Firms, Labor Shares, and Capital–Labor Substitution (Chapter 4)

Online Appendix to “Not a Typical Firm: The Joint Dynamics of Firms, Labor Shares, and Capital–Labor Substitution” Joachim Hubmer and Pascual Restrepo March 22, 2021 A A.1 Proofs for the CES-demand model Lemmas and propositions in the main text This section provides proofs for Lemma 1 and Propositions 1–3 in the main text. Proof of Lemma 1. We first prove that optimal firm choices are given by α′ = max{α̂t (z), α}. Consider a firm that sets α′ above α at time t. The first-order condition pinning down this choice is given by ⎡ RRR ⎤⎥ ⎢ ∂πt+1 (α′ , zt+1 ) 1 ⎢ ca ⋅ y t = E + ca ⋅ yt+1 RRRRRzt ⎥⎥ . 1 + r ⎢⎢ ∂α′ RRR ⎥ ⎣ ⎦ We can rewrite this as ⎡ ′ 1−σ RRR ⎤ ⎢ −σ RRRz ⎥⎥ . σ−1 σ−1 ∂c̃t+1 (α ) ⎢ ca ⋅ ((1 + r)yt − yt+1 ) = E ⎢σ (σ − 1) ⋅ yt+1 ⋅ zt+1 ⋅ RRR t ⎥ ∂α′ ⎢ RR ⎥⎦ ⎣ By assumption, the left hand side is a positive constant. Moreover, the right-hand side is increasing in zt and decreasing in α′ . It follows that the first-order condition implicitly defines the optimal choice of α′ as an increasing function of zt , α̂t (z). The restriction that α′ ≥ α then implies that α′ = max{α̂t (z), α} as wanted. We now turn to the limiting properties of α̂t (z). Since z follows an AR1 in logs, given σ−1 ρz > 0 it holds that limzt →0 E [zt+1 ] = 0. Thus, the period t + 1 benefit of automation converges to zero as zt converges to zero. For any fixed cost ca > 0, automating a positive measure of tasks ∆α > 0 in period t cannot be optimal, as doing so in t + 1 would reduce the present discounted value of automation costs by a discrete amount: ca yt+1 ∆α 1+r < ca yt ∆α, which holds given the assumption that aggregate output yt grows at a rate below r. σ−1 ∗ Likewise, since limzt →∞ E [zt+1 ] = ∞, for any α < αt+1 the marginal automation benefit exceeds the (finite) marginal automation cost for large enough zt . Proof of Proposition 1. Suppose that qt (x) = q(x). Let’s consider an allocation in which α = α∗ for all firms. Incumbent firms set α′ = max{α̂t (z), α}. Because α̂t (z) ≤ α∗ , 51 incumbents keep α′ = α∗ . Finally, entrants start with α = α0t = α∗ , which shows that this allocation is a stationary equilibrium. Note: because of the fixed cost structure, the economy admits multiple stationary equilibria. In particular, an allocation with all firms having a common automation level α ∈ [α∗ , 1] also corresponds to an equilibrium. However, if we start from an allocation with α ≤ α∗ for all firms, the economy will converge to the stationary equilibrium with α = α∗ for all firms. Proof of Proposition 2. To account for a proportional increase in q(x), let’s write q(x) = q ⋅ q0 (x). Consider a steady state, where α = α∗ for all firms. We are interested in the comparative statics of the stationary equilibrium with aggregate equilibrium objects (w, y, α∗ ) as q changes by d ln q. A firm’s cost function can be written as c(z; w, q, α∗ ) = 1 ⋅ c̃(w, q, α∗ ), z where 1 c̃(w, q, α∗ ) = (Ψk (α∗ ) ⋅ q η−1 + Ψ` (α∗ ) ⋅ w1−η ) 1−η is the unit cost function of a firm with unitary productivity. We claim that c̃ is unchanged across steady states. To see this, observe that if the claim is true, then firm profits are proportional to y: π(z; y) ∝ z σ−1 ⋅ y. Since fixed costs are also scaled by y, the value function is linear in y, and across steady states entry and exit decisions as a function of z are unchanged. Consequently, the distribution of firm productivities with pdf f (z) does not change across steady states. We now turn to the ideal-price index condition in equation (3), which can be written in a stationary equilibrium as ∫z µ 1−σ ⋅ z σ−1 ⋅ c̃(w, q, α∗ )1−σ ⋅ f (z) ⋅ dz = 1, and implies that the unit cost c̃(w, q, α∗ ) is constant, verifying our claim. An application of Shephard’s lemma implies that d ln c̃ = ε` ⋅ d ln w − εk ⋅ d ln q. The envelope theorem ensures that the effect of changes in α∗ on c̃ are second order and 52 can be ignored. Because d ln c̃ = 0, we can solve for the change in wages as d ln w = εk 1 − ε` ⋅ d ln q = ⋅ d ln q > 0. ε` ε` We now turn to the behavior of cost shares. Along the stationary equilibrium, all firms have the same labor cost share, which is given by ε` = Ψ` (α∗ ) ⋅ w1−η . Ψk (α∗ ) ⋅ q η−1 + Ψ` (α∗ ) ⋅ w1−η This common cost share for labor will vary with prices and α∗ . Equation (5) implies that the change in the optimal threshold α∗ satisfies d ln α∗ = 1 ∂ ln ψ` (α∗ )/(q(α∗ ) ⋅ ψk (α∗ ))/∂ ln α (d ln q + d ln w). Using this expression for d ln α∗ and the definition of ηLR , we can compute the change in the cost share of labor as d ln ε` = (1 − ε` )d ln ε` εk ∂ ln Ψ` (α∗ )/Ψk (α∗ ) d ln α∗ ∂ ln α ∂ ln Ψ` (α∗ )/Ψk (α∗ )/∂ ln α = (1 − ε` )(1 − η)(d ln w + d ln q) + (1 − ε` ) (d ln q + d ln w) ∂ ln ψ` (α∗ )/(q(α∗ ) ⋅ ψk (α∗ ))/∂ ln α = (1 − ε` )(1 − η)(d ln w + d ln q) + (1 − ε` ) = (1 − ε` )(1 − η)(d ln w + d ln q) + (1 − ε` )(η − ηLR )(d ln q + d ln w) = (1 − ε` )(1 − ηLR )(d ln q + d ln w), which using the formula above for the change in wages can be written as d ln ε` = 1 − ε` (1 − ηLR )d ln q. ε` Finally, for a firm that is small enough such