Economic growth and economic development 658

Introduction to Modern Economic Growth strictly positive Repeating this argument successively, we have that if δ n0 +1 − δ n0 ≥ 0, then δ n+1 − δ n > for all n ≥ n0 However, we know from Proposition 14.6 that {vn }∞ n=0 is strictly increasing and converges to a constant v∞ This implies that δ n ↓ 0, which contradicts the hypothesis that δ n+1 − δ n ≥ for all n ≥ n0 ≥ 0, ∗ ≤ zn∗ To see that the inequality is strict when zn∗ > 0, it and establishes that zn+1 suffices to note that we have already established (14.66), i.e., δ n+1 − δ n < 0, thus if ∗ < zn∗ equation (14.56) has a positive solution, then we necessarily have zn+1 ∗ : (14.54) can be written as Proof of z0∗ ≥ z−1 (14.69) ρv0 = −ω ∗ G (z0∗ ) + z0∗ [v−1 + v1 − 2v0 ] We have v0 ≥ from Proposition 14.6 Suppose v0 > Then (14.69) implies z0∗ > and (14.70) v−1 + v1 − 2v0 > v1 − v0 > v0 − v−1 ∗ This inequality combined with (14.58) and (14.57) yields z0∗ > z−1 Suppose next that v0 = Inequality (14.70) now holds as a weak inequality and implies that ∗ Moreover, since G (·) is strictly convex and z0∗ is given by (14.58), (14.69) z0∗ ≥ z−1 ∗ = then implies z0∗ = and thus z−1 Proof of z0 > z1 : See Exercise 14.24 Ô This proposition therefore shows that the highest amount of R&D is undertaken in neck-and-neck industries This explains why composition effects can increase aggregate innovation Exercise 14.25 shows how a relaxation of intellectual property rights protection can increase the growth rate in the economy So far, we have not provided a closed-form solution for the growth rate in this economy It turns out that this is generally not possible, because of the endogenous market structure in these types of models Nevertheless, it can be proved that a steady state equilibrium exists in this economy, though the proof is somewhat more involved and does not generate additional insights for our purposes (see Acemoglu and Akcigit, 2006) An important feature of this model is that equilibrium markups are endogenous and evolve over time as a function of competition between the firms producing in the 644