The Joint Dynamics of Firms, Labor Shares, and Capital–Labor Substitution

Not a Typical Firm: The Joint Dynamics of Firms, Labor Shares, and Capital–Labor Substitution Abstract The decline in the U.S. labor share is far from uniform across firms. While the aggregate labor share has declined, especially in manufacturing, retail, and wholesale, the labor share of a typical firm in these industries has risen. This paper studies the dynamics of the substitution of capital for labor at the firm level and its implications for market structure and labor shares. We introduce a model of firm dynamics where firms make costly upfront investments to adopt the capitalintensive technologies required to automate additional tasks. Following a decline in the price of capital, large firms automate more of their tasks and become more capital intensive; while the median firm continues to operate a more labor-intensive technology. In line with firm-level data, our model generates transitions in which the labor share of the median firm increases at the same time as the aggregate labor share declines. We use an extension of our model that allows for endogenous markups to study the role of rising competition and reallocation towards more productive and higher-markup firms as another driver of the decline in the labor share during 1982–2012. We provide a quantitative decomposition showing that reallocation played a minor role in explaining the decline in the labor share in U.S. manufacturing but an important role in retail and other sectors. The substitution of labor with cheaper capital in a widening range of tasks played a more dominant role in explaining the decline of the manufacturing labor share. One of the most striking facts of the recent past is the fall of labor’s share of GDP in the United States and many other countries.1 After being stable throughout much of the last century, the U.S. labor share declined from a peak of 66% in the 1980s to 60% in 2014, as can be seen in Figure 1.2 The decline of the labor share is not driven by changes in the industry composition of the U.S. economy. Figure 1 shows that, if anything, holding industry shares in GDP constant at their 1987 levels would have led to a more pronounced and earlier decline in the labor share.3 Indeed, most of the decline in the U.S. labor share is driven by a sizable decline dating back to the mid 80s in the share of value added accruing to labor in retail and wholesale trade, and in particular in manufacturing. Labor share, BLS data for 1987−2016 80 75 70 retail and wholesale trade aggregate labor share 65 60 holding industry shares in GDP fixed at 1987 55 50 manufacturing 45 1985 1990 1995 2000 2005 2010 2015 Figure 1: Labor share in the U.S. The blue line plots the headline BLS measure for the U.S. non-farm business sector. The gold line plots a counterfactual labor share holding industry shares in GDP constant at their 1987 values. The red and purple lines plot the labor shares for manufacturing and for retail and wholesale trade. Data from the BLS Multifactor Productivity Tables. Despite growing consensus on these facts, there is an ongoing debate about the underlying causes of the decline. One set of explanations points to the increased substitution of capital for labor. In these accounts, the development of new or more efficient capital1 See for instance Elsby, Hobijn and Şahin (2013); Karabarbounis and Neiman (2013); Piketty (2014); Dao, Das and Koczan (2019). For a different perspective arguing that the decline in the labor share is exclusively a U.S. phenomenon, see Gutierrez and Piton (2020). 2 The figure plots the labor share in the U.S. from the BLS. Although there is some consensus about the decline in the labor share, there is more debate on the exact magnitude of the decline and how it is affected by different measurement issues such as the treatment of self-employment (Elsby, Hobijn and Şahin, 2013), housing (Rognlie, 2015), depreciation (Bridgman, 2018), income shifting by business owners (Smith et al., 2019), and intangible capital (Koh, Santaeulalia-Llopis and Zheng, 2020). 3 See also Elsby, Hobijn and Şahin (2013); Hubmer (2020). 1 intensive technologies—as evidenced by the decline in the price of capital equipment and software—leads to the substitution of capital for labor and a decline in the labor share. This substitution can take place along an aggregate production function with an elasticity of substitution greater than one (Karabarbounis and Neiman, 2013; Eden and Gaggl, 2018; Hubmer, 2020), or within tasks in a widening range of tasks that can now be profitably automated (Acemoglu and Restrepo, 2018). The fact that the decline in the labor share is more pronounced in manufacturing, and within that sector in industries and firms adopting new automation technologies or that are more capital-intensive (Acemoglu and Restrepo, 2020; Acemoglu, Lelarge and Restrepo, 2020; Hubmer, 2020) lends some credence to these explanations. Labor share decomposition in manufacturing Kehrig & Vincent (2020) 75th percentile 90 80 50th percentile 70 60 25th percentile 50 manufacturing labor share 40 1967 1972 1977 1982 1987 1992 1997 2002 2007 2012 Figure 2: Labor share in U.S. manufacturing firms. The