that it does not automate along the transition, the capital–labor elasticity is equal to the task-level substitution elasticity η. For this firm, we have d ln ε` (θ) =(1 − ε` (θ))d ln ε` (θ) = (1 − ε` )(1 − η)(d ln w + d ln q), εk (θ) 53 which using the formula above for the change in wages can be written as d ln ε` (θ) = 1 − ε` (θ) (1 − η)d ln q. ε` (θ) Proof of Proposition 3. The reasoning is analogous to 2. To account for a proportional increase in q(x) for x > α∗ , let’s write q(x) = q⋅q0 (x) for x > α∗ and q(x) = q0 (x) otherwise. We are interested in the comparative statics of the stationary equilibrium with aggregate equilibrium objects (w, y, α∗ ) as q changes from 1 by d ln q. First, recall that c̃(w, q, α∗ ) is the minimum cost of production given w and q. An increase in q thus reduces c̃ once we account for changes in α∗ , which implies that w increases. Thus, we have d ln w > 0. Note that the first-order approximation used in the proof of Proposition 2 yields d ln w = 0. This is because the increase in wages is second order but positive nonetheless. We now turn to the behavior of cost shares. Along the stationary equilibrium, all firms have the same labor cost share, which is given by ε` = Ψ` (α∗ ) ⋅ w1−η . Ψk (α∗ ) + Ψ` (α∗ ) ⋅ w1−η This common cost share for labor will vary with prices and α∗ . Equation (5) implies that the change in the optimal threshold α∗ satisfies d ln α∗ = 1 ∂ ln ψ` (α∗ )/(q(α∗ ) ⋅ ψk (α∗ ))/∂ ln α (d ln q + d ln w). Using this expression for d ln α∗ and the definition of ηLR , we can compute the change in the cost share of labor as d ln ε` = (1 − ε` )d ln ε` εk ∂ ln Ψ` (α∗ )/Ψk (α∗ ) d ln α∗ ∂ ln α ∂ ln Ψ` (α∗ )/Ψk (α∗ )/∂ ln α (d ln q + d ln w) = (1 − ε` )(1 − η)d ln w + (1 − ε` ) ∂ ln ψ` (α∗ )/(q(α∗ ) ⋅ ψk (α∗ ))/∂ ln α = (1 − ε` )(1 − η)d ln w + (1 − ε` ) = (1 − ε` )(1 − η)d ln w + (1 − ε` )(η − ηLR )(d ln q + d ln w) = −(1 − ε` )(ηLR − η)d ln q + (1 − ε` )(1 − ηLR )d ln w. Finally, for a firm that is small enough such that it does not automate along the 54 transition, the capital–labor elasticity is equal to the task-level substitution elasticity η. For this firm, we have the same expression derived above: d ln ε` (θ) = (1 − ε` (θ))(1 − η)(d ln w + d ln q). A.2 Additional results and formulas This section derives the formula for the long-run elasticity of substitution provided in the main text. It also derives the approximation for the productivity gains from automation given in footnote 18. Long-run K-L elasticity of substitution ηLR : As usual, we define this elasticity as the partial equilibrium response of the capital–labor ratio to a change in the relative price w̃ ≡ w 1/q ; here taking into account the change in α∗ . This long-run elasticity of substitution is therefore defined implicitly by the identity d ln ε` ≡ (1 − ηLR ) ⋅ d ln w̃. εk We have that (8) d ln Ψ` (α) ⋅ w1−η ∂ ln Ψ` (α)/Ψk (α) ∂ ln α∗ ε` = d ln = ⋅ ⋅ d ln w̃ + (1 − η) ⋅ d ln w̃. εk Ψk (α) ⋅ q η−1 ∂ ln α ∂ ln w̃ Equation (5) implies that −1 ∂ ln α∗ ∂ ln ψ` (α)/(q(α) ⋅ ψk (α)) =( ) . ∂ ln w̃ ∂ ln α Plugging this in equation (8) we obtain ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ε` ⎢ ∂ ln Ψk (α)/Ψ` (α) ∂ ln ψ` (α)/(q(α) ⋅ ψk (α)) ⎥⎥ d ln = ⎢1 − (η + / )⎥ d ln w̃ εk ⎢ ∂ ln α ∂ ln α ⎥ ⎢ ´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 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 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 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 ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶⎥ ⎢ ⎥ ≡ηLR ⎣ ⎦ which gives the formula for ηLR provided in the main text. Productivity gains from automation: Consider an increase in the automation threshold of dα > 0 for a firm with α < α∗ . The contribution of this technology to a firm’s 55 TFP is then given by d ln TFPα ∶= −d ln c(θ, w) = − Ψ′ (α) Ψ′ (α) 1 [ε` (α, w) k + ε` (α, w) ` ] dα. 1−η Ψk (α) Ψ` (α) We can rewrite this expression as follows: 1 (q(α) ⋅ ψk (α))η−1 ψ` (α)η−1 [ε` (α, w) − ε` (α, w) ] ⋅ dα 1−η Ψk (α) Ψ` (α) (q(α) ⋅ ψk (α))η−1 ψ` (α)η−1 w1−η 1 [ =− − ] ⋅ dα ≥ 0. ̃ ̃ 1−η c1−η c1−η d ln TFPα = − Log-linearizing the function 1 1−η , 1−η x we can approximate the productivity gains when α → α∗ as d ln TFPα ≈ [ln ( w 1 (w/ψ` (α))1−η ) − ln ( )] ⋅ ⋅ dα ≥ 0, ̃ ψ` (α) q(α) ⋅ ψk (α) c1−η which yields the formula in the main text. B B.1 Proofs for the model with variable markups Properties of the Klenow–Willis specification As a convenient functional form for the Kimball (1995) aggregator H(⋅) we use the specification from Klenow and Willis (2016), defined as ν 1 σ 1 σ ȳ σ σ H(ȳ) ≡ 1 + (σ − 1) ⋅ exp ( ) ⋅ ν ν −1 ⋅ [Γ ( , ) − Γ ( , )] , ν ν ν ν ν where ȳ = y(θ) λ⋅y is the relative quantity of a variety, and Γ(⋅, ⋅) is the upper incomplete Gamma function, ∞ Γ(s, x) ≡ ∫ ts−1 ⋅ exp(−t)dt. x 56 This specification gives rise to the following (relative) demand function D−1 = H ′ : σ ν σ )) , D(p̄) = (1 − ν ⋅ ln (p̄ ⋅ σ−1 σ −1 ν σ σ ′ , D (p̄) = ⋅ (1 − ν ⋅ ln (p̄ ⋅ )) p̄ σ−1 where p̄ = p(θ) ρ is the relative price of a variety. Thus, the price elasticity of demand is (9) − D′ (p̄) ⋅ p̄ σ ν −σ , = = σ ⋅ D(p̄) σ D(p̄) ) 1 − ν ⋅ ln (p̄ ⋅ σ−1 which reduces to the constant σ if ν = 0 (the benchmark case of a CES aggregator). In general, equation (9) shows that under this parametrization, the supra-elasticity of demand is equal to the constant − σν . Static profit maximization: As usual, static optimal pricing relates the demand elasticity to the optimal markup µ = p/c as µ D′ (p̄) ⋅ p̄ =− . µ−1 D(p̄) Since p̄ = p ρ = µ⋅c ρ , we can re-write this equation using (6) such that it implicitly defines the optimal markup µ∗ as a function of the relative cost ĉ = c/ρ, σ ⋅ (1 − 1 µ∗ (ĉ) ) = 1 − ν ⋅ ln ( σ ⋅ ĉ ⋅ µ⋆ (ĉ)) . σ−1 Since the left-hand-side is an increasing function of µ⋆ (ĉ) and the right-hand-side a decreasing function of ĉ⋅µ⋆ (ĉ), the optimal markup is implicitly defined as a strictly decreasing function of (relative) cost. Moreover, setting µ∗ = 1, we can solve for the (relative) choke price as p̄choke = 1 σ−1 ⋅ exp ( ) , σ ν implying that firms with cost c < p̄choke ⋅ ρ do not produce. Marshall’s second laws: Equation (6) shows that the demand elasticity is increasing in the relative price and greater than 1 (Marshall’s weak second law), imposing the restriction that σ > 1 and ν > 0. To see that the strong law holds as well, write the logarithm of 57 marginal revenue as ln (p̄ + D(p̄) D(p̄) ) = ln p̄ + ln (1 + ′ ) ′ D (p̄) D (p̄) ⋅ p̄ = ln p̄ + ln σ ) − 1⎞ ⎛ σ + ν ⋅ ln (p̄) + ν ⋅ ln ( σ−1 , σ ⎝ ⎠ which is a concave function of ln p̄ as desired. B.2 Proofs and derivations Before turning to the proofs of the propositions in the text, we derive the equations used in the characterization of the equilibrium. We will ignore time subscripts as long as it causes no confusion. The demand for each variety is obtained by solving the following cost minimization problem: min ∫ p(θ) ⋅ y(θ) ⋅ m(θ) ⋅ dθ y(θ) θ y(θ) s.t: ∫ λ ⋅ H ( ) m(θ)dθ = 1. λ⋅y θ Let ρ ⋅ y denote the Lagrange multiplier on the constraint. The first-order condition for the choice of y(θ) is then p(θ) = ρ ⋅ H ′ ( y(θ) ) λ⋅y y(θ) = y ⋅ λ ⋅ D ( ⇒ p(θ) ). ρ Moreover, because the price of the final good is normalized to 1, we must have y = ∫ p(θ) ⋅ y(θ) ⋅ m(θ) ⋅ dθ θ ⇒ 1 = ∫ λ ⋅ p(θ) ⋅ D ( θ p(θ) ) ⋅ m(θ) ⋅ dθ, ρ which is the ideal-price index condition given in the text. Finally, plugging the demand for each variety in the constraint, we obtain p(θ) ∫θ λ ⋅ H (D ( ρ )) m(θ)dθ = 1, which pins down the competitors’ price index ρ. The labor market clearing condition follows from an application of Roy’s lemma, which implies that cw (θ, w) yields the demand of labor per unit of goods sold. 58 Proof of Proposition 4. Prices are given by p p(c) = arg max y ⋅ λ ⋅ D ( ) ⋅ (p − c). ρ p This problem has increasing differences in p and c, which implies that p(c) is increasing in c. Moreover, the first order condition for this problem is 1 p p − D′ ( ) ⋅ (p − c) = D ( ) ρ ρ ρ ⇒ ′ p(c) µ(c) p(c) D ( ρ ) =− . µ(c) − 1 ρ D ( p(c) ) ρ Marshall’s weak second law combined with the fact that p(c) increases in c implies that the right-hand side of the above equation increases in c. The left-hand side is a decreasing function of µ(c), which therefore implies that µ(c) is decreasing in c as wanted. We can rewrite the first-order condition for prices as p(c) D(p(c)/ρ) c + ′ = . ρ D (p(c)/ρ) ρ Differentiating this expression yields p(c) ∂ ln p(c) = 1/n ( ), ∂ ln c ρ where n(x) = ∂ ln (x + D(x)/D′ (x)) ∂ ln x is a decreasing function according to Marshall’s strong second law. It follows that ln p(c) is a convex function in ln c as wanted. Moreover, ln µ(c) = ln p(c) − ln c will inherit this convexity. Turning to sales, we have that s(c) can be written as s(c) = h(p(c)), where h(x) = xD(x) is a log-concave and decreasing function of x (from Marshall’s weak second law). Thus, s(c) is the composition of a log-concave and decreasing function (h(x)) with a log-convex and increasing function p(c), which results in a log-concave and 59 decreasing function. Proof of Proposition 5. In the next proofs, it will be convenient to define c̄ ∶= ̃ c p(c) p̄ ∶= , ρ ρ where recall that ̃ c is the unit cost for a firm with unitary productivity. We can rewrite the firm problem as 1 1 p̄ ( c̄) = arg max D(p̄) ⋅ (p̄ − c̄) . p̄ z z Proposition 4 has three implications. First, we can write prices as p(z) = ρ ⋅ p̄ ( z1 c̄), where p̄(.) is an increasing function. Second, the resulting markup µ(z) is given by 1 µ(z) = µ̄ ( c̄) , z where the function µ̄(.) is decreasing and log-convex. Finally, sales are given by 1 1 1 s(z) = s̄ ( c̄) = λ ⋅ ρ ⋅ p̄ ( c̄) ⋅ D (p̄ ( c̄)) , z z z where s̄(.) is a decreasing and log-concave function. The implicit definition of the competitors’ price index implies 1 ∫z λ ⋅ H (D (p̄ ( z c̄))) f (z)dz = 1. Thus, as λ increases, c̄ must increase. This implies that the effect of an increase in λ on prices, markups, and sales shares is isomorphic to that of an increase in c̄. We now turn to characterizing the effects of an increase in c̄. First, we have that for a given z, µ(z) = µ̄ ( z1 c̄) will be decreasing in c̄, as wanted. Second, because the function µ̄ is log-convex, we have that, for z > z ′ , 1 1 ln µ(z) − ln µ(z ′ ) = ln µ̄ ( c̄) − ln µ̄ ( ′ c̄) z z is decreasing in c̄. 