Problem Set 4
SOLUTIONS
Due Thursday, Oct. 22nd, by 10pm
1.
Consider the production function
f
(
l, k
) =
l
α
+
k
α
.
(a)
For what values of
α >
0
is the production function quasiconcave?
Quasiconcave production functions imply that the isoquants have convex upper contour
sets, or that averages are at least as good as extremes. If
α >
1 then the isoquants bend
away from the origin, implying that averages are worst than extremes. If
α
= 1 then the
isoquants are linear lines, which means that averages are always as good as extremes.
If
α <
1 then the isoquants bend towards the origin, producing convex upper contour
sets, which imply the production function is quasiconcave. So the production function
is quasiconcave if
α <
= 1.
For the rest of this problem, assume
α
is such that the production function
is quasiconcave.
(b)
Find the conditional input demands for
l
and
k
.
If
α
= 1 then the firm will use either only labor or only capital, whichever is cheaper
(except for the case where they have the same price, then the firm is indifferent between
mixing).
The answers to the rest of the problem when
α
= 1 are trivial, so for the
remainder of the problem we will focus on when
α <
1.
When
α <
1 we have to use lagrangians.
To find the conditional input demands we set up the expenditure minimization problem.
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 Fall '09
 CUR
 Microeconomics, Supply And Demand, conditional input demands

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