red line plots the aggregate labor share in manufacturing. The gray lines plot the labor shares for firms at the 25th, 50th, and 75th percentile of the labor share distribution. Data from Kehrig and Vincent (2020). This evidence notwithstanding, recent studies using firm-level data have revealed that the decline in labor shares is far from uniform across firms in an industry. While the aggregate labor share has declined, the labor share of the typical U.S. firm has risen (Autor et al., 2020; Kehrig and Vincent, 2020). For example, Figure 2, which we reproduced using data from Kehrig and Vincent (2020), shows that, in manufacturing, the firms located at the 25th, 50th, and 75th percentile of the labor share distribution saw a mild increase in their labor shares over time at the same time as the aggregate labor share in the sector plummeted. As shown by Autor et al. (2020), a similar pattern holds across all U.S. sectors and is also evident for other countries. At a first glance, these facts cast some doubt on explanations based on capital–labor substitution. After all, a simple 2 version of these theories in which firms face similar prices and have access to the same technologies would imply a more uniform decline in labor shares. Instead, these facts seem to favor a second set of explanations that emphasize the role of rising concentration and the reallocation of economic activity towards the top firms in an industry (Barkai, 2020; De Loecker, Eeckhout and Unger, 2020; Autor et al., 2020; Baqaee and Farhi, 2020b). This reallocation, which could be the result of increased competition or winner-takes-all dynamics, reduces the aggregate labor share because the top firms in each sector have higher markups. In these accounts, the decline in the labor share reflects an increase in the aggregate markup resulting from reallocation and does not represent technological changes favoring the substitution of capital for labor.4 This paper starts from the observation that the adoption of modern automation technologies facilitating the substitution of capital for labor is quite uneven and concentrates on large firms. Our main point is that, once we account for this form of heterogeneity in technology adoption, one can make sense of the dynamics of the labor share decline across firms and industries as a response to lower capital prices. As a result, explanations of the labor share decline based on capital–labor substitution are consistent, both qualitatively and quantitatively, with the new firm level evidence put forth by Autor et al. (2020) and Kehrig and Vincent (2020). Moreover, accounting for these differences in technology adoption leads to different conclusions regarding the relative contribution of markups and technology to the decline in the labor share across industries. In particular, we provide a series of decompositions that account for the uneven adoption of automation technologies, and show that capital-labor substitution played a key role in explaining the decline in the manufacturing labor share during 1982–2012; whereas rising competition and reallocation towards firms with lower labor shares played a key role in retail and other sectors. We develop these points via three exercises. First, we revisit the firm labor share evidence through the lens of a standard firm-dynamics model with a CES demand structure augmented with capital–labor substitution decisions across tasks (as in Zeira, 1998; Acemoglu and Restrepo, 2018). The key assumption in our model is that firms must make costly upfront investments to adopt the capital-intensive technologies required to automate tasks. This assumption is motivated by micro evidence pointing to the skewed 4 A variant of these explanations emphasizes the role of technologies that favor larger firms (see Unger, 2019; Lashkari, Bauer and Boussard, 2019; De Ridder, 2020), or the possibility that lower overhead costs facilitate the expansion of large firms across multiple markets, leading to tighter competition at the local level (see Aghion et al., 2019). Another branch of this literature emphasizes the lack of technology diffusion from large firms to followers, which can affect concentration and markups (see Akcigit and Ates, 2019). These models are different from our mechanism, which emphasizes the fact that new automation technologies have a strong factor bias and change the production function of large firms, making it less labor-intensive. 