Economic growth and economic development 652

Introduction to Modern Economic Growth Standard arguments imply that the value function for a firm that is n steps ahead (or −n steps behind) is given by (14.51) r (t) Vn (t) − V˙ n (t) = max{[Πn (t) − w∗ (t) G (zn (t))] zn (t) ¡ ∗ ¢ +zn (t) [Vn+1 (t) − Vn (t)] + z−n (t) + κ [V0 (t) − Vn (t)]} In steady state, the net present value of a firm that is n steps ahead, Vn (t), will also grow at a constant rate g ∗ for all n ∈ Z+ Let us then define normalized values as (t) ≡ (14.52) Vn (t) Y (t) for all n which will be independent of time in steady state, i.e., (t) = Using (14.52) and the fact that from (14.32), r (t) = g (t) + ρ, the steady-state value function (14.51) can be written as: (14.53) ¡ ¢ ρvn = max{ − λ−n − ω ∗ G (zn ) + zn [vn+1 − ] , zn £ Ô + zn + [v0 ]} for all n ≥ 1, ∗ is the equilibrium value of R&D by a follower that is n steps behind, and where z−n ω ∗ is the steady-state labor share (while zn is now explicitly chosen to maximize ) Similarly the value for neck-and-neck firms is (14.54) ρv0 = max {−ω ∗ G (z0 ) + z0 [v1 − v0 ] + z0∗ [v−1 − v0 ]} , z0 while the values for followers are given by ρv−n = max {−ω ∗ G (z−n ) + [z−n + κ] [v0 − v−n ]} z−n It is clear that these value functions and profit-maximizing R&D decision for followers should not depend on how many steps behind the leader they are, since a single innovation is sufficient to catch-up with the leader Therefore, we can write (14.55) ρv−1 = max {−ω ∗ G (z−1 ) + [z−1 + κ] [v0 − v−1 ]} , z−1 638