The race between man and machine: implications of technology for growth, factor shares, and employment

May combine two claims.

In the static model with fixed capital and exogenous technology, automation reduces employment and the labor share, and may even reduce wages.

“In a static version where capital is fixed and technology is exogenous, automation reduces employment and the labor share, and may even reduce wages, while the creation of new tasks has the opposite effects.”

• Also stated in: § By Daron Acemoglu and Pascual Restrepo* · hedged

  “Automation always reduces the labor share and employment, and may even reduce wages.”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.
• AssumptionThe authors state this as an assumption: The capital stock is not too large: K < K_bar, where K_bar is such that R = W/γ(N).

The quote only partly states this claim.

Otherwise, there exists a stable balanced growth path in which automation and creation of new tasks advance at equal rates.

“Otherwise, there exists a stable balanced growth path in which the two types of innovations go hand-in-hand.”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.

May combine two claims.

An increase in equilibrium automation I* increases the share of capital and reduces the share of labor in the aggregate production function, while creation of new tasks does the opposite.

“An increase in I * , which corresponds to greater equilibrium automation , increases the share of capital and reduces the share of labor in this aggregate production function, while the creation of new tasks does the opposite.”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.
• AssumptionThe authors state this as an assumption: The capital stock is not too large: K < K_bar, where K_bar is such that R = W/γ(N).

May combine two claims.

If I* = ~I < I, new tasks increase productivity, but additional automation technologies do not.

“That is, new tasks increase productivity, but additional automation technologies do not.”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.
• AssumptionThe authors state this as an assumption: The capital stock is not too large: K < K_bar, where K_bar is such that R = W/γ(N).

May combine two claims.

An increase in N (creation of new tasks) raises W/R, the labor share, and employment.

“An increase in N ( the creation of new tasks) raises W/R , the labor share, and employment.”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.
• AssumptionThe authors state this as an assumption: The capital stock is not too large: K < K_bar, where K_bar is such that R = W/γ(N).

May combine two claims.

If I* = ~I < I, an increase in N always increases the equilibrium wage but may reduce the rental rate, while an increase in I has no effect on factor prices.

“That is, an increase in N (more new tasks) always increases the equilibrium wage but may reduce the rental rate, while an increase in I (greater technological automation) has no effect on factor prices.”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.
• AssumptionThe authors state this as an assumption: The capital stock is not too large: K < K_bar, where K_bar is such that R = W/γ(N).

May combine two claims.

Both automation and creation of new tasks increase inequality in the short run, but standardization limits the increase in inequality in the long run.

“This extension formalizes the idea that both automation and the creation of new tasks increase inequality in the short run but standardization limits the increase in inequality in the long run.”

• AssumptionThe authors state this as an assumption: Skilled labor has a comparative advantage in new tasks.

May combine two claims.

In the dynamic model, although the real wage increases in the long run due to the productivity effect, automation still reduces the labor share and employment.

“Although the real wage in the long run increases because of this productivity effect, automation still reduces the labor share and employment.”

May combine two claims.

Both automation and creation of new tasks increase productivity.

“That is, both technologies increase productivity.”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.
• AssumptionThe authors state this as an assumption: The capital stock is not too large: K < K_bar, where K_bar is such that R = W/γ(N).

May combine two claims.

The technology-constrained elasticity of substitution between capital and labor, σ_hat, differs from the technology-free elasticity, σ_free, and σ_free > σ_hat.

“This additional margin of adjustment implies that σ free > ˆ σ .”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.
• AssumptionThe authors state this as an assumption: The capital stock is not too large: K < K_bar, where K_bar is such that R = W/γ(N).

May combine two claims.

A higher N always increases the equilibrium wage but may reduce the rental rate of capital, while a higher I always increases the rental rate of capital but may reduce the equilibrium wage.

“That is, a higher N always increases the equilibrium wage but may reduce the rental rate of capital, while a higher I always increases the rental rate of capital but may reduce the equilibrium wage.”

• AssumptionThe authors state this as an assumption: Labor has a strict comparative advantage in tasks with a higher index (and in new and more complex tasks).
• AssumptionThe authors state this as an assumption: The demand for labor and capital is homothetic, ensured by either η→0 or ζ=1.
• AssumptionThe authors state this as an assumption: The capital stock is not too large: K < K_bar, where K_bar is such that R = W/γ(N).

May combine two claims.

In the second extension with creative destruction, results are similar, but conditions for uniqueness and stability of the balanced growth path are more demanding.

“The results in this case are similar, but the conditions for uniqueness and stability of the balanced growth path are more demanding.”

May combine two claims.

In the extension with heterogeneous skills, automation directly takes jobs away from unskilled labor and increases inequality, while new tasks directly benefit skilled workers and at first increase inequality as well.

“Because of this pattern of comparative advantage, automation directly takes jobs away from unskilled labor and increases inequality, while new tasks directly benefit skilled workers and at first increase inequality as well.”

• AssumptionThe authors state this as an assumption: Skilled labor has a comparative advantage in new tasks.

May combine two claims.

In the two interior BGPs, automation and introduction of new tasks go hand-in-hand, and n*(t) is constant at some value between 0 and 1, implying both capital and labor perform a fixed measure of tasks.

“More important for our focus are the two interior BGPs where automation and the introduction of new tasks go hand-in-hand, and as a result, n * (t) is constant at some value between 0 and 1; this implies that both capital and labor perform a fixed measure of tasks.”

May combine two claims.

In the model, automation makes the production process more capital intensive and less labor intensive.

“In our model, automation shifts the range of tasks performed by capital and labor: it makes the production process more capital intensive and less labor intensive”

May combine two claims.

The displacement effect dominates the productivity effect when the gap between W/γ(I*) and R is small, which is guaranteed when K < ~K.

“The displacement effect dominates the productivity effect when the gap between W/γ(I * ) and R is small: which is guaranteed when K < ˜ K .”

May combine two claims.

In the interior BGP with eventual automation (case iii), n*(t)=_n(ρ) and I*(t)=~I(t)>I(t).

