Rice University
Economics 370
Professor Diamond
Microeconomic Theory
SOLUTION TO PROBLEM SET 1: INTRODUCTION
AND APARTMENT SUPPLY AND DEMAND
I.
Suppose the production function for the output produced by a firm is
f
(
L
,
K
) = 40
L
3 / 4
K
1 / 4
,
where L is labor and K is capital.
1.
What is the marginal product of labor? What is the marginal product of capital?
The marginal product of labor is
1
∂
f
(
L
,
K
)
∂
(40
L
3/ 4
K
1/ 4
=
)
= 40
·
3
L
3/ 4
−
1
K
1/ 4
= 30(
K
)
1/ 4
.
∂
L
∂
L
4
L
The marginal product of capital is
∂
f
(
L
,
K
)
∂
(40
L
3/ 4
K
1/ 4
=
)
= 40
·
1
L
K
1/ 4
−
1
= 10(
L
)
.
∂
K
∂
K
4
K
2.
Suppose the rental rate of capital in the market is
r
=2, the wage of labor is
w
=10, the
price of output is
p
=1/3 and the amount of capital is fixed at 16 units (in the short run).
Assuming that the firm acts to maximize profit, what is the optimal level of labor?
Show
that both the first and second order conditions for a maximum are satisfied.
What is the
associated optimal output level?
Since profit is pf
(
L
,
K
)
−
(
wL
+
rK
)
and the capital stock K is fixed, the optimization
problem can be expressed as a function of a single variable, labor L:
Max
1
·
40
L
16
1/ 4
−
10
L
−
2
·
16
L
3
⇒
Max
1
·
40
L
16
1/ 4
−
10
L
L
3
F
.
O
.
C
.
⇒
2
·
40
·
3
L
−
1
−
10 = 0
3
4
⇒
20
L
−
1/ 4
= 10
⇒
2 =
L
1/ 4
⇒
L
* = 16
S
.
O
.
C
.
⇒
20(
−
1
)
L
−
1 / 4
−
1
=
−
5
L
−
5 / 4
< 0 ,
so L*
=16
maximizes profit.
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 Spring '08
 DIAMOND
 Economics, Supply And Demand, Marginal product, total rents

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