University of California, Davis
Date:
June 25, 2007
Department of Agricultural and Resource Economics
Department of Economics
Time:
5 hours
Microeconomics
Reading Time:
20 minutes
PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE
ANSWER KEY
Part 1 Insurance
(a)-(b)
Given
q
and
α
, the consumer’s consumption contingent to the bad state is
α
−
+
ω
=
]
1
[
1
1
q
x
, and her consumption contingent to the good state is
α
−
ω
=
q
x
2
2
. The
consumer solves the following optimization problem.
Consumer’s Problem
. Given
q
, choose
α
in order to maximize
)
]
1
[
(
)
1
(
)
]
1
[
(
1
1
α
−
+
ω
π
−
+
α
−
+
ω
π
q
u
q
u
subject to
]
,
0
[
1
2
ω
−
ω
∈
α
.
Because it is explicitly assumed that the consumer always chooses a positive
α
, we
disregard the nonnegativity constraint on
α
and consider only the constraint
1
2
ω
−
ω
≤
α
. The
Lagrangian now becomes
L
(
α
,
λ
) =
)]
(
[
)
(
)
1
(
)
]
1
[
(
1
2
2
1
ω
−
ω
−
α
λ
−
α
−
ω
π
−
+
α
−
+
ω
π
q
u
q
u
,
with Kuhn-Tucker conditions
(KT-1)
0
]
)[
(
'
)
1
(
]
1
)[
]
1
[
(
'
2
1
=
λ
−
−
α
−
ω
π
−
+
−
α
−
+
ω
π
q
q
u
q
q
u
,
(KT-2)
0
)]
(
[
1
2
=
ω
−
ω
−
α
λ
,
(KT-3)
0
≥
λ
.
If
α
were equal to
1
2
ω
−
ω
, then one would have
x
q
q
≡
α
−
ω
=
α
−
+
ω
2
1
)
]
1
[
, and (KT-1)
would become
0
)
(
'
)
1
(
]
1
)[
(
'
=
λ
−
π
−
−
−
π
q
x
u
q
x
u
, i.e.,
0
)
(
'
)
1
(
]
1
)[
(
'
≥
π
−
−
−
π
q
x
u
q
x
u
(by KT-
3), or
q
q
−
≥
π
−
π
1
1
, contradicting the assumption that
q
>
π
. Hence,
1
2
ω
−
ω
<
α
, and the consumer
chooses not to fully insure, and hence to bear some risk, i. e.,
2
2
1
1
]
1
[
x
q
q
x
≡
α
−
ω
<
α
−
+
ω
≡
.
(
1
)
It follows from (2) that
λ
= 0, and therefore (KT-1) becomes the FOC equality
F
(
α
,
π
)
≡
0
)
(
'
)
1
(
]
1
)[
]
1
[
(
'
2
1
=
α
−
ω
π
−
−
−
α
−
+
ω
π
q
q
u
q
q
u
,
(2)