, x
*
2
) was
optimal.
Thus we have two conditions that characterize the set of optimal bundles (
x
*
1
, x
*
2
):
p
1
x
*
1
+
p
2
x
*
2
=
w,
x
*
1
=
x
*
2
.
Solving this system of linear equations yields
x
*
(
p, w
) =
w
p
1
+
p
2
,
w
p
1
+
p
2
.
In particular,
x
*
(
p, w
) is a singleton in
R
2
+
for each
p
∈
R
2
++
and
w
∈
R
+
.
•
Prove that
x
*
(
p, w
) (viewed as a function) is continuous.
Page 2 of 9
Econ 201A
Fall 2010
Problem Set 2 Suggested Solutions
4. Suppose the consumer lives for two periods, 1 and 2, and can consume a weakly positive
amount of a single good in both periods. The price of the good is $1 in both periods.
Before period 1, she is endowed with wealth
w >
0. Any wealth she does not spend in
period 1 is saved and accrues interest at rate
r
. Her utility for lifetime consumption is
the discounted sum of her perperiod utilities, i.e.
U
(
x
1
, x
2
) =
u
(
x
1
) +
δu
(
x
2
), where
0
< δ <
1. The perperiod utility function
u
meets the Inada conditions:
u
(0) = 0;
u
is continuously differentiable everywhere;
u
0
(
x
)
>
0 for all
x
;
u
00
(
x
)
<
0 for all
x
; and
lim
x
→∞
u
0
(
x
) =
∞
.
(a) Formally describe this consumer’s Walrasian budget set.
(b) Prove that
U
is strictly monotone and strictly quasiconcave in
R
2
+
.
Page 3 of 9
Econ 201A
Fall 2010
Problem Set 2 Suggested Solutions
U
(
x
) =
u
(
x
1
) +
δu
(
x
2
)
> u
(
y
1
) +
δu
(
y
2
) =
U
(
y
). The other case is similar. So
U
is strictly monotone.
To prove strict quasiconcavity we prove something stronger. Here we prove that
the function is strictly concave, which implies that it is strictly quasiconcave. A
twice differentiable function
U
on
R
2
+
is strictly concave if and only if the Hessian
matrix is negative definite.
1
H
=
"
∂
2
U
∂x
2
1
∂
2
U
∂x
1
∂x
2
∂
2
U
∂x
2
∂x
1
∂
2
U
∂x
2
2
#
=
u
00
0
0
δu
00
This is clearly negative definite since
u
00
(
x
)
<
0 for any
x
.
Note:
A more direct (but also more tedious) proof that relies only on the defini
tion of quasiconcavity given in class (and the fact that a function with a strictly
negative second derivative is strictly concave) is possible as well:
Suppose
U
(
x
)
≥
U
(
y
) and
x
6
=
y
. Let
α
∈
(0
,
1). By definition, we have
u
(
x
1
) +
δu
(
x
2
)
≥
u
(
y
1
) +
δu
(
y
2
)
.
Given that
u
(
·
) is strictly concave, we have
u
(
αx
1
+ (1

α
)
y
1
)
≥
αu
(
x
1
) + (1

α
)
u
(
y
1
)
(= iff
x
1
=
y
1
)
,
(3)
δu
(
αx
2
+ (1

α
)
y