4. The higher the population growth rate is, the lower the steady-state level of capital per
worker, and therefore there is a lower level of steady-state 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 steady-state level of cap-
ital per worker is lower.
In a model with no technological change, the steady-state 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 per-worker 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, break-even investment
Capital per worker
k
k
2
*
k
1
*
Figure 7–1
Y
L
KL
L
=
12 12
.