3 adoption of new capital-intensive technologies, including robotics, dedicated machinery, and specialized software. For instance, Acemoglu, Lelarge and Restrepo (2020) show that, in France, only 1% of manufacturing firms purchased industrial robots between 2010 and 2015, but these firms accounted for over 20% of manufacturing value added. The Annual Business Survey reveals a similar fact for the U.S., where the adoption of these technologies is also highly concentrated among large firms (Zolas et al., 2020).5 This evidence suggest that large firms not only differ in their productivity (and potentially in their markups) but also in the extent to which their production relies on capital-intensive technologies. Using this simple version of our model in which markups are constant, we find that, following a persistent decline in the price of capital—as the one observed in the US data since the 1980s—, the resulting dynamics of the labor share across firms match the evidence put forth by Autor et al. (2020) and Kehrig and Vincent (2020). Large and growing firms automate more of their tasks and become more capital intensive. In these firms, capital and labor become substitutes, driving the decline in the aggregate labor share. Instead, the median firm will not automate additional tasks and will continue to operate a more labor-intensive technology. For the median firm, capital and labor remain complements, explaining why the labor share rises in the typical firm.6 Moreover, as in the Melitz-Polanec decomposition conducted by Autor et al. (2020), the decline in the labor share is driven by a more negative covariance between the market share of a firm and their labor share, and not by the change in the unweighted average of labor shares among surviving firms. Importantly, the negative covariance term reflects the fact that automation efforts in our model are episodic and coincide with periods of rapid firm expansion in terms of value added, which aligns with the joint dynamics of changes in value added and labor shares for manufacturing firms documented by Kehrig and Vincent (2020). Our model also accounts for the positive contribution of entry and the negative 5 Dinlersoz and Wolf (2018) document a similar phenomenon for advanced manufacturing technologies using the Survey of Manufacturing Technologies from 1993. A series of recent papers document that the adoption of industrial robots also concentrates among the top firms in each manufacturing industry (see Koch, Manuylov and Smolka, 2019; Humlum, 2019; Bonfiglioli et al., 2020). 6 A related mechanism going back to Houthakker (1955) emphasizes the possibility that a decline in capital prices will reallocate economic activity towards more capital-intensive firms. For example, Oberfield and Raval (2014) show that this reallocation has a small but positive contribution to the aggregate substitution between capital and labor in response to lower capital prices. More recently, Kaymak and Schott (2018) argue that this mechanism can explain up to a third of the decline in the manufacturing labor share in response to lower corporate taxes—which are isomorphic to lower capital prices. This form of reallocation to ex-ante more capital intensive firms is different from the main mechanism in our model, which emphasizes the fact that new automation technologies are unevenly adopted by large firms which then become more capital intensive. 4 contribution of exit to changes in the labor share and generates an endogenous rise in sales concentration and productivity dispersion during episodes of capital–labor substitution that are quantitatively in line with empirical estimates.7 In the second step of our analysis, we extend our model to a non-CES demand system where demand is log-concave and markups increase with firm size—Marshall’s second law of demand.8 This extension allows us to quantify the effects of increased competition on concentration and the labor share. We show that one can use data on the decline in the aggregate labor share, the change in the median labor share, and the change in sales concentration in each sector of the economy to decompose the decline in the labor share into a part driven by rising competition and a part driven by capital–labor substitution. Our model-based decomposition suggests that in retail and other sectors outside of manufacturing, rising competition and the ensuing concentration of economic activity at top firms account for up to 60% of the decline in the labor share during 1982– 2012. However, when looking at manufacturing this pattern reverses and capital–labor substitution explains over 90% of the decline in the labor share during this period. The reason why rising competition plays a minor role in manufacturing is that concentration in this sector increased mildly (as evident from the data in Autor et al., 2020), and that, based on the distribution of firm sales, we estimate a distribution of firm productivity that is more log-convex than but not too far from Pareto. When the productivity distribution is Pareto, the decline in the labor share due to reallocation to high markup firms and the decline in within firm markups generated by rising competition exactly offset each other. Our quantitative exercise focuses entirely on differences in markups associated with firm size (or productivity), which is what existing explanations based on the rise of superstar firms emphasize. However, the association between size and markups is imperfect. Increased competition might reallocate economic activity to high-markup firms of any size, including smaller ones. We address this possibility in our third exercise, where we provide model-free bounds for the contribution of reallocation to high-markup firms to the decline of the labor share by sector. These bounds come at the expense of additional data requirements, and rely heavily on empirical estimates of markups and variable-input 7 While we focus on falling capital prices and automation, in our framework, globalization and increased offshoring as emphasized by Elsby, Hobijn and Şahin (2013) would have similar implications for domestic labor share and market share dynamics if offshoring decisions were likewise correlated with firm size. As we are unaware of systematic firm-level evidence on offshoring by firm size, we do not address this channel in this paper. 