Economic growth and economic development 657
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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∗ > 0. Furthermore, z0∗ > z1∗ and z0∗ ≥ z−1
.
n ≥ 1 and moreover, zn+1
Proof. From equation (14.56),
δ n+1 ≡ vn+1 − vn < vn − vn−1 ≡ δ n
(14.66)
∗
is sufficient to establish that zn+1
≤ zn∗ .
Let us write:
(14.67)
where
vn = max
zn
+ z1 + .
Ă
 ê
âĂ
Â
+ κ v0 ,
1 − λ−n − ω ∗ G (zn ) + zn [vn+1 − vn ] + z−1
∗
∗
Since zn+1
, zn∗ and zn−1
are maximizers of the value functions
vn+1 , vn and vn−1 , (14.67) implies:
¡ ∗ ¢
¡ ∗
¢
∗
ρ¯vn+1 = 1 − λ−n−1 − ω ∗ G zn+1
[vn+2 − vn+1 ] + z−1
+ κ v0 ,
+ zn+1
(14.68)
¡ ∗ ¢
¡ ∗
¢
∗
ρ¯vn ≥ 1 − λ−n − ω∗ G zn+1
[vn+1 − vn ] + z−1
+ κ v0 , ,
+ zn+1
¡ ∗ ¢
¡ ∗
¢
∗
ρ¯vn ≥ 1 − λ−n − ω∗ G zn−1
[vn+1 − vn ] + z−1
+ κ v0 , ,
+ zn−1
¡ ∗ ¢
¡ ∗
¢
∗
ρ¯vn−1 = 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 1 − λ−1 + zn+1
(δ n+2 − δ n+1 )
¡
¢
∗
ρ¯δ n ≥ λ−n+1 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 > 0. Now to obtain a contradiction, suppose that
δ n+1 − δ n ≥ 0. From the previous equation, this implies δ n+2 − δ n+1 > 0 since kn is
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