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1)
Suppose the production function for x is: x = L
a
K
b
with 0 < a <1 and 0 < b < 1 and a + b = 1.
a)
Show that each input, L and K, has a positive marginal product.
ANS:
Calculate the marginal products, so take the partial derivatives with respect to each input and
show that this expression is positive.
So:
mp
L
x
= aL
(a1)
K
b
which has to be a positive number and
likewise, mp
K
x
= bL
a
K
(b1)
which also has to be a positive number.
b)
Show that each input, L and K, has diminishing marginal productivity.
ANS:
Now take the partial of each marginal product you just got above with respect to that input and
show that what you get it negative, i.e., that the value of marginal product is getting smaller and smaller.
So:
∂
mp
L
x
/
∂
L = (a1)aL
(a2)
K
b
which is negative since (a

1)<1.
Do the same thing again for K and get
∂
mp
K
x
/
∂
K = (b1)bL
a
K
(b2)
which will be negative since (b1)<1.
c)
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This homework help was uploaded on 10/31/2007 for the course ECON 3130 taught by Professor Masson during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 MASSON
 Microeconomics

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