60 Third, because the function s̄ is log-concave, we have that, for z > z ′ , 1 1 ln s(z) − ln s(z ′ ) = ln s̄ ( c̄) − ln s̄ ( ′ c̄) z z is increasing in c̄. Note that there is a common λ term multiplying both s(z) and s(z ′ ) which cancels when computing the difference ln s(z) − ln s(z ′ ). Thus, for relative shares, the effect of λ continues to be isomorphic to that of an increase in c̄. Proof of Proposition 6. As before, we investigate the implications of an increase in c̄. We can write the aggregate markup as 1 1 1 1 =∫ c̄) ⋅ f (z) ⋅ dz/ s̄ ( c̄) ⋅ f (z) ⋅ dz. ⋅ s̄ ( ∫ µ z z z µ̄ ( 1 c̄) z z With the change of variable x = z1 c̄, we can rewrite this as 1 1 =∫ ⋅ g(x, c̄) ⋅ dx, µ x µ̄ (x) where g(x, c̄) is a density function given by g(x, c̄) = s̄ (x) ⋅ f (c̄/x) ⋅ dx/ ∫ s̄ (x) ⋅ f (c̄/x) ⋅ dx. x First, suppose that f (z) is log-concave. This implies that ln s̄ (x) + ln f (c̄/x) has increasing differences in x and c̄. This is equivalent to the following monotone likelihood ratio property (MLRP): g(x, c̄) increasing in c̄ for x > x′ . g(x′ , c̄) The MLRP property implies that an increase in c̄ generates a shift up (in the first-order stochastic dominance sense) in g(x, c̄). Because the function aggregate markup µ is decreasing in c̄ as wanted. Second, suppose that f (z) is log-convex. This implies that ln s̄ (x) + ln f (c̄/x) 61 1 µ̄(x) is increasing in x, the has decreasing differences in x and c̄. This is equivalent to the following monotone likelihood ratio property (MLRP): g(x, c̄) decreasing in c̄ for x > x′ . g(x′ , c̄) The MLRP property implies that an increase in c̄ generates a shift down (in the first-order stochastic dominance sense) in g(x, c̄). Because the function 1 µ̄(x) is increasing in x, the aggregate markup µ is increasing in c̄ as wanted. Finally, suppose that f (z) is log-linear. This implies that ln s̄ (x) + ln f (c̄/x) is a linear function in c̄. Equivalently, g(x, c̄) independent of c̄. g(x′ , c̄) Thus, the integral defining µ is independent of c̄ as wanted. C Robustness of model-based results In this section, we discuss the robustness of our quantitative findings in Section 1 to the timing of automation decisions, as well as to different values for the short- and long-run capital–labor elasticity. C.1 Timing of automation decisions In the main text, we assume that firms invest in the adoption of new capital technologies (raising their α) in period t before the realization of their productivity in period t + 1, when the new technology first becomes operative. We have experimented with other timing assumptions and found that our results are robust on this dimension. Specifically, column (3) in Table 7 shows the calibration for an alternative model version where firms decide in the beginning of period t+1, after their new productivity draw has materialized, whether to pay the operating fixed cost and whether to adopt new capital technologies, which are immediately operative. Column (3) in Table 8 shows the quantitative results over the transition period. We find that the results largely agree with our findings from the benchmark model, which are re-produced in column (2). As expected, since firms 62 with a high z (i.e., large firms) can raise their α immediately, the cross-dynamics term is somewhat larger in absolute value, going from -0.143 to -0.157. The inferred automation fixed cost ca falls from 0.23 to 0.18, which reflects a lower option value of automation for small firms. Table 7: Model robustness: Steady state calibration Data (1) ln q0 σ σz cf ξ µe η γ I. Parameters Inverse capital price Demand elasticity Std. dev. of ln z innovations Minimum fixed cost (×10−6 ) Dispersion fixed cost Entrant productivity Task substitution elasticity Comparative advantage* II. Moments Manufacturing labor share Aggregate markup Top 4 firms’ sales share Entry (=exit) rate Size of exiters Size of entrants Short-run K–L elasticity Long-run K–L elasticity 60.1% 1.10 40.0% 0.062 0.490 0.600 Model Benchmark (2) Altern. timing (3) Lower η ηLR < 1 (4) (5) −5.35 11.0 0.094 6.0 0.330 0.935 0.40 0.95 −5.36 11.0 0.094 9.0 0.350 0.944 0.40 0.95 −5.33 11.0 0.094 6.0 0.330 0.936 0.20 1.15 3.82 11.0 0.094 6.0 0.330 0.935 0.40 5.0* 60.2% 1.10 40.1% 0.063 0.491 0.598 0.40 1.35 60.1% 1.10 40.1% 0.063 0.490 0.600 0.40 1.35 60.1% 1.10 40.1% 0.063 0.491 0.602 0.20 1.35 60.1% 1.10 40.1% 0.063 0.491 0.598 0.40 0.90* Notes: See Section 1 and Table 1 for the calibration of the benchmark model (corresponding to the manufacturing sector and using a CES demand system). Column (3) differs from the benchmark model insofar as firms decide in the current period, after current productivity is realized, whether to pay the fixed cost and whether (and how much) to automate, such that new capital technologies are immediately productive. Column (4) differs from the benchmark insofar as the task-substitution elasticity is lower (while, again, other parameters are re-calibrated to match the same data targets). Column (5) features a different parameterization