“In this BGP, n * (t) = _ n (ρ) and I * (t) = ~ I (t) > I(t) .”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Competition among prospective technology monopolists to hire scientists implies W_I^S(t)=κ_I V_I(t) and W_N^S(t)=κ_N V_N(t).

“Competition among prospective technology monopolists to hire scientists implies that W I S (t ) = κ I V I (t ) and W N S (t ) = κ N V N (t) .”

Probably not a substantial claim: may not pass S4 (based).

There exists _S such that, when S < _S, Proposition 6 cases hold.

“There exists _ S such that, when S < _ S , we have: 31 (i) Full Automation: For ρ < _ ρ , there is a BGP in which n(t ) = 0 and thus all tasks are produced with capital”

May combine two claims.

For n < _n(ρ), n*(t)=_n(ρ), w_I(n)=w_I(_n(ρ)), w_N(n)=w_N(_n(ρ)), and small changes in n do not affect paths of effective wages, employment, and labor share.

“For n < _ n (ρ) , we have that n * (t) = _ n (ρ) , w I (n) = w I ( _ n (ρ)) , and w N (n) = w N ( _ n (ρ)) . In this region, small changes in n do not affect the paths of effective wages, employment, and the labor share;”

Classified as probably not a substantial claim.

For an interior BGP with n∈(0,1), κ_I v_I(n)=κ_N v_N(n).

“and for an interior BGP with n ∈ (0, 1) we need (29) κ I v I (n) = κ N v N (n).”

Probably not a substantial claim: may not pass S4 (based).

The value of developing new automation technologies depends on the gap between the cost of producing with labor (effective wage w(τ)) and the rental rate of capital.

“The value of developing new automation technologies depends on the gap between the cost of producing with labor (given by the effective wage, w(τ) ) and the rental rate of capital”

May combine two claims.

In the long run, automation reduces employment and the labor share, but it always increases the wage.

“In the long run, automation reduces employment and the labor share, but it always increases the wage.”

May combine two claims.

Equation (28) implies n ̇(t)>0 iff κ_N V_N(t)>κ_I V_I(t), and n ̇(t)<0 iff κ_N V_N(t)<κ_I V_I(t).

“Equation (28) implies that n ̇ (t ) > 0 if and only if κ N V N (t ) > κ I V I (t) , and n ̇ (t ) < 0 if and only if κ N V N (t ) < κ I V I (t ) .”

May combine two claims.

For n > _n(ρ), n*(t)=n, w_I(n) is increasing and w_N(n) is decreasing in n.

“For n > _ n (ρ) , we have that n * (t) = n , and w I (n) is increasing and w N (n) is decreasing in n .”

Classified as probably not a substantial claim.

When the wage is higher, V_I(t) increases and technology monopolists have greater incentives to introduce new automation technologies to substitute capital for more expensive labor.

“When the wage is higher, V I (t) increases and technology monopolists have greater incentives to introduce new automation technologies to substitute capital for the more expensive labor.”

May combine two claims.

Because the short-run impact of new tasks on the rental rate of capital is ambiguous, the response of capital accumulation is also ambiguous.

“Because the short-run impact of new tasks on the rental rate of capital is ambiguous, so is the response of capital accumulation.”

May combine two claims. The quote only partly states this claim.

Along a path with I ̇=N ̇=Δ and n(t)=n≥_n(ρ), n*(t)=n and g(t)=AΔ.

“Along such a path, n * (t) = n and g(t) = AΔ .”

May combine two claims.

The unique equilibrium price for all types of intermediates is a limit price of ψ, yielding a per unit profit of (1-μ)ψ>0 for technology monopolists.

“This ensures that the unique equilibrium price for all types of intermediates is a limit price of ψ , and yields a per unit profit of (1 -μ) ψ > 0 for technology monopolists.”

• AssumptionThe authors state this as an assumption: Intellectual property rights to each intermediate are held by a technology monopolist who can produce at marginal cost μψ, with μ∈(0,1), ψ>0; a competitive fringe can replicate at higher marginal cost ψ; μ is such that the unconstrained monopoly price exceeds ψ.

May combine two claims.

A permanent increase in N increases the wage, aggregate output, employment, and the labor share, both in the short and long run.

“In this case, new tasks increase the wage (because w I (n) is increasing in n ), aggregate output, employment, and the labor share, both in the short and the long run.”

Classified as probably not a substantial claim.

V_N(t) is greater when the gap between the rental rate of capital and the cost of producing new tasks with labor (w(τ)/γ(n(t))) is larger.

“The expression for V N (t) has an analogous interpretation, and is greater when the gap between the rental rate of capital and the cost of producing new tasks with labor ( w(τ )/γ(n(t )) ) is larger.”

Classified as probably not a substantial claim. May combine two claims.

When ρ < _ρ and n < ~n(ρ), w_N(n) > ρ+δ+θg, implying labor is too expensive relative to capital and creating new tasks is not profitable.

“Intuitively, as also shown in Figure 4, when ρ < _ ρ and n < ~ n (ρ) , we have w N (n) > ρ + δ + θg , which implies that labor is too expensive relative to capital.Utilizing and thus creating new tasks is not profitable.”

Probably not a substantial claim: may not pass S4 (based).

There exists _κ ≥ κ_ > 0 such that if κ_I/κ_N > _κ there exists a unique interior BGP with n*(t)=n(t)=n∈(_n(ρ),1) and κ_N v_N(n)=κ_I v_I(n).

“if κ I / κ N > _ κ there exists a unique BGP.In this BGP we have n * (t ) = n(t ) = n ∈ ( _ n ( ρ), 1) and κ N v N (n) = κ I v I (n) .”

May combine two claims.

For κI/κN > κbar, the unique interior BGP is globally stable if θ = 0 and locally stable if θ > 0.

“Proposition 6 also shows that for κ I / κ N > _ κ , the unique interior BGP is globally stable provided that the intertemporal elasticity of substitution is infinite (i.e., θ = 0 ), and locally stable otherwise (i.e., when θ > 0 ).”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

When κ_I/κ_N > _κ, the curve κ_I v_I(n) is sufficiently steep relative to κ_N v_N(n) so that the two curves necessarily intersect and can only intersect once; hence there exists a unique interior BGP.

