Firms, Labor Shares, and Capital–Labor Substitution (Chapter 3)

Retail: We now conduct the same model-based decomposition for the U.S. retail sector. Table 6 summarizes the retail data and presents the inferred shocks and their implications.29 In retail, we observe a decline in the labor share of 10.2 percentage points and a rise in sales concentration among the top 4 firms of 14 percentage points. This represents a vast increase in sales concentration, given that the top 4 firms represent only 0.023% of all firms in a retail industry, on average. For comparison, in manufacturing, the top 4 firms represent 1.1% of the firms in each industry, and their share of sales only increased by 6 percentage points. Our model can explain the observed patterns for retail with a decline in the price of capital of 75 log points and an increase in competition of 41%—an order of magnitude larger than in manufacturing. As before, a small fixed cost of automating tasks is enough to ensure a rise in the labor share of the typical firm. The counterfactual scenarios in Columns 3 and 4 suggest an important role for both the relative price of capital as well as the competition shock in explaining the observed trends in retail. Moreover, the results point to a significant interaction between these two shocks. For example, the estimated q-shock by itself causes the aggregate labor share to decline by 4.2 pp, while the estimated λ-shock generates a 2.0 pp decline. Yet, in combination the two shocks generate a decline of 10.3 pp. Thus, their interaction accounts for 4.1 percentage points—40% of the total decline. Two mechanisms are responsible for this interaction. First, rising competition reallocates activity towards more automated firms, since these are the firms who experience less cost inflation as competitors bid up wages. This form of reallocation also contributes to the decline in the labor share. Second, rising competition implies that more productive firms will now account for a greater share of sales, generating extra incentives for automation among these firms and giving a higher weight to those firms in determining the aggregate labor share. Turning to the predicted behavior of markups, our model predicts an increase in the aggregate markup in retail of 3.3% (from 1.15 to 1.19). This is the result of a 2.5% decrease in the within-firm component and a 5.8% increase driven by improvements in allocative efficiency. Thus, our model predicts that rising competition improved allocative efficiency in retail, and this generated an increase in TFP in that sector of 5.8%. Other sectors: We also conducted a similar decomposition for other economic sectors, including wholesale, as well as utilities & transportation sectors.30 We relegate the de29 For retail, our best proxy for the behavior of the labor share of the typical firm is the change in the mean unweighted payroll share (rather than the median labor share) reported by Autor et al. (2020). For aggregates in this sector, we target aggregate labor share data from the BLS, both in levels and changes. 30 We omit the finance and services sectors. In the former, measuring the labor share of valued added is conceptually difficult, while the latter did not experience a decline in its labor share. 36 Table 6: Transitional dynamics and decomposition of the retail labor share using a non-CES demand system (1982–2012) Model d ln q d ln λ ca 37 ∆ ∆ ∆ ∆ aggregate labor share unweighted mean labor share Top 4 firms’ sales share Top 20 firms’ sales share ∆ log productivity dispersion ∆ log aggregate markup Within firm change in markups Reallocation Only effects of d ln q (3) Only effects of d ln λ (4) I. Parameters and inferred aggregate shocks . 0.75 0.75 . 0.41 0 . 0.15 0.15 0 0.41 0.15 Data Benchmark (1) (2) II. Targeted moments, 1982–2012 −0.103 0.042 0.115 0.204 −0.042 −0.023 0.001 0.003 −0.020 0.014 0.108 0.188 . III. Other moments, 1982–2012 0.014 0.001 0.000 . . . IV. Markups, 1982–2012 0.033 −0.025 0.058 0.001 −0.011 0.012 0.029 −0.024 0.053 −0.102 0.044 0.140 0.163 Notes: Column (2) reports the findings from our benchmark model, which calibrates a uniform decline in the capital price (over 1982–2012), an increase in competition (over 1982–2012), as well as the automation fixed cost to replicate the change in the aggregate labor share in retail (BLS), the unweighted mean labor share change among surviving firms (Autor et al., 2020, Table 5), and the increase in the top 4 as well as top 20 firms’ sales share (Autor et al., 2020, Table 1), all for the retail sector. Column (3) shows results when shutting down the competition shock, and column (4) when shutting down instead the price of capital shock. Panel IV displays the log change in the aggregate markup, as well as a decomposition into within firm and reallocation components as in Baqaee and Farhi (2020b). tails behind the calibration and the data used to Appendix D. Figure 6 summarizes our findings and for reference contrasts them with our results for manufacturing and retail. In wholesale as well as utilities & transportation, the estimated q-shock by itself accounts for roughly two thirds of the aggregate labor share decline, while the remaining one third is almost entirely due to the interaction of the two shocks, which reinforce each other. 