Economic growth and economic development 655

Introduction to Modern Economic Growth
Subtracting ln Q (t), dividing by ∆t and taking the limit as ∆t → 0 gives (14.63). Ô
This proposition clarifies that the steady-state growth comes from two sources:
(1) R&D decisions of leaders or of firms in neck-and-neck industries.
(2) The distribution of industries across different technology gaps, µ∗ ≡ {µ∗n }∞
n=0 .
The latter channel reflects the composition effect discussed above. This type
of composition effect implies that the relationship between competition and growth
(or intellectual property rights protection and growth) is more complex than in the
models we have seen so far, because such policies will change the equilibrium market
structure (i.e., the composition of industries).
Definition 14.1. A steady-state equilibrium is given by hµ∗ , v, z∗ , ω ∗ , g ∗ i

such that the distribution of industries µ∗ satisfy (14.59), (14.60) and (14.61), the
∗
values v ≡ {vn }∞
n=−∞ satisfy (14.53), (14.54) and (14.55), the R&D decisions z

are given by, (14.56), (14.57) and (14.58), the steady-state labor share ω ∗ satisfies

(14.62) and the steady-state growth rate g ∗ is given by (14.63).
We next provide a characterization of the steady-state equilibrium. The first
result is a technical one that is necessary for this characterization.
Proposition 14.6. In a steady state equilibrium, we have v−1 ≤ v0 and {vn }∞
n=0

forms a bounded and strictly increasing sequence converging to some positive value
v∞ .
∞
Proof. Let {zn }∞
n=−1 be the R&D decisions of a firm and {vn }n=−1 be the

sequence of values, taking the decisions of other firms and the industry distributions,
∗ ∞
∗
∗
{zn∗ }∞
n=−1 , {µn }n=−1 , ω and g , as given. By choosing zn = 0 for all n ≥ −1, the

firm guarantees vn ≥ 0 for all n ≥ −1. Moreover, since flow profits satisfy π n ≤ 1
for all n ≥ −1, we have vn ≤ 1/ρ for all n ≥ −1, establishing that {vn }∞
n=−1 is a

bounded sequence, with vn ∈ [0, 1/ρ] for all n ≥ −1.

Proof of v1 > v0 : Suppose, first, v1 ≤ v0 , then (14.58) implies z0∗ = 0, and by the

symmetry of the problem in equilibrium, (14.54) implies v0 = v1 = 0. As a result,
∗
∗
= 0. Equation (14.53) then implies that when z−1
= 0,
from (14.57) we obtain z−1
¡
¢
−1
/ (ρ + κ) > 0, yielding a contradiction and proving that v1 > v0 .
v1 ≥ 1 − λ

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