Economic growth and economic development 657

Introduction to Modern Economic Growth greater values of the technology gap translate into smaller increases in the equilibrium markup (recall (14.44)) The fact that leaders that are sufficiently ahead of their competitors undertake little R&D is the main reason why composition effects play an important role in this model For example, all else equal, closing the technology gap across a range of industries will increase R&D spending and equilibrium growth (though, as discussed in the previous section, this may not always increase welfare, especially if there is a strong business stealing effect) ∗ ≤ zn∗ for all Proposition 14.8 In any steady-state equilibrium, we have zn+1 ∗ ∗ < zn∗ if zn∗ > Furthermore, z0∗ > z1∗ and z0∗ ≥ z−1 n ≥ and moreover, zn+1 Proof From equation (14.56), δ n+1 ≡ vn+1 − < − vn−1 ≡ δ n (14.66) ∗ is sufficient to establish that zn+1 ≤ zn∗ Let us write: (14.67) where ρ¯ ≡ ρ¯vn = max zn ∗ ρ + z1 + Ă Â ê âĂ Â + κ v0 , − λ−n − ω ∗ G (zn ) + zn [vn+1 − ] + z−1 ∗ ∗ Since zn+1 , zn∗ and zn−1 are maximizers of the value functions vn+1 , and vn−1 , (14.67) implies: ¡ ∗ ¢ ¡ ∗ ¢ ∗ ρ¯vn+1 = − λ−n−1 − ω ∗ G zn+1 [vn+2 − vn+1 ] + z−1 + κ v0 , + zn+1 (14.68) ¡ ∗ ¢ ¡ ∗ ¢ ∗ ρ¯vn ≥ − λ−n − ω∗ G zn+1 [vn+1 − ] + z−1 + κ v0 , , + zn+1 ¡ ∗ ¢ ¡ ∗ ¢ ∗ ρ¯vn ≥ − λ−n − ω∗ G zn−1 [vn+1 − ] + z−1 + κ v0 , , + zn−1 ¡ ∗ ¢ ¡ ∗ ¢ ∗ ρ¯vn−1 = − λ−n+1 − ω ∗ G zn−1 + zn−1 [vn − vn−1 ] + z−1 + κ v0 Now taking differences with ρ¯vn and using the definition of δ n ’s, we obtain ¡ ¢ ∗ ρ¯δn+1 ≤ λ−n − λ−1 + zn+1 (δ n+2 − δ n+1 ) ¡ ¢ ∗ ρ¯δ n ≥ λ−n+1 − λ−1 + zn−1 (δ n+1 − δ n ) Therefore, ¢ ¡ ∗ ∗ (δ n+2 − δ n+1 ) , (δ n+1 − δ n ) ≤ −kn + zn+1 ρ¯ + zn−1 where kn ≡ (λ − 1)2 λ−n−1 > Now to obtain a contradiction, suppose that δ n+1 − δ n ≥ From the previous equation, this implies δ n+2 − δ n+1 > since kn is 643