Economic growth and economic development 529

Introduction to Modern Economic Growth
µh :
c (t)1−θ − 1
H (a, h, c, ih , µa , µk ) =
+ µa (t) [(r (t) − n)a (t) + w (t) h (t) − c (t) − ih (t)]
1−θ
+µh (t) [ih (t) − δ h h (t)] .
Now the necessary conditions of this optimization problem imply the following (see
Exercise 11.8):
(11.25)

µa (t) = µh (t) = µ (t) for all t
w (t) − δh = r (t) − n for all t
1
c˙ (t)
=
(r (t) − ρ) for all t.
c (t)
θ

Combining these with (11.24), we obtain that
f 0 (k (t)) − δ k − n = f (k (t)) − k (t) f 0 (k (t)) − δ h for all t.
Since the left-hand side is decreasing in k (t), while the right-hand side is increasing,
this implies that the effective capital-labor ratio must satisfy
k (t) = k∗ for all t.
We can then prove the following proposition:
Proposition 11.3. Consider the above-described AK economy with physical and
human capital, with a representative household with preferences given by (11.1), and
the production technology given by (11.21). Let k∗ be given by
(11.26)

f 0 (k∗ ) − δ k − n = f (k ∗ ) − k∗ f 0 (k∗ ) − δ h .

Suppose that f 0 (k∗ ) > ρ + δ k > (1 − θ) (f 0 (k∗ ) − δ) + nθ + δ k . Then, in this economy

there exists a unique equilibrium path in which consumption, capital and output

all grow at the same rate g ∗ ≡ (f 0 (k ∗ ) − δ k − ρ)/θ > 0 starting from any initial

conditions, where k∗ is given by (11.26).The share of capital in national income is

constant at all times.
Proof. See Exercise 11.9

Ô

The advantage of the economy studied here, especially as compared to the baseline AK model is that, it generates a stable factor distribution of income, with a
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