Economics 104A
Solution for Final Exam
Winter 2007
1. Jack lives in SB and is considering a move to LA because prices for some goods
are lower there. In all other respects, he is indifferent between the two cities. Jack
consumes two goods: good 1 and good 2. Good 1 and good 2 are priced in SB twice
as high as they are priced in LA. Jack has a utility function
u
(
x
1
, x
2
) = ln
x
1
+ ln
x
2
and his weekly income is
I
, which will not change if he moves.
a) (4pt) At least by how much must Jack’s income increase in SB to make him
willing to remain there?
Answer:
The optimal bundle is always an interior bundle regardless of what the
prices and income are because of the logarithmic form of the utility function.
Thus, given prices
p
1
, p
2
and given income
I
, the optimal bundle is determined
by
x
2
x
1
=
p
1
p
2
(1)
and
p
1
x
1
+
p
2
x
2
=
I.
(2)
Solve (1) and (2), we have
x
1
(
p, I
) =
I/
2
p
1
and
x
2
(
p, I
) =
I/
2
p
2
(3)
as the demand functions for Jack.
Let
p
L
1
and
p
L
2
denote the prices in LA and
p
B
1
and
p
B
2
those in SB. Then, using
demand functions in (3), Jack’s utility level in LA is
u
L
= ln
I
2
2
1
p
L
1
p
L
2
.
Denote by
I
S
the least income that makes Jack willing to remain in SB. Then,
we must have
u
S
= ln
I
S
2
2
1
p
S
1
p
S
2
= ln
I
2
2
1
p
L
1
p
L
2
.
Since
p
S
1
= 2
p
L
1
and
p
S
2
= 2
p
L
2
, the above equation implies
I
S
= 2
I
. So, Jack’s
income must be doubled in order to make him willing to stay in SB.
1
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b) (4pt) What would his income in LA have to be to make him indifferent between
moving to LA and staying in SB?
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 Fall '08
 York
 Supply And Demand, Jack, SB, utility level

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