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- elasticities of capital. Theorem 2 from (Farmer et al., 2008, 15) claims that for Ñ < Ñ, at the lower steady state solution, kL , two eigenvalues of the Jacobian are larger than one and one eigenvalue equals αK < 1, while at the larger steady state solution of k = F(k) two eigenvalues are less than one and one eigenvalue is larger than one. Hence, the lower steady state is saddle–path unstable while the larger steady state is saddle–path stable.
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- In considering the general case αK 6= α∗ K, we focus again at a sufficiently small difference between αK and α∗ K. Under this assumption, Theorem 2 of (Farmer et al., 2008, 15) can be generalized as the following Theorem 2. Theorem 2 For every parameter set É ∈ Ω with |αK −α∗ K| sufficiently small some (non– unique) b, b∗ > 0 exist such that for all b ∈ (0, b) and b∗ ∈ (0, b∗ ) the larger strictly positive solution of k = F(k) + ∆(k) is saddle–path stable.
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- Next, consider the case of identical production technologies, i.e. αK = α∗ K. Clearly, ∆(k) = 0 and (49) reduces to k = F(k) ≡ (1 − αK)Ã (1 + i) αK − ÑÃ(1 + i) − Ñ(1 − Ã), with Ñ ≡ (ζb + (1 − ζ)b∗ ˜ S−1 ). Theorem 1 from Farmer et al. (2008, 13–14), which is reproduced as Lemma 1 below, provides sufficient conditions for exactly two strictly positive solutions of equation k = F(k). Lemma 1 Let the parameter vector É = (αK, αP , β, ζ, M, S, Ñ) be an element of the parameter space Ω = [0, 1]4 × R3 +. For any É ∈ Ω there exists Ñ ∈ R++ such that 1. for Ñ < Ñ there are one trivial (k = 0) and two non–trivial steady states kL and kH with 0 < kL < kH < k, 2. for Ñ = Ñ there are one trivial and one non–trivial steady state, and 3. for Ñ > Ñ there is only the trivial steady state.
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- Proof 3 For αK = α∗ K see the proof to Theorem 2 of (Farmer et al., 2008, 32–33) Again, since k = F(k) + ∆(k) depends continuously on α∗ , there is some interval Λ1 = (α1 −, α1 +) such that for all α∗ K ∈ Λ1 ⊂ Λ the larger solution kH of k = F(k) + ∆(k) is saddle–path stable.△ A.3 Comparative Statics Effects To derive the signs of the comparative statics effects of a shock in S on h and k, we generalize again the effects for the symmetrical case αK = α∗ K. Theorem 3 For |α∗ K − αK| sufficiently small we know that dh dS < 0 and S k dk dS >
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