Economic growth and economic development 653
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Introduction to Modern Economic Growth
where v−1 represents the value of any follower (irrespective of how many steps behind
it is). The maximization problems involved in the value functions are straightforward and immediately yield the following profit-maximizing R&D decisions
à
ả ắ
ẵ
[vn+1 vn ]
01
(14.56)
,0
zn = max G
à
ả ắ
ẵ
[v0 vn ]
01
zn = max G
(14.57)
,0
à
ả ắ
ẵ
[v1 v0 ]
01
z0 = max G
(14.58)
,0 ,
where G01 (·) is the inverse of the derivative of the G function, and since G is twice
continuously differentiable and strictly concave, G0−1 is continuously differentiable
and strictly increasing. These equations therefore imply that innovation rates, the
zn∗ ’s, are increasing in the incremental value of moving to the next step and decreasing in the cost of R&D, as measured by the normalized wage rate, ω∗ . Note also
that since G0 (0) > 0, these R&D levels can be equal to zero, which is taken care of
by the max operator.
The response of innovation rates, zn∗ , to the increments in values, vn+1 − vn , is
the key economic force in this model. For example, a policy that reduces the patent
protection of leaders that are n + 1 steps ahead (by increasing κ) will make being
n + 1 steps ahead less profitable, thus reduce vn+1 − vn and zn∗ . This corresponds to
the standard disincentive effect of relaxing IPR protection. However, relaxing IPR
protection may also create a beneficial composition effect; this is because, typically,
∗
∗
{vn+1 − vn }∞
n=0 is a decreasing sequence, which implies that zn−1 is higher than zn
for n ≥ 1 (see Proposition 14.8 below). Weaker patent protection (in the form of
shorter patent lengths) will shift more industries into the neck-and-neck state and
potentially increase the equilibrium level of R&D in the economy.
Given the equilibrium R&D decisions, the steady-state distribution of industries
across states µ∗ has to satisfy the following accounting identities:
¡ ∗
¢
∗
(14.59)
zn+1 + z−1
+ κ µ∗n+1 = zn∗ µ∗n for n ≥ 1,
(14.60)
(14.61)
¡ ∗
¢
∗
z1 + z−1
+ κ µ∗1 = 2z0∗ µ∗0 ,
∗
2z0∗ µ∗0 = z−1
+ κ.
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