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Solutions to Problems
Chapter 32
1.
The economy does not conform to the one-third rule. It conforms to a one-half rule.
In this economy, an x percent increase in the capital stock per hour of work leads to a 0.5x percent increase in real GDP per
hour of work. You can confirm this fact by calculating the percentage change in capital and real GDP at each of the levels
provided in the table and then dividing the percentage change in real GDP by the percentage change in capital (see table 1).
Table 1  Problem 1
Longland’s Production Function
Capital per
hour of work
$
10
20
30
40
50
60
70
80
Real GDP
per hour of
work
$
3.80
5.70
7.13
8.31
9.35
10.29
11.14
11.94
Y/K
$
%K
%Y
%Y/%
K
0.19
0.14
0.12
0.10
0.09
0.09
0.08
100.0
50.0
33.3
25.0
20.0
16.7
14.3
50.0
25.1
16.5
12.5
10.1
8.3
7.2
0.5
0.5
0.5
0.5
0.5
0.5
0.5
3a. Yes, Longland experiences diminishing returns.
Diminishing returns are present if the marginal product of capital diminishes as capital
increases, holding technology constant (see table 1). The increase in real GDP per hour of
work that occurred in the question resulted from an increase in capital and an advance in
technology. We know this because to produce $10.29 in 1999 would have required a capital
stock of $60 per hour of work, and in 2001, this output can be produced by a capital stock of
$50. The change in real GDP divided by the change in capital is not the marginal product of
capital because technology is not constant.
3b. The contribution of the change in capital is $1.04.
Along the productivity function in table 1, when capital per hour of work increases from $40
to $50, real GDP per hour of work increases from $8.31 to $9.35, a difference of $1.04. This
number is also calculated as the percentage increase in real GDP that is equal to one-half the
percentage increase in capital.
1 10 $8.31
x x

2 40
1
 $1.04
contribution of capital change 
3c. The contribution of technological change is $0.94.
This number is calculated as the change in real GDP minus the contribution of the change in
capital to the growth of productivity, which is $0.94.
contribution of techno log ical change  change in realGDP  contribution of capital change
 $1.98  $1.04
 $0.94
5a. Employment is 6 billion hours per year and the real wage rate is $7 an hour.
The labour market is n equilibrium at the real wage rate at which the quantity demanded equals the quantity supplied (see
figure 1). That real wage rate is $7 an hour.
5b. The real wage rate rises to $8.
A technological advance increases the productivity of labour and the demand for labour increases. The increase in demand
for labour moves the demand curve to the right from LD to LD/ in figure 1 and the real wage rate rises to $8.
5c. The population begins to grow.
The reason for the population growth is that the real wage rate exceeds the subsistence level.
5d. Employment is 7 billion hours a year.
Cape Despair – Problem 5
20
Real interest %
Real wage $
In long-run equilibrium, employment equals the quantity of labour demanded at the subsistence real wage rate of $7 an hour.
Only when the population has grown by enough to make the quantity of labour supplied equal 7 billion hours a year does the
population stop growing.
18
16
14
LS
12
Martha’s Island – Problem 6
12
10
8
LS/
10
6
b
8
a
•
6
•
•b
Subsistence
wage
c
•
4
4
Target rate
LD/
2
LD
2
0
•
c
KD/
KD
4
5
6
7
8
9
10
11
0
1
2
3
4
5
6
7
8
9
10
11
Quantity of capital (billions of hours)
Quantity of labour (billions of hours)
7.
a
0
3
Figure 1
•
Figure 2
When the demand for capital raises the real interest rate above the target rate, the capital stock and real GDP begin to
grow and keep on growing. In contrast, in the neoclassical Martha’s Island, as the capital stock grows, the real interest
rate falls (because of diminishing returns) and growth eventually ends.