8 Non-CES demand systems have been widely used in trade (see Melitz and Ottaviano, 2008; Amiti, Itskhoki and Konings, 2019; Arkolakis et al., 2018). A recent and growing literature uses non-CES demand systems to quantify the distortions introduced by markups (see Edmond, Midrigan and Xu, 2018). See also Baqaee and Farhi (2020a) for a thorough discussion on these models and Marshall’s laws. 5 elasticities from Compustat. We first provide estimates of firm-level markups and output elasticities that account for differences in the capital-intensity of technology across firms. In line with the key mechanism in our model, we find that over time, the largest firms in each industry have experienced a large drop in their elasticity of output with respect to variable inputs, suggesting that their production processes are becoming less labor-intensive. On the other hand, we find positive or no changes for smaller firms. Using these empirical estimates of firm markups, we reach a conclusion that supports our quantitative model-based results: empirically, reallocation towards high-markup firms can explain at most a small share of the labor share decline in manufacturing, but plays a more important role in other sectors.9 The rest of the paper is organized as follows. Section 1 introduces our model for the special case of a CES demand aggregator, where markups are constant, and shows that the model matches firm-level evidence on the decline of the labor share. Section 2 considers a model with variable markups, and decomposes the decline in the labor share into a part driven by reallocation and tougher competition, and a part driven by capital–labor substitution. Section 3 provides empirical bounds on the contribution of reallocation to the decline of the labor share by sector using Compustat firm-level data. The appendix contains all proofs, as well as additional details and robustness exercises. 1 1.1 Model with CES Demand Description of the model We use a standard firm-dynamics model (as in Hopenhayn, 1992; Clementi and Palazzo, 2016), and augment it to include firms’ decisions to automate tasks by adopting costly capital-intensive technologies (as in Acemoglu and Restrepo, 2018). We consider an economy in discrete time and omit time subscripts unless needed. Existing firms differ in their productivity z and in the fraction of production processes or tasks which they have 9 Autor et al. (2020) provide a related exercise when assessing the magnitudes of their superstar firms’ mechanism. In particular, they use empirical estimates of markups across firms and compute the extent to which the relative expansion of high markup firms contributed to the decline of the manufacturing labor share. Their markup estimates are based on production function elasticities as in De Loecker, Eeckhout and Unger (2020), or measured as revenue over total costs. Baqaee and Farhi (2020b) conduct a similar exercise, but in their case they are interested in computing the contribution of reallocation to high-markup firms to TFP growth, which they show is equal to the contribution of reallocation to the decline in the labor share. They also provide estimates based on accounting approaches and a production function approach. We follow the production function approach, but allow production functions to vary more flexibly across firms of different sizes. 6 automated α ∈ [0, 1]. A firm of type θ = (z, α) produces output y(θ) by combining a continuum of tasks indexed by x via a CES production function with task substitution elasticity η ≥ 0: 1 y(θ) = z ⋅ (∫ Y(θ, x) η−1 η η η−1 dx) . 0 Tasks in [0, α] are automated and can be produced by capital or labor; whereas nonautomated tasks in (α, 1] must be produced by labor: ⎧ ⎪ ⎪ ψk (x)k(θ, x) + ψ` (x)`(θ, x) Y(θ, x) = ⎨ ⎪ ⎪ ⎩ ψ` (x)`(θ, x) if x ∈ [0, α] if x ∈ (α, 1] Here, Y(θ, x) denotes the quantity of task x, and k(θ, x) and `(θ, x) denote the utilization of capital and labor in producing task x by a firm of type θ. Firms face a competitive market for inputs. There is a fixed supply of labor ` rented to firms at a wage rate w. On the other hand, the capital used for task x is produced from the final good at a rate q(x) and rented to firms at a rate 1/q(x). Without loss of generality, we assume that tasks are ordered so that ψ` (x) is increasing in x, q(x) ⋅ ψk (x) which implies that labor has a comparative advantage at high-indexed tasks. We start with a CES demand system. Existing firms produce differentiated varieties that are then combined via a CES aggregator to produce the final good y: y = (∫ y(θ) σ−1 σ m(θ)dθ) σ σ−1 , θ where m(θ) denotes the mass of firms of type θ, and σ > 1 denotes the elasticity of substitution across varieties. In what follows, we normalize the price of y to 1. Incumbent firms begin a period with productivity z and automation level α. They then make optimal employment and capital utilization decisions and collect profits π(θ). Subsequently, firms draw a fixed operating cost cf ⋅y, where cf ∼ G(cf ), and decide whether to continue operating.10 If they continue, they can automate additional tasks in (α, α′ ] at a cost ca ⋅ y ⋅ (α′ − α), and draw next period’s productivity level z ′ , which follows an 10 We specify that all fixed costs are paid in units of the final good and scaled by aggregate output. This normalization ensures that the model can generate a balanced growth path if labor productivity increases uniformly. 