975) The long-term structure of production, factor demand, and factor productivity in U.S. manufacturing industries. National Bureau of Economic Research Conference. New Developments in Productivity. NBER Income and Wealth Series, forthcoming. New York: National Bureau of Economic Research. Russell, Robert R. (1975) Functional separability and partial elasticities of substitution. The Review of Economic Studies 42(January):74-84. Stone, R., and Brown, J. A. C. (1962) A Computable Model ok Ecc'''o'?'ic Growth: A Programme for Growth. Vol. 10. London: Chapman and Hull. Waldorf, William (1973) Quality of labor in manufacturing. Revien, oJ Economic Statistics 55(August):284-291.

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