1
7th December 2007
Instructions
: Answer 6 of the following 8 questions. All questions are of equal
weight. Indicate clearly on the
fi
rst page which questions you want marked.
1. A job candidate from a big US university is presenting his work which empha-
sizes a “wonderful” new functional form for estimating ordinary demand equations,
based on the utility maximizing approach.
Two of the observations on one of his
households are that the household bought: (a)
(1
,
1)
at prices
(1
,
2)
and (b)
(3
/
2
,
1
/
2)
at prices
(2
,
2)
. What question would you ask him?
ANSWER
Cost of these bundles
at these
prices
1
2
Deductions
1
3
5
/
2
1
"
2
2
4
4
2
"
1
These data are inconsistent with WARP so it is nonsensical to
fi
t demand equa-
tions based on utility-maxmizing approach to the data. The natural question is “Why
are you doing this – it makes no sense?”
2. Consider a risk-averse individual with initial wealth
w
0
and VNM utility func-
tion
u
(
·
)
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
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.
If the loss (
L
) increased would this individual buy more or less insurance at the
point where insurance is fair? Justify your answer carefully.
ANSWER
The individual’s expected utility will be
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
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2
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.
The question asks for the sign of
dx/dL
at the point where insurance is fair. Since
we have just proved that, with fair insurance,
x
=
L
,
dx
dL
= 1
>
0
.
In words, if the person’s loss in the event of an accident increases he or she will
buy one more dollar of insurance for each dollar loss increases.
3. Consider the two-state world of Andy and Brian. In the good state each has a
wealth of
100
; in the bad state each has a wealth of
50
, assuming they do not trade
with each other. Let the probability of the good state be
2
/
3
.
The only way in which
they di
ff
er is Andy is risk averse and Brian is risk neutral.
Describe as precisely
as you can the Pareto e
ffi
cient allocations for these two people. Now suppose there
are many Andys and an equal number Brians. Describe as precisely as you can the

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- Fall '08
- Burbidge,John
- Game Theory, good driver, TV ads, good state
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