of the capital and labor productivity schedules, which allows for a lower long-run capital-labor elasticity ηLR (the parameter γ refers to a different object and is not directly comparable to other model versions). ηLR is not a constant in this exercise. Locally, at the initial steady state, it equals 0.90, and it falls only slightly to 0.88 over the range considered in this exercise. C.2 Short-run capital-labor elasticity We argued in the main text that we view the calibrated short-run elasticity of η = 0.4 as an upper bound on the elasticity of substitution between tasks, since the firm-level evidence in Oberfield and Raval (2014) refers to cross-sectional variation, which might be interpreted as incorporating adjustments in task allocations within firms. Column (4) in Table 7 describes an alternative calibration with η = 0.2. For this exercise, we hold the long-run elasticity constant, which requires increasing the comparative advantage 63 Table 8: Model robustness: Transitional dynamics Model d ln q ca ∆ aggregate labor share ∆ median labor share ∆ Top 4 firms’ sales share ∆ Top 20 firms’ sales share ∆ log productivity dispersion ∆ aggregate labor share ∆ unweighted mean Exit Entry Covariance term Market share dynamics Labor share by size dynamics Joint dynamics Altern. timing (3) Lower η ηLR < 1 (4) (5) I. Parameters and inferred aggregate shocks . 1.40 1.40 1.39 . 0.23 0.18 0.24 3.80* 0.35 Data Benchmark (1) (2) −0.178 0.030 0.060 0.052 0.050 II. Targeted moments, 1982–2012 −0.176 −0.181 0.026 0.031 III. Other moments, 0.043 0.070 0.057 IV. Melitz–Polanec decomposition −0.185 −0.176 −0.002 0.032 −0.055 −0.003 0.059 0.006 −0.187 −0.212 1982–2012 0.045 0.074 0.058 −0.176 0.029 −0.174 0.016 0.029 0.049 0.059 0.222 0.207 0.015 from Autor et al. (2020) −0.181 −0.176 0.038 0.035 −0.003 −0.003 0.005 0.006 −0.221 −0.215 V. Covariance decomposition from Kehrig and Vincent (2020) 0.047 0 0 0 −0.043 −0.036 −0.027 −0.036 −0.232 −0.143 −0.157 −0.144 −0.174 0.002 −0.010 0.016 −0.183 0 −0.041 −0.140 Notes: See Section 1 and Table 2 for details on the benchmark model (corresponding to the manufacturing sector and using a CES demand system). Column (3) differs from the benchmark model insofar as firms decide in the current period, after current productivity is realized, whether to pay the fixed cost and whether (and how much) to automate, such that new capital technologies are immediately productive. Column (4) differs from the benchmark insofar as the task-substitution elasticity is lower. Column (5) features a different parameterization of the capital and labor productivity schedules, which allows for a lower long-run capital-labor elasticity. The calibrated shock d ln q is not comparable to other model versions, as it does not raise investment efficiency uniformly across all tasks (see details in text). parameter γ from 1.35 − 0.4 = 0.95 to 1.35 − 0.2 = 1.15. Comparing columns (2) and (4) in Table 8 reveals that the results virtually coincide, implying that our findings are not sensitive to the value of η within a reasonable range. C.3 Long-run capital-labor elasticity In the main text, we studied the effects of a uniform decline in the price of capital goods. This shock requires an above one long-run capital-labor elasticity (ηLR > 1) to match the decline in the labor share observed in the data. Following Proposition 3, this appendix shows that even if ηLR < 1, we can also generate a decline in the aggregate labor share and an increase in the labor share of the typical median firm by introducing declines in the price of capital at marginal tasks. 64 Implementing this exercise requires using a different parametrization of the schedules of capital and labor productivity, since the one used in the main text does not allow for ηLR < 1.35 Specifically, we use the parametrization ⎧ γ ⎪ ⎪ ⎪(xL − x) ψk (x) = ⎨ ⎪ ⎪0 ⎪ ⎩ if x ≤ xL if x > xL ψ` (x) = xγ , where xL ∈ (0, 1] denotes a cut-off such that all tasks in (xL , 1] can only be performed by labor, and γ denotes the comparative advantage parameter. Setting xL = 0.9 and γ = 5 generates a long-run elasticity that ranges between 0.88 and 0.90 along the transitional dynamics studied in this exercise. Column (5) in Table 7 describes the full steady state calibration. With this parametrization, the aggregate labor share would increase in both the short and long run in response to a uniform increase in q(x). Instead, we consider a task-specific increase in q(x) of the following form: 1. From 1982, when the economy is in the initial steady state, until 1992, ln q(x) ∗ increases linearly for all x ≥ α1982 . ∗ 2. Over 1992-2002, ln q(x) increases linearly for all x ≥ α1992 . ∗ 3. Over 2002-2012, ln q(x) increases linearly for all x ≥ α2002 . The particular formulation of the shock is not crucial; the main point is that we want to engineer an increase in capital productivity at marginal tasks while limiting a reduction of capital prices at infra-marginal tasks that are already performed by capital. This reduction reduces the capital share at those tasks, and is the reason why a uniform decline in the price of capital across all tasks requires ηLR > 1 to generate a decline in the aggregate labor share. Column (5) in Table 8 describes the transitional dynamics in this case. For some tasks, we infer a change in log investment efficiency of up to 3.8, but one should keep in mind that this does not map to the observed decline in the price of any broad category of capital goods.36 The inferred fixed cost of automation needs to be around 50% higher than in 35 The specification in the main text is the unique specification with the property that ηLR is a constant, but it requires ηLR > 1. All other specifications, and by implication all specifications that allow for ηLR < 1, are such that ηLR is not a constant. 