ECO 310, Fall 2007
Midterm Examination Solutions
October 25
Question 1 [50 points]
If necessary, you may use log 2
≈
0
.
69, log 3
≈
1
.
10, log 4
≈
1
.
39, log 5
≈
1
.
61,
log 6
≈
1
.
79, and log(1 +
z
)
≈
z
when
z
is close to 0.
Consider the following quasilinear utility function:
U
(
x, y
) = log(
x
+ 1) +
y.
(a)
[2 points]
Does
U
*
(
x, y
) =
U
(
x, y
+ 1) represent the same preference
relation as
U
(
x, y
)?
Yes.
Since
U
*
(
x, y
) + 1 =
U
(
x, y
+ 1) is a monotonic tranformation of
U
*
(
x, y
)
(b)
[2 points]
Does
U
**
(
x, y
) =
U
(
x
+ 1
, y
) represent the same preference
relation as
U
(
x, y
)?
No, adding to x changes the relationship between x and y, and so is a
change in preferences. One way to see this is to calculate the MRS between
x and y: (
∂U/∂x
)
/
(
∂U/∂y
). In the previous question this was the same
for the utilities. Here for a given x and y it will be different. This implies,
among other things, that for the same prices and income, two people with
these utility functions will demand different amounts.
A consumer chooses
x
and
y
to maximize
U
(
x, y
) subject to the budget and
nonnegativity constraints:
px
+
y
≤
I, x
≥
0
, y
≥
0
.
Note that the price of good
y
is normalized to 1. In other words, the price of
good
x
and her income are nominated in good
y
.
(c)
[12 points]
Compute her Marshallian (uncompensated) demand func
tions. Note that boundary solutions may arise.
We have to solve the maximization proble: Max log(
x
+1)+
y
s.t.
px
+
y
≤
I
,
x
≥
0
, y
≥
0.
As is usual, we assume that the budget constraints bind. Let us find the
solutions to the Lagrangian problem (without the inequality constraints)
first.
(4 points for setting up the problem.)
L
= log(
x
+ 1) +
y
+
λ
(
I

px

y
)
1
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First order conditions:
1
x
+ 1

λp
= 0
1

λ
= 0
From these, we get
p
(
x
+ 1) = 1 or
x
=
1
p

1. Substituting in the budget
constraint,
y
=
I

1 +
p
.
We find that
x
=
1
p

1
≥
0 if and only if
p
≤
1. Therefore for
p >
1,
x
= 0
is binding. We also find that
y
=
I

1 +
p
≥
0 if and only if 1

I
≤
p
.
Therfore, for
p <
1

I
,
y
= 0 is binding.
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 '08
 StephenE.Morris
 Economics, Utility

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