Input demand elasticities for heterogeneous labor: firm-level estimates and an investigation into the effects of aggregation.

I. Introduction

Labor economists have paid increasing attention to the estimation of demand elasticities for heterogeneous labor.(1) The majority of these studies investigate the effect on wage rates resulting from exogenous changes in factor quantities; i.e., the measurement of elasticities of complementarity. Examples of these studies abound. Borjas, for example, investigates the impact of differences in labor market representation on the earnings of racial groups [8]. In another paper, Borjas defines immigrants as a separate labor group and is able to determine the impact of immigration on the wages earned by natives [9]. Berger defines labor inputs along education and gender lines, thereby allowing for the investigation of changes in the age/education composition of the work force on wage rates [5]. And, Grant and Hamermesh study the effect of increasing female labor force participation on the wages earned by youth and adults [17].

Less attention has been paid to the study of the responsiveness of employment to exogenous changes in wage rates; i.e., the measurement of elasticities of substitution. Less attention has been focused on this issue because the data requirements are more stringent since, in order to presume that wage rates are exogenously determined, the unit of observation must be small. Ideally, establishment- or firm-level data would be used. The relative inaccessibility of these type of data has led researchers to use more aggregated figures, such as SMSA-level or industry-level data, to address micro- issues. As long as wage rates are determined exogenously, the unit of observation chosen is appropriate. There are real concerns, however, about the adequacy of more aggregated units to describe the behavior of firms.(2)

This study addresses the issue of whether micro-level data are, in fact, needed to accurately measure elasticities of substitution among labor groups. A firm-level data set that includes input levels and prices for 1550 large U.S. companies in 1980 is used. In keeping with the spirit of the recent literature, labor groups are defined along race and gender lines.(3)

The aggregation issue is tackled in a novel, yet straight-forward, manner. To determine the impact of aggregation on the ability of data to reflect firm-level behavior, input demand systems are estimated for firms aggregated into three-digit and two-digit industry groups. This paper finds that the use of the more aggregate data results in parameter estimates that are significantly different from the firm-level estimates. For this sample of firms and for the empirical specification employed, industry-level data do not adequately describe the input hiring behavior of firms.

The paper is organized in the following way. The next section presents the methodology used to estimate input demand elasticities. Section III describes the data used. Firm-level input demand elasticities are shown in section IV. Section V includes the discussion of the effects on demand elasticities from aggregation. Conclusions are provided in section VI.

II. Theoretical and Methodological Framework

It is assumed that the production technology used by firms is embodied in a translog cost function.(4) A cost function is estimated rather than a production function because the firm-level data used in this study imply that factor prices, rather than factor quantities, should be treated as exogenous. In estimating a cost function it is also assumed that the output level is fixed. The resulting parameter estimates, therefore, represent substitution possibilities along an isoquant and ignore scale effects. The translog cost function is:

ln C = [[Alpha].sub.o] + [[Alpha].sub.Y](ln Y) + (1/2)[[Beta].sub.YY][(ln Y).sup.2] + [[Sigma].sub.i][[Beta].sub.Yi](ln Y)(ln [W.sub.i]) + [[Sigma].sub.i][[Alpha].sub.i](ln [W.sub.i]) + (1/2)[[Sigma].sub.i][[Sigma].sub.j][[Beta].sub.ij](ln [W.sub.i])(ln [W.sub.j]), (1)

where C is total cost, Y is output and [W.sub.i] is the price of factor i, i = 1 ... n. The cost function is constrained to be symmetric and homogeneous of degree one in factor prices. Further, it is assumed that firms face perfectly competitive input and output markets.

The partial derivative of a cost function with respect to a factor price is the constant-output demand equation for the factor, [Delta]C/[Delta][W.sub.i] = [X.sub.i], where [X.sub.i] is the quantity of factor i hired. In logarithmic form,

[Delta](In C)/[Delta](ln [W.sub.i]) = [W.sub.i][X.sub.i]/C = [S.sub.i], (2)

where [S.sub.i] is the share of factor i in total cost.

