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- Solution to firm’s R&D problem in Section 3.2 The firm faces a discounted infinite-horizon optimal control problem of the type studied in Section 7.5 of Acemoglu (2009). The current-value Hamiltonian is: H(Æ, lR , λ) =  1 − β β βpjsÇR js ws ! 1 1−β Æ − lR   ws + λ Æ 1 − β h ÈBsÆ−γj(1−β) lR α − (δ + gj) i ,
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- Suppose the growth rates gj for j = 1, . . . , J are known. To prove that f(w) = 0 implies a unique solution for wages I use results from Allen, Arkolakis and Li (2015). For all s = 1, . . . , S define the scaffold function F : RS+1 ++ → RS by: Fs (w̃, ws) = j à+ ζ ζ + βà+ αà(δ + gj) à+ ζ + γj (δ + gj) w−Ã s BsΨ γj(1−β) 1+κj −α js !(Ã−1)(1+κj) γj PS ŝ=1 w̃1−Ã ŝ BŝΨ γj(1−β) 1+κj −α jŝ !(Ã−1)(1+κj) γj − Ls.
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- The solution to (28) and (29) is a candidate for a solution to the firm’s problem. To show it is in fact the unique solution we can use Theorem 7.14 in Acemoglu (2009). Suppose λ is the current-value costate variable obtained from the solution to (28) and (29). Equation (54) implies λ is always strictly positive. Therefore, given any path for Æ on which Æ is always positive we have limt̃→∞ h e−(ÃÂ+ζ)(t̃−t)λÆ i ≥ 0. Now define: H(Æ, λ) = max lR H(Æ, lR , λ), =  1 − β β βpjsÇR js ws ! 1 1−β ws − λ (δ + gj) 1 − β   Æ + 1 − α α w −α 1−α s
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- which is negative by Assumption 1. This means the Jacobian has one strictly negative and one strictly positive eigenvalue. Therefore, by Theorem 7.19 in Acemoglu (2009), the steady state is locally saddle-path stable. There exists an open neighborhood of the steady state such that if the firm’s initial Æ lies within this neighborhood, the system of differential equations given by (28) and (29) has a unique solution. The solution converges to the steady state along the stable arm of the system as shown in Figure 1 in the paper. From equation (56) it follows that λ̇ → 0 as the solution converges to the steady state. Since à+ ζ > 0 this implies the solution satisfies the transversality condition (55).
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