Economic growth and economic development 299

Introduction to Modern Economic Growth
Example 6.4. Consider the following optimal growth, with log preferences, CobbDouglas technology and full depreciation of capital stock
max

{c(t),k(t+1)}∞
t=0

subject to

∞
X

β t ln c (t)

t=0

k (t + 1) = [k (t)]α − c (t)
k (0) = k0 > 0,
where, as usual, β ∈ (0, 1), k denotes the capital-labor ratio (capital stock), and

the resource constraint follows from the production function K α L1−α , written in per
capita terms.
This is one of the canonical examples which admits an explicit-form characteriza-

tion. To derive this, let us follow Example 6.1 and set up the maximization problem
in its recursive form as
V (x) = max {ln (xα − y) + βV (y)} ,
y≥0

with x corresponding to today’s capital stock and y to tomorrow’s capital stock. Our
main objective is to find the policy function y = π (x), which determines tomorrow’s

capital stock as a function of today’s capital stock. Once this is done, we can easily
determine the level of consumption as a function of today’s capital stock from the
resource constraint.
It can be verified that this problem satisfies Assumptions 6.1-6.5. The only
non-obvious feature here is whether x and y indeed belong to a compact set. The
argument used in Section 6.6 for Proposition 6.1 can be used to verify that this
is the case, and we will not repeat the argument here. Consequently, Theorems
6.1-6.6 apply. In particular, since V (·) is differentiable, the Euler equation for the
one-dimensional case, (6.22), implies
xα

1
= βV 0 (y) .
−y

The envelope condition, (6.23), gives:
αxα−1
.
xα − y
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V 0 (x) =