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Economic Growth and Development
Prof. Murphy
EC 375
Problem Set 4 Answers
Chapter 9 # 1, 4 (on pages 271272).
1. The annual growth rate of productivity is given by the following equation:
ˆ
A
=
ˆ
y
+
!
ˆ
L
.
We are given a value of 1/3 for
β
and 0 for
ˆ
y
,
leaving the growth rate of the population,
ˆ
L
,
as the only unknown. To solve for
ˆ
L
,
we use the standard growth equation with the initial
population as 4 million and the final population after 10,000 years as 170 million. The
equation is:
4(1
+
ˆ
L
)
10,000
=
170
ˆ
L
=
(170 / 4)
(1/10,000)
!
1
=
0.000375.
Now we substitute to find our growth rate of productivity over this period:
ˆ
A
=
0
+
(1/ 3)(0.000375)
=
0.000125.
That is, the growth rate of productivity over this period was roughly 0.0125 percent per year.
4. a.
In any given year, the production of bread must equal the production of cheese in this
economy. That is,
Y
b
=
Y
c
,
always. Knowing that the productivity of each good is equal
at this point in time, we can solve for the quantity of labor devoted to each sector as
follows.
Y
b
=
Y
c
,
A
b
L
b
=
A
c
L
c
,
L
b
=
L
c
.
Since
L
b
+
L
c
=
L
,
L
b
=
L
c
=
L
/ 2.
The labor force will be equally split between the two sectors.
b.
To calculate the growth rate of total output, we first calculate the growth rates of each
sector by taking the natural log of both sides and differentiating with respect to time. For
the bread sector:
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ln(
Y
b
)
=
ln(
A
b
)
+
ln(
L
b
),
d
dt
ln(
Y
b
)
=
d
dt
ln(
A
b
)
+
ln(
L
b
),
ˆ
Y
b
=
ˆ
A
b
+
ˆ
L
b
.
Similarly, for the cheese sector, we get,
ˆ
Y
c
=
ˆ
A
c
+
ˆ
L
c
.
We know the value for the growth rate of productivity in both sectors. Furthermore, in our
answer to Part (a), we found that labor is currently equally divided among the two
sectors. Thus, the growth of labor in one sector must be offset by the growth of labor in
the other
ˆ
L
b
=
(
!
ˆ
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 Spring '10
 Jannett

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