0 change in sectoral labor share -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 manufacturing retail wholesale utilities and transportation Figure 6: Model-based decomposition of sectoral labor share changes. For each sector, this figure shows the observed change in the aggregate labor share, the benchmark model with q- and λshocks jointly calibrated to match observed labor share and concentration changes, the two counterfactuals restricting to a single shock at a time, and the residual interaction term. The data sources are Kehrig and Vincent (2020) for manufacturing, and BLS MFP Tables for all other sectors. See also Tables 5, 6 and 10 for more detailed information . We conclude that while the fall in the manufacturing labor share is almost entirely attributed to automation through the lens of our model-based decomposition exercise, in non-manufacturing sectors there is an important role for rising competition and the way in which it interacts with differences in capital–labor substitution across firms. 3 Bounding the effects of reallocation and changes in markups on the labor share Our model-based decomposition accounts for the role of rising competition generating a reallocation towards the most productive firms, which also happen to be the ones with largest markups. However, the relationship between productivity (or size) and markups might not be perfect, and one could also imagine that rising competition reallocates economic activity towards firms with high markups that are not necessarily large or the most productive. This form of reallocation would also contribute to a rise in the aggregate 38 markup and a decline in the labor share. In this section, we explore the role of this more general form of reallocation empirically. The greater generality comes at the expense of requiring estimates of markups at the firm level. In what follows, we explain our approach to estimating markups and compare them with the existing literature. We then use our estimates to conduct several accounting exercises designed to illustrate the contribution of more general forms of reallocation and changes in markups to the labor share of manufacturing and other economic sectors. 3.1 Output elasticities by firm size We estimate markups for a firm f in industry i at time t as µf t = εvf t , svf t where εvf t is the elasticity of gross output quantities with respect to the quantity of variable inputs used, and svf t denotes the share of variable inputs in the firm’s revenue. The elasticity εvf t is not observed and must be estimated from firm-level data on revenue (y), expenditures in variable inputs (v), capital (k), and investment (x). Following Olley and Pakes (1996) and Ackerberg, Caves and Frazer (2015), we make the following assumptions: A1 differences across firms in the price of variable inputs reflect quality, which implies that we can treat expenditures in variable inputs as a measure of their qualityadjusted quantity A2 revenue is given by a revenue production function of the form R ln yf t = zf t + εR vc(f )t ⋅ ln vf t + εkc(f )t ⋅ ln kf t + f t , where c(f ) denotes groups of firms with a common technological bias and the same process for their revenue productivity, and f t is an i.i.d. ex-post shock orthogonal to kf t and vf t A3 unobserved productivity zf t evolves according to a Markov process of the form zf t = g(zf t−1 ) + ζf t , where ζf t is orthogonal to kf t and vf t−1 , and 39 A4 the gross output production function exhibits constant returns to scale in capital and variable inputs, which implies that quantity elasticities are given by R R εv,f t = εR vc(f )t / (εvc(f )t + εkc(f )t ) . (7) Assumptions A1–A3 are standard in the literature. Assumption A4 is added to deal with the fact that we don’t observe prices, and so the estimation procedure yields revenue elasticities, not the quantity elasticities that are relevant for computing markups (Bond et al., 2020).31 Under these assumptions, and given a grouping of firms c(f ), we can estimate revenue elasticities following the usual approach from Ackerberg, Caves and Frazer (2015), which uses investment as a proxy variable for unobserved productivity. This requires a first-stage regression where we first compute “true” output as h ln ỹf t = E[ln yf t ∣ ln xf t , ln kf t , ln vf t , t, c(f )] = h(ln xf t , ln kf t , ln vf t ; θc(f )t ). h Here θc(f is a parametrization for a flexible function h that might vary over time and )t between groups of firms. For any pair of revenue elasticities εR and εR , one can vc(f )t kc(f )t then compute R z̃f t = ln ỹf t − εR vc(f )t ⋅ ln vf t − εkc(f )t ⋅ ln kf t , estimate the flexible model g z̃f t = g(z̃f t−1 ; θc(f ) + ζ̃f t , )t g where θc(f is a parametrization for a flexible function g, and form the following moment )t conditions that identify the revenue elasticities: E [ζf t ⊗ (ln kf t , ln vf t−1 )] = 0. Suppose that revenue is given by y = p(q)q, where p(q) is the inverse demand curve. Quantity elasticities and revenue elasticities are then linked according to 31 εR v =( p′ (q)q + 1) ⋅ εv p(q) εR k =( p′ (q)q + 1) ⋅ εk , p(q) (q)q where 1/µ = ( pp(q) + 1). Moreover, firms will demand variable inputs until εR v = sv . Assuming constant ′ R R returns to scale implies that εv = εR v /(εv + εk ), as wanted. 