This preview shows pages 1–3. Sign up to view the full content.
Answer Keys for 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
)
.
MRS
=
MU
x
y
=
∂U
∂x
∂y
=
Aαx
α
−
1
y
1
−
α
A
(1
−
α
)
x
α
y
−
α
=
α
1
−
α
y
x
(b) Describe the consumer’s utility maximaiztion problem using the information provided.
max
(
x,y
)
Ax
α
y
1
−
α
s.t.
P
X
x
+
P
Y
y
=
I
(c) Illustrate graphically where the optimal consumption bundle will lie with appropriately drawn
indi
f
erence curve and budget line.
X
Y
I/P
X
I/P
Y
x
*
y
*
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document(d) Write out the Lagrangian function for the consumer’s maximization problem.
L
=
Ax
α
y
1
−
α
+
λ
(
I
−
P
X
x
−
P
Y
y
)
(e) What are the
f
rst order conditions ?(Hint: There are three.)
∂L
∂x
=
Aαx
α
−
1
y
1
−
α
−
P
X
λ
=0
∂y
=
A
(1
−
α
)
x
α
y
−
α
−
P
Y
λ
∂λ
=
I
−
P
X
x
−
P
Y
y
(f) Use the
f
rst order conditions to solve for the optimal quantity of
x
and
y.
X
(
P
X
,P
Y
,I
)=
αI
P
X
Y
(
P
X
Y
(1
−
α
)
I
P
Y
(g) If
α
.
25
,
I
= 100
, and both prices are equal to
2
,
f
nd the optimal consumption bundle of
apples and bananas.
x
∗
=
0
.
25 (100)
2
=12
.
5
y
∗
=
0
.
75 (100)
2
=37
.
5
(h) Now assume that the income of the household is $80 and the prices for apples and bananas stay
the same. What’s the new demand (utility maximizing amount) for apples and bananas?
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 WOMER
 Microeconomics

Click to edit the document details