Economic growth and economic development 656

Introduction to Modern Economic Growth Proof of v−1 ≤ v0 : Suppose, to obtain a contradiction, that v−1 > v0 Then, ∗ = 0, which leads to v−1 = κv0 / (ρ + κ), contradicting v−1 > v0 (14.57) implies z−1 since κ/ (ρ + κ) < (given that κ < ∞) Proof of < vn+1 : Suppose, to obtain a contradiction, that ≥ vn+1 Now (14.56) implies zn∗ = 0, and (14.53) becomes ¡ ¢ ∗ (14.64) ρvn = − λ−n + z−1 [v0 − ] + κ [v0 − ] Also from (14.53), the value for state n + satisfies ¡ ¢ ∗ [v0 − vn+1 ] + κ [v0 − vn+1 ] (14.65) ρvn+1 ≥ − λ−n−1 + z−1 Combining the two previous expressions, we obtain ¡ ¢ ∗ [v0 − ] + κ [v0 − ] − λ−n + z−1 ∗ ≥ − λ−n−1 + z−1 [v0 − vn+1 ] + κ [v0 − vn+1 ] Since λ−n−1 < λ−n , this implies < vn+1 , contradicting the hypothesis that ≥ vn+1 , and establishing the desired result, < vn+1 ∞ Consequently, {vn }∞ n=−1 is nondecreasing and {vn }n=0 is (strictly) increasing Since a nondecreasing sequence in a compact set must converge, {vn }∞ n=−1 converges to its limit point, v∞ , which must be strictly positive, since {vn }∞ n=0 is strictly increasing and has a nonnegative initial value This completes the proof ¤ A potential difficulty in the analysis of the current model is that we have to determine R&D levels and values for an infinite number of firms, since the technology gap between the leader and the follower can, in principle, take any value However, the next result shows that we can restrict attention to a finite sequence of values: Proposition 14.7 There exists n∗ ≥ such that zn∗ = for all n ≥ n∗ Proof See Exercise 14.23 Ô The next proposition provides the most important economic insights of this model and shows that z∗ ≡ {zn∗ }∞ n=0 is a decreasing sequence, which implies that technological leaders that are further ahead undertake less R&D Intuitively, the benefits of further R&D investments are decreasing in the technology gap, since 642