4.
The higher the population growth rate is, the lower the steadystate level of capital per
worker, and therefore there is a lower level of steadystate income per worker. For
example, Figure 7–1 shows the steady state for two levels of population growth, a low
level
n
1
and a higher level
n
2
. The higher population growth
n
2
means that the line rep
resenting population growth and depreciation is higher, so the steadystate level of cap
ital per worker is lower.
In a model with no technological change, the steadystate growth rate of total income is
n
: the higher the population growth rate
n
is, the higher the growth rate of total
income. Income per worker, however, grows at rate
zero
in steady state and, thus, is
not affected by population growth.
Problems and Applications
1.
a.
A production function has constant returns to scale if increasing all factors of pro
duction by an equal percentage causes output to increase by the same percentage.
Mathematically, a production function has constant returns to scale if
zY
=
F
(
zK
,
zL
) for any positive number
z
. That is, if we multiply both the amount of capital
and the amount of labor by some amount
z
, then the amount of output is multi
plied by
z
. For example, if we double the amounts of capital and labor we use (set
ting
z
= 2), then output also doubles.
To see if the production function
Y
=
F
(
K
,
L
) =
K
1/2
L
1/2
has constant returns to
scale, we write:
F
(
zK
,
zL
) = (
zK
)
1/2
(
zL
)
1/2
=
zK
1/2
L
1/2
=
zY
.
Therefore, the production function
Y
=
K
1/2
L
1/2
has constant returns to scale.
b.
To find the perworker production function, divide the production function
Y
=
K
1/2
L
1/2
by
L
:
If we define
y
=
Y/L
, we can rewrite the above expression as:
y
=
K
1/2
/
L
1/2
.
Defining
k
=
K/L
, we can rewrite the above expression as:
y
=
k
1/2
.
58
Answers to Textbook Questions and Problems
(
δ
+
n
2
)
k
(
δ
+
n
1
)
k
sf
(
k
)
Investment, breakeven investment
Capital per worker
k
k
2
*
k
1
*
Figure 7–1
Y
L
K
L
L
=
1 2
1 2
.
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c.
We know the following facts about countries A and B:
δ
= depreciation rate = 0.05,
s
a
= saving rate of country A = 0.1,
s
b
= saving rate of country B = 0.2, and
y
=
k
1/2
is the perworker production function derived
in part (b) for countries A and B.
The growth of the capital stock
Δ
k
equals the amount of investment
sf
(
k
), less
the amount of depreciation
δ
k
. That is,
Δ
k
=
sf
(
k
) –
δ
k
. In steady state, the capital
stock does not grow, so we can write this as
sf
(
k
) =
δ
k
.
To find the steadystate level of capital per worker, plug the perworker pro
duction function into the steadystate investment condition, and solve for
k
*
:
sk
1/2
=
δ
k
.
Rewriting this:
k
1/2
=
s
/
δ
k
= (
s
/
δ
)
2
.
To find the steadystate level of capital per worker
k
*
, plug the saving rate for
each country into the above formula:
Country A:
k
= (
s
a
/
δ
)
2
= (0.1/0.05)
2
= 4.
Country B:
k
= (
s
b
/
δ
)
2
= (0.2/0.05)
2
= 16.
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 Spring '08
 Staff
 Economics, Steady State, labor force, Δk, k*, Textbook Questions and Problems

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