40 This approach requires the choices of variable inputs to be correlated over time, which we view as a reasonable requirement. We implement this approach using data from Compustat. Appendix F describes our sample selection and definitions. We parametrize the functions h and g using quadratic polynomials and conduct our estimation over 10-year rolling windows. More importantly, and in line with the emphasis in our model that large and growing firms might operate different technologies and have different output elasticities, we group firms by quintiles of sales in each industry. Thus, our estimation provides output elasticities that vary over time, by industry, and by quintiles of firm size in each industry. This represents a significant deviation from previous papers which assume that all firms in a given industry share a common bias of technology. 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 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 7: Output elasticity with respect to variable inputs. The left panel presents estimates obtained for all Compustat firms. The right panel presents estimates obtained for Compustat firms in manufacturing. Figure 7 summarizes our findings for all Compustat firms and for manufacturing. In the 60s and 70s, firms had similar variable input elasticities. However, from there on, we estimate a pronounced decline in the variable input elasticity that concentrates among the firms in the top quintiles of the sales distribution. This clockwise rotation is precisely what our model predicts and coincides with the motivating evidence discussed in the introduction and which pointed to the uneven adoption of capital-intensive technologies by large firms. Appendix F provides alternative estimates assuming that: i. h and g are given by cubic polynomials; ii. there are no ex-post shocks  (so that no proxy variables 41 are needed and we can treat yf t as true revenue); or iii. assuming that zf t follows a linear Markov process, so that we can conduct the estimation using a dynamic panel approach. All these sets of alternative assumptions deliver similar results. We recognize that estimating markups at the firm level relies on strong assumptions. Moreover, we only conduct this exercise for firms in Compustat which are not a representative sample of the U.S. economy. Our estimates must be interpreted with the same caution required to approach previous empirical estimates of markups relying on these data and on similar methods. 3.2 Markups and comparison to previous papers Figure 8 plots the implied time series for markups using our estimates of output elasticities described in the previous section. The black line provides the aggregate markup, which we compute as a sales-weighted harmonic mean of firm-level markups. As discussed above, this is the relevant notion of an aggregate markup for the behavior of the aggregate labor share in an industry or the economy, as well as for welfare (Edmond, Midrigan and Xu, 2018; Baqaee and Farhi, 2020b). Our estimates for markups suggest that they have been quite stable over time, fluctuating around 1.2. For comparison, the red line in the figure plots the aggregate markup that would result if we grouped firms by industry only and assumed that all firms in the same industry operated technologies with the same capital intensity. This series reveals a mild secular increase in the aggregate markup from 1.25 in 1960 and 1.2 in 1980 to 1.3 in recent years, which is broadly in agreement with the harmonic mean (or cost weighted) estimates in Edmond, Midrigan and Xu (2018). Finally, the dashed red line reports estimates of an arithmetic mean of sales-weighted markups obtained under the assumption that all firms in a given industry operate technologies with the same capital intensity. These estimates are in line with those reported by De Loecker, Eeckhout and Unger (2020), but as we will see, are not relevant for understanding the contribution of markups to the decline in the labor share.32 32 Appendix F shows that the arithmetic mean of sales-weighted markups computed under the assumption that all firms in an industry operate the same technology will spuriously increase when some firms adopt technologies with different capital intensities. The harmonic mean does not suffer from this mechanical bias, providing an additional reason for looking at the harmonic mean rather than the arithmetic sales-weighted mean of firm markups, especially if one believes that firms are increasingly diverging in their use of technology, as is the case in the data. 42 1.5 Markups, estimated for firms in Compustat 1.4 Common technology and sales weighted 1.3 Common technology 1.1 1.2 Our estimates 1960 1970 1980 year 1990 2000 2010 Figure 8: Evolution of markups. The figure presents the aggregate markup estimated for firms in Compustat. The black line provides our baseline estimates. The red line provides estimates obatined under the assumption of common revenue elasticities across firms in the same industry. The red dotted line provides a version of the later estimates where the aggregate markup is computed as the arithmetic sales weighted mean of firm-level markups. 