Most of the problems in this chapter focus on illustrating the concept of risk aversion. They
assume that individuals have concave utility of wealth functions and therefore dislike variance in
their wealth. For some of these problems (especially the later ones), students will need to review
the material on mathematical statistics in Chapter 2.
Comments on Problems
7.1 This problem reverses the risk-aversion logic to show that observed behavior can be used
to place bounds on subjective probability estimates.
7.2 This problem provides a graphical introduction to the idea of risk-taking behavior. The
Friedman–Savage analysis of coexisting insurance purchases and gambling could be
presented here.
7.3 This is a nice, homey problem about diversification. The problem can be done
graphically, but instructors could introduce variances into the problem if desired.
7.4 This problem is a graphical introduction to the economics of health insurance that
examines cost-sharing provisions. Health insurance is discussed in more detail in Chapter
18.
7.5 This problem provides some simple numerical calculations involving risk aversion and
insurance when utility is logarithmic.
7.6 This is a rather difficult problem as written. It can be simplified by using a particular
utility function (e.g.,
( ) lnU W W=
). With the logarithmic utility function, one cannot use
the Taylor approximation until after differentiation, however. If the approximation is
applied before differentiation, concavity (and risk aversion) is lost. This problem can,
with specific numbers, also be done graphically, if desired. The notion that fines are more
effective can be contrasted with the criminologist’s view that apprehension of law-
breakers is more effective and some shortcomings of the economic argument (i.e., no
disutility from apprehension) might be mentioned.
7.7 This problem makes some numerical estimates of willingness to avoid specific risks. It
also shows how these values depend on wealth.
CHAPTER 7:
Uncertainty
Chapter 7: Uncertainty and Information
57
7.8 This problem is an illustration of diversification. The problem also shows how insurance
provisions can affect diversification.
7.9 This is a new problem on diversification, here applied to investing in financial assets. The
problem illustrates a case in which it is optimal to diversify into an asset with obviously
lower expected returns. The problem shows that diversification can be beneficial with
independent asset returns and even more so with negatively correlated returns.
7.10 This problem covers option values. It is similar to Example 7.5, just tweaking some of the
functions. The similarity to the text example is useful to allow students to master the
fairly difficult concepts and calculations involved. The new functions do provide some
economic insight as well: an increase in the value of one of the choices can reduce option
value because just committing to the single enhanced choice provides a lot of utility.
Also, working through the case with risk aversion provides a somewhat surprising
example in which risk aversion reduces option value.
Analytical Problems
7.11 HARA utility. This problem shows that the harmonic absolute risk aversion utility
function is compatible with other frequently used forms. These other forms are just
special cases of the HARA function.
7.12 More on the CRRA function. This problem stresses the close connection between the
relative risk-aversion parameter and the elasticity of substitution. It is a good problem for
building an intuitive understanding of risk aversion in the state preference model. Part
(d) uses the CRRA utility function to examine the “equity-premium puzzle.”
7.13 Graphing risky investments. This problem provides an illustration of investment theory
in the state preference framework.
7.14 The portfolio problem with a Normally distributed risky asset. This problem shows
how the portfolio problem can be solved explicitly if asset returns are Normal.
Behavioral Problem
7.15 Prospect theory. A good problem to assign for instructors interested in integrating
behavioral economics into their course. It covers one of the most influential models in
behavioral economics, which Kahneman (Nobel Prize winner) and Tversky applied to
explain the results of their lab experiments. Actual experimental results are cited in the
problem.
Solutions
Chapter 7: Uncertainty and Information
58
7.1 The expected utility with the bet must be greater than or equal to that without the bet. So,
p
must satisfy:
7.2 See graph.
7.3 a.
Strategy 1
Strategy 2
Chapter 7: Uncertainty and Information
59
b.
7.4 a. The insurance company has a 50% chance of paying out $10,000. Its cost is thus
7.6 Expected utility is
Chapter 7: Uncertainty and Information
60
where the last inequality follows from the formula given for the Taylor series
approximation. So, a fine is more effective.
The calculations are even more transparent in the special case of logarithmic
Chapter 7: Uncertainty and Information
61
Calculate
p
when
10W=
Risk premium is higher when the level of initial wealth is lower. The greater the
7.8 a. The farmer will plant corn since
c. Let
percent in wheat.
=
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62
7.9 a. (1)
A
has an expected return of 8 and
B
of 4.5. Maria’s expected utility from
investing the whole $1 in
A
B
is
(2) Dividing the investment
a
in
A
and
1a−
in
B
A
B
leads to four possible
outcomes: the assets both turn out to yield a positive return, generating
One could try to maximize this with respect to
, but it is simpler to graph
Chapter 7: Uncertainty and Information
63
puts more in the asset (
A
) with the higher expected return.
b. (1) With perfect negative correlation, and half invested in each asset, there are
only two possible outcomes:
A
has a positive return and
B
nothing,
22
This is greater than the expected utility from an equal split when asset
returns were independent from part (a1).
Chapter 7: Uncertainty and Information
64
b. We have
0 1 3
Let’s compute these integrals by substitution separately. To compute the first
integral, substitute
1.r x F= − −
Then
Chapter 7: Uncertainty and Information
65
Putting these integrals together,
Setting this expression equal to the utility 0.94 from the best option gives the
option value. Unfortunately, this complicated equation is difficult to solve
Analytical Problems
7.11 HARA utility
a. We have
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66
b. When
0
=
and
c. From part (a),
Chapter 7: Uncertainty and Information
67
d. Let
e. If
1,
=−
f. For certain values of the parameters, utility is still unbounded, so the St.
7.12 More on the CRRA function
a. A high value for
1R−
implies a low elasticity of substitution between states of
Chapter 7: Uncertainty and Information
68
c. A rise in
b
p
rotates the budget constraint counterclockwise about the
g
W
intercept. Both substitution and income effects cause
b
W
to fall. There is a
7.13 Graphing risky investments
a. See graph.
Chapter 7: Uncertainty and Information
69
7.14 The portfolio problem with a Normally distributed asset
From Example 7.3,
For the portfolio allocation, we are looking to allocate
k
to the risky asset and
0–Wk
to
the risk-free one. Since the risky asset
r
is normally distributed with the distribution
Maximizing this with respect to
k
yields
Behavioral Problem
7.15 Prospect theory
Scenario 1
Gamble
Expected Wealth
A
( )
1,000 1 2 (1,000 0) 1,500+ + =
B
1,000 500 1,500+=
Chapter 7: Uncertainty and Information
70
Scenario 2
d. (1) Pete should make the same choices as the majority of experimental
(2)
Utility
Scenario 2