Questions

1. Financial planners and investment advisors often
instruct their clients to hold a broad portfolio of
investments to reduce the overall risk. Having a large
number of uncorrelated or negatively correlated
investments in one’s portfolio reduces the variance of
the return on investment. At first blush, it might appear
that the advisor is suggesting that investments are more
attractive when grouped together. Contrast this with
the risk aggregation bias discussed in this chapter. Is
this diversification a good idea? If the investor were
not allowed to diversify, should she still be willing to
buy any single investment in the portfolio?
2. Expected utility theory suggests that all risk preferences
are due to diminishing marginal utility of
wealth. We have briefly discussed some reasons for
doubting this hypothesis. Why might diminishing
marginal utility of wealth be related to risk preferences?
What other explanations for risk behavior
can you think of? How would these alternative motives
suggest behavior that is different from diminishing
marginal utility of wealth?
3. Many small farms sell their vegetable crops through
cooperative arrangements. A subscriber pays in
advance for a certain portion of the crop. When the
crop is harvested, the subscriber receives (usually
weekly) deliveries of produce. The produce is composed
of the particular crops the farmer has decided to
grow that year. Consider yourself as a potential subscriber
to this system. You can subscribe for a fixed fee
or you can purchase your vegetables as needed
throughout the year. Suppose someone considering
subscribing before the season starts brackets broadly,
and someone purchasing vegetables throughout the
season brackets narrowly. How will bracketing affect
the number and types of vegetables purchased? If you
were marketing such a subscription, how could you use
bracketing to encourage purchases?
4. (a) Suppose that Schuyler faces the choice of whether
to take a gamble that results in $120 with
probability 0.50, and −$100 with 0.50 probability.
Suppose that Schuyler’s preferences can be represented
by the value function in (6.15). Would she
take the gamble? Would she be willing to take four
of these gambles?
(b) Suppose that Sydney would turn down a single
case of this gamble. Consider a gamble that results
in a loss of $600 with probability 0.5 and a gain of
x with the remaining probability. If we knew
Sydney behaved according to expected utility,
how much would x need to be before he could
possibly be willing to take the new gamble? Use
equation 6.23 to produce your answer.
5. Suppose Rosario faces a two-period time-allocation
problem. Rosario can allocate 10 hours of time in each
period between two activities: work and family. The
utility function for the first time period is u1 x1,
10− x1 = x0.5
1 10− x1
0.5, where x is the amount of
time spent at work, 10−x is the amount of time spent
with family, and the subscript refers to time period.
Thus the marginal utility of time at work in the first
period is u1 x1 =0.5x− 0.5
1 10− x1
0.5 − 0.5x0.5
1
10−x1
− 0.5. In the second period, the utility function
is given by u2 x2, 10− x2, x1, 10− x1 = x1x0.5
2 +
10−x1 10− x2
0.5. Marginal utility of time at work
in the second period is given by u2
x2
=0.5x1x −0.5
2
−0.5 10− x1 10 −x2
−0.5. Thus, total utility for both
periods is given by u1 +u2 = x0.5
1 10 −x1
0.5 +
x1x0.5
2 + 10− x1 10− x2
0.5, resulting in total marginal
utility of work in period 1 of u1 x1 =
0.5x −0.5
1 10 − x1
0.5 − 0.5x0.5
1 10 − x1
−0.5 + x0.5
2 −
10 −x2
0.5, and total marginal utility of work in period
2 of u2
x2
= 0.5x1x− 0.5
2 −0.5 10− x1 10 −x2
−0.5. The
optimal allocation can be found by setting marginal
utility of time at work in each period to zero. If Rosario
brackets broadly, what will be the optimal allocation of
time between work and family in both time periods?
How will Rosario allocate her time if she displays
melioration? What is the level of utility in each solution?