Homework 2, Spring 08
(Total 20 points)
Q1
(2
×
4): Ken’s preference is represented by
u
(
x
1
,x
2
) = 2ln
x
1
+ ln
x
2
. Denote the
prices by (
p
1
,p
2
) and income by
m
.
(a) Obtain the marginal rate of substitution of good 2 for good 1 (as you did before).
(b) Obtain the demand function,
x
1
(
p
1
,p
2
,m
) and
x
2
(
p
1
,p
2
,m
).
(c) Redo (a)(b) for another representation
v
(
x
1
,x
2
) =
x
2
3
1
x
1
3
2
.
Answer
: (a)
MU
1
(
x
1
,x
2
) =
∂
∂x
1
(2ln
x
1
+ ln
x
2
) =
2
x
1
, MU
2
(
x
1
,x
2
) =
∂
∂x
2
(2ln
x
1
+ ln
x
2
) =
1
x
2
.
MRS
(
x
1
,x
2
) =
MU
1
(
x
1
,x
2
)
MU
2
(
x
1
,x
2
)
=
2
x
1
1
x
2
=
2
x
2
x
1
(b) From the tangency condition
MRS
(
x
1
,x
2
) =
p
1
p
2
, we have
2
x
2
x
1
=
p
1
p
2
, hence
x
2
=
p
1
2
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
·
p
1
2
p
2
·
x
1
=
3
p
1
2
·
x
1
=
m,
which yields
x
1
=
2
m
3
p
1
. Plug this to the previous formula, then
x
2
=
m
3
p
2
Summing up,
x
1
(
p
1
,p
2
,m
) =
2
m
3
p