7 exogenous AR1 process in logs: ln z ′ = ρz ln z + ε, where ρz ∈ (0, 1), ε ∼ N (µz , σz ), and µz = −(1 − ρz ) ⋅ σz2 /(1 − ρ2z ) so that the long-run mean of z is normalized to one.11 We also allow for entry of new firms. In particular, every period a unit mass of potential entrants decides whether to enter the market. Entrants draw a productivity signal z from a distribution H(z) and start with a common level of automation α0 . After observing z, α0 , as well as the realization of the fixed operating cost cf , entrants decide whether to pay the fixed cost and enter. After paying this cost, entrants become incumbents and face the same problem described above. Finally, we assume that automation technologies also have some exogenous diffusion through entrants, and that α0 equals the unweighted average of α among incumbents. This is a common specification used in models of technology diffusion (for a recent example, see Perla, Tonetti and Waugh, 2021). Both incumbents and new entrants discount the future at a constant interest rate r, which we take as exogenous. Throughout, we assume that r > g, where g is the growth rate of output between two consecutive periods. The timing and treatment of entry and exit in this model follows the standard firmdynamics literature. Our key innovation is to incorporate heterogeneity in the extent to which firms have automated their production process, and to endogenize the evolution of these decisions over time as determined by the payment of a fixed cost per task ca . This fixed cost of automation plays a crucial role in our results and ensures that firms will automate more tasks as they grow and reach the required scale. The assumption that automating a task entails some fixed cost is plausible and intuitive. For example, consider a car-manufacturing firm contemplating the possibility of automating welding. Besides purchasing the industrial robots required to complete these tasks (k(θ, x) in our model), the firm must also hire a team of engineers and integrators to reorganize its plant and production process, and in some cases to redesign some of their products so that they are more standardized, so that the robots can be integrated seamlessly. In the case of industrial robots, these upfront investments and payments to integrators far exceed the cost of the robot system itself (Acemoglu and Restrepo, 2020). Likewise, a firm contemplating to deploy a new software to automate its logistics and inventory management decisions must pay a fixed cost for developing the software and 11 In Appendix C.1, we demonstrate that our findings are not sensitive to different timing assumptions. In particular, we find similar results if we let firms observe z ′ before making their automation decisions. 8 rearranging its operations (ca in our model) and then continue to pay a variable cost for the operation of the system ( q(x)ψ1 k (x) in our model).12 Besides the costly adoption decision by firms to automate additional tasks, we also assume that these technologies diffuse gradually through the entry of new firms. This is a natural starting point and a simple way to get automation technologies to fully diffuse over time, reflecting the standardization of these production techniques and the associated organizational and productive changes required to deploy them. 1.2 Equilibrium and transitional dynamics To save on notation, we focus on an equilibrium where firms produce all tasks in [0, α] with capital. This will be the relevant scenario following a reduction in the cost of capital. The production function above implies that the unit cost for a firm of type θ is c(θ, w) = (1) 1 1 ⋅ (Ψk (α) + Ψ` (α) ⋅ w1−η ) 1−η . z This is the usual CES price index, with the difference that the share parameters Ψk (α) and Ψ` (α) are endogenous and depend on the mass of tasks that are automated: α Ψk (α) = ∫ 0 1 (q(x) ⋅ ψk (x))η−1 dx, Ψ` (α) = ∫ α ψ` (x)η−1 dx. The share of capital in cost εk (α, w)—which equals the elasticity of output with respect to capital—and the share of labor in cost ε` (α, w)—which equals the elasticity of output with respect to labor—are then given by ε` (α, w) = Ψk (α) , Ψk (α) + Ψ` (α)w1−η ε` (α, w) = Ψ` (α) ⋅ w1−η . Ψk (α) + Ψ` (α) ⋅ w1−η As these equations show, by investing in automation, firms increase the share of capital in cost and reduce the share of labor, becoming more capital-intensive. Finally, firms charge a constant markup µ ∶= σ/(σ − 1) > 1 and earn profits (2) π(θ, w) = σ −σ (σ − 1)σ−1 ⋅ y ⋅ c(θ, w)1−σ . We focus on perfect foresight equilibria. Given a path for investment productivities 12 One can think of these fixed costs as intangible investment, which would contribute to firms’ intangible capital (see, e.g., Corrado, Hulten and Sichel, 2009, on the increasing importance of intangibles). In our model, firms not only differ in their capital stock, but also in their ability to use capital productively for additional tasks—an intangible asset. 