Differentiating (1) with respect to the log of factor prices yields

[S.sub.i] = [[Alpha].sub.i][[Sigma].sub.j][[Beta].sub.ij](ln [W.sub.j]) + [[Beta].sub.Yi](ln Y). (3)

Both the cost function, (1), and the share equations, (3), yield information about the parameters of interest, so they are estimated jointly using iterated seemingly unrelated regression (ITSUR). One share equation is dropped, making the covariance matrix of the disturbances non-singular. ITSUR estimation guarantees that the parameters estimated are invariant to the share equation dropped.

Parameter estimates obtained from (1) and (3) can be transformed into cross-wage elasticities,

[[Eta].sub.ij] = ([[Beta].sub.ij]/[S.sub.i]) + [S.sub.j], for i [not equal to] j (4)

[Mathematical Expression Omitted].

The cross-wage elasticities measure the responsiveness of factor quantifies to changes in factor prices, holding output and other factor prices constant and is the appropriate elasticity measure when firm-level data are used. Factors i and j are substitutes if [[Eta].sub.ij] [greater than] 0 and are complements if [[Eta].sub.ij] [less than] 0.(5)

Equations (1) and (3) yield estimates of the parameters determining the cross-wage elasticities between factors, (4), and own-wage elasticities, (5). Two sets of point estimates are derived. One set evaluates the point estimates for the elasticities at the mean shares for each factor. Standard errors for these elasticities are derived assuming that the mean shares are non-stochastic.(6) The second set evaluates the mean elasticity for the sample; i.e., elasticities are derived for each firm (or industry, depending on the level of aggregation) and the mean of each elasticity for the sample is reported. These two measures will not generate identical estimates due to the nonlinear nature of (4) and (5). Both sets of estimates are presented as a means of improving the robustness of the conclusions drawn from the estimation procedure.

The standard errors for the elasticities evaluated at the mean shares are:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is the mean cost share for factor i in the sample.

The standard errors for the mean elasticities are

S.E. ([[Eta].sub.i]) = S.E. ([[Beta].sub.ii]) [[Sigma].sub.k](1/[([S.sub.i]).sub.k])/n, S.E. ([[Eta].sub.ij]) = S.E. ([[Beta].sub.ij]) [[Sigma].sub.k](1/[([S.sub.i]).sub.k])/n, (7)

k = 1 ... n.

III. Data Description

The data used in this study are firm-level input and output measures for 1550 relatively large firms in the year 1980. These firms employed in excess of 15 million workers at the time the data were gathered.

Employment data for rums are obtained from annual reports that virtually all large private-sector firms must file with the Equal Employment Opportunity Commission (EEOC). These reports, which are called EEO-1 reports, detail the composition of a firm's labor force along race, gender and occupation lines and are used to monitor a firm's compliance with Title VII of the Civil Rights Act of 1964.(7) The employment data, however, do not include employee characteristics other than race, gender and occupation. Age, experience, education and earnings are among the more important employee characteristics that are not available in the EEO-1 data.

Output and capital stock measures are computed using data contained in the Compustat data series, which includes the financial characteristics of most publicly traded firms. The firm's capital stock is computed as the value of the firm's net property, plant and equipment.(8) A firm's output is derived as a firm's net sales plus the change in the firm's inventories from the past to the current year.(9)

A firm is included in the sample if it filed an identifiable EEO-1 report and if it appeared in the Compustat data set with identifiable output and capital stock measures in 1980. Of the more than 15000 firms filing EEO-1 reports and 5800 firms included in the Compustat data, 1550 firms could be successfully matched.

In order to investigate the theory outlined in the previous section, factor prices for the labor inputs and capital are necessary. Unfortunately, neither the EEO-1 nor Compustat data contain reliable wage data. It is necessary, therefore, to use another data source to generate wage data. The A Sample (5%) of the 1980 Census is used to calculate wage rates for each labor group. Industry-specific wage rates for race, gender and occupation groups are computed. In order to have a reasonable number of observations per cell, it is necessary to reduce the number of potential race/gender/occupation/industry combinations.(10) The wage data, as a result, are aggregated into 744 cells, composed of 3 race, 2 gender, 4 occupation and 31 industry groups.(11)