36 ∗ ∗ ∗ For x ∈ [0, α1982 ], q(x) does not change at all; for x ∈ [α1982 , α1992 ], ln q(x) increases linearly by 1.27 65 the benchmark to generate a slight increase in median and unweighted mean labor shares, since the lack of an increase in capital productivity for tasks that are initially performed by capital implies that the labor shares of non-automating firms increases by little. While the labor share decompositions are comparable to the main exercise, productivity dispersion and sales concentration (Panel III) react differently: Since capital productivity increases primarily in marginal tasks, the productivity benefit of automation is smaller. Thus, first, the rise in productivity dispersion amounts to merely a quarter to one third of the effect generated in the benchmark exercise (which roughly matches the data). Second, the fraction of automating firms is much smaller, such that only the top firms automate at all. As a result, the difference between the top and the typical firm is much bigger, and the rise in sales concentration is much stronger. We conclude from this exercise that, even when the long-run elasticity of substitution is below 1, we can generate the observed decline in the aggregate labor share and the rise in the labor share of the typical firm as a response to a reduction of capital prices at marginal tasks. D Calibration of the non-CES demand model for other sectors Table 9 summarizes the steady state calibration of the model with size-dependent markups in the wholesale as well as the utilities & transportation sector. The calibration strategy is identical to manufacturing and retail, which we describe in the main text. The logconvexity of the z-distribution is rather mild in these two sectors (n only slightly below 1), more in line with manufacturing than with retail. Table 10 shows the model-based decomposition exercise, where we follow the same strategy as for manufacturing and retail. In wholesale as well as in utilities & transportation, the labor share decline is of the same magnitude as in retail, while the observed increase in concentration is not as drastic. Consequently, the inferred decline in the price of capital (d ln q) is of similar magnitude, while the inferred increase in competition (d ln λ) is weaker than in retail. The inferred automation fixed costs (ca ) are small, both in absolute terms and relative to manufacturing and retail. Comparing the various model versions, we find that the declining capital price caused 8.1pp or 70% of the sectoral labor share decline in wholesale, and 5.2pp or 72% of the sectoral labor share decline in utilities & transportation. For both sectors, the remainder is almost entirely due to the non∗ ∗ from 1982 to 1992; for x ∈ [α1992 , α2002 ], ln q(x) increases linearly by 2.53 from 1982 to 2002; and for ∗ x ∈ [α2002 , 1], ln q(x) increases linearly by 3.8 from 1982 to 2012. 66 Table 9: Steady state calibration of the non-CES demand model: Wholesale, Utilities & Transportation Parameter Moment ln q0 σ ν/σ ζ n cf ξ µe I. Wholesale: steady state parameters and moments (1982) Inverse capital price −5.97 Aggregate labor share Demand elasticity 9.2 Aggregate markup Demand supra-elasticity 0.20 Median labor share ratio Weibull scale 0.098 Top 20 firms’ sales share Weibull shape 0.83 Top 4 firms’ sales share Minimum fixed cost 6.5 ⋅ 10−8 Entry (=exit) rate Dispersion fixed cost 0.210 Size of exiters Entrant productivity 0.888 Size of entrants ln q0 σ ν/σ ζ n cf ξ µe II. Utilities & Transportation: steady Inverse capital price −5.38 Demand elasticity 10.2 Demand supra-elasticity 0.16 Weibull scale 0.143 Weibull shape 0.95 Minimum fixed cost 7.0 ⋅ 10−9 Dispersion fixed cost 0.180 Entrant productivity 0.888 state parameters and moments Aggregate labor share Aggregate markup Median labor share ratio Top 20 firms’ sales share Top 4 firms’ sales share Entry (=exit) rate Size of exiters Size of entrants Data Model 58.6% 1.15 1.169 42.9% 22.3% 0.062 0.490 0.600 58.6% 1.15 1.130 42.9% 22.3% 0.063 0.485 0.596 (1992) 52.0% 1.15 1.169 59.1% 30.4% 0.062 0.490 0.600 52.0% 1.15 1.133 56.1% 33.1% 0.062 0.488 0.600 Notes: The aggregate labor shares correspond to the BLS Multifactor Productivity Tables estimate for the wholesale as well as utilities & transportation sector. The ratio median-to-aggregate is from Kehrig and Vincent (2020), which