9 qt (x) and an initial distribution of firms m0 (θ), an equilibrium is given by a path for wages wt and aggregate output yt , as well as a path for the distribution of firms mt (θ), such that for all t ≥ 0: 1. the ideal-price index condition holds (3) ∫θ µ 1−σ ⋅ ct (θ, wt )1−σ ⋅ mt (θ) ⋅ dθ = 1, 2. the labor market clears (4) ∫θ yt ⋅ µ −σ ⋅ ct (θ, wt )−σ ⋅ ∂ct (θ, wt ) ⋅ mt (θ) ⋅ dθ = `, ∂wt 3. automation and exit decisions maximize the value function of incumbents Vt (θ) =πt (θ)+ 1 {−ca ⋅ yt ⋅ (α′ − α) + E [Vt+1 (θ′ )∣z, α′ ]}} dG(cf ), ∫ max {0, −cf ⋅ yt + αmax ′ ∈[α,1] 1+r where θ′ = (z ′ , α′ ) denotes the vector of technology next period, 4. entry decisions maximize the value of entrants Vte (z) = ∫ max {0, −cf ⋅ yt + ′max {−ca ⋅ yt ⋅ (α′ − α0t ) + α ∈[α0t ,1] 1 E [Vt+1 (θ′ )∣z, α′ ]}} dG(cf ), 1+r where z denotes an entrant’s productivity signal, and α0t ≡ ∫θ α⋅mt (θ)dθ , ∫θ mt (θ)dθ 5. starting from m0 (θ), the evolution of mt (θ) is governed by the exogenous process for z, the endogenous process for α, and optimal entry and exit decisions of firms. To characterize equilibrium automation decisions, it is useful to define αt∗ as the level of automation that would minimize production costs if tasks could be automated at no cost. This involves automating tasks up to the point at which the unit cost of producing a task with labor equals that of producing it with capital, which defines αt∗ as (5) ψ` (αt∗ ) = wt . qt (αt∗ ) ⋅ ψk (αt∗ ) The following lemma characterizes automation decisions and shows that, in line with the existence of fixed costs per task, more productive firms automate more of their tasks. 10 Lemma 1 A firm with technology (α, z) sets α′ = max{α̂t (z), α}, where α̂t (z) is an increasing function of z with lim α̂t (z) = 0, ∗ lim α̂t (z) = αt+1 . z→∞ z→0 Our first proposition characterizes the steady state of this economy. Due to the diffusion of automation technology, we have that all firms eventually achieve the same level of automation, and that this is the cost-minimizing level of automation defined above: Proposition 1 Suppose that qt (x) → q(x). The economy admits a stationary equilibrium where α = α∗ for all firms. We now turn to a characterization of the transitional dynamics in response to improvements in the efficiency with which the economy can produce capital goods qt (x). Suppose that the economy is in a steady state with q0 (x) and experiences a uniform proportional increase in q0 (x) by ∆ ln q > 0. To analyze the response of the economy, it is useful to define two elasticities of substitution. On the one hand, we have the elasticity of substitution between capital and labor holding the level of automation constant—the short-run elasticity of substitution. In our model, this coincides with the elasticity of substitution across tasks η. On the other hand, we have the elasticity of substitution between capital and labor that accounts for the endogenous shifts in technology over time captured by α∗ . This long-run elasticity of substitution is given by ηLR = η + ∂ ln Ψk (α)/Ψ` (α) ∂ ln ψ` (α)/(q(α) ⋅ ψk (α)) / . ∂ ln α ∂ ln α Because optimal automation decisions are increasing in the wage and the level of q(x) (as can be seen from (5)), we have that this long-run elasticity always exceeds η. Proposition 2 Following a uniform proportional increase in q0 (x) by d ln q > 0, the economy converges to a new steady state with higher output and wages, and a higher level of automation α∗ . The aggregate share of labor in costs (or in value added) changes by d ln ε` = (1 − ε` ) ⋅ (1 − ηLR ) ⋅ (d ln w + d ln q) = 1 − ε` ⋅ (1 − ηLR ) ⋅ d ln q. ε` At the same time, for an incumbent firm with low realizations of z along the transition, 11 the share of labor in costs (or in value added) changes by d ln ε` (θ) = (1 − ε` (θ)) ⋅ (1 − η) ⋅ (d ln w + d ln q) = 1 − ε` (θ) ⋅ (1 − η) ⋅ d ln q. ε` (θ) One case that will be particularly relevant for our analysis emerges when the short-run elasticity is less than 1, and at the same time, the long-run elasticity is greater than 1. In this case, Proposition 2 implies that the aggregate labor share is governed by the longrun elasticity of substitution and that it will decline as capital goods become cheaper and wages increase. At the same time, firms that do not reach a big enough scale to justify investments in automating additional tasks will keep their α fixed at its initial value. For these firms, capital and labor are complements, and their labor share rises as capital goods become cheaper and wages increase. The previous proposition suggest that, one way to explain the dynamics of the labor share in the data, is to have a parametrization of our model where the long-run elasticity of substitution exceeds 1 and the short-run elasticity is below 1. Although this case is intuitive, and some evidence supports is as a natural starting point, our model can still generate the dynamics that we see in the data even if capital