6. Solow Model With Technological Growth. Cobb

ECON 385. Intermediate
Macroeconomic Theory II. Solow
Model With Technological Progress.
Cobb-Douglas Example
Instructor: Dmytro Hryshko
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Equilibrium allocations
Let production function be of Cobb-Douglas type
Y = K α (EL)1−α .
It is CRS in K and L:
F (zK, zL, E) = (zK)α (EzL)1−α = z α z 1−α K α (EL)1−α = zY.
{z
}
|
=Y
1
to obtain ypew = (kpew )α , where kpew =
Define z ≡ EL
Y
ypew = EL .
K
EL
and
The steady-state equilibrium in this economy is defined from
∗
∗
s(kpew
)α = (n + g + δ)kpew
.
Thus,
∗
kpew
=
s
n+g+δ
1
1−α
.
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Furthermore,
α
1−α
s
=
=
,
n+g+δ
α
1−α
s
∗
cpew = (1 − s)
.
n+g+δ
∗
ypew
∗
(kpew
)α
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Equilibrium prices
We know that
w = FL = (1 − α)K α E 1−α L−α = (1 − α)
K α (EL)1−α
L
Y
= (1 − α)ypw = (1 − α)ypew E.
L
The rental price of capital
= (1 − α)
K α (EL)1−α
K
ypw
ypew
Y
Y /L
Y /(EL)
=α =α
=α
=α
=α
.
K
K/L
kpw
K/(EL)
kpew
R = FK = αK α−1 (EL)1−α = α
The real interest rate is equal to
r = FK − δ = α
ypew
− δ.
kpew
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Prices in the steady state equilibrium
∗ E(t) and
In the steady-state equilibrium, w∗ (t) = (1 − α)ypew
y∗
R∗ = α kpew
.
∗
pew
For our example,
∗
w (t) = (1 − α)
∗
R = α
s
n+g+δ
s
n+g+δ
s
n+g+δ
α
1−α
α
1−α
1
1−α
=α
s
n+g+δ
E(0)(1 + g)t ,
−1
=α
n+g+δ
,
s
and
r∗ = α
n+g+δ
− δ.
s
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Growth rates in the steady state—1
∗
∗
Note that kpew
and ypew
are constant in the steady state. What
about K and Y , and kpw and ypw ? By definition,
K = kpew EL.
Therefore,
∆kpew
∆K
∆E ∆L
=
+
+
.
K
kpew
E
L
In the steady state,
∗
∆kpew
∗
kpew
= 0 and so
∆K
∆E ∆L
=
+
=g+n
K
E
L
Aggregate capital grows at a constant rate equal to (g + n).
The same can be shown for aggregate output, Y .
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Growth rates in the steady state—2
ypw and kpw will grow in the steady state at the rate g.
kpw = kpew E.
Therefore,
∆kpw
∆kpew
∆E
=
+
.
kpw
kpew
E
In the steady state,
∗
∆kpew
∗
kpew
= 0 and so
∆kpw
∆E
=
=g
kpw
E
Capital per worker grows at a constant rate equal to g. The
same can be shown for the growth rate of output per worker,
ypw .
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Golden rule steady state
If Y = K α (EL)1−α , then the golden-rule savings rate in the
economy is equal to α, the share of capital income in total
income.
For the economy with technological progress, the golden-rule
∗ ) = n + g + δ.
capital per worker is obtained from M P K(kgold
For this production function,
M P K = FK (K, L) = αK α−1 (EL)1−α = α
K α−1
EL
α−1 .
= αkpew
Thus, the golden rule capital per worker is obtained from
α−1
αkpew,gold
= n + g + δ.
and so kpew,gold =
α
n+g+δ
1
1−α
.
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Golden rule steady state—2
If the economy saves its capital income, the total savings in the
economy are αY , the per effective worker savings are αypew .
For this economy, the steady state occurs when
∗
∗
)α = (n + g + δ)kpew
,
α(kpew
i.e., when
∗
kpew
=
α
n+g+δ
1
1−α
,
which is exactly equal to the golden rule capital per worker
we’ve just found.
9/9

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