“Conversely, when κ I / κ N > _ κ , the curve κ I v I (n) is sufficiently steep relative to κ N v N (n) so that the two curves necessarily intersect and can only intersect once.Hence, there exists a unique interior BGP”

Probably not a substantial claim: may not pass S4 (based).

At the unique intersection, a further increase in n raises the value of automating an additional task vI(n) more than the value of creating a new task vN(n).

“At this point, a further increase in n always raises the value of automating an additional task, v I (n) , more than the value of creating a new task, v N (n) .”

Classified as probably not a substantial claim. May combine two claims.

When ρ > _ρ, the two curves can only intersect for n∈(n̅(ρ),1], implying that in any BGP newly automated tasks will be immediately produced with capital.

“The two curves can only intersect for n ∈ ( n ̅ ( ρ), 1] , implying that in any BGP, newly automated tasks will be immediately produced with capital.”

May combine two claims.

When ρ < _ρ, κ_I v_I(0) is above κ_N v_N(0) for n < ~n(ρ); in this region it is not optimal to create new tasks, and there exists a BGP with full automation where all tasks are automated and produced with capital.

“Panel A of Figure 8 illustrates the first part of Proposition 6 (which parallels the first part of Proposition 4): when ρ < _ ρ , κ I v I (0) is above κ N v N (0) for n < ~ n ( ρ) .In this region it is not optimal to create new tasks.Consequently, there exists a BGP with full automation, meaning that all tasks will be automated and produced with capital.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

When κ_I/κ_N < κ_, κ_I v_I(n) is not sufficiently steep relative to κ_N v_N(n), and the two never intersect; even at n=1, it is not profitable to create new automation technologies, and all tasks will be produced with labor.

“This means that even at n = 1 , it is not profitable to create new automation technologies, and all tasks will be produced with labor.”

Probably not a substantial claim: may not pass S4 (based).

For ρ > _ρ, all BGPs feature n(t)=n > _n(ρ).

“For ρ > _ ρ , all BGPs feature n(t ) = n > _ n (ρ) .”

Classified as probably not a substantial claim. May combine two claims.

A lower value of n reduces w_I(n); w_I(n) is increasing in n.

“A lower value of n , which corresponds to additional automation, reduces w I (n) : in other words w I (n) is increasing in n .”

Classified as probably not a substantial claim.

Because κI vI(n) starts below κN vN(n) at nbar(ρ), the unique intersection must have the κI vI(n) curve steeper than the κN vN(n) curve.

“Because κ I v I (n) starts below κ N v N (n) at n ̅ (ρ) (reflecting the fact that at this point, new automation technologies are not immediately adopted and thus the value of creating these technologies is 0), the unique intersection must have the former curve being steeper than the former.”

May combine two claims.

If θ=0, the equilibrium is unique everywhere and the BGP is globally saddle-path stable; if θ>0, the equilibrium is unique in the neighborhood of the BGP and asymptotically saddle-path stable.

“If, in addition, θ = 0 , then the equilibrium is unique everywhere and the BGP is globally (saddle-path) stable.If θ > 0 , then the equilibrium is unique in the neighborhood of the BGP and is asymptotically (saddle-path) stable;”

Probably not a substantial claim: may not pass S4 (based).

When ξ = 1, the relative productivity of low-skill workers converges to 1 for tasks that have been around for a long time.

“when ξ = 1 , the relative productivity of low-skill workers converges to 1 for tasks that have been around for a long time.”

Probably not a substantial claim: may not pass S4 (based).

The structure of comparative advantage ensures there exists a threshold task M such that high-skill labor performs tasks in [M,N], low-skill labor performs tasks in (I*,M), and capital performs tasks in [N-1,I*].

“The structure of comparative advantage ensures that there exists a threshold task M such that high-skill labor performs tasks in [ M, N ] , low-skill labor performs tasks in ( I * , M ) , and capital performs tasks in [N -1, I * ] .”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

If ξ = 1, in the unique BGP WH and WL grow at the same rate as the economy, the wage gap WH/WL remains constant, and capital, low-skill, and high-skill workers perform constant shares of tasks.

“If ξ = 1 , in the unique BGP W H (t) and W L (t) grow at the same rate as the economy, the wage gap, W H (t )/ W L (t ) , remains constant, and capital, low-skill, and high-skill workers perform constant shares of tasks.”

Probably not a substantial claim: may not pass S4 (based).

If ξ = 1, lim WH/WL is decreasing in n.

“is decreasing in n .”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

When ξ < 1, the productivity of low-skill workers relative to high-skill workers converges to γL(i,t)/γH(i) = γH(i)^{ξ-1}, and limits to zero as more advanced tasks are introduced.

“When ξ < 1 , the productivity of low-skill workers relative to high-skill workers converges to γ L (i, t)/ γ H (i) = γ H (i) ξ-1 , and limits to zero as more and more advanced tasks are introduced.”

May combine two claims.

The transition process following an increase in κI/κN involves a slower rate of increase of N and a more rapid rate of increase of I than the BGP.

“The transition process will involve a slower rate of increase of N and a more rapid rate of increase of I than the BGP.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

If ξ < 1, in the unique BGP lim WH/WL = ∞, the share of tasks performed by low-skill workers converges to zero, and capital and high-skill workers perform constant shares of tasks.

“If ξ < 1 , in the unique BGP we have lim t→∞ W H (t )/ W L (t ) = ∞ , the share of tasks performed by low-skill workers converges to zero, and capital and high-skill workers perform constant shares of tasks.”

May combine two claims.

Creative destruction introduces a new source of instability: vI^CD(n) is decreasing in n.

“The major difference with our previous analysis is that creative destruction introduces a new source of instability.Unlike the previous case with no creative destruction, we now have that v I CD (n) is decreasing in n .”

May combine two claims. The quote only partly states this claim.