refers to manufacturing, since in these sectors the census data used, e.g., by Autor et al. (2020) does not allow to compute the labor share of value added. The two concentration measures are from Autor et al. (2020) and correspond to these two sectors in 1982, respectively 1992. The model equivalents refer to the top 0.074% and top 0.369% of firms ranked by sales in wholesale (since there are on average 5,420 firms per 4-digit wholesale industry). For utilities & transportation, the model equivalents correspond to the top 0.100% and top 0.499% of firms ranked by sales (since there are on average 4,010 firms per 4-digit industry in this sector). The data moments and calibration strategy on entry rates, size of exiters and of entrants, follow the model with CES demand, see Table 1. In each of the two sectors, the eight parameters are jointly calibrated to match the eight data moments. linearity or interaction effect of the two shocks: rising competition increases automation incentives for the top firms, and automation increases labor share differentials, magnifying the effect of reallocation on the aggregate labor share decline. We conclude that while for these two sectors, both shocks are important to account for the labor share decline, the inferred q-shock by itself is relatively more important than in retail, and less than in manufacturing. 67 Table 10: Transitional dynamics and decomposition of the labor share using a non-CES demand system: Wholesale, Utilities & Transportation Model Data Benchmark (1) (2) Only effects of d ln q (3) Only effects of d ln λ (4) A. Wholesale (1982–2012) d ln q d ln λ ca ∆ ∆ ∆ ∆ aggregate labor share unweighted mean labor share Top 4 firms’ sales share Top 20 firms’ sales share I. Parameters and inferred aggregate shocks . 0.74 0.74 . 0.27 0 . 0.05 0.05 II. Targeted −0.114 0.047 0.047 0.101 ∆ log productivity dispersion moments, 1982–2012 −0.115 −0.081 0.050 0.024 0.063 0.002 0.086 0.003 −0.001 0.007 0.058 0.080 III. Other moments, 1982–2012 . 0.026 0.014 0.000 IV. Markups, 1982–2012 0.003 0.001 −0.023 −0.010 0.025 0.011 0.001 −0.018 0.019 . . . ∆ log aggregate markup Within firm change in markups Reallocation 0 0.27 0.05 B. Utilities & Transportation (1992–2012) d ln q d ln λ ca ∆ ∆ ∆ ∆ aggregate labor share unweighted mean labor share Top 4 firms’ sales share Top 20 firms’ sales share I. Parameters and inferred aggregate shocks . 0.40 0.40 . 0.22 0 . 0.002 0.002 II. Targeted −0.073 0.004 0.046 0.047 ∆ log productivity dispersion moments, 1992–2012 −0.072 −0.052 0.007 −0.002 0.045 0.000 0.049 −0.001 0.000 0.003 0.044 0.050 III. Other moments, 1992–2012 . 0.015 0.007 0.000 IV. Markups, 1992–2012 0.000 0.000 −0.011 −0.004 0.011 0.004 0.000 −0.010 0.010 . . . ∆ log aggregate markup Within firm change in markups Reallocation 0 0.22 0.002 Notes: Column (2) reports the findings from our benchmark model, which calibrates a uniform decline in the capital price, an increase in competition, as well as the automation fixed cost to replicate the change in the aggregate sectoral labor share (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), for both sectors. Due to data availability, the transition is over 1982–2012 for wholesale, resp. 1992–2012 for utilities & transportation. 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). 68 E Comparing estimated shocks to data This section provides additional motivation for our focus on the 1982–2012 period and benchmarks the inferred shocks and calibrated model parameters to the available data. Historical behavior of payroll shares: As a starting point, Figure 10 provide data on payroll shares by sector for 1947–1987 and 1987–2012 from the BEA industry accounts. We split the data in these two periods due to changes in industry definitions introduced by the BEA in 1987, as it switched from the Standard Industry Classification to the North American Industry Classification System. We also focus on payroll shares, since labor shares of value added (which also include non-wage compensation), are not available for the earlier period. Figure 1 in the Introduction already provided the BLS data for labor shares of value added for the later period. Payroll share, BEA data for 1947−1987 Payroll share, BEA data for 1987−216 80 65 manufacturing 75 60 70 manufacturing 55 65 60 55 50 retail and wholesale trade retail and wholesale trade aggregate holding industry shares in GDP fixed at 1987 50 aggregate holding industry shares in GDP fixed at 1987 45 45 40 1947 1952 1957 1962 1967 1972 1977 1982 1987 1987 1992 1997 2002 2007 2012 2017 Figure 10: Payroll share in the U.S. for 1947–2016. The figure plots the payroll share of value added for 1947–1987, both for some specific sectors and the economy as a whole. Data from the BEA industry accounts. As discussed in the main text, the figure shows that payroll shares were constant or increasing up to 1982, and then started a sharp decline both in manufacturing, retail and wholesale. This motivates our focus on the 1982–2012 period and supports our choice of 1982 as the steady state of the model. Benchmarking the