Questions

1. In this chapter and the last, we have presented three
models of intertemporal decision making: naïve,
sophisticated, and partially naïve. As presented here
and in the literature, naïve decision makers seem
doomed to repeat their mistakes over and over. This
seems unrealistic. Alternatively, sophisticates anticipate
their procrastination problem perfectly and avoid
procrastinating wherever possible. Thus, sophisticates
would never procrastinate unless they anticipated that
they would. This, too, seems unrealistic. Finally, the
third model allows misperception of how one might act
in the future, allowing unexpected procrastination. But
this model also allows people to procrastinate forever,
never learning from their mistakes.
(a) Write about an experience you have had with
procrastination and how the behavior may be
explained by one of these models. (Write a mathematical
model if you can.)
(b) Write also about what parts of your behavior could
not be explained by any of these three models. Is
there a way to modify one of these models to
include a description of this behavior?
2. Consider the savings club problem from Example 13.1.
Suppose again that Guadalupe earns $30 each week but
that the time period is only three weeks. In weeks 1 and 2,
instantaneous utility of consumption (for a week of
consumption) is given by u c = c, so that marginal
utility of instantaneous consumption is given
by 0.5c− 05. In week 3, Guadalupe has utility given by
u3 χ =0.371χ, where χ is the amount spent on
Christmas gifts (there is no other consumption in week
3). The instantaneous marginal utility of Christmas is
given by 0.371. Savings beyond the third period leads to
no additional utility. The regular savings account offers
an annual interest rate of 5 percent, compounded weekly,
whereas placing money in a savings club offers an
annual interest rate of r. For the following, it may be
useful to use the formulas derived in Example 13.1.
(a) Suppose that δ=0.97. Solve for the optimal consumption
and savings decision in each period,
supposing that the decision maker has time-consistent
preferences. To do this, solve for the
amount of consumption in weeks 1 and 2 and the
amount of gifts in week 3 that yield equal discounted
marginal utilities.
(b) Suppose that β = 0.5. Solve for the optimal savings
and consumption decision supposing the decision
maker is a naïf.
(c) Now suppose that the decision maker is a sophisticate.
Solve for the optimal savings and consumption
decisions.
(d) Finally, suppose that the decision maker is a partial
naïf, with β =0.8. Now solve for the optimal
savings and consumption decisions.
(e) Solve for the r that would be necessary to induce
the time-consistent decision maker, the naïf, the
sophisticate, and the partial naïf to commit to
the Christmas club. What is the optimal savings
and consumption profile in this case?