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frqrplfv
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
fi
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
the price of insurance at the point where 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
fi
rm’s expected pro
fi
t on each dollar of
insurance is zero, so
π
p
+ (1
−
π
) (
p
−
1)
=
0
or
p
=
1
−
π
.
To
fi
nd the optimal value of
x
given the parameters of the problem set
∂
f
(
x, p,
π
, w
0
, L
)
∂
x
= 0
and solve for
x
at
p
= 1
−
π
. Thus
(
−
p
)
π
u
±
(
w
0
−
px
) + (1
−
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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E
frqrplfv
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
)
, where
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
fi
rstorder condition for this problem is
∂
f
(
x, a
)
∂
x
= 0
which implicitly de
fi
nes the optimal level of
x
as a function of
a
. Taking a total
di
ff
erential of this equation obtain
∂
2
f
(
x, a
)
∂
x
2
dx
+
∂
2
f
(
x, a
)
∂
a
∂
x
da
=
0
or
dx
da
=
−
∂
2
f
(
x, a
) /
∂
a
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 Fall '08
 Burbidge,John
 Economics, Game Theory, Utility, good driver

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