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Managerial Economics and Business Strategy, 5e
Page 1
Chapter 5: Answers to Questions and Problems
1.
a.
When
K = 16
and
L = 16
,
() ()
0.75
0.25
16
16
16
Q
=
=
. Thus,
AP
L
= Q/L = 16/16 =
1.
When
K = 16
and
L = 81
,
( ) ( ) ()
()
0.75
0.25
16
81
8 3
24
Q
=
==
. Thus,
AP
L
=
24/81 = 8/27
.
b.
The marginal product of labor is
34
2
L
MP
L
−
=
. When
L = 16
,
216
1
/4
L
MP
−
. When
L = 81
,
281
2
/27
L
MP
−
. Thus, as the
number of units of labor hired increases, the marginal product of labor decreases
( )
16
1/ 4
2 / 27
81
LL
MP
MP
=>
=
, holding the level of capital fixed.
c.
We must equate the value marginal product of labor equal to the wage and solve
for L. Here,
( )
3/4
$100 2
200
VMP
P
MP
L
L
−−
=
. Setting this
equal to the wage of $25 gives
200
25
L
−
=
. Solving for
L
, the optimal
quantity of labor is
L = 16.
2.
See Table 51.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Capital
Labor
Output
Marginal
Product of
Average
Value Marginal
KL
Q
MP
K
AP
K
L
VMP
K
02
00

 



12
0
5
0
5
0
5
0
2
.
5
0
1
0
0
2
20
150
100
75
7.50
200
3
300
15
4
400
5
450
50
90
22.50
6
475
25
79.17
23.75
7
0
67.86
0
82
0
4
5
0

2
5
56.25
50
9
44.44
100
10
30
200
11
150
13.64
300
Table 51
a.
Labor is the fixed input while capital is the variable input.
b.
Fixed costs are 20($15) = $300.
c.
To produce 475 units in the leastcost manner requires 6 units of capital, which
cost $75 each. Thus, variable costs are ($75)(6) = $450.
d.
Using the
VMP
K
=
r
rule,
K = 5
maximizes profits.
e.
The maximum profits are
$2(450)
$15(20)
$75(5)
$225
−
−=
.
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Michael R. Baye
f.
There are increasing marginal returns when
K
is between 0 and 3.
g.
There are decreasing marginal returns when
K
is between 3 and 11.
h.
There are negative marginal returns when
K
is greater than 7.
3.
The law of diminishing marginal returns is the decline in marginal productivity
experienced when input usage increases, holding all other inputs constant. In contrast,
the law of diminishing marginal rate of technical substitution is a property of a
production function stating that as less of one input is used, increasing amounts of
another input must be employed to produce the same level of output.
4.
a.
FC = 50
b.
() () () ()
23
10
25 10
30 10
5 10
$8,250
VC
=+
+=
.
c.
() () ( ) ( )
300
,
8
$
10
5
10
30
10
25
50
10
3
2
=
+
+
+
=
C
.
d.
()
5
$
10
50
$
10
=
=
AFC
.
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 Spring '11
 adamcheung

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