The compensation rates that firms pay in employing workers include wages and non-wage benefit costs. In order to account for all employment costs, the Wage rate for each group is multiplied by the ratio of compensation to wages in each job group and industry using data from the National Income and Product Accounts [26]. The resulting figures reflect the total employment costs per worker to a firm. The wage costs used in this study are derived from the cell averages of compensation costs for the race, gender, occupation and industry groups described above.(12,13)

Many previous labor demand studies have aggregated labor groups along race and/or gender lines because of the increased attention paid of late to the changing labor market conditions facing minorities and women.(14) Some sort of aggregation must be conducted in this study as well because of the empirical intractability of a model that includes 24 labor groups. The 24 race/gender/occupation cells derived so far are aggregated into six race/gender groups, where the wage and quantity indices for each group are weighted averages of the individual component groups.(15) Since each firm uses a different mix of each race/gender/occupation, each firm faces a different effective compensation rate for each race/gender group. This heterogeneity between firms transforms the industry-specific race/gender/occupation compensation rates into firm-specific race/gender compensation rates.(16)

The use of a wage index to transform industry-level compensation rates into firm-level compensation rates is consistent with the assumption that firms face perfectly competitive input markets.(17) The overall compensation costs to a firm are then simply an appropriately weighted average of (exogenously determined) compensation rates facing a firm for each input.

The correct choice of aggregation methodology is determined by the production technology that is assumed. When the underlying technology is embodied in a translog cost function the occupation groups should be aggregated in a way that is consistent with this functional form. Diewert has shown that the Tornqvist wage index is consistent (exact) when a translog cost function represents the underlying technology [11].(18)

The Tornqvist wage index is

[W.sub.jk] = [[Pi].sub.i]([W.sub.ijk]/[W.sub.ijo]) exp((1/2)[S.sub.ijo] + (1/2)[S.sub.ijk]), (8)

where j indexes the race/gender group, k the firm, i the occupation group within the race/gender classification, o the mean for the entire sample and S the share of total costs for the group. Associated with the Tornqvist wage index is a quantity index,

[L.sub.jk] = [C.sub.jk]/([C.sub.jo][W.sub.jk]) (9)

for each race/gender group. These firm-specific Tornqvist wage and quantity indices are used to represent compensation costs and employment levels for each firm.

The cost of capital to a firm is equal to the rate of return on capital, where the rate of return reflects both the income generated by the capital stock and the depreciation rate of the capital stock. The Department of Commerce has published estimates of the rate of return on capital for many two-digit industries [27]. These rates of return are used as capital prices for each firm within an industry.

In summary, the data used in this paper are mostly firm-level data containing information on the firms' labor force compositions and labor costs, the firms' capital stocks and capital rental rates, from which are derived the firms' total costs of hiring inputs, and the firms' output levels for 1980.(19)

IV. Firm-Level Input Demand Elasticity Estimates

The labor demand literature, according to Hamermesh [21; 22], has yielded several important findings. These include the results that own-price demand elasticities are less elastic the more general human capital is possessed by a group of workers and the more skilled is a group of workers, the more complementary the group is with capital.(20)

The results presented here address these points to the extent that race/gender groups can be differentiated on the basis of the amount of general human capital that groups possess or on the average skill levels of members of each group. Because of the well-known differences in occupational distributions among labor groups, care must be exercised in interpreting own-wage and cross-wage elasticities since the elasticities may, in fact, be capturing elasticities between occupation groups instead of between race/gender groups.

The labor demand elasticities of interest are presented in Table I. The own- and cross-wage elasticities are derived from the estimated parameters using equations (4) and (5). Column 1 shows the elasticities calculated at the mean shares for the sample. Column 2 shows the mean of the elasticities, when elasticities are calculated for each firm. For example, in Column 1, the own-wage elasticity for white males is [Mathematical Expression Omitted]. The standard error for this statistic is [Mathematical Expression Omitted]. In Column 2, the own-wage elasticity for white males is [Mathematical Expression Omitted]. This statistic has a standard error of S.E. ([[Beta].sub.wm]) [[Sigma].sub.i](1/[([S.sub.wm]).sub.i])/n.

