## 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?