fixed cost of automating tasks: As discussed in the main text, the fixed cost of automating tasks can be thought of as an investment in R&D required 69 to design and integrate automation equipment or software. As such, these fixed costs will contribute to rising R&D expenditures in the economy. The left panel in Figure 11 compares the inferred behavior of automation fixed costs to the available data on R&D spending. In particular, the panel displays the time series of automation cost spending as a share of output in the model, ∫ ca ⋅ max{0, α̂t (z) − α} ⋅ mt (θ)dθ, both for manufacturing and for retail. For simplicity, we focus on the non-CES version of the model calibrated in Section 2.4. As the price of capital declines from 1982 to 2012 and stays constant thereafter, automation cost spending in manufacturing reaches a peak of roughly 0.7% of output around 2005, and declines to zero eventually as the model economy converges to the new steady state. In retail, automation cost spending is much lower, around 0.1% of output in 2005. The black line provides the behavior of R&D as a share of GDP for the entire US economy. This share rose from a level of 1% before 1980 to a current level of 2.5% of GDP. Because not all R&D expenditure is due to automation, we see this series as an upper bound for automation cost spending. In line with this view, our calibration implies that about 25% of manufacturing R&D after 1980 corresponds to investments in automation fixed costs, and that rising automation cost spending since 1980 explain about 0.6 percentage points of the 1 percentage point increase in R&D observed from 1980–2005. This comparison shows that our estimated fixed costs are of a reasonable magnitude, and that they do not generate a counterfactual increase in R&D spending. Benchmarking the inferred decline in capital prices and rising competition: The right panel of Figure 11 compares the inferred decline in the price of capital d ln q to data. As the empirical counterpart for these series, we use the percent decline in the price of equipment and software over the time period 1982–2012 from the BEA’s Fixed Asset Tables, which we deflate using the PCE index. We focus on software and equipment because these are most relevant for capital–labor substitution in our framework. In addition, we display a series by DiCecio (2009), which attempts to correct the BEA series for missing quality-adjustment in the spirit of Gordon (1990), following the imputation procedure in Cummins and Violante (2002). The inferred decline in the price of capital in our model is well within the range of the empirical counterparts. In particular, the decline in the price of equipment and software needed to explain the decline of the manufacturing labor share is of 140 log points, which is comparable to the observed decline of 174 log points in the data (112 without Gordon’s quality adjustment). 70 R&D investment share vs. fixed cost share Relative price of equipment & software 0 2.5 data: R&D investment / GDP model: automation fixed cost / output (manufacturing) model: automation fixed cost / output (retail) -0.2 2 -0.4 log, 1982=0 -0.6 in % 1.5 1 -0.8 -1 -1.2 -1.4 0.5 -1.6 0 1960 -1.8 1970 1980 1990 2000 2010 2020 1985 1990 1995 2000 2005 2010 Figure 11: Automation fixed costs and capital price declines. The left panel displays the time series of automation fixed cost spending relative to aggregate output for manufacturing and retail using our estimates from Section 2.4. Before 1982, the model is in steady state, and automation spending is zero. The black line corresponds to the ratio of R&D investment spending relative to U.S. GDP. The right panel displays the calibrated decline in capital prices −d ln q for manufacturing and retail using our estimates from Section 2.4. The black line plots the observed decline in the price of equipment and software capital for 1982–2012, deflated by the PCE price index (source: BEA Fixed Asset Tables). In addition, the gray line plots another series for the relative price of equipment and software due to DiCecio (2009), which builds on work by Gordon (1990) and Cummins and Violante (2002) and imputes missing quality-adjustment. Turning to the inferred measure of rising competition, it is difficult to find an empirical counterpart to λ in the data. Our model infers a particular strong increase in retail (d ln λ = 0.41), followed by wholesale (0.27), utilities/transportation (0.22), and manufacturing (0.04). The type of changes that λ proxies for include in particular a rise in the effective market size caused by the widespread availability of internet search engines, and in general breakthroughs in information and communications technology, which reduce information frictions. For example, Akerman, Leuven and Mogstad (2021) provide evidence showing that broadband availability is causally associated with an expansion of the choice set of importers and exporters in Norway. At a qualitative level, it is reasonable that this type of technological change most strongly affected retail trade, followed by wholesale trade. 71