With labor market frictions where quasi-labor supply is increasing, has elasticity εL > 0, and is binding (W/C > ν'(L)), automation can reduce welfare (dW/dI ≶ 0), while new tasks increase welfare (dW/dN > 0).

“Suppose also that the quasi-labor supply schedule L qs (ω) is increasing in ω , has an elasticity ~ ε L > 0 , and is binding in the sense that W/C > ν ′ (L) .”

May combine two claims.

vN^CD(n) continues to be increasing in n because the productivity effect ensures the effective wage in new tasks wN(n) is decreasing in n, and because newly-created tasks are automated after n/Δ units of time, an increase in n increases the present discounted value of profits from new tasks.

“v N CD (n) continues to be increasing in n .This is for two reasons: first, as before, the productivity effect ensures that the effective wage in new tasks, w N (n) , is decreasing in n ; and second, because newly-created tasks are automated after n/Δ units of time, an increase in n increases the present discounted value of profits from new tasks.”

The quote only partly states this claim.

Under the same conditions, if ι(n) is constant, any stable BGP involves n(t) → 0 or n(t) → 1.

“Any stable BGP involves n(t ) → 0 or n(t ) → 1 .”

Probably not a substantial claim: may not pass S4 (based).

Under the same conditions, if ρ < ρmin, for all n ∈ [0,1] we have wI(n) ≥ wN(n) > ρ+δ+θg, which implies new tasks do not increase aggregate output and will not be adopted for any n ∈ [0,1].

“(iv) If ρ < ρ min , for all n ∈ [0, 1] we have w I (n ) ≥ w N (n ) > ρ + δ + θg , which implies that new tasks do not increase aggregate output and will not be adopted for any n ∈ [0, 1] .”

May combine two claims.

For ρ > ρ_max, ρ+δ+θg > w_I(n) > w_N(n), and n* = 1, and no task will be produced with capital.

“Thus, ρ + δ + θg > w I ( n ) > w N (n) , and for all these values we have n * = 1 , and no task will be produced with capital.”

May combine two claims.

For ρ < ρ_min, ρ+δ+θg < w_N(1) < w_I(1).

“To prove part (iv), note that for ρ < ρ min , we have ρ + δ + θg < w N (1) < w I (1).”

May combine two claims.

_n(ρ_max)=1 and _n(_ρ)=0, so _n(·) is well defined for all ρ ≥ _ρ.

“Moreover, _ n ( ρ max ) = 1 and _ n ( _ ρ ) = 0 , so _ n ( ⋅ ) is well defined for all ρ ≥ _ ρ .”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Under the same conditions, if ρ ∈ [ρbar, ρmax], there is an increasing function nbar(ρ) such that for all n > nbar(ρ), wI(n) > ρ+δ+θg > wN(n) and ρ+δ+θg = wI(nbar(ρ)); nbar(ρmax)=1 and nbar(ρbar)=0.

“(ii) If ρ ∈ [ _ ρ , ρ max ] , there is an increasing function _ n ( ρ ) : [ _ ρ , ρ max ] → (0, 1] such that for all n > _ n (ρ) , we have w I ( n ) > ρ + δ + θg > w N (n) and ρ + δ + θg = w I ( _ n ( ρ )) .”

May combine two claims.

For ρ < ρ_min, ρ+δ+θg < w_N(n) < w_I(n) and new tasks do not raise aggregate output.

“Thus, ρ + δ + θg < w N (n) < w I (n) and for these values of ρ , new tasks do not raise aggregate output.”

Probably not a substantial claim: may not pass S4 (based).

Under the same conditions, if ρ > ρmax, for all n ∈ [0,1] we have ρ+δ+θg > wI(n) ≥ wN(n), which implies automation is not profitable for any n ∈ [0,1].

“(iii) If ρ > ρ max , for all n ∈ [0, 1] we have ρ + δ + θg > w I ( n ) ≥ w N ( n ) , which implies that automation is not profitable for any n ∈ [0, 1] .”

May combine two claims.

For all n > ~n(ρ), w_I(n) > w_I(~n(ρ)) > ρ+δ+θg = w_N(~n(ρ)) > w_N(n).

“Thus, for all n > ~ n (ρ) , we have w I (n) > w I ( ~ n ( ρ )) > ρ + δ + θg = w N ( ~ n ( ρ )) > w N (n), as claimed.”

Probably not a substantial claim: may not pass S4 (based).

The unique intersection of the relative demand and ideal price curves determines the equilibrium factor prices for given I*, N, and K.

“The unique intersection of the relative demand and ideal price curves determines the equilibrium factor prices for given I * , N , and K .”

May combine two claims.

For ρ ∈ [_ρ, ρ_max] and starting at _n(ρ), w_N(n) is decreasing in n and w_I(n) is increasing in n.

“Thus, the formulas for w I ′ (n) and w N ′ (n) show that, for ρ ∈ [ _ ρ , ρ max ] and starting at _ n (ρ) , w N (n) is decreasing in n and w I (n) is increasing in n .”

May combine two claims.

Lemma A1: Suppose Assumption 2′ holds, K < Kbar and I* ≤ I~. Then ω(I*,N,K) is decreasing in I* and increasing in N.

“LEMMA A1: Suppose that Assumption 2′ holds, K < _ K and I * ≤ ~ I .Then ω( I * , N, K ) is decreasing in I * and is increasing in N .”

May combine two claims.

For n = ~n(ρ), w_I(~n(ρ)) > ρ+δ+θg = w_N(~n(ρ)).

“For n = ~ n (ρ) , we have w I ( ~ n (ρ )) > ρ + δ + θg = w N ( ~ n ( ρ)) .”

May combine two claims.

In the region n < _n(ρ), n* = _n(ρ) > n, and not all automated tasks are produced with capital.

“In this region we have n * = _ n ( ρ) > n , and not all automated tasks are produced with capital.”

Classified as probably not a substantial claim. May combine two claims.

~n(ρ_min)=1 and ~n(_ρ)=0, so ~n(·) is well defined for ρ ∈ [ρ_min, _ρ].