and labor are complements in the long run and ηLR < 1. In particular, the aggregate labor share might decline following improvements in technology that reduce the cost of capital used at marginal tasks (those around α∗ ), instead of uniformly across all tasks as in Proposition 2. Proposition 3 Following an increase in q0 (x) for all x > α∗ by d ln q > 0, the economy converges to a new steady state with higher output and wages, and a higher level of automation α∗ . The aggregate share of labor in costs (or in value added) changes by d ln ε` = −(1 − ε` ) ⋅ (ηLR − η) ⋅ d ln q + (1 − ε` ) ⋅ (1 − ηLR ) ⋅ d ln w At the same time, for an incumbent firm with low realizations of z along the transition, the share of labor in costs (or in value added) changes by d ln ε` (θ) = (1 − ε` (θ)) ⋅ (1 − η) ⋅ d ln w. The proposition clarifies that an improvement in the productivity of capital at marginal tasks could reduce the aggregate labor share even if ηLR < 1. For example, if ηLR approaches 1 from below, the aggregate labor share will fall while the labor share of some typical firms will increase slightly. The reason is that these marginal changes lead to the reallocation of tasks away from labor and towards capital. At the same time, there is no 12 change in the price of tasks below α∗ , which otherwise would reduce the share of capital (assuming η < 1). This is the key difference relative to a uniform increase in q(x), which also reduces the share of infra-marginal tasks produced by capital in value added. This also clarifies that, while ηLR is an interesting object, it only mediates the response of the labor share to uniform changes in prices, but not to task-specific changes of the sort described in Proposition 3. Under the lens of a task model, the observed decline in the labor share in response to a decline in equipment prices can have one of two interpretations. Either this decline is uniform across tasks and ηLR > 1, as emphasized in much of the existing literature and in Proposition 2. Or this decline is more pronounced for new types of capital used at tasks where humans had a stronger comparative advantage, as emphasized in Proposition 3. In this last case we could have a long-run elasticity of substitution above or below 1. 1.3 Calibration and quantitative results This section presents results from a calibrated version of our model and shows that it is able to reproduce the firm-level dynamics found in the U.S. manufacturing sector. We study the effects of a uniform decline in the price of capital goods in the main text, and leave an analysis of task-specific changes to Appendix C.3. Initial steady state: We first discuss the calibration of the parameters governing the initial steady state of the economy. We calibrate the model under the assumption that industries were in their steady state in 1982, and that at this point in time, there was a uniform (inverse) price of capital q(x) = q0 , which we choose to match the observed labor share in manufacturing. This choice is motivated by the fact that the decline of the manufacturing labor share of value added starts in 1982, after being roughly constant in the decades preceding it. The period after 1982 also coincided with a pronounced decline in the price of equipment and software, which will be the main driving force behind the labor share decline in our exercise.13 Likewise, we set the demand elasticity to σ = 11, which generates a common markup of 1.1. Finally, we pick the dispersion of firm productivity to match the observed sales concentration in manufacturing in 1982. Specifically, we target the share of sales among the top 4 firms in each manufacturing industry reported by Autor et al. (2020), which roughly corresponds to the top 1.1% of 13 Appendix E shows that this is also the case for most economic sectors. After having labor shares that were roughly stable for the 1947–1982 period (and in some cases until 1987), some sectors experienced a decline in their labor share of value added after this period. This Appendix also provides data for capital prices and shows a sharp decline for the 1982–2012 period (see Hubmer, 2020, for more on capital prices) 13 firms.14 Panel I in Table 1 summarizes the obtained parameters and the corresponding data moments that we used to calibrate them. Table 1: Calibration of the CES-demand model for manufacturing Parameter ln q0 σ σz cf ξ µe η γ Moment I. Parameters governing steady state in 1982 Inverse capital Manufacturing labor share from −5.35 price Kehrig and Vincent (2020) Demand elasticity 11.0 Aggregate markup Std. dev. of ln z Top 4 firms’ sales share in 1982 0.094 innovations from Autor et al. (2020) Data Model 60.1% 60.2% 1.10 1.10 40.0% 40.1% 0.062 0.063 0.490 0.491 0.600 0.598 0.40 0.40 1.35 1.35 II. Parameters governing firm dynamics Minimum fixed Entry (=exit) rate from Lee and 6.0 ⋅ 10−6 cost Mukoyama (2015) Dispersion fixed Size of exiters from Lee and 0.330 cost Mukoyama (2015) Entrant Size of entrants from Lee and 0.935 productivity Mukoyama (2015) III. Parameters related to the elasticity of substitution Task substitution Short-run K–L elasticity from 0.4 elasticity Oberfield and Raval (2014) Comparative Long-run K–L elasticity from 0.95 advantage