3.3 Contribution to the decline in the labor share As discussed above, the labor share in an industry at time t can be written as s`t ∶= ε`t , µt where ε`t denotes the share of labor in total industry costs, and µt is the industry markup, defined as a sales-weighted harmonic mean of firms’ markups: 1 1 ∶= ∑ ωf t ⋅ . µt µf t f The effects of technology, changes in factor prices, and substitution across firms with different factor intensities is captured by the cost share of labor in the industry, ε`t . In what follows, we investigate the contribution of changes in markups and reallocation across firms with different markups to the labor share by looking at the percent change 43 over time in the inverse of the aggregate markup 1/µt . We focus on the period from 1980 to 2012, which matches the period studied in our model, and provide a number of extensions and robustness checks in Appendix F. Figure 9 plots the cumulative percent change in 1/µt between 1960 and each year aggregated across all industries. The figure presents our results separately for manufacturing firms and for non-manufacturing. Furthermore, following the exercises in Section 2.4, we decompose the contribution of markups into within-firm changes and allocative changes, which capture the contribution of reallocation towards firms with higher markups over time.33 In line with what one would expect from an increase in competition since 1980, the reallocation component generates a negative contribution to the labor share, while within firm changes in markups generate a positive contribution. .1 Other sectors, size-dependent technology .1 Manufacturing, size-dependent technology Contribution Within firm .05 0 -.05 -.1 -.1 -.05 0 .05 Reallocation 1980 1990 2000 2010 1980 1990 2000 2010 Figure 9: 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. The left panel provides the decomposition for manufacturing firms in Compustat. The right panel provides the decomposition for Compustat firms in other economic sectors. The patterns in Figure 9 suggest that the reallocation towards high-markup firms played a minor role in explaining the decline of the manufacturing labor share. In this sector, the reallocation component accounts for a decline of the labor share of 7% between 1980 and 2012—a quarter of the observed decline. These estimates provide an upper bound on the contribution of rising competition to the decline in the labor share of the 33 In particular, we compute the within-firm contribution to the percent change in markups in each year as − ∑f ωf ∆ ln µf and the reallocation component as − ln µt + ∑f ωf ∆ ln µf . We then compute the cumulative contribution of within firm changes and the reallocation component over time. 44 sector, since we would expect the reallocation to higher markup firms induced by rising competition to be accompanied by a decrease in firm markups. Indeed, once we factor in the observed changes in firm markups, we find that the total contribution from markups to the labor share in manufacturing has been essentially zero. Outside of manufacturing, we find that the reallocation component explained up to 6% of the decline in the labor share since 1980—about one half of the observed decline. In these sectors, the within component has been much weaker and close to zero. As a whole, the estimates support the idea that in these sectors, rising competition might have reduced the labor share via allocative changes towards high markup firms without bringing a similar offsetting reduction in firm markups. As a whole, we view this exercise as supporting the quantitative conclusions derived in our model-based decomposition. In particular, the estimates here point to a crucial role for technology in the form of increased capital–labor substitution in explaining the decline in the labor share in manufacturing. Although more noisy, the estimates also suggest that rising competition and allocative changes might explain up to one half of the decline in the labor share outside of manufacturing. This decomposition also highlights the importance of accounting for differences in technology across firms. If we counterfactually assumed that output elasticities are common to all firms in a given industry, as is commonly done in the literature, we would estimate a larger negative contribution of markups to the labor share, especially in manufacturing. Accounting for differences in technology modifies these conclusions and shows that previous estimates confound differences in the factor intensity of technology across firms in a given industry with differences in markups.34 34 An alternative approach to estimating markups assumes constant returns to scale (as we do) but tries to directly measure the user cost of capital as R = r + δ − πk , where r is a required rate of return inclusive of an industry-specific risk premium, δ is the depreciation rate, and πk is the expected change over time in capital prices. One can then compute markups as revenue divided by total cost (= V + RK). The user-cost formula, which goes back to Hall and Jorgenson (1967) requires common and frictionless capital markets and assumes no adjustment costs for capital. In other words, this formula assumes that at all points in time, the marginal product of capital is equalized across all firms in a given industry. This strikes us as restrictive when thinking about firms undergoing a costly automation process. Instead, the approach used above makes no assumption about the marginal product of capital across firms, or the importance of adjustment costs. In any case, Baqaee and Farhi (2020b) report that, when using the user-cost formula above, the contribution of the between component to the decline of the labor share during 1985–2015 is 5%, about half of the observed decline. 