Economics 154a
Fall 2005
Bj¨orn Br¨ugemann
Problem Set 2 Solution
Analytical Problem 1
1. The derivative of the production function with respect to
N
is given by
∂Y
∂N
= (1

α
)
AK
α
N
−
α
.
To obtain this result, remember that for a function
f
(
x
) =
x
a
the first derivative is
given by
f
′
(
x
) =
ax
a
−
1
. Notice that
K
has the exponent
α
while
N
is the exponent

α
, so we can combine the two and write
∂Y
∂N
= (1

α
)
A
parenleftbigg
K
N
parenrightbigg
α
.
Thus the marginal product of labor (MPN) is a function of the capital labor ratio
K
N
. The higher the capital labor ratio, i.e. the more machines there are per worker,
the higher the marginal product of labor. Conversely, the more workers there are
per machine, the lower the MPN. Thus as long as
α >
0, the production function
has decreasing returns to labor. Using the assumptions that
A
= 1,
K
= 1 and
α
= 0
.
3, we obtain
MPN
= 0
.
7
N
−
0
.
3
.
2. Setting the MPN equal to the real wage
w
yields the condition
(1

α
)
A
parenleftbigg
K
N
parenrightbigg
α
=
w.
(1)
The intuition for this condition is as follows. The left hand side (the MPN) is the
additional output generated by hiring an additional unit of labor. The left hand
side is the real wage (here
real
means that it is the wage measured in terms of units
of output) that must be paid to this additional unit of labor.
If at the current
level of employment the MPN exceeds the real wage, i.e. the left hand side exceeds
the right hand side, then it is profitable for the firm to hire more units of labor.
Conversely, if at the current level of employment the MPN is below the real wage,
i.e. the right hand side exceeds the left hand side, then it is profitable for the firm
to reduce the number of units of labor it hires. At the profit maximizing level of
employment it must be that the left hand side equals the right hand side.
3. The graph is shown in Figure 1.
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Figure 1:
Labor Demand Curve
0
0
.
5
1
.
0
0
0
.
5
1
.
0
N
w
N
d
4. Substituting for
C
and
L
yields
U
=
wN
+
λ
(
N
max

N
)
.
It will be useful to collect terms involving
N
, which yields
U
= (
w

λ
)
N
+
λN
max
.
(2)
5. It is clear from equation (2) that in the case
w > λ
utility is strictly increasing in
N
over the entire range [0
, N
max
]. Thus the worker will choose to work as much as
possible, that is
N
max
.
6. It is clear from equation (2) that in the case
w < λ
utility is strictly decreasing in
N
over the entire range [0
, N
max
]. Thus the worker will choose to work as little as
possible, that is 0.
7. In the case
w
=
λ
utility is given by
U
=
λN
max
no matter how many units of labor
the worker decides to supply. Thus she is indifferent about how much she works.
8. Figure 2 adds the labor supply curve. Notice that at
w
=
λ
the labor supply curve
is horizontal since the worker is willing to work any amount between 0 and
N
max
at that wage.
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 Fall '07
 BjoernBruegemann
 Economics, Macroeconomics, Supply And Demand, labor supply curve, Nmax

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