Firms, Labor Shares, and Capital–Labor Substitution (Chapter 5)

F Compustat data and additional empirical results regarding markups and output elasticities F.1 Data description, sample, and definitions We use data from Compustat from 1960 to 2016. We use the following variable definitions and conventions: • Revenue yf t : we measure revenue using firm sales—SALES in Compustat. • Expenditures in variable inputs vf t : we measure these expenditures using the cost of goods sold— COGS in Compustat. • Stock of capital kf t : we measure capital using the gross value of property, plants, and equipment—PPEGT in Compustat. • Investment xf t : we measure investment as the percent change in capital; that is, ln xf t = ln kf t+1 − ln kf t • Industry and firm groupings c(f ): we conduct our estimation separately for 23 NAICS industries, roughly defined at the 2-digit level. When grouping firms into size quintiles, we do so for each year and within each 3-digit NAICS industry. We also experimented with the classification of industries based on SIC codes used in Baqaee and Farhi (2020b) and obtained very similar results. • Sample definition and trimming: following De Loecker, Eeckhout and Unger (2020), we trim the sample by removing firms in the bottom 5th and top 5th percentiles of the COGS -to-SALES distribution. In addition, following Baqaee and Farhi (2020b), we exclude firms in farm and agriculture, construction, real estate, finance, and utilities from our markup and labor share calculations in Figures 7 to 9. • Winsorizing: we winsorize the obtained revenue elasticities at zero. Moreover, following Baqaee and Farhi (2020b), we winsorize our markup estimates at the 5th and 95th percentile of their distribution. F.2 Robustness checks This sub-section provides estimates for firm markups using different sets of assumptions. 72 Estimates parametrizing g and h using cubic polynomials We estimate elasticities under the same assumptions outlined in the main text, but parametrize g and h using cubic polynomials. Figure 12 plots the behavior of the resulting output elasticities over time by firm size quintile. Figure 13 reports the contribution of within-firm changes in markups and between-firm reallocation to (percent) changes in the labor share. Estimates assuming there are no ex-post shocks  In the absence of ex-post shocks, we can treat observed revenue as true revenue and there is no need to use a proxy variable to recover productivity. Instead, we can compute productivity directly as R zf t = ln yf t − εR vc(f )t ⋅ ln vf t − εkc(f )t ⋅ ln kf t , and proceed with the rest of the estimation in the same way as before. Figure 14 plots the behavior of the resulting output elasticities over time by firm size quintile. Figure 15 reports the contribution of within-firm changes in markups and between-firm reallocation to (percent) changes in the labor share. Estimates assuming a linear Markov process for productivity Suppose that productivity follows a linear Markov process zf t = βzf t−1 + ζf t . Define υf t ∶= zf t + f t . Because ex-post shocks are i.i.d, we have that υf t also follows a linear Markov process υf t = βυf t−1 + ζf t + f t − βf t−1 . ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ∶=ιf t Estimation proceeds as follows. First, we can compute υf t directly as R υ̃f t = ln yf t − εR vc(f )t ⋅ ln vf t − εkc(f )t ⋅ ln kf t . Then we estimate the linear model υ˜f t = β υ̃f t−1 + ι̃f t , 73 and base estimation on the moment conditions E [ιf t ⊗ (ln kf t , ln vf t−1 )] = E [(ζf t + f t − βf t−1 ) ⊗ (ln kf t , ln vf t−1 )] = 0. Figure 16 plots the behavior of the resulting output elasticities over time by firm size quintile. Figure 17 reports the contribution of within-firm changes in markups and between-firm reallocation to (percent) changes in the labor share. F.3 Biases of a sales weighted average of markups This subsection demonstrates that, when there are differences in output elasticities across firms that are not taken into account, a sales-weighted average of firm-level markups provides a biased assessment of the contribution of changes in markups to the decline in the labor share. First, as demonstrated in the main text (see footnote 27), the true contribution of markups to the labor share is given by µt ∶= 1/ ∑ ωf t ⋅ f 1 . µf t Suppose that, as is typically done in the literature, we compute firm-level markups under the assumption that all firms operate a technology with common output elasticities, εvt . This implies that the measured firm-level markups µm f t and the true markups µf t satisfy µm ft = εvt ⋅ µf t , εvf t where εvf t is the firm-level elasticity of output with respect to variable input. This implies that measured firm-level markups will over-state the true markups when εvt > εvf t and under-state them otherwise. Moreover, the average elasticity is given by εvt = ∑ ωf t ⋅ εvf t . f We now show that these firm-level biases do not cancel when one computes an arithmetic sales-weighted average of markups, and that this problem aggravates when there dispersion of εvf t rises. 