fin-401-07fa - 1 7th December 2007 Instructions Answer 6 of...

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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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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