Examination of the estimates in Column 1 shows that the results strongly support previous findings. All of the estimated own-price elasticities are negative. The white male, white female, black female and capital own-price elasticities are the only estimates significantly different from zero, however. The observation that factors that possess more general human capital are less elastically demanded appears to be supported by these results. Although measures of labor market experience are not included in this data, males have, on average, more labor market experience than females. The elasticity estimates for males are all less elastic than those of females and, perhaps more interestingly, the relative inelasticity of male own-price elasticities holds within races as well.

Somewhat surprising is the observation that the demand for minority males is less elastic than for white males. The discussion in the previous paragraph would suggest that, since the average white male has more labor market experience than the average minority male, the demand for white male labor should be less elastic than for minority male labor. If, however, firms face [TABULAR DATA FOR TABLE I OMITTED] exogenously-imposed hiring constraints that encourage some minimum level of minority representation, such as Affirmative Action or Title VII, then the effect may be to lower the demand elasticities for minority labor groups. The presence of less elastic demands for minority males than for white males, and of minority females than white females, is consistent with this hypothesis.(21)

The cross-wage elasticity estimates display several findings:

* White males and white females are strong substitutes in production.

* White females and minority males are also substitutes in production.

* Minority males and minority females are complementary factors of production.

The capital-skill complementarity hypothesis is supported. White males, the most skilled labor group, on average, are the labor group that is least substitutable for capital, with the puzzling, but not statistically significant, exception of Hispanic females. Less skilled labor groups, black males, for example, are found to be strong substitutes for capital.(22)

The average elasticities shown in Column 2 present very similar qualitative conclusions. The results are, therefore, quite robust to the method used in evaluating elasticities. The main difference between the two sets of estimates is that some mean estimates are not statistically significant (Column 2) while their estimates at the mean (Column 1) are. This is true for the black female own-wage elasticity, and the black female-capital and black male-capital cross-wage elasticities.

The estimates in Table I are in agreement with much of the existing literature. As such, this corroboration of earlier results for heterogeneous labor using micro data is important. The methodology used in this study is not immune from criticism, however. One issue is that there may be systematic differences between the input hiring decisions made by the firms used in this paper and the representative firm in the economy. These differences may arise because it may be easier for larger firms to respond to changes in the price of any one input because larger firms typically employ a broader range of occupation types than do smaller firms. A second point is that all of the firms used in this paper are subject to the anti-discrimination guidelines of Title VII of the Civil Rights Act of 1964, and roughly half are subject further to the (even stricter) regulations imposed by affirmative action. Griffin demonstrates that compliance with affirmative action hampers the ability of firms to substitute between inputs [18].

These concerns are not testable with the data available. The estimates shown in this paper are descriptive of the representative firm in the economy to the extent that the net impact on estimated elasticities arising from the nonrandom selection of the firms into the sample is small.(23)

V. Industry-Level Input Demand Elasticity Estimates

The second point of focus in this paper is to determine the impact on input demand elasticity estimates from aggregating firms into broader units of observation. It is of clear importance to ascertain the magnitude and direction of the biases introduced by aggregation. If the elasticity estimates change significantly upon aggregation then the bulk of the research conducted using aggregated units to estimate micro-behavior must be viewed with concern. On the other hand, if the estimates are robust to changes in the unit of observation then researchers that have access only to data for aggregated units can have confidence that their estimates do, indeed, capture input demand behavior by firms.

The analysis proceeds in two phases. First, the 1550 firms in the sample are aggregated into 243 three-digit industry groups. Input demand elasticities from this relatively disaggregated sample are presented and tests to determine the validity of aggregation are conducted. Second, the firms are aggregated into 62 two-digit industry groups and the elasticities of concern are estimated from this relatively aggregated sample. Similar aggregation tests are then conducted for the two-digit industry sample.

Table II presents input demand elasticities estimated from the sample of firms aggregated into three-digit industry groups.(24) The two sets of elasticity estimates are derived in the same way as in Table I; i.e., Column 1 shows elasticities evaluated at the sample means and Column 2 presents the means of each elasticity calculated for each of the 243 industry groups. Starting with Column 1 of Table II, perhaps the most striking aspect of the estimates is the close correspondence with the firm-level estimates in Column 1 of Table I. Also notice that input demand elasticities are estimated with much less precision when using three-digit industry aggregates.

