## 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?