Homework 2, Spring 08
(Total 20 points)
Q1
(2
×
4): (a)
MU
1
(
x
1
, x
2
) =
∂
∂x
1
(3 ln
x
1
+5 ln
x
2
) =
3
x
1
, MU
2
(
x
1
, x
2
) =
∂
∂x
2
(3 ln
x
1
+5 ln
x
2
) =
5
x
2
.
MRS
(
x
1
, x
2
) =
MU
1
(
x
1
, x
2
)
MU
2
(
x
1
, x
2
)
=
3
x
1
5
x
2
=
3
x
2
5
x
1
(b) From the tangency condition
MRS
(
x
1
, x
2
) =
p
1
p
2
, we have
3
x
2
5
x
1
=
p
1
p
2
, hence
x
2
=
5
p
1
3
p
2
·
x
1
. Plug this into the budget equation, then
p
1
x
1
+
p
2
x
2
=
p
1
x
1
+
p
2
·
5
p
1
3
p
2
·
x
1
=
8
p
1
3
·
x
1
=
m,
which yields
x
1
=
3
m
8
p
1
. Plug this to the previous formula, then
x
2
=
5
m
8
p
2
Summing up,
x
1
(
p
1
, p
2
, m
) =
3
m
8
p
1
,
x
2
(
p
1
, p
2
, m
) =
5
m
8
p
2
(c): (a)
MU
1
(
x
1
, x
2
) =
∂
∂x
1
x
3
8
1
x
5
8
2
=
3
8
x

5
8
1
x
5
8
2
,
MU
2
(
x
1
, x
2
) =
∂
∂x
2
x
3
8
1
x
5
8
2
=
5
8
x
3
8
1
x

3
8
2
.
MRS
(
x
1
, x
2
) =
MU
1
(
x
1
, x
2
)
MU
2
(
x
1
, x
2
)
=
3
8
x

5
8
1
x
5
8
2
5
8
x
3
8
1
x

3
8
2
=
3
x
2
5
x
1
(b): same as above.
Warning
: Many students calculated MRS by
MU
2
MU
1
.
Our MRS is defined by the local slope of indifference curve

Δ
x
2
Δ
x
1
where 1 comes in
the denominator and 2 comes in the numerator. But in terms of marginal utilities it
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 Spring '10
 d
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