Economic growth and economic development 656

Introduction to Modern Economic Growth
Proof of v−1 ≤ v0 : Suppose, to obtain a contradiction, that v−1 > v0 . Then,

∗
= 0, which leads to v−1 = κv0 / (ρ + κ), contradicting v−1 > v0
(14.57) implies z−1

since κ/ (ρ + κ) < 1 (given that κ < ∞).

Proof of vn < vn+1 : Suppose, to obtain a contradiction, that vn ≥ vn+1 . Now

(14.56) implies zn∗ = 0, and (14.53) becomes
¡
¢
∗
(14.64)
ρvn = 1 − λ−n + z−1
[v0 − vn ] + κ [v0 − vn ] .

Also from (14.53), the value for state n + 1 satisfies
¡
¢
∗
[v0 − vn+1 ] + κ [v0 − vn+1 ] .
(14.65)
ρvn+1 ≥ 1 − λ−n−1 + z−1
Combining the two previous expressions, we obtain
¡
¢
∗
[v0 − vn ] + κ [v0 − vn ]

1 − λ−n + z−1

∗
≥ 1 − λ−n−1 + z−1
[v0 − vn+1 ] + κ [v0 − vn+1 ] .

Since λ−n−1 < λ−n , this implies vn < vn+1 , contradicting the hypothesis that vn ≥
vn+1 , and establishing the desired result, vn < vn+1 .

∞
Consequently, {vn }∞
n=−1 is nondecreasing and {vn }n=0 is (strictly) increasing.

Since a nondecreasing sequence in a compact set must converge, {vn }∞
n=−1 converges

to its limit point, v∞ , which must be strictly positive, since {vn }∞
n=0 is strictly
increasing and has a nonnegative initial value. This completes the proof.

Ô

A potential diculty in the analysis of the current model is that we have to
determine R&D levels and values for an infinite number of firms, since the technology
gap between the leader and the follower can, in principle, take any value. However,
the next result shows that we can restrict attention to a finite sequence of values:
Proposition 14.7. There exists n∗ ≥ 1 such that zn∗ = 0 for all n ≥ n .
Proof. See Exercise 14.23.

Ô

The next proposition provides the most important economic insights of this
model and shows that z∗ ≡ {zn∗ }∞
n=0 is a decreasing sequence, which implies that

technological leaders that are further ahead undertake less R&D. Intuitively, the
benefits of further R&D investments are decreasing in the technology gap, since
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