Economic growth and economic development 654

Introduction to Modern Economic Growth
The first expression equates exit from state n+1 (which takes the form of the leader
going one more step ahead or the follower catching up for surpassing the leader) to
entry into this state (which takes the form of a leader from the state n making one
more innovation). The second equation, (14.60), performs the same accounting for
state 1, taking into account that entry into this state comes from innovation by
either of the two firms that are competing neck-and-neck. Finally, equation (14.61)
equates exit from state 0 with entry into this state, which comes from innovation
by a follower in any industry with n ≥ 1.

The labor market clearing condition in steady state can then be written as

á

X
Ă Â
1


àn n + G (zn ) + G z−n
and ω ∗ ≥ 0,
(14.62)
1≥
ω
λ
n=0
with complementary slackness.

The next proposition characterizes the steady-state growth rate in this economy:
Proposition 14.5. The steady-state growth rate is given by
#

"
∞
X
(14.63)
g ∗ = ln λ 2µ∗0 z0∗ +
µ∗n zn∗ .
n=1

Proof. Equations (14.48) and (14.50) imply
P∞

∗

w (t)
Q (t) λ− n=0 nµn (t)
Y (t) =
=
.
ω (t)
ω (t)
Since ω (t) = ω ∗ and {µ∗n }∞
n=0 are constant in steady state, Y (t) grows at the same
rate as Q (t). Therefore,

ln Q (t + ∆t) − ln Q (t)
.
∆t→0
∆t
During an interval of length ∆t, we have that in the fraction µ∗n of the industries
g ∗ = lim

with technology gap n ≥ 1 the leaders innovate at a rate zn∗ ∆t + o (∆t) and in the

fraction µ∗0 of the industries with technology gap of n = 0, both firms innovate, so

that the total innovation rate is 2z0∗ ∆t + o (∆t)). Since each innovation increases
productivity by a factor λ, we obtain the preceding equation. Combining these
observations, we have
ln Q (t + ∆t) = ln Q (t) + ln λ

"

2µ∗0 z0∗ ∆t

640

+

∞
X
n=1

µ∗n zn∗ ∆t

#

+ o (∆t) .