r K a L b b f x ð Þ b ² 1 b ² 1 ð Þ @ f x ð Þ @ x 2 = f x ð Þ " ! þ @ 2 f x ð Þ @ x @ x # (11) where: @ f x ð Þ @ x ¼ x r þ l 1 ² x ð Þ r ½ 1 = r ² 1 x r ² 1 ² l 1 ² x ð Þ r ² 1 µ ¶ ¼ x r þ l 1 ² x ð Þ r ½ 1 = r ² 1 x r ² 1 1 ² l 1 ² x x · ¸ r ² 1 ! (12) and, most importantly, the second order derivative of the CES index is: @ 2 f x ð Þ @ x @ x ¼ r ² 1 ð Þ x r þ l 1 ² x ð Þ r ½ 1 = r ² 2 l 1 ² x ð Þ r ² 2 x r ² 2 (13) The sign of (11) is entirely determined by those of parameters β and ρ . The first parameter is nothing but the output elasticity with respect to total labour of the Cobb-Douglas part of the production function. And, presumably, in the presence of capital, it is inferior to 1. This means a diminishing marginal productivity for total labour ( L ). And, by extension, that law also applies to any quality-adjusted labour aggregate à-la HN. Assuming that labour types ’ marginal productivity differ significantly (i.e. in expression (12) l 1 ² x = x µ ¶ r ² 1 a 1 or just that λ ≠ 1 in case of perfect substitutability ( ρ ¼ 1)), then changes in the value of x amounts to changing the overall level of (quality-adjusted) labour. That logically translates into a fall of marginal productivity, captured in expression (11) by the term premultiplied by β − 1. The more interesting question is what happens with parameter ρ conditional on a certain value of β ; or, said differently, to determine whether the law of diminishing marginal productivity is positively (or negatively) affected by the diversity of the labour force. And that amounts to determining if ρ ≠ 1. If ρ o 1 we would conclude that diversity is good for efficiency ceteris paribus . If ρ W 1 then diversity is a bad thing for efficiency. And if ρ ¼ 1 diversity becomes irrelevant. 3. Econometric analysis 3.1 Data The empirical results of this paper derive from the analysis of two panels. The first one contains around 8,000 + firms with more than 20 employees. These firms are largely representative of the Belgian private economy in terms of sector/industry, and are well documented as to the capital they used and, their productivity performance. Using 258 IJM 37,2 Downloaded by UNIVERSITI TEKNOLOGI MALAYSIA At 19:25 23 December 2016 (PT)
firm identifiers, we have been able to add social security information on the age and gender of (all) workers employed by these firms, for a period running from 1998 to 2006. Table I presents the descriptive statistics. Of particular importance are the ones describing age and gender. Note in that age and gender diversity (as captured by the Herfindahl index) seems to have risen between 1998 and 2006. The second panel contains information about the educational attainment of the workforce. It comprises a slightly smaller number of firms (4,000 + ); also from all sectors forming the Belgian private economy. It runs from 2008 to 2012. Firms are also well documented in terms of sector, overall size of the labour force, capital used, and productivity (value added). But there is no information on the age and gender of the workforce that would allow a more refined breakdown of educational categories.
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