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

• Author: Griffin, Peter B..

1. 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.

2. 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.

3. 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...