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601
1
Uncertainty questions
1. Consider a riskaverse individual with initial wealth
w
0
and a Bernoulli utility
function
u
(
w
)
who must decide whether and for how much to insure his car. Assume
the probability that he will not have an accident is
π
. In the event of an accident,
he incurs a loss of
$
L<w
0
in damages. Suppose that insurance is available at
an actuarially fair price, that is, one that yields insurance companies zero expected
pro
f
ts; denote the price of
$1
worth of insurance coverage by
p.
Write down the individual’s expected utility if he purchases
x
dollars of insurance.
How much insurance will this individual purchase?
Does this this individual’s demand for insurance slope downward with respect to
thep
r
iceo
fin
su
rancea
tthepo
in
twhe
re insurance is priced fairly? Defend your
answer carefully.
ANSWER
The individual’s expected utility is
f
(
x,p,
π
,w
0
,L
)
≡
π
u
(
w
0
−
px
)+(1
−
π
)
u
(
w
0
−
px
−
L
+
x
)
If the insurance is priced “fairly” the
f
rm’s expected pro
f
tonea
chdo
l
la
ro
f
insuranceiszero
,so
π
p
+(1
−
π
)(
p
−
1) = 0
or
p
=1
−
π
.
To
f
ndtheopt
ima
lva
lueo
f
x
given the parameters of the problem set
∂
f
(
x, p,
π
0
)
∂
x
=0
and solve for
x
at
p
−
π
.Thu
s
(
−
p
)
π
u
±
(
w
0
−
px
−
π
)(1
−
p
)
u
±
(
w
0
−
px
−
L
+
x
)=0
or
u
±
(
w
0
−
px
)=
u
±
(
w
0
−
px
−
L
+
x
)
and thus
w
0
−
px
=
w
0
−
px
−
L
+
x
or
x
=
L.
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601
2
We want to know
Sign
dx
dp
±
±
±
±
±
p
=1
−
π
********************************************************************
Aside on the comparative statics method:
Suppose an agent is maximizing her objective function
f
(
x, a
)
,wh
e
r
e
x
is the
choice variable and
a
is a parameter. Assuming the maximization problem is well
behaved, and all we are interested in is the sign of
dx/da
then the method of com
parative statics (the implicit function theorem) tells us that
Sign
dx
da
=
Sign
∂
2
f
(
x,a
)
∂
a
∂
x
.
Why? The
f
rstorder condition for this problem is
∂
f
(
)
∂
x
=0
which implicitly de
f
nes the optimal level of
x
as a function of
a
. Taking a total
di
f
erential of this equation obtain
∂
2
f
(
)
∂
x
2
dx
+
∂
2
f
(
)
∂
a
∂
x
da
or
dx
da
=
−
∂
2
f
(
x, a
)/
∂
a
∂
x
∂
2
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 Fall '08
 Burbidge,John
 Utility

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