Economics 101
Problem Set 2 Solutions
January 25, 2011
Lingwen Huang
1. Andrew consumes positive amounts of goods 1 and 2. He thinks that
3 units of good 1 is always a perfect substitute for 1 unit of good
2. Which of the following utility functions represent Andrew’s prefer
ences?
(a)
U
(
x
1
, x
2
) = min
{
x
1
,
3
x
2
}
.
(b)
U
(
x
1
, x
2
) = 9
x
2
1
+ 6
x
1
x
2
+
x
2
2
.
(c)
U
(
x
1
, x
2
) =
√
5
x
1
+ 15
x
2
+ 100.
(d)
U
(
x
1
, x
2
) =
x
1
+ 3
x
2
+ 500.
Soln:
The preference represented by the utility function in 1(a) is perfect
complementary. The utility function in 1(b) represents the same pref
erence as a utility function
W
(
x
1
, x
2
) = 3
x
1
+
x
2
(because
W
=
√
U
),
so 3 units of good 1 is always a perfect substitute for 9 units of good
2. Both the utility functions in 1(c) and 1(d) represent the same pref
erence as the utility function
W
(
x
1
, x
2
) =
x
1
+ 3
x
2
(make sure you
understand why). Since with utility function
W
(
x
1
, x
2
) =
x
1
+ 3
x
2
, 3
units of good 1 is always a perfect substitute for 1 unit of good 2 (you
can calculate that
MRS
21
=
∂U/∂x
2
∂U/∂x
1
= 3 ), utility functions 1(c) and
1(d) represent Andrew’s preferences.
2. Bruce has preferences,
, that are complete and transitive. Suppose
there are four bundles
x
1
,
x
2
,
x
3
, and
x
4
, with the property that
(a)
x
1
v
x
2
,
(b)
x
3
v
x
4
, and
(c)
x
1
x
3
.
How do Bruce’s preferences rank
x
2
and
x
4
?
Soln:
1
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From 2(a) and 2(c), transitivity implies that
x
2
v
x
1
x
3
.
Then,
from 2(b), we have
x
2
x
3
v
x
4
. Therefore,
x
2
x
4
.
3. Anthony’s utility function for good 1 and good 2 is given as the fol
lowing. Graph his indifference curves for different utility functions in
the commodity space,
R
2
+
, and discuss whether these preferences are
complete, transitive, monotonic and convex. Justify your answer.
(a)
U
(
x
1
, x
2
) = (2
x
1
+ 5
x
2
)
2

50.
(b)
U
(
x
1
, x
2
) =

(
x
1

2)
2

(
x
2

3)
2
+ 10.
(c)
U
(
x
1
, x
2
) = min
{
2
x
1

x
2
,
2
x
2

x
1
}
.
Soln:
We first argue that all these utility functions represent preferences,
which are complete and transitive (in fact, if a preference has util
ity representation, it must be complete and transitive). For any two
bundles
x
and
x
0
, we calculate
U
(
x
) and
U
(
x
0
). Since we can always
compare
U
(
x
) with
U
(
x
0
), Anthony can always compare
x
and
x
0
.
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 DANNICATAMBAY
 Microeconomics, Utility, Monotonic function, Convex function, soln

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