“Moreover, ~ n ( ρ min ) = 1 and ~ n ( _ ρ ) = 0 , so ~ n ( ⋅ ) is well defined for ρ ∈ [ ρ min , _ ρ ] .”

May combine two claims.

For ρ ∈ [ρ_min, _ρ] and starting at ~n(ρ), w_N(n) is decreasing in n and w_I(n) is increasing in n.

“Thus, the formulas for w I ′ (n) and w N ′ (N) show that, for ρ ∈ [ ρ min , _ ρ ] and starting at ~ n (ρ) , the curve w N (n) is decreasing in n and the curve w I (n) is increasing in n .”

Probably not a substantial claim: may not pass S4 (based).

Because the relative demand curve is upward-sloping and the ideal price index curve is downward-sloping, there can be at most one intersection.

“Because the relative demand curve is upward-sloping and the ideal price index curve is downward-sloping, there can be at most one intersection.”

May combine two claims.

For ρ > ρ_max, ρ+δ+θg > w_I(1) > w_N(1).

“To prove part (iii), note that for ρ > ρ max , we have ρ + δ + θg > w I (1) > w N (1).”

May combine two claims.

For all n > _n(ρ), w_I(n) > w_I(_n(ρ)) = ρ+δ+θg > w_N(_n(ρ)) > w_N(n).

“Thus, for all n > _ n ( ρ ) , we have w I (n) > w I ( _ n ( ρ )) = ρ + δ + θg > w N ( _ n ( ρ )) > w N (n ), as claimed.”

May combine two claims.

In all-tasks-automated case, F_L = 0, so representative household supplies zero labor.

“Because all tasks are produced with capital, we also have F L = 0 , and thus the representative household supplies zero labor.”

May combine two claims.

In this path economy converges to unique BGP with C(t) and K(t) growing at rate (A_K - δ - ρ)/θ, establishing uniqueness and global stability.

“In this path the economy converges to a unique BGP in which C(t ) and K(t ) grow at a constant rate A K -δ -ρ ______ θ , thus also establishing uniqueness and global stability.”

May combine two claims.

In that case, net output is A_K K, and economy grows at rate (A_K - δ - ρ)/θ.

“In this part of the parameter space, net output is given by A K K , and the economy grows at the rate A K -δ -ρ ______ θ .”

May combine two claims.

The differential equation for ~c has a stable rest point at zero and an unstable rest point at c_B > 0.

“This differential equation has a stable rest point at zero and an unstable rest point at c B = ( A k -δ -1 _ θ ( A K -δ -ρ) ) e ν (0) 1-θ _ θ > 0 .”

Classified as probably not a substantial claim.

Net output is F(k,L;n).

“Substituting the market-clearing conditions for capital and labor, (A3) and (A4), into (1), ( 2), and (3) and then subtracting the costs of intermediates, we obtain net output as F(k, L; n) .”

May combine two claims.

F(k,L;n) exhibits constant returns to scale, and R(t)=F_K, w(t)=F_L.

“F(k, L; n) exhibits constant returns to scale, and because factor markets are competitive, we also have R(t) = F K (k(t), L(t); n) and w(t) = F L (k (t), L(t); n) .”

Classified as probably not a substantial claim.

Dynamic equilibrium can be characterized by system of differential equations.

“Consequently, the dynamic equilibrium can be characterized as the solution to the system of differential equations”

May combine two claims.

First possibility satisfies transversality condition and yields equilibrium path.

“The first possibility, on the other hand, satisfies the transversality condition (since asymptotically it involves r > g ), and yields an equilibrium path.”

May combine two claims.

In the all-tasks-automated case (n*=0), ρ < _ρ and I(t)=N(t).

“Because in this case capital performs all tasks, Lemma A2 implies that we must have ρ < _ ρ and I(t ) = N(t) .”

May combine two claims.

In interior equilibrium with n*(t)=n(t)=n∈(0,1), n > max{_n(ρ), ~n(ρ)} and Ṅ(t)=İ(t).

“Because capital performs all automated tasks, Lemma A2 implies that n > max { _ n ( ρ), ~ n ( ρ)} and N ˙ (t) = I ˙ (t) .”

Classified as probably not a substantial claim.

Lemma A2 verifies n*(t)=n > _n(ρ).

“Lemma A2 then verifies that n * (t) = n > _ n (ρ) .”

May combine two claims.

Three possible equilibrium paths for ~c(t): jump to c_B; start at [0,c_B) and converge to zero; start at (c_B,∞) and diverge.

“There are therefore three possible equilibrium paths for ~ c (t) : (i) it immediately jumps to c B and stays there; (ii) it starts at [ 0, c B ) and converges to zero; (iii) it starts at ( c B , ∞) and diverges.”

May combine two claims.

Second and third possibilities violate transversality condition and resource constraint.

“The second and third possibilities violate, respectively, the transversality condition (19) (because the capital stock would grow at the rate A k -δ , implying r = g ), and the resource constraint (when ~ c (t ) = ∞ ).”

May combine two claims.

There is a unique solution to the system, converging to full automation BGP.

“We next show that there is a unique solution to the system, and this solution converges to the full automation BGP described in the proposition.”

May combine two claims.

Generalized ideal price index condition implies R = c_u^{-1}(B^{1-σ}/(1-σ)), and thus A_K = c_u^{-1}(B^{1-σ}/(1-σ)).

“The generalized ideal price index condition, equation (A5), then implies that R = c u -1 ( B 1-ˆ σ ____ 1-σ ) , and thus A K = c u -1 ( B 1-ˆ σ ____ 1-σ ) .”

May combine two claims.

In candidate BGP, R(t) constant implies W(t)/γ(I(t)) constant, only possible if İ(t)=Δ.

“Moreover, because in this candidate BGP R(t) is constant, the general form of the generalized ideal price index condition, (A5), implies that W(t)/γ (I(t)) must be constant too, and this is only possible if I ˙ (t) = Δ .”

May combine two claims.

