Problem Set 3
Microeconomics
Spring 2008
1. A household consumes only apples (
X
) and bananas (
Y
). We denote a consumption bundle consisting
of
x
bags of apples and
y
bags of bananas is denoted by
(
x, y
)
. The preferences of the household are
given by the utility function
U
(
x, y
)=
Ax
α
y
1
−
α
where
A>
0
and
0
<α<
1
.
Suppose that prices for apples and bananas are
P
X
and
P
Y
and let
I
denote the income of the household.
(a) Determine the marginal rate of substitution (MRS) for any consumption bundle
(
x, y
)
.
(b) Describe the consumer’s utility maximization problem using the information provided.
(c) Illustrate graphically where the optimal consumption bundle will lie with appropriately drawn
indi
f
erence curve and budget line.
(d) Write out the Lagrangian function for the consumer’s maximization problem.
(e) What are the
f
rst order conditions?(Hint: There are three.)
(f) Use the
f
rst order conditions to solve for the optimal quantity of
x
and
y.
(g) If
α
=0
.
25
,
I
= 100
, and both prices are equal to
2
,
f
nd the optimal consumption bundle of
apples and bananas.
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 Spring '08
 WOMER
 Microeconomics, Engel, Miss Baker, qo, Thom

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