12
Thus we have
Wealth
=
$15
,
000
with insurance
$10
,
000 or $20
,
000
without insurance
We know that a riskaverse individual has a convex utility function. Let us graph a convex
utility function to show that a riskaverse individual prefers insurance. (Refer to Figure 3
in part (b)).
b) Suppose two types of insurance policies were available:
•
a fair policy covering the complete loss; and
•
a fair policy covering only half of any loss incurred.
Calculate the cost of the second type of policy and show that the individual will generally
regard it as inferior to the first.
e. Ms. Fogg is planning an aroundtheworld trip on which she plans to spend $10,000. The utility
from the trip is a function of how much she actually spends on it (
Y
), given by
U
(
Y
) =
lnY
13
a) If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip,
what is the trip’s expected utility?
b) Suppose that Ms. Fogg can buy insurance against losing the $1,000 (say, by purchasing
traveler’s checks) at an “actuarially fair” premium of $250. Show that her expected utility
is higher if she purchases this insurance than if she faces the chance of losing the $1,000
without insurance.
c) What is the maximum amount that Ms. Fogg would be willing to pay to insure her $1,000?
f. For the CRRA utility function (Equation 7.42), we showed that the degree of risk aversion is
measured by (1

R
). In Chapter 3 we showed that the elasticity of substitution for the same
function is given by
1
(1

R
)
Hence the measures are reciprocals of each other. Using this result,
discuss the following questions.
a) Why is risk aversion related to an individual’s willingness to substitute wealth between
states of the world? What phenomenon is being captured by both concepts?