All of the own-wage elasticities are negative, but only three are statistically significant. The point estimates of the cross-wage elasticities show similarity to firm-level estimates as well. White males and white females are still strong substitutes in production. White females and Hispanic males are also substitutes, but the substitutability between white females and black males is no longer apparent. Minority females and Hispanic males are complements, as before. The complementarity between black males and black females found at the firm-level, however, is not present at this level of aggregation.

An examination of the mean elasticities (Column 2) yields similar conclusions. There are two estimates (the capital own-wage elasticity and the black female-capital cross-wage elasticity) that are not statistically significant but that are significant when evaluated at the means. A comparison of the mean elasticities estimated for the three-digit industry groups and the firm-level estimates (Table I, Column 2) shows that the main difference by aggregation is loss of precision.

These observations lead to the following conclusion: All of the statistically significant elasticities found at the three-digit industry level are also found at the firm level. But, some of the statistically significant elasticities found using firm-level data are not discovered by the three-digit industry data. It is the case, therefore, that the use of three-digit industry-level data will not lead the researcher to make incorrect conclusions regarding significant elasticities at the firm level. The use of the aggregated data, however, does not allow the researcher to identify all of the significant elasticities at the firm level.

A formal test of the effect of aggregation on parameter estimates is to compare the residuals from the firm-level translog cost system when unconstrained to the residuals derived when the three-digit industry-level estimates are imposed. The appropriate test, outlined in Gallant [16, 367-70], is a likelihood ratio test. The test statistic is derived by calculating the difference in the determinant of the covariance matrix of the residuals between the constrained and unconstrained [TABULAR DATA FOR TABLE II OMITTED] models and multiplying by the number of observations. The critical value for the test statistic, which is distributed [[Chi].sup.2] with 21 degrees of freedom, is 38.9 at the 99 percent level of confidence. The actual value of the test statistic is 60.55. It must be concluded, therefore, that the aggregation of firms into three-digit industry groups for the purpose of estimating input demand elasticities is rejected by the data.

Table III shows input demand elasticities estimated from the sample of firms aggregated into 62 two-digit industry groups. These estimates differ more from the firm-level results than do the elasticities derived from the three-digit industry sample. Many of the same qualitative comments that were made about the firm-level results, however, apply to the results from the two-digit industry sample as well.

An analysis of the estimation of elasticities at sample means shows that all own-wage elasticities for men, both within and across races, are less elastic than for females. White men and white females, white men and black females, white men and Hispanic females, white females and Hispanic males, white females and capital, and black females and capital are all substitutable factors. [TABULAR DATA FOR TABLE III OMITTED] Hispanic males and Hispanic females are complements. And, the capital-skill complementarity hypothesis, if anything, is more strongly supported.

With minor exception, the same observations apply to the mean elasticity estimates in Column 2.

As was the case with the three-digit industry-level data, but to a greater extent, the aggregation of firms to the two-digit industry level masks some of the relationships between inputs found at the firm level. Part of this is due to the lack of precision of the estimates induced by the much smaller sample size of the aggregated sample while part is due to the inability of the aggregated data to reveal these relationships. Aggregation to the two-digit level does not, however, lead to the misinterpretation of the relationships between any inputs; i.e., all inputs found to be substitutes at the two-digit level are also found to be substitutes at the firm level. The same can be said for factors found to be complements.

The aggregation of firm-level data to the two-digit industry level is rejected by the data. The calculation of the likelihood ratio test statistic described above yields a value of 163.69. Two-digit industry-level data is not, therefore, a statistically valid substitute for firm-level data when used to estimate input demand elasticities.(25)

VI. Conclusion

This paper uses firm-level input and output data to estimate input demand elasticities for heterogeneous labor (defined by race and gender groups) and to determine the effect on the demand elasticity estimates from aggregating firms into broader units. The firm-level data used here represent a significant improvement over the more aggregated, and, therefore, less appropriate, data used in many existing studies that explore the responsiveness of firms' hiring decisions to exogenous changes in factor prices.