45 4 Concluding remarks This paper starts from the observation that the adoption of modern automation technologies concentrates at large firms. Our main point is that, once we account for this vast heterogeneity in technology adoption, one can make sense of the dynamics of the labor share decline across firms both qualitatively and quantitatively. Moreover, accounting for these differences in technology adoption leads to different conclusions regarding the relative contribution of markups vs. technology to the decline in the labor share across industries. We made this point in three related exercises: 1. First, we developed a model of firm dynamics with costly automation decisions to study the dynamics of labor shares, market shares, and capital–labor substitution across firms and industries. We first show that a special case of our model with a simple demand side—a standard CES aggregator that implies constant markups— produces firm-level labor share dynamics in response to falling capital prices that are qualitatively and quantitatively in line with the observed firm-level data in the U.S. manufacturing sector. In particular, we find that the model reproduces the striking fact that while the sectoral labor share declined drastically, the labor share of the median firm increased slightly. The model also explains a range of other related observations, ranging from specific labor share decompositions proposed in the literature to the observed increases in concentration and productivity dispersion. 2. Second, we extended our model to allow for a more general demand side—a logconcave demand system that implies markups which are increasing in firm size and productivity—, which allows us to account for the effects of rising competition and the ensuing reallocation towards larger high-markup firms. We used this model to quantitatively decompose the fall in the labor share and the rise in sales concentration into a component driven by lower capital prices and a component driven by rising competition. We find that the main drivers of the decline in the labor share vary by sector. In manufacturing, capital–labor substitution driven by falling capital prices accounts for the majority of the sectoral labor share decline. Instead, we find that the rise in competition is more important in other sectors, in particular in retail, where it accounts for up to 60% of the falling labor share. 3. Third, we empirically estimated the contribution of a more general form of reallocation to high-markup firms, which are not necessarily large in the data. This 46 extra flexibility comes at the expense of requiring firm-level estimates of markups and output elasticities. Using standard techniques to estimate production functions but allowing for technology to vary across firms of different size, we estimate that the variable input elasticity of large firms has declined over time. While this finding is also in line with the costly automation mechanism at the core of our paper, the main implication here is that accounting for technological heterogeneity substantially weakens the rise in the aggregate markup. This exercise largely confirms our model-based findings: reallocation to high-markup firms can explain only a small fraction of the labor share decline in manufacturing, while playing a significant role in other sectors. Our paper motivates several avenues for future research. On the empirical front, we need more direct evidence on the causes and consequences of the heterogeneous adoption of modern capital-intensive technologies. The new technology modules in the US Census Annual Business Survey provide a promising tool for studying these questions. Moreover, our model points to the importance of developing estimates for markups and production function that can account in a flexible way for heterogeneity in technology, automation, and factor intensity across firms. On the theory side, we need more work to understand the root causes of rising competition as well as more flexible quantitative models to account for more general specifications of markups and demand. Although the non-CES demand systems used here are gaining traction in macroeconomics, they only allow markups and passthroughs to be functions of firm size. Finally, in our work all firms are ex-ante equal and have the same efficiency at using capital. However, it would be interesting to explore the implications of allowing for permanent differences in capital efficiency in firm dynamics models, and the response of the economy to lower capital prices. References Acemoglu, D., C. Lelarge, and P. Restrepo (2020): “Competing with Robots: FirmLevel Evidence from France,” in AEA Papers and Proceedings, vol. 110, pp. 383–88. Acemoglu, D., and P. Restrepo (2018): “The Race between Man and Machine: Implications of Technology for Growth, Factor Shares, and Employment,” American Economic Review, 108(6), 1488–1542. (2020): “Robots and jobs: Evidence from US labor markets,” Journal of Political Economy, 128(6), 2188–2244. Ackerberg, D. A., K. Caves, and G. Frazer (2015): “Identification Properties of Recent Production Function Estimators,” Econometrica, 83(6), 2411–2451. 47