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Solutions to Problems
Chapter 31
1a. The graph (figure 1a) plots leisure on the x-axis and real GDP on the y-axis. As leisure increases from zero to 12 hours a day,
real GDP decreases from $30 to $0 a day.
1b. The table replaces leisure with labour and labour equals 12 hours a day minus leisure hours. The graph (figure 1b) plots labour
on the x-axis and real GDP on the y-axis. As labour increases from zero to 12 hours a day, real GDP increases from $0 to $30 a
day.
Table 1  Problem 1
Crusoe’s Production Function
Leisure Real GDP
Possibility (hours/day)
($/day)
a
12
0
MPL
(Y/L)
Labour
(hours/day)
0
b
10
10
2
5
c
8
18
4
4
d
6
24
6
3
e
4
28
8
2
f
2
30
10
1
g
0
30
12
0
1c. Marginal product is the change in real GDP divided by the change in labour hours.
When labour increases there is a diminishing marginal product of labour because the most productive labour is used first and as
more labour is used, it is less and less productive (see table 1).
Real GDP (dollars per day)
Real GDP (dollars per day)
Crusoe’s PPF– Problem 1
35
g
f
30
e
d
25
20
c
15
Crusoe’s Production Function – Problem 1
35
30
f
g
10
12
e
d
25
20
c
15
b
10
b
10
5
5
a
0
0
2
4
6
8
10
12
a
0
0
Leisure (hours per day)
Figure 1a
2
4
6
8
Labour (hours per day)
Figure 1b
3a. Crusoe’s demand for labour schedule in table 3 is the same as his marginal product of labour schedule in table1. It is the quantity
of goods and services that an hours work can produce.
The graph (figure 3) plots a marginal product of $5 at 1 hour and a marginal product of $1 at 9 hours of labour and is a straight
line between these points.
Table 3  Problem 3
Crusoe’s labour schedules
Real wage
($ per
hour)
Labour
demand
Labour
supply
(hours/day)
(hours/day)
6
0
5
4
3
2
1
1
3
5
7
9
12
0
0
0
0
0
Real wage ($ per hour)
Crusoe’s real wage – Problem 3
LS
6
5
3b.
4
3
2
1
LD
0
0
2
4
6
8
10
Quantity of labour (hours per day)
Crusoe is willing to supply labour is $6.00 an
This is his subsistence real wage. Crusoe’s supply curve is horizontal at $6.00 an hour.
For
each
hour,
the
wage
rate
at
whic
h
hour.
3c. There is no equilibrium real wage. Crusoe needs to produce at least $6 real GDP per hour for 12 hours in order to survive. Unless
he can either find or build new capital to improve his productivity he is going to perish. Working with current stocks of capital
will simply delay the inevitable. He would work zero hours.
Note: Six dollars is probably a poor choice for a supply wage. I suggest you set another if using this question, say $4.50. In this case
the full-employment equilibrium real wage rate would be $4.50 an hour because Crusoe is willing to work any number of hours
at this wage rate. The equilibrium level of employment would be 2 hours a day because this is the number of hours at which
Crusoe’s marginal product of labour is $4.50 an hour.
3d. At $6 an hour potential GDP is zero.
(At $4.50 an hour potential GDP would be $10 a day. Potential GDP is $10 a day because this quantity of real GDP is produced
when labour is 2 hours a day.)
5a. At full employment the quantity of labour employed is 6 billion hours and the real wage is $7 per hour.
Equilibrium occurs in the labour market at a real wage where the quantity of labour demanded is equal to the quantity of labour
supplied. This equilibrium is full employment equilibrium. The quantity of labour supplied equals the quantity demanded when
the real wage is $7 per hour and the labour employed is 6 billion hours per year.
5b. If the GDP deflator is 120 the money wage will be $8.40.
W
7
P
W
7
120
100
W  7 x1.20
 $8.40
5c. The long-run aggregate supply is $98 billion per year.
The long-run aggregate supply is the level of real GDP produced with full employment labour. The full employment level of
labour, 6 billion hours per year, will produce $98 billion of output (see figure 5). The LAS curve is vertical at $98 billion real
GDP.
5d. In the short run money wages are sticky. If the GDP deflator rises to 130 the real wage falls to $6.46 per hour. Employment and
real GDP will rise (see figure 5).
W
P
8.40 100

x
1 130
 $6.46
real wage 
Nominal wages rise to $9.10 per hour in the long run
Real wage ($/ hour )
In the long run money wages are flexible and the real wage is fixed. If the price level rises money wages will increase
proportionately and the real wage and employment will stay at full employment equilibrium. Nominal wages rise to $9.10 per
hour in the long run.
Longland – Problem 5
20
18
LS
16
14
12
10
8
6
4
2
LD
0
0
1
2
3
4
5
6
7
8
9
10
11
Real GDP ($ billions )
Quantity of labour (billions of hours)
120
PF
100
80
60
40
20
0
1
2
3
4
5
6
7
8
Quantity of labour (billions of hours)
W
7
P
W
7
130
100
W  7 x1.30
 $9.10
Figure 5
7a. Real wage rate is $3 an hour and employment is 3,000 hours a day.
This wage rate and quantity of labour are at the intersection of the demand curve and the supply curve in the figure.
7b. Potential GDP is $13,500 a day.
To calculate potential GDP, use the fact that the real wage rate on the demand for labour curve is the marginal product of labour.
Add up the marginal products across the employment range zero to 3,000. Or equivalently, calculate the area under the demand
for labour curve. This area is $13,500 a day.
7c. The natural rate of unemployment is 25 percent.
The natural rate of unemployment occurs at full employment when job search is 1,000 hours. Employment is 3,000 hours, and
job search, which is unemployment, is 1,000 hours. The labour force (employment plus unemployment) is 4,000 hours. The
natural rate of unemployment is 25 percent (1,000/4,000 multiplied by 100).