Hubmer (2020) Notes: The labor share corresponds to the time average over 1967–1982 in Kehrig and Vincent (2020). The relative capital price is identified only jointly with the level of the capital productivity schedule ψk (x), which we normalize to 1. The model equivalent to the top 4 firms’ sales share refers to the top 1.1% of firms, since there are on average 364 firms per 4-digit industry in the manufacturing sector as reported in Autor et al. (2020). The annual entry rate, as well as relative sizes of entrants and exiters, are from Lee and Mukoyama (2015) and based on the Annual Survey of Manufactures. The six parameters listed in Panels I and II are all jointly calibrated to match the six corresponding moments. The parameters governing firm dynamics in our model are taken from the literature or calibrated to match entry rates, exit rates, and firm-size distributions.15 Panel II of Table 1 lists the jointly calibrated parameters and corresponding model and data moments. The calibration of the schedule of capital and labor productivities across tasks is more novel and important for our purposes. We adopt the following functional form: ψ` (x) = (x 1−η−γ γ − 1) 1 1−η−γ , ψk (x) =1. 14 Although not targeted, the model roughly matches the share of sales accounted for by the top 20 firms in an industry (roughly corresponding to the top 5.5% sales share), which according to Autor et al. (2020) averages 69.7% across manufacturing industries, while the model equivalent is 68.0%. 15 We follow Clementi and Palazzo (2016) in parametrizing mean and dispersion of the fixed cost distribution G(cf ) to match the overall frequency and relative size of exiting firms. Specifically, we impose a Pareto distribution for the operating cost with scale parameter cf and tail coefficient ξ. We also match the relative size of entrants by modeling the entrant distribution H(z) as a log-normal that differs from the long-run distribution of z only insofar as it has a lower mean µe . We fix the persistence of productivity ρz at 0.9, which is in the range of estimates for various TFP measures and demand shocks in Foster, Haltiwanger and Syverson (2008). 14 This specification implies that the short run elasticity of substitution is given by η and the long-run elasticity of substitution is constant and equal to ηLR = η + γ, where γ > 0 is an inverse measure of the strength of the comparative advantage of labor at higher index tasks. We set η = 0.4 to match the estimate of the firm-level elasticity of substitution between capital and labor in Oberfield and Raval (2014), which is based on cross-regional wage variation. Although this is a reasonable starting point, we also explored the implications of using lower values for η in Appendix C.2. The reason is that wage differences across regions might persist, which implies that their estimates also account for some changes in the allocation of tasks and should be seen as an upper bound on the elasticity of substitution between tasks η. On the other hand, we set γ = 0.95 to match the available estimates of the aggregate long-run elasticity of substitution between capital and labor in response to a fall in capital prices, which place it around 1.35 (see Karabarbounis and Neiman, 2013; Hubmer, 2020).16 As emphasized in Proposition 3 and demonstrated in Appendix C.3, we can also generate the observed behavior of the manufacturing labor share through a decline in the price of capital at higher-indexed tasks even if the long-run elasticity of substitution ηLR is less than 1. Transitional dynamics: We now explore the adjustment of the economy following a decrease in the price of capital. In particular, we ask if our model can generate (i) the observed decline in the manufacturing labor share at the same time as (ii) an increase in the median labor share in response to a uniform increase in q(x). To this end, we calibrate the decline in the price of capital and the fixed cost of automation by targeting (i) and (ii). Intuitively, the decline in the price of capital controls the aggregate labor share, and the fixed cost of automation controls the behavior of the median labor share. Column 1 in Table 2 summarizes the manufacturing data, and Column 2 displays the inferred shocks as well as model moments corresponding to our benchmark calibration. Column 2 shows that our model with CES demand and a fixed cost of automation per task of 0.23 generates the observed trends in (i)–(ii) as the response to a decline in the price of capital of 140 log points.17 Thus, the observation that the labor share has risen for the 16 Whether the strategy pursued by Oberfield and Raval (2014) identifies a long-run capital–labor elasticity or a short-run task-level elasticity of substitution depends on the extent to which the regional wage differences used for identification are permanent. Presumably, some of the variation in regional wages is more transitory than the time-series variation in capital prices exploited in Karabarbounis and Neiman (2013); Hubmer (2020), which could reconcile these divergent estimates in the literature. 17 We assume that the capital price declines at a constant rate over the sample period 1982–2012; subsequently, it remains constant. 15