Hyperbolic discounting : Some
old results; some recent results;
some interpretation
Daniel Read (WBS)
Yael Grushka-Cockayne (Darden)
Lisheng He (WBS)
Umar Taj (WBS)
Intertemporal choices
• Choices between outcomes that occur at different
times.
• Usually a trade off between smaller-sooner (SS)
and larger-later (LL) outcomes.
NOW
SOME YEARS
HENCE
Diversity of intertemporal choices
•
•
•
•
•
•
•
Employment now or Education now and work later
Misery now or Purgatory later
Submit research now or wait for definitive findings
Children now or Children later
Carbon reduction now or mitigation later
Smoke now or Live longer
Dessert now or Slender Later
• £100 now or £150 in one year
Typical intertemporal choice question
€200 Today
€____ in 2 years
What would you demand in 2 years in exchange
for giving up € 200 today?
Write down your answer
Measuring your annual “discount rate”
% (Up to)
2%
4%
8%
16%
24%
48%
> 48%
€
208
216
233
269
307
438
>438
N
£ 200 versus £ ______ in two years
Write down
your discount
rate.
What are your reasons?
What are your reasons?
• “I am impulsive, and 2 years is a long time to
wait.”
• “I could use it to buy my
wife/husband/SO/child a gift.”
• “I need the money now;” “I don’t need the
money now.”
• “I can earn more by investing the money.”
Normative theory: Intertemporal
choices for money and market goods
• Determine the opportunity cost of money?
• The interest rate currently being earned (for savers) or
paid (by borrowers)
• If offered a return more than you are earning in your
savings account, you should take the later outcome.
Otherwise take the earlier one.
Inconsistent preferences due to time
I had a reduction schedule carefully worked out. It was
supposed to take twelve days. I had the junk [heroin] in
solution, and in another bottle distilled water. Every time
I took a dropper of solution out to use it, I put the same
amount of distilled water in the junk solution bottle.
Eventually I would be shooting plain water.
From Junky
Four days later in Cincinnati, I was out of junk and
immobilized. I have never known one of these selfadministered reduction cures to work. You find reasons
to make each shot an exception that calls for a little
extra junk. Finally, the junk is all gone and you still have
your habit.
A “classic” study
• Office workers chose between junk food or a
piece of fruit.
• One week before, and then immediately before,
consumption.
530 kCal
per 100 g
95 kCal per
100 g
10
The results
100%
75%
50%
25%
0%
Male
Female
One week early
Read & Van Leeuwen, 1998; See also Kirby
and Herrnstein, 1995.
Male
Female
Now
11
Distribution of choice patterns
0.6
P(H|Switch) =
93%
0.4
0.2
P(A|Switch) =
7%
0
Healthy now, Healthy Healthy now, Junk
Later
Later
Junk Now, Healthy Junk Now Junk Later
Later
Hyperbolic discounting
Virtue
(LL)
Vicious
Impulse
Decision
utility
Good
intentions
Vice
(SS)
Preference
reversal
Earlier Time  Later
But does that study provide evidence
for hyperbolic discounting?
• Both kinds of food available at the same time.
• We must assume a difference in reward
distribution for the Junk and Healthy food.
Assumption underlying the hyperbolic
discounting interpretation
• The utility distribution from junk food is more
‘front loaded’ and adds up to less than the
distribution from fruit.
Instantaneous
utility / value
Time
Thaler’s (1981) example
A thought experiment: Often replicated … in thought
Our study
Results: When the blue bar is higher
we have the “apple” effect.
90%
85%
80%
75%
Now
One year
70%
65%
60%
Now-One year
One year-Now
Apple questions first
Now-One year
One year-Now
Apple questions second
Choice pattern groups
400
Apple Choices
350
300
250
200
150
p(A|Switch)
= .47
100
p(H|Switch)
= .53
50
0
One One (Impatient)
Two One (Antihyper)
One Two
(Hyperbolic)
Two Two (Patient)
Two Conclusions
• Average apple choice: 78% now, 80% in one
year. Very weak evidence of hyperbolic
discounting.
• Many people want one apple in a year.
• Thought experiments are not always the best
way to learn about the world
Studies using Google Consumer
Surveys
EVERY study in first wave showed antihyperbolic discounting
Not the presence of an “indifference”
option
Not simply unusually large magnitudes
or delays
What do people do when they are “indifferent”?
They disproportionately choose SS.
Replicated with UK sample
provided by marketing firm.
Amount or Delay within
subjects?
This made no difference!
Percent choosing LL – “anti-hyperbolic” effect is very large.
Effect of Cognitive ability (CRT score)
Effect of age?
More on age
Now-1 year
1 year – 2 year
Pakistan panel:
N=2028, National representative sample
• Suppose that you were to receive some
money. If you receive the money today you
will get Rs. 500, but if you wait one month you
will receive Rs. 750.
• If you receive the money in six months you
will get Rs. 500, but if you wait seven months
you will receive Rs. 750.
Pakistan Wave 1
Pakistan Wave 2
Conclusion
• Impulsivity is observed when trading off “virtues” and
“vices”
• Hyperbolic discounting explains this effect, but only
occasionally choices for money.
– Anti-hyperbolic discounting common
– No effect of delay to outcome common
– Common experience of temptation or impulsivity requires
explanation
• Also, discounting varies among outcomes. We are
patient about library books, less patient about heroin.
(If we want both heroin and library books)
What produces “hyperbolic”
discounting?
1. Emotions/”visceral” factors
2. Psychophysics
3. Multi-criteria discounting
1. Visceral arousal/emotions
Act
available
now?
No
Yes
Triggering stimulus present
or effective?
No
Yes
No Desire
Desire
No Action
No Action
No Desire
Desire
No Action
Action
2. Psychophysics
• Psychophysics: Intervals seem smaller when
delayed, and can become “peanuts”.
time perception
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
45
50
Psychophysics for other outcomes is
uncontroversial
• Diminishing marginal utility/marginal value for
money
amount perception (value)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.00 €
5.00 €
10.00 €
15.00 €
20.00 €
25.00 €
30.00 €
35.00 €
40.00 €
45.00 €
50.00 €
3. Multi-dimensional discounting
• Outcomes are typically multi-dimensional.
• The value of each dimension changes as a
function of delay at different rates.
• Implication: Time inconsistency
• If the value change is negative (discounting)
• Implication: Hyperbolic discounting
𝒕
𝒕
=𝜹 𝒑+𝜸 𝒒
(Mixed)
Taste
Health
𝒕
p
𝜹𝒑
q
𝒕
𝜸𝒒
𝒕
𝒕
=𝜹 𝒑+𝜹 𝒒
(Uniform)
Money
Money
𝒕
p
𝜹𝒑
q
𝒕
𝜹𝒒
Preference reversals caused by
dimension-dependent discounting
Any questions?
• I still don’t know what causes “anti”
hyperbolic discounting.
How do students compare to the rest
of the UK?