Economics 105
Midterm 1
Solutions
1. Lady Gaga values outlandish clothes (
x
1
) and special effects (
x
2
). Here utility function is given
by:
u
(
x
1
, x
2
) =
x
α
1
+
βx
α
2
where 0
< α <
1 and 0
< β
.
(a)
Calculate Lady Gaga’s marginal rate of substitution (2 pts)
MRS
=

∂U/∂x
1
∂U/∂x
2
=

αx
α

1
1
βαx
α

1
2
=

1
β
parenleftbigg
x
1
x
2
parenrightbigg
α

1
(b)
Calculate Lady Gaga’s elasticity of substitution (6 pts)
MRS
=

1
β
parenleftbigg
x
1
x
2
parenrightbigg
α

1
=

1
β
parenleftbigg
x
2
x
1
parenrightbigg
1

α
ln
(

MRS

) =
ln
parenleftBigg
1
β
parenleftbigg
x
2
x
1
parenrightbigg
1

α
parenrightBigg
=
ln
parenleftbigg
1
β
parenrightbigg
+ (1

α
)
ln
parenleftbigg
x
2
x
1
parenrightbigg
dln
(

MRS

) = (1

α
)
dln
parenleftbigg
x
2
x
1
parenrightbigg
dln
parenleftBig
x
2
x
1
parenrightBig
dln
(

MRS

)
=
σ
=
1
1

α
(c)
How does Lady Gaga’s elasticity of substitution depend upon
α
and
β
? (2 pts)
∂σ/∂α
=

1
/
(1

α
)
2
<
0. It does not depend on
β
(d)
Suppose Lady Gaga’s elasticity of substitution was greater than 1.
What would
this tell you about how an increase in
p
1
affected demand for
x
2
?
Explain your
answer. (2 pts)
If
σ >
1 then, then the substitution effect dominates the income effect and
∂x
2
/∂p
1
>
0
1
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2. Stunned by Kanye’s appearance, Taylor forgets her preferences. All she can remember is that
her expenditure function is (
p
1
+ 4
p
2
)
u
.
(a)
Find Taylor’s indirect utility function.
Verify that it is homogeneous of degree
in
(3 pts)
The indirect utility function is the inverse of the expenditure function. So:
v
(
p
1
, p
2
, I
) =
I
p
1
+ 4
p
2
Indirect utility functions are homogeneous of degree zero in income and prices
v
(
τp
1
, τp
2
, τI
) =
τI
τp
1
+ 4
τp
2
=
I
p
1
+ 4
p
2
=
v
(
p
1
, p
2
, I
)
(b)
Find Taylor’s hicksian demands. Verify that
h
1
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 Fall '09
 CUR
 Microeconomics, Utility, The Land, Kanye, p1

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