Midterm 2 Answer Key
April 8, 2010
1) Graphically, &gure 6.2 in the text is an example of a production func
tion with constant returns to scale, and diminishing returns to both labor and
capital.
Using calculus it is apparent that any CobbDouglass production function
in which the exponents add up to one has these properties.
For example, a
production function q=5L
:
3
K
:
7
demonstrates diminishing returns to labor and
diminishing returns to capital. Taking second derivatives,
@MP
L
@L
=
@
2
q
@L
2
=
1
:
05
L
1
:
7
K
:
7
@MP
K
@K
=
@
2
q
@K
2
=
1
:
05
L
:
3
K
1
:
3
(1)
Since both second derivatives are less than zero, this production function
demonstrates diminishing marginal returns for each factor.
The function also demonstrates constant returns to scale.
2
q
=
10
L
:
3
K
:
7
q
(2
L;
2
K
)
=
5(2
L
)
:
3
(2
K
)
:
7
= 5
±
2
:
3+
:
7
L
:
3
K
:
7
= 10
L
:
3
K
:
7
(2)
2
q
=
q
(2
L;
2
K
)
(3)
2) The wage/rent ratio is 300/1000. The marginal rate of technical sub
stitution = the marginal product of labor (50)/the marginal product of capital
(200). The wage/rent ratio of 6/20 is greater than the MRTS of 5/20. Because
If marginal
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 Spring '10
 PERLOFF
 Economics, Supply And Demand, Economics of production

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