Econ 301
Professor Severinov
Answer Key
Problem Set 2
1. Question for Review 5, Chapter 5.
Why do people often want to insure fully against uncertain situations even when the premium paid
exceeds the expected value of the loss being insured against?
Risk averse people have declining marginal utility, and this means that the pain of a loss
increases at an increasing rate as the size of the loss increases.
As a result, they are willing
to pay more than the expected value of the loss to insure against suffering the loss.
For
example, suppose a homeowner does not insure his house that is worth $200,000.
Also
suppose there is a small .001 probability that the house will burn to the ground and be a
total loss.
This means there is a high probability of .999 that there will be no loss.
The
expected loss is .001(200,000) + .999(0) = $200.
Many risk averse homeowners would be
willing to pay a lot more than $200 (like $400 or $500) to buy insurance that will replace
the house if it burns.
They do this because the disutility of losing their $200,000 house is
more than 1000 times larger than the disutility of paying the insurance premium.
2. Exercise 1, Chapter 5.
1.
Consider a lottery with three possible outcomes:
•
$125 will be received with probability .2
•
$100 will be received with probability .3
•
$50 will be received with probability .5
a.
What is the expected value of the lottery?
The expected value,
EV
, of the lottery is equal to the sum of the returns weighted by their
probabilities:
EV
= (0.2)($125) + (0.3)($100) + (0.5)($50) = $80.
b.
What is the variance of the outcomes?
The variance,
σ
2
, is the sum of the squared deviations from the mean, $80, weighted by
their probabilities:
σ
2
= (0.2)(125  80)
2
+ (0.3)(100  80)
2
+ (0.5)(50  80)
2
= $975.
c.
What would a riskneutral person pay to play the lottery?
A riskneutral person would pay the expected value of the lottery: $80.
3. Exercise 6, Chapter 5.
Suppose that Natasha’s utility function is given by
u
(
I
)
=
10
I
, where
I
represents annual income in
thousands of dollars.
a.
Is Natasha risk loving, risk neutral, or risk averse?
Explain.
Natasha is risk averse.
To show this, assume that she has $10,000 and is offered a gamble
of a $1000 gain with 50 percent probability and a $1000 loss with 50 percent probability.
Her utility of $10,000 is
u
(10) =
)
10
(
10
= 10.
Her expected utility with the gamble is:
EU
= (0.5)
)
11
(
10
+ (0.5)
)
9
(
10
= 9.987 < 10.
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She would avoid the gamble.
If she were risk neutral, she would be indifferent between the
$10,000 and the gamble, and if she were risk loving, she would prefer the gamble.
You can also see that she is risk averse by noting that the square root function increases at a
decreasing rate (the second derivative is negative), implying diminishing marginal utility.
b.
Suppose that Natasha is currently earning an income of $40,000 (
I
= 40) and can earn that
income next year with certainty.
She is offered a chance to take a new job that offers a .6
probability of earning $44,000 and a .4 probability of earning $33,000.
Should she take the
new job?
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 Spring '08
 CHAPPLE
 Economics, Utility, Probability theory, Marginal product, Returns to scale

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