74 The measured sales-weighted markup is given by m µ̄m t = ∑ ωf t ⋅ µf t . f After some algebra, one can show that µ̄m t µt ∑ f,f ′ yf t ⋅ yf ′ t ⋅ ( =1+ √s vf ′ t svf t − √s 2 vf t svf ′ t ) 2 . (∑f yf t ) This implies that the measured sales weighted markup exceeds the true markup, with equality if and only if there is no dispersion in factor shares svf t = svf ′ t . Moreover, this expression shows that the gap between the measured sales-weighted markup and the real markup rises with the dispersion in factor shares across firms, even if this dispersion is fully driven by differences in technology. In particular, if all firms had µf t = 1 and charged no markups, a sales-weighted average of markups computed under the assumption of common technology would be equal to µ̄m t =1+ ∑f,f ′ yf t ⋅ yf ′ t ⋅ ( √ε vf ′ t εvf t 2 − √ε vf t εvf ′ t 2 ) . (∑f yf t ) This sales-weighted markup not only exceeds 1, but would also increase spuriously with the dispersion in output elasticities, εvf t . Thus, the uneven adoption of capital intensive technologies, like the one produced by our model along the transition, can generate a spurious increase in the sales-weighted markup, even if there are no changes in market power. 75 Output elasticity wrt variable inputs, estimated for manufacturing firms in Compustat .95 .95 Output elasticity wrt variable inputs, estimated for firms in Compustat .9 1960-1979 .85 .85 .9 1960-1979 .8 1980-1999 .75 .8 1980-1999 .75 .7 2000-2016 1 2 3 4 quintiles of firm size 5 2000-2016 1 2 3 4 quintiles of firm size 5 Figure 12: Output elasticity with respect to variable inputs estimated using a cubic parametrization of g and h. The left panel presents estimates obtained for all Compustat firms. The right panel presents estimates obtained for Compustat firms in manufacturing. See figure 7 in the main text for our baseline estimates used in the results reported in the paper. .1 Other sectors, size-dependent technology .1 Manufacturing, size-dependent technology Contribution Within firm 1980 .05 0 -.05 -.1 -.1 -.05 0 .05 Reallocation 1990 2000 2010 1980 1990 2000 2010 Figure 13: Decomposition of the contribution of within-firm changes in markups and between-firm reallocation to (percent) changes in the labor share. See the main text for details on this decomposition. Firm-level markups are estimated using a cubic parametrization for g and h, as explained in Appendix F.2. The left panel provides the decomposition for manufacturing firms in Compustat. The right panel provides the decomposition for Compustat firms in other economic sectors. See figure 9 in the main text for our baseline estimates reported in the paper. 76 Output elasticity wrt variable inputs, estimated for manufacturing firms in Compustat .95 .95 Output elasticity wrt variable inputs, estimated for firms in Compustat .9 1960-1979 .85 .85 .9 1960-1979 1980-1999 .75 .8 .8 1980-1999 2000-2016 .75 .7 2000-2016 1 2 3 4 quintiles of firm size 5 1 2 3 4 quintiles of firm size 5 Figure 14: Output elasticity with respect to variable inputs estimated under the assumption that there are no ex-post shocks. The left panel presents estimates obtained for all Compustat firms. The right panel presents estimates obtained for Compustat firms in manufacturing. See figure 7 in the main text for our baseline estimates used in the results reported in the paper. Manufacturing, size-dependent technology Other sectors, size-dependent technology .1 .1 Contribution Within firm 1980 .05 0 -.05 -.1 -.1 -.05 0 .05 Reallocation 1990 2000 2010 1980 1990 2000 2010 Figure 15: Decomposition of the contribution of within-firm changes in markups and between-firm reallocation to (percent) changes in the labor share. See the main text for details on this decomposition. Firm-level markups are estimated under the assumption of no ex-post shocks, as explained in Appendix F.2. The left panel provides the decomposition for manufacturing firms in Compustat. The right panel provides the decomposition for Compustat firms in other economic sectors. See figure 9 in the main text for our baseline estimates reported in the paper. 77 Output elasticity wrt variable inputs, estimated for manufacturing firms in Compustat 1 .95 Output elasticity wrt variable inputs, estimated for firms in Compustat .9 1960-1979 .8 .85 .9 1960-1979 1980-1999 .7 .8 1980-1999 2000-2016 1 2 3 4 quintiles of firm size .6 .75 2000-2016 5 1 2 3 4 quintiles of firm size 5 Figure 16: Output elasticity with respect to variable inputs estimated under the assumption that productivity follows a linear Markov process. The left panel presents estimates obtained for all Compustat firms. The right panel presents estimates obtained for Compustat firms in manufacturing. See figure 7 in the main text for our baseline estimates used in the results reported in the paper. .15 Other sectors, size-dependent technology .15 Manufacturing, size-dependent technology Contribution Within firm 1980 .1 .05 0 -.05 -.1 -.1 -.05 0 .05 .1 Reallocation 1990 2000 2010 1980 1990 2000 2010 Figure 17: Decomposition of the contribution of within-firm changes in markups and between-firm reallocation to (percent) changes in the labor share. See the main text for details on this decomposition. Firm-level markups are estimated under the assumption that productivity follows a linear Markov process, as explained in Appendix F.2. The left panel provides the decomposition for manufacturing firms in Compustat. The right panel provides the decomposition for Compustat firms in other economic sectors. See figure 9 in the main text for our baseline estimates reported in the paper. 78