This candidate BGP satisfies transversality condition (19) and is unique BGP.

“Moreover, in view of the fact that ρ + (θ -1 ) AΔ > 0 , this candidate BGP satisfies the transversality condition (19), and is indeed the unique BGP.”

May combine two claims.

This allocation satisfies transversality condition, defines unique BGP.

“Because this allocation also satisfies the transversality condition (in view of the fact that ρ + (θ -1)AΔ > 0 ), it defines a unique BGP.”

May combine two claims.

Because n*(t)=_n(ρ), Ṅ(t)=Δ.

“Because n * (t) = _ n ( ρ) , this also implies that N ˙ (t) = Δ .”

May combine two claims.

To generate constant growth of wages, İ*(t)=Δ ≤ I(t), so growth rate is AΔ.

“Thus, to generate constant growth of wages we must have I ˙ * (t) = Δ ≤ I(t) , so that the growth rate of the economy is given by AΔ .”

May combine two claims.

For ρ<_ρ and n<~n(ρ), v_I(n) is positive and decreasing in n, and v_N(n) is negative and increasing in n.

“As depicted in panel A of Figure 8 and shown in Lemma A3, in this region v I (n) is positive and decreasing in n , and v N (n) is negative and increasing in n .”

Probably not a substantial claim: may not pass S3 (load-bearing).

Proof of global stability is similar to neoclassical growth model with endogenous labor supply.

“The proof of the global stability of this unique BGP is similar to the analysis of global stability of the neoclassical growth model with endogenous labor supply, and for completeness, we provide the details in online Appendix B.”

May combine two claims.

This ensures unique L_B>0, and k_B=φ L_B.

“This ensures that there exists a unique value L B > 0 that satisfies this equation, and also pins down the value of the normalized capital stock as k B = ϕ L B .”

May combine two claims.

In all-tasks-labor case n*(t)=1, ρ>ρ_max and n(t)≥1, and W(t)/γ(N(t)) constant.

“Because labor performs all tasks, Lemma A2 now implies ρ > ρ max and n(t) ≥ 1 , while the ideal price index condition, (A5), imposes that W(t)/γ (N(t)) must be constant.”

May combine two claims.

Uniqueness and global stability follow from identical arguments to part (ii), with _n(ρ) playing role of n.

“The uniqueness and global stability of the BGP follow from an identical arguments to part (ii), with the only modification that _ n ( ρ) plays the role of n in the preceding proof.”

May combine two claims.

Euler equation implies R_B = ρ+δ+θg, and since R_B=F_K(k_B,L_B;n), k_B/L_B=φ, where φ is unique solution to F_K(φ,1;n)=ρ+δ+θg.

“Then, the Euler equation, (17), implies R B = ρ + δ + θg , and because R B = F K ( k B , L B ; n) , we must also have k B / L B = ϕ , where ϕ is the unique solution to F K (ϕ, 1 ; n ) = ρ + δ + θg.”

Classified as probably not a substantial claim.

Then we would have ν'(0)e^{(θ-1)/θ ν(0)} < F_L(φ,1;n)/c_B, which contradicts labor supply optimality condition (18).

“But then we would have ν ′ (0) e θ-1 ___ θ ν(0) < F L (ϕ, 1; n) ______ c B , which contradicts the labor supply optimality condition, (18).”

May combine two claims.

This implies K(t)=0 and C(t)=w_B γ(N(t)-1)L(t).

“This implies K(t) = 0 and C(t) = w B γ (N(t) -1) L(t) .”

May combine two claims.

In interior equilibrium with n*(t)=_n(ρ)>n(t), n(t)<_n(ρ) and ρ>_ρ.

“Because capital does not immediately perform all automated tasks, Lemma A2 implies that n(t) < _ n ( ρ) and ρ > _ ρ .”

Classified as probably not a substantial claim. May combine two claims.

BGP variables w_B, k_B, c_B, L_B, R_B are all constant.

“These variables are, by definition, all constant.”

May combine two claims.

Left-hand side is linear and increasing in L, right-hand side decreasing in L.

“The left-hand side of this equation is linear and increasing in L (the concavity of F in k implies that F(ϕ, 1; n) > ϕ F K (ϕ, 1; n) > (δ + g )ϕ ), while the right-hand side is decreasing in L .”

May combine two claims.

If n < ~n(ρ) (and ρ<_ρ), κ_I v_I(n) > 0 > κ_N v_N(n). Moreover, in this region, v_I(n) is decreasing and v_N(n) is increasing in n.

“• If n < ~ n (ρ) (and ρ < _ ρ ), we have κ I v I (n) > 0 > κ N v N (n).Moreover, in this region, v I (n) is decreasing and v N (n) is increasing in n .”

May combine two claims.

Net output is F(k,L;n)=w_B γ(N(t)-1)L(t), depends linearly on labor and independent of capital.

“Consequently, net output is given by F(k, L; n ) = w B γ (N(t) -1) L(t) , and thus depends linearly on labor and is independent of capital.”

May combine two claims.

Laws of motion imply existence of unique stable arm, and no equilibrium paths not along stable arm.

“The laws of motion of the two variables, v and n , take the form shown in the phase diagram.34 This implies the existence of the unique stable arm, and also establishes 34 This can also be verified locally from the fact that the behavior of n and v near the BGP can be approximated by the linear system n ̇ = -( κ N + κ I ) G ′ (0 ) Sv and v ̇ = ρv -Q, where Q > 0 denotes the derivative of -M κ I c”

May combine two claims.

All paths above stable arm feature v̇>0 and eventually n→0 and v→∞, v_I→∞, violating transversality condition (27).

“In particular, all paths above the stable arm feature v ˙ > 0 and eventually n → 0 and v → ∞ , and since v N is positive, v I → ∞ .But this violates the transversality condition, (27).”

Classified as probably not a substantial claim.

This contradicts n(t)<_n(ρ).

“But this contradicts n(t) < _ n ( ρ ) .”

May combine two claims.

