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 utilitymaxmizing approach to the data. The natural question is “Why
are you doing this – it makes no sense?”
2. Consider a riskaverse 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 twostate 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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