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Ani Guerdjikova
Fall 2006
Department of Economics
Intermediate Microeconomics
Cornell University
ECON 313
Problem Set 4
Problem 15
∗
(Utility maximization, CobDouglas utility function)
A household consumes only apples and bananas. We denote a consumption bundle consisting
of
x
a
bags of apples and
x
b
bags of bananas is denoted by
(
x
a
;
x
b
)
. The preferences of the
household are given by the utility function
U
(
x
a
;
x
b
)=
x
1
/
2
a
·
x
1
/
2
b
a) Suppose that prices for apples and bananas are
p
a
=4
and
p
b
=2
and let the income of the
household be
$20
. Derive the optimization problem of the household and represent it in a graph.
.
b) Determine the marginal rate of substitution
(
MRS
)
for any consumption bundle
(
x
a
;
x
b
)
.Does
the optimization problem of the household have a corner solution? Determine the household’s
optimum graphically and analytically.
c) Now assume that the income of the household is
y
dollars and the prices
p
a
and
p
b
are positive.
Derive the demand for apples
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 Fall '06
 MASSON
 Microeconomics, Utility

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