For ρ>_ρ and n(t)≤_n(ρ), n*(t)=_n(ρ) and v_N(n)=v_N(_n(ρ)), v_I(n)=v_I(_n(ρ)).

“Suppose n(t) ≤ _ n (ρ) .Then, we have n * (t ) = _ n (ρ) and there- fore v N (n ) = v N ( _ n ( ρ )) and v I (n) = v I ( _ n (ρ)) .”

May combine two claims.

There exists threshold ˇS such that BGP is locally stable for S<ˇS, and conclusions follow setting _S = min{~S, ˆS, S ̆, ˇS}.

“Online Appendix B shows that there exists a threshold ˇ S such that the BGP in this case is locally stable for S < ˇ S , and thus the conclusions of the proposition follow setting _ S = min { ~ S , ˆ S , S ˘ , ˇ S } .”

Classified as probably not a substantial claim.

This follows from κ_I v_I(n) cuts κ_N v_N(n) from below at n_B.

“This follows from the fact that κ I v I (n) cuts κ N v N (n) from below at n B as shown in Figure 8.”

May combine two claims. The quote only partly states this claim.

Therefore, all scientists employed to create new tasks, so ṅ>0 (uniformly bounded away from zero).

“Therefore, in this region this region we always have that all scientists will be employed to create new tasks, and thus n ̇ > 0 (and is uniformly bounded away from zero).”

May combine two claims.

All paths below stable arm feature v̇<0 and eventually n→1 and v→-∞, v_N→∞, violating transversality condition.

“Similarly, all paths below the stable arm feature v ˙ < 0 and eventually n → 1 and v → -∞ , and thus v N → ∞ , once again violating the transversality condition.”

May combine two claims.

When θ=0, R=ρ+δ, capital adjusts immediately and equilibrium stock only depends on n, unique state variable.

“Because θ = 0 , we also have R = ρ + δ , and capital adjusts immediately and its equilibrium stock only depends on n , which becomes the unique state variable of the model.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The relationship shown in Figure 1 controls for the demographic composition of employment in the occupation in 1980.

“The relationship shown in Figure 1 controls for the demographic composition of employment in the occupation in 1980.”

May combine two claims.

A wave of automation pushes down the effective cost of producing with labor, discouraging further efforts to automate additional tasks and encouraging creation of new tasks.

“This stability result highlights a crucial new force: a wave of automation pushes down the effective cost of producing with labor, discouraging further efforts to automate additional tasks and encouraging the creation of new tasks.”

May combine two claims.

The same relationship between new job titles and employment growth holds for 1990-2015 and 2000-2015, and is present without controls and when controlling for average education and structural changes in the US economy.

“In online Appendix B, we show that the same relationship holds between the share of new job titles in 1990 (in 2000) and employment growth from 1990 to 2015 (from 2000 to 2015), and that these patterns are present without any controls and when we control for average education in the occupation and the structural changes in the US economy as well.”

Probably not a substantial claim: may not pass S4 (based).

Constant growth requires productivity gains from new tasks to be exponential.

“constant growth requires productivity gains from new tasks to be exponential.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

Alternatively, faster automation might be driven by a deceleration in N(t), in which case there would also be low growth of aggregate output and wages.

“Alternatively, it might be driven by a deceleration in the rate at which N(t) advances, in which case we would also have low growth of aggregate output and wages.”

Probably not a substantial claim: may not pass S4 (based).

The asymptotic stability of the interior BGP implies powerful market forces push the economy toward balanced growth.

“The asymptotic stability of the interior BGP implies that there are powerful market forces pushing the economy toward balanced growth.”

Classified as probably not a substantial claim. May combine two claims.

Increases in n beyond its BGP value trigger further automation, while lower values of n encourage creation of new tasks, ensuring stability of the unique BGP.

“This ensures that increases in n beyond its BGP value trigger further automation, while lower values of n encourage the creation of new tasks, ensuring the stability of the unique BGP.”

Classified as probably not a substantial claim. May combine two claims.

In the no automation BGP, capital becomes redundant, and growth is driven by endogenous technological change increasing labor's productivity as in standard quality ladder models.

“In this BGP, capital becomes redundant, and growth is driven by endogenous technological change increasing labor's productivity as in the standard quality ladder models such as Aghion and Howitt (1992) or Grossman and Helpman (1991).”

• cites Aghion and Howitt (1992)Not checked yet

  Not checked in the free run.

• cites Grossman and Helpman (1991)Not checked yet

  Not checked in the free run.

May combine two claims.

The fact that v_N(n) is increasing in n creates a force toward multiplicity of BGPs, while v_I(n) increasing pushes toward uniqueness and stability.

“We will see next that the fact that v N (n) is increasing in n creates a force toward multiplicity of BGPs, while the fact that v I (n) is increasing in n pushes toward uniqueness and stability.”

Probably not a substantial claim: may not pass S4 (based).

Certain changes in the environment, such as a large increase in B or a decline in ρ, can shift the economy from the region with a unique interior BGP to the region with full automation, with disastrous consequences for labor.

“certain changes in the environment (for example, a large increase in B or a decline in ρ ) can shift the economy from the region in which there is a unique interior BGP to the region with full automation, with disastrous consequences for labor.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

The ratio γH(i)/γL(i,t) is increasing in i, which implies high-skill workers have a comparative advantage in higher-indexed tasks.

“The ratio γ H (i)/ γ L (i, t) is increasing in i , which implies that high-skill workers have a comparative advantage in higher-indexed tasks.”

May combine two claims.

The alternative structure with creative destruction has similar implications for the BGP but necessitates more demanding conditions to guarantee its uniqueness and stability.

“We will see that this alternative structure has similar implications for the BGP, but necessitates more demanding conditions to guarantee its uniqueness and stability.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

When ξ = 1, the creation of new tasks, which leads to a permanently higher level of n, increases inequality in the short run but reduces it in the long run.

“The creation of new tasks, which leads to a permanently higher level of n , increases inequality in the short run but reduces it in the long run.”

Probably not a substantial claim: may not pass S4 (based).

