14.451 Lecture Notes Economic Growth

• Straightforward algebra gives
We infer
14.451 Lecture Notes
Γ 0 (z) = sφ 0 (k ∗ e z )k ∗ e z < 0
φ 0 (k) = f 0 (k)k − f(k)
k 2
sf(k ∗ ) = (δ + n)k ∗
= −
Γ 0 (0) = −(1 − εK)(δ + n) < 0
·
1 − f 0 ¸
(k)k f(k)
f(k) k2 where εK ≡ FKK/F = f 0 (k)k/f(k) is the elasticity of production with respect to
capital, evaluated at the steady-state k.
• We conclude that
where
˙k
k
µ
k
= λ ln
k∗
λ = −(1 − εK)(δ + n) < 0
The quantity −λ is called the convergence rate.
• Note that, around the steady state
and
It follows that
˙y
y
˙y
y = εK · ˙ k
k
y
y ∗ = εK · k
k ∗
µ
y
= λ ln
y∗
Thus, −λ is the convergence rate for either capital or output.
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