Adv Behav Econ pdf

173
T I M E D I S C O U N T I N G
in 1 month, 1 year, and 10 years to make them indifferent to receiving $15 now.
The median responses—$20, $50, $100—imply an average (annual) discount rate
of 345% over a one-month horizon, 120% over a 1-year horizon, and 19% over 
a 10-year horizon.
11
Other researchers have found a similar pattern (Benzion,
Rapoport, and Yagil 1989; Chapman 1996; Chapman and Elstein 1995; Pender
1996; Redelmeier and Heller 1993).
Second, when mathematical functions are explicitly fit to such data, a hyper-
bolic functional form, which imposes declining discount rates, fits the data better
than the exponential functional form, which imposes constant discount rates
(Kirby 1997; Kirby and Marakovic 1995; Myerson and Green 1995; Rachlin,
Raineri, and Cross 1991).
12
Third, researchers have shown that preferences between two delayed rewards
can reverse in favor of the more proximate reward as the time to both rewards 
diminishes—for example, someone may prefer $110 in 31 days over $100 in 
30 days, but also prefer $100 now over $110 tomorrow. Such “preference rever-
sals” have been observed both in humans (Green, Fristoe, and Myerson 1994;
Kirby and Herrnstein 1995; Millar and Navarick 1984; Solnick et al. 1980) and in
pigeons (Ainslie and Herrnstein 1981; Green et al. 1981).
13
Fourth, the pattern of declining discount rates suggested by these studies is also
evident 
across 
studies. Figure 6.1a plots the average estimated discount factor
(
5
1/(1 
1
discount rate)) from each of these studies against the average time
horizon for that study.
14
As the regression line reflects, the estimated discount fac-
tor increases with the time horizon, which means that the discount rate declines.
We note, however, that after excluding studies with very short time horizons (one
year or less) from the analysis (see figure 6.1b), there is no evidence that discount
11
That is, $15
5
$20
3
(
e
2
(3.45)(1/12)
)
5
$50
3
(
e
2
(1.20)(1)
)
5
$100
3
(
e
2
(0.19)(10)
). While most
empirical studies report average discount rates over a given horizon, it is sometimes more useful to
discuss average “per-period” discount rates. Framed in these terms, Thaler’s results imply an average
(annual) discount rate of 345 percent between now and one month from now, 100% between 1 month
from now and 1 year from now, and 7.7% between 1 year from now and 10 years from now. That 
is, $15
5
$20*(
e
2
(3.45)(1/12)
)
5
$50*(
e
2
(3.45)(1/12)
e
2
(1.00)(11/12)
)
5
$100*(
e
2
(3.45)(1/12)
e
2
(1.00)(11/12)
e
2
(0.077)(9)
).
12
Several hyperbolic functional forms have been proposed: Ainslie (1975) suggested the function
D
(
t
)
5
1
/t
, Herrnstein (1981) and Mazur (1987) suggested 
D
(
t
)
5
1
/
(1
1
a
t
), and Loewenstein and
Prelec (1992) suggested 
D
(
t
)
5
1
/
(1
1
a
t
)
b
/
a
.
13
These studies all demonstrate preference reversals in the synchronic sense—subjects simulta-
neously prefer $100 now over $110 tomorrow and prefer $110 in 31 days over $100 in 30 days, which
is consistent with hyperbolic discounting. Yet there seems to be an implicit belief that such preference
reversals would also hold in the diachronic sense—that if subjects who currently prefer $110 in 
31 days over $100 in 30 days were brought back to the lab 30 days later, they would prefer $100 at that
time over $110 one day later. Under the assumption of stationary discounting (as discussed earlier),
synchronic preference reversals imply diachronic preference reversals. To the extent that subjects an-
ticipate diachronic reversals and want to avoid them, evidence of a preference for commitment could
also be interpreted as evidence for hyperbolic discounting (to be discussed).
14
In some cases, the discount rates were computed from the median respondent. In other cases, the
mean discount rate was used.

---