If ξ = 1, a permanent increase in I raises the wage gap WH/WL in both the short and the long run.

“a permanent increase in I raises the wage gap in both the short and the long run.”

Probably not a substantial claim: may not pass S4 (based).

When ξ = 1, automation increases inequality both in the short and the long run.

“Consequently, automation increases inequality both in the short and the long run.”

May combine two claims.

Permanent changes in the ability of society to create new automation technologies trigger self-correcting dynamics but take the economy toward a new BGP with lower employment and labor share, and may also involve slower productivity growth.

“Permanent changes in the ability of society to create new automation technologies trigger self-correcting dynamics as well, but these will take us toward a new BGP with lower employment and labor share, and may also involve slower productivity growth in the process.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

When ξ < 1, low-skill workers are progressively squeezed into a smaller and smaller set of tasks, and wage inequality grows without bound.

“As a result, low-skill workers are progressively squeezed into a smaller and smaller set of tasks, and wage inequality grows without bound.”

Probably not a substantial claim: may not pass S4 (based).

When ξ = 1, in the long run inequality is decreasing in n because a higher n translates into a greater range of tasks for low-skill workers.

“In the long run, inequality is decreasing in n (because a higher n translates into a greater range of tasks for low-skill workers).”

Probably not a substantial claim: may not pass S4 (based).

If ξ = 1, a permanent increase in N raises the wage gap WH/WL in the short run but reduces it in the long run.

“a permanent increase in N raises the wage gap W H (t )/ W L (t) in the short run, but reduces it in the long run”

Classified as probably not a substantial claim.

Inequality may be high following a period of adjustment in which the labor share first declines due to increases in automation and then recovers due to the introduction and later standardization of new tasks.

“These observations suggest that inequality may be high following a period of adjustment in which the labor share first declines (due to increases in automation), and then recovers (due to the introduction and later standardization of new tasks).”

Classified as probably not a substantial claim.

Starting from the interior BGP, the effects of small increases in automation technology reverse themselves over time, restoring employment and the labor share back to their initial values.

“Starting from the interior BGP, the effects of (small) increases in automation technology will reverse themselves over time, restoring employment and the labor share back to their initial values.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

When ξ = 1, automation and the creation of new tasks still reduce the relative earnings of low-skill workers in the short run, but their long-run implications are very different.

“In this case, automation and the creation of new tasks still reduce the relative earnings of low-skill workers in the short run, but their long-run implications are very different.”

Probably not a substantial claim: may not pass S4 (based).

When ξ = 1, because new tasks become standardized, they can over time be as productively used by low-skill workers.

“Because new tasks become standardized, they can over time be as productively used by low-skill workers.”

May combine two claims.

By reducing the effective cost of labor in the least complex tasks, automation discourages further automation and generates a self-correcting force toward stability.

“by reducing the effective cost of labor in the least complex tasks, automation discourages further automation and generates a self-correcting force toward stability.”

May combine two claims.

In our model, because the supply of labor is elastic, automation tends to reduce employment, while the creation of new tasks increases employment.

“In our model, because the supply of labor is elastic, automation also tends to reduce employment, while the creation of new tasks increases employment.”

May combine two claims.

If in the long run capital is very cheap relative to labor, automation technologies will advance rapidly and labor will become redundant.

“If in the long run capital is very cheap relative to labor, automation technologies will advance rapidly and labor will become redundant.”

May combine two claims.

Even though automation expands productivity, a force which always raises welfare, it also reduces employment.

“Even though automation expands productivity, a force which always raises welfare, it also reduces employment.”

May combine two claims.

When the equilibrium range of tasks allocated to capital increases, for example as a result of automation, the wage relative to the rental rate and the labor share decline, and the equilibrium wage rate may also fall.

“when the equilibrium range of tasks allocated to capital increases (for example, as a result of automation), the wage relative to the rental rate and the labor share decline, and the equilibrium wage rate may also fall.”

May combine two claims.

The task-based framework implies that as a factor becomes cheaper, this not only influences the range of tasks allocated to it, but also generates incentives for the introduction of technologies that allow firms to utilize this factor more intensively.

“The task-based framework, differently from the standard models of directed technological change based on factor-augmenting technologies , implies that as a factor becomes cheaper, this not only influences the range of tasks allocated to it, but also generates incentives for the introduction of technologies that allow firms to utilize this factor more intensively.”

May combine two claims.

In this case, under reasonable assumptions, the dynamic equilibrium is unique and converges to the BGP.

“Moreover in this case, under reasonable assumptions, the dynamic equilibrium is unique and converges to the BGP.”

May combine two claims.

As more tasks are automated, the rental rate remains unchanged and newly-automated tasks will be replaced less frequently, so automating more tasks renders further automation more profitable.

“As a result, automating more tasks renders further automation more profitable.”

May combine two claims.

Assuming labor market frictions also constrain the social planner's choices, the decentralized equilibrium involves too much effort devoted to improving automation relative to what the social planner would like, because the social planner recognizes that additional automation has a negative effect through employment.

“assuming that labor market frictions also constrain the social planner's choices, the decentralized equilibrium involves too much effort being devoted to improving automation relative to what she would like , because the social planner recognizes that additional automation has a negative effect through employment.”

May combine two claims.

With labor market frictions, the reduction in employment resulting from automation has a negative impact on welfare, and this negative effect can exceed the positive impact from productivity gains, turning automation, on net, into a negative for welfare.

“Crucially, the reduction in employment resulting from automation now has a negative impact on welfare, and this negative effect can exceed the positive impact following from the productivity gains, turning automation, on net, into a negative for welfare.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

In practice, the acquisition of new skills may need to go hand-in-hand with workers shifting to newer tasks, and the inability of the educational system to adapt to the requirements of these new tasks could become a bottleneck and prevent the rebound in the demand for labor following a wave of automation.

“In practice, the acquisition of new skills may need to go hand-in-hand with workers shifting to newer tasks, and the inability of the educational system to adapt to the requirements of these new tasks could become a bottleneck and prevent the rebound in the demand for labor following a wave of automation.”