The labor demand elasticities derived in this study are obtained from the estimation of a translog cost system. The estimated firm-level elasticities reinforce, for the most part, the results of earlier work. In particular, most estimated own-price elasticities are negative with, in general, factors possessing more general human capital being demanded less elastically than factors with less general human capital.

The capital-skill complementarity hypothesis is supported; i.e., the more skilled a labor group is, the more complementary it is with capital.

Several notable substitutability relationships are found. White males and white females are very substitutable in the production process. White females and minority male groups are also close substitutes in production. Minority females and minority males, on the other hand, are strongly complementary inputs.

The firm-level data are then aggregated into broader units of observation to determine the impact of aggregation on elasticity estimates. The estimates generated by systems in which firms are aggregated into three-digit industry groups and two-digit industry groups share a common characteristic: All of the own-wage and cross-wage elasticities that are found to be statistically significant by the aggregated data are also significant at the firm level. There are many statistically significant elasticities at the firm level that are not found by the aggregated data, however. These observations imply that studies that use three- or two-digit industry-level data to estimate own-and cross-wage elasticities will not generate false conclusions about estimates that are found to be statistically significant but will falsely conclude that many input demand relationships are not significant.

The likelihood ratio tests conducted to determine whether firm-level parameter estimates are statistically similar to the estimates made by aggregated data reject the aggregation of firm-level data to both the three-digit industry level and the two-digit industry level.

The author wishes to thank Stephen Bronars, Philip Ganderton, Jane Gau, Lisa Grobar, Daniel Hamermesh, Jack Hou, Judith Roberts and Stephen Trejo for helpful comments. Finis Welch, Unicon Research, granted me access to the EEOC data. Scott Hayes provided helpful research assistance. All errors are mine.

1. See Hamermesh [21; 22] for thorough descriptions of the state of the literature.

2. There are many studies that use aggregated data to estimate elasticities of substitution. A list is provided in Hamermesh [22, Chapter 3]. Two examples are Freeman and Medoff [15] and Field and Grehenstein [14]. Both use 2-digit SIC data, by state.

3. There are an increasing number of studies that use micro-data to estimate input demand elasticities. Again, a thorough list is provided in Hamermesh [22, Chapter 3]. Most papers cover specific industries. To my knowledge, this is the first study to use firm-level data on a representative cross-section of industries in the United States.

4. There is a large literature of studies that estimate translog cost functions to derive substitutability relationships between inputs. Many are from the energy literature, including Berndt and Wood [6], Morrison and Berndt [24], and Baltagi and J. Griffin [4], for example. Other applications using translog cost function estimates include Allen [2; 3], Gyapong and Brempong [20] and Diewert and Wales [13].

5. Cross-wage elasticities, instead of (Allen) elasticities of substitution, are reported because of the concerns raised in Blackorby and Russell [7] about the adequacy of the elasticity of substitution to measure ease of substitution between factors of production.

6. The assumption that mean factor shares are non-stochastic is a simplification that linearizes the calculation of standard errors. This assumption implies that income shares for each race/gender group in the economy are fixed. An alternative is to present the T-statistics for the parameters estimated from equations (1) and (3). The significance levels implied by the latter are quite similar to those implied by the former.

7. See U.S. Commission on Civil Rights [25] for a description of the EEOC, EEO-1 reports and other federal anti-discrimination programs.

8. As with all capital measures, a capital utilization rate would improve the accuracy of this variable since "capital services" is the true input into the production process. The value of the capital stock is a helpful proxy, particularly if there is little variation in capital utilization rates across firms.

9. The output measure is actually the total revenue for the firm. The data do not allow for the separation of prices and quantities.

10. 30 observations is chosen as the minimum acceptable number of observations per call.

11. The races are white, which include Asians and American Indians, blacks and Hispanics. The occupations include two white collar and two blue collar groups. The nine census occupations are aggregated together on the basis of similarity of skills and responsibilities inherent in each occupation. The 89 two-digit industries are aggregated into 31 composite industries on the basis of the similarity of products produced in the industries.

