1. Many have erroneously described hyperbolic discounting
as an extreme bias toward current consumption.
Describe why this is a false statement. Explain
intuitively what hyperbolic discounting does to decisions
involving intertemporal choice.
2. Naïve hyperbolic discounting leads people to make
plans that are never executed. However, there are many
reasons people might not execute plans. What other
reasons might lead someone to abandon a plan for the
future? What distinguishes plans that are not executed
owing to hyperbolic discounting from alternative
explanations for not executing plans? Do hyperbolic
discounters regret not executing their plans?
3. Many people display something like hyperbolic discounting.
Some businesses thrive on supporting this
sort of short-term excess. For example, several establishments
offer payday loans—short-term loans with
ultrahigh interest rates designed to be paid off the next
time the person is paid.
(a) Suppose you were considering opening such a
payday loan establishment. Given that hyperbolic
discounters often fail to follow through on plans,
how could you structure the loans to ensure payment?
Use the quasi-hyperbolic model to make
(b) The absolute-magnitude effect suggests that people
are much closer to time consistency with regard to
larger amounts. How might this explain the difference
in the structure of consumer credit (or shortterm
loans) and banks that make larger loans?
(c) Lotteries often offer winners an option of receiving
either an annual payment of a relatively small
amount that adds up to the full prize over a number
of years or a one-time payment at a steep discount.
Describe how time inconsistency might affect a
lottery winner’s decision. How might the lottery
winner view her decision after the passage of time?
4. Harper is spending a three-day weekend at a beach
property. Upon arrival, Harper bought a quart of ice
cream and must divide consumption of the quart over
each of the three days. Her instantaneous utility of ice
cream consumption is given by U c =c0.5, where c is
measured in quarts, so that the instantaneous marginal
utility is given by 0.5c− 0.5.
(a) Suppose Harper discounts future consumption
according to the fully additive model, with the
daily discount factor δ= 0.8. Solve for the optimal
consumption plan over the course of the three days
by finding the amounts that equate the discounted
marginal utility of consumption for each of the
three days, with the amounts summing to 1.
(b) Now suppose that Harper discounts future utility
according to the quasi-hyperbolic discounting
model, with β =0.5, and δ=0.8. Describe the
optimal consumption plan as of the first day of the
weekend. How will the consumption plan change
on day two and day three?
(c) The model thus far eliminates the possibility that
Harper will purchase more ice cream. In reality, if
consumption on the last day is too low, Harper
might begin to consider another ice cream purchase.
Discuss the overall impact of hyperbolic
discounting on food consumption or on the consumption
of other limited resources.
5. Consider the diet problem of Example 12.3. Let
δ=0.99, ul =2, uh =1, γi =1 180 for all i, and
w=140. Suppose that initial weight in the first period
is 200. How high does β need to be before the person
will actually go on a diet rather than just planning to
in the future? Use geometric series to solve this
analytically.