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Department of Economics
W3211
Columbia University
Fall 2011
SOLUTIONS TO
Problem Set 4
Intermediate Microeconomics
Prof. Seyhan E Arkonac
1.
Bob's utility function is shown in the Figure below. He currently has $100 worth of property,
but there is a 50% chance that all of it will be stolen. An insurance company offers to
reimburse Bob for his loss if the money is stolen. What is the most that Bob would pay for
such a policy? Explain.
Answer:
A risky life leaves Bob with expected utility that could be had from a certain $30.
Thus he is willing to part with $70 to insure himself against such a loss.
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An individual has an initial wealth of $35,000 and might incur a loss of $10,000 with
probability
p
.
Insurance is available that charges $
gK
to purchase $
K
of coverage.
What
value of
g
will make the insurance actuarially
fair
?
If she is risk averse and insurance is fair,
what is the optimal amount of coverage?
Answer:
The insurance company’s expected payoff is:
p(gK – K) + (1–p)(gK)
Fair insurance requires:
p(gK – K) + (1–p)(gK)=0
Which means the
g=p
If she is risk averse, she will purchase full coverage (
K = 10,000
).
Formally, she will choose
K
to maximize her expected utility:
EU =
p
*U(25,000 +(1–
p
)K) + (1–
p
)*U(35,000 –
pK
)
The Necessary Condition for Maximum is:
U’(25000+(1
p
)
K
) = U’(35,000 –
pK
)
which requires that:
25000 + (1–
p
)K = 35000 –
pK.
Solving yields K=10,000.
For this to be a maximum (not a minimum), the expected utility function must be concave, which
is assured from the fact that the utility function is concave (risk averse).
3.
Derive the ArrowPratt measure of
absolute
risk aversion for the following utility functions.
Which represents the greatest level of risk aversion according to the measure?
a.
U(X) =
b.
U(X) = e
x
c.
U(X) = 1  1/X
Answer:
a.
U'(X) = .5/X
.5
, U''(X) = .25/X
1.5
, so
!
(X) = .5/X
b.
U'(X) = e
x
, U"(X) = e
x
, so
!
(X) = 1.
c.
U'(X) = 1/X
2
, U"(X) = 2/X
3
, so
!
(X) = 2/X
(c) has the greatest level of risk aversion if x<2 and (b) has the greatest if x>2
4.
Suppose an individual has $100 to invest.
Two assets are available.
One asset will yield a
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This note was uploaded on 01/16/2012 for the course ECON W3211 taught by Professor Elmes during the Fall '09 term at Columbia.
 Fall '09
 Elmes
 Microeconomics, Utility

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