12. A complete description of the construction of the compensation rates is available from the author upon request.

13. In order for the calculated wages to be valid for estimating firm-level labor demand relationships, observed differences in wages across industries must be exogenously determined. If wage differences are endogenous, due to differences in employee characteristics across industries, for example, then wage rates across industries will not reflect the cost of hiring the same quality worker and the estimation process will be biased.

14. This study follows the existing line of research on public policy issues by making groupings by race and gender. Borjas [8], Merrilees [23], and Grant and Hamermesh [17] are examples of similar studies.

15. Researchers must sort heterogenous laborers into labor groups. The only strictly valid labor group classifications are those for which all workers within each group are identical in their productive capacities. The use of labor groups defined by race and gender is acceptable to the extent that group members are similar in their labor market skills.

16. Heterogeneity between firms in a two-digit industry may arise from (small) differences in production technology, for example.

17. To the extent that the wages imputed to firms from industry-level aggregates are biased, the parameters estimated from the translog cost system will likely be biased downward (in absolute value terms). This, in turn, will underestimate the labor demand elasticities of interest.

18. See Diewert [11; 12] and Allen [1] for information on index number theory.

19. It is of interest to determine whether the data collected are consistent with cost-minimization. Given that the data set is constructed from a variety of sources, there is a possibility that the variables generated are not comparable. A non-parametric test of the Strong Axiom of Cost Minimization (SACM), as described by Varian [28], is conducted. It is found that the data are consistent with cost minimization for 87.4% of the comparisons made between firms. These results give the impression that the data generated do capture, in large part, optimizing behavior by firms in the sample.

20. An example of a study using firm-level data which supports these findings is Denny and Fuss [10]. Their data is a time-series for a single firm, however, so the applicability of their results to other samples is of interest.

21. See Griffin [18; 19] for supporting evidence.

22. As in any labor demand study, the measurement of the capital input and its price is problematic. The capital price measure used in this study is properly specified but is available only at the industry level. The capital input is measured as the dollar value of the equipment of the firm but incorporates no utilization rate of the equipment. Raising further concerns is that when factor demand elasticities are obtained from the estimation of a system of equations, errors in the measurement of one factor price can bias the elasticity estimates for all of the factors.

For these reasons, it is worthwhile to test if capital can be omitted from the estimation process without biasing the results. Capital can be omitted from the system if labor inputs are (weakly) separable from capital. It is found that the data do not reject the separability of labor inputs from capital.

The separability of labor from capital, therefore, allows for the estimation of labor demand elasticities for a system in which labor inputs are the only factors of production. The results for the labor-only model are quite similar to the system in which capital is included. The similarity in outcomes implies that the inclusion of capital into the system does not systematically affect the parameters estimated.

23. Another potential criticism of the methodology used in this paper is that no allowances are made in the estimation process for the industry to which each firm belongs. All firms are, therefore, assumed to operate using the same technology. While this is clearly not the case in actual production processes, the important matter is whether the assumption of identical technologies biases the elasticity estimates obtained.

There is substantial evidence that the imposition of a common technology does not significantly alter the results presented. Three pieces of evidence are given here: One, the inclusion of industry dummy variables does not qualitatively alter parameter estimates. It should be noted that the dummies are entered additively to the share equations and cost function, thereby imposing a rather strict form of technology differences. Two, results for a sample of 824 manufacturing firms are quite similar to the results presented. And, three, tests of the Strong Axiom of Cost Minimization (SACM) across industries yield similar success rates. Large technology differences would result in significantly higher rejection rates of SACM across industries than within industries. These differences are not found. In fact, the acceptance rate for SACM for across industries is 87.7% while the acceptance rate within industries is 89.7%.

24. Wages and employment levels are derived for both the three-digit and two-digit industry groups using the Tornqvist wage and quantity indexes (8) and (9).

25. It is worth noting that the data also reject the aggregation of firm-level data in systems in which capital is excluded. The critical value of the test statistic, with 15 degrees freedom for these tests, is 30.58 at the 99 percent level. The test statistic for aggregation to the three-digit level is 107.17 and for aggregation to the two-digit level is 147.82.

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