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Uncertainty, Waiting Costs, and Hyperbolic Discounting

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Uncertainty and Hyperbolic
Discounting
P. Dasgupta
E. Maskin
Strong evidence in behavioral ecology and
economics that animals and humans place
less weight on future than current payoffs,
i.e., they discount the future
more intriguingly: considerable evidence that
discount rates increase as time to payoffs
shrinks
(experimental evidence better for birds—
pigeons and starlings— than for humans)
2
Strotz (1956)
• People attempt to lay aside money for
Christmas early in the year
• But later on, spend this money on summer
vacations or back-to-school clothes
They have become more “impatient.”
3
O’Donoghue and Rabin (1999)
• If offered choice in February between
– painful 5-hour task on April 1
– painful 8-hour task on April 15
people choose April 1.
• As April 1 approaches, people apt to postpone to
April 15
That is, they discount later pain more as time grows
short
4
Both examples accord with hyperbolic
discounting (discount rates increase as
payoffs approach, so more distant payoffs
are discounted at lower per-period rate)
has attracted attention in economics literature
because sheds light on important economic
phenomena (e.g., saving behavior)
5
• Offer evolutionary explanation for hyperbolic
discounting
• Based on hypothesis that human or animal
preferences—as manifested in cravings, urges, or
instincts—selected to induce animal to make
“right” choice in typical situation it faces
“right” = survival-maximizing
• Will show: if typical situation entails uncertainty
about when payoffs are realized, get preference
reversals consistent with hyperbolic discounting
6
Rough intuition:
• suppose DM offered choice between:
prospect P - - small payoff, relatively early realization
and
prospect P п‚ў - - large payoff, relatively late realization
• small but positive probability of early realization
• assume DM initially chooses P 
– bigger payoff
– might be realized early
• with time, chance of early realization of P 
declines
– true of P too, but pays off early anyway
• so DM may switch to P - - à la hyperbolic
discounting
7
• Although preference reversal here looks like
hyperbolic discounting, actually dynamically
consistent
– optimal for DM to switch from P  to P
• But suppose DM faces atypical choice problem
– urge to switch may lead her astray
• If problem recurs, DM may learn how to
overcome impatience
8
Why discount at all?
• Conventional answer: future payoffs may
disappear or depreciate
• For example, blackbird waits for fruit on raspberry
bush to ripen
– crows may devour raspberries before ripe
– blackbird should discount the payoff of getting fruit
– discount rate = hazard rate
пЂЅ
1
п‚ґ (probability that crows arrive
пЃ„t
betw een t and t пЂ« пЃ„ t )
9
• Apparently cheap explanation for
hyperbolic discounting: hazard rate
declining in time
• But no particular reason why this should be
so (if crow-arrival time is Poisson, hazard
rate and hence discount rate is constant—
i.e., discounting is “exponential”)
• Doesn’t explain preference reversals: if P 
over P at time 0, will do so at time t
10
• Another reason for discounting: waiting
costs
– waiting entails using up energy (physiological
cost)
– waiting entails passing up other prospects
(opportunity cost)
• Waiting costs immediately explain another
empirical regularity: large payoffs
discounted less
11
Two prospects: пЂЁ V , T пЂ© an d пЂЁ V п‚ў, T п‚ў пЂ©
V пЂјVп‚ў
payoffs
T пЂјTп‚ў
realizatio n tim es
Suppose пЂЁ V , T пЂ© favored over пЂЁ V п‚ў, T п‚ў пЂ©
Evidence suggest that, for k >1 big enough,
пЂЁ kV п‚ў, T п‚ў пЂ©
favored over пЂЁ kV , T
пЂ©
12
• Phenomenon follows immediately from
waiting costs
• Let c be waiting cost per unit time,
• Suppose V  cT  V   cT  (earlier prospect preferred)
• Then
kV пЂ­ cT пЂј kV п‚ў пЂ­ cT п‚ў (later prospect preferred
for k big enough
when payoffs increased).
• Both hazard-rate and waiting-cost stories
important in practice
• Henceforth, stick with hazard rates
13
Introduce uncertainty about realization time
• Blackbird may be pretty sure raspberries
will ripen by tomorrow morning
• But some chance they will ripen earlier (or
later)
14
Prospect of raspberries п‚« P пЂЅ пЂЁ V , T пЂ©
V = calories in ripe raspberries
T = likely ripening (realization) time, say,
tomorrow morning
15
• At any time t < T, there is probability q  t
that V will be realized in пЂЁ t , t пЂ« пЃ„ t пЂ©
• q = probability density of “early realization”
• r = hazard rate (rate of crow arrival)
• expected payoff =
пѓІ
T
qe
0
пЂ­ rt
d tV пЂ« пЂЁ 1 пЂ­ q T пЂ© e
пЂ­ rT
V
16
• Suppose there is another prospect P    V , T  
(blackberry bush)
• bird can monitor only one bush at a time
(must choose P o r P п‚ў )
• VV
Tп‚ўпЂѕT
(more calories/volume in blackberries)
(blackberries likely to ripen later)
• Some chance of early ripening (density q )
17
Proposition 1: Assume that there exists t пЂЄ пЂј T
such that decision maker (DM) indifferent
between P пЂЅ пЂЁ V , T пЂ© and P п‚ў пЂЅ пЂЁ V п‚ў, T п‚ў пЂ© at tim e t пЂЄ
with 0 пЂј V пЂј V п‚ў, T пЂј T п‚ў,
and probability density q of early arrival. Then
DM prefers
пЂЄ
пЂЄ
P п‚ў to P at t пЂј t and P to P п‚ў at t пЂѕ t .
Implies “hyperbolic discounting”: when much time
remains пЂЁ t пЂј t пЂЄ пЂ© , DM willing to wait for bigger
reward. But when horizon short пЂЁ t пЂѕ t пЂЄ пЂ© ,
becomes impatient.
18
Proof :
Expected payoffs from
пѓІ
(1)
T
e
P an d P п‚ў at t пЂј T :
пЂ­ rпЃґ
пЂ­ rT
qˆ  t  d  V   1  qˆ  t   T  t   e V
пЂ­ rпЃґ
qˆ  t  d  V   1  qˆ  t   T   t   e
t
and
пѓІ
(2)
Tп‚ў
e
t
V , w here qˆ  t  
q
пЂ­ rT п‚ў
1 пЂ­ qt
Time derivatives:
пЂ­e
пЂ­ rt
qˆ  t  V  qˆ  t  e
пЂ­ rT
пЂ«
пЂ­e
qˆ  t  V   qˆ  t  e
e
пЂ­ rпЃґ
t
and
пЂ­ rt
пЂЁпѓІ
T
пЂ­ rT п‚ў
пЂ«
пЂЁпѓІ
Tп‚ў
t
e
dпЃґ пЂ­ пЂЁT пЂ­ t пЂ© e
пЂ­ rпЃґ
пЂ­ rT
dпЃґ пЂ­ пЂЁT п‚ў пЂ­ t пЂ© e
пЂ©
V
пЂ­ rT п‚ў
dqˆ
dt
пЂ©
Vп‚ў
пЂЁt пЂ©
dqˆ
dt
пЂЁt пЂ©
19
пѓІ
(1)
and
(2)
пѓІ
T
e
пЂ­ rпЃґ
пЂ­ rT
qˆ  t  d  V   1  qˆ  t   T  t   e V
пЂ­ rпЃґ
пЂ­ rT
qˆ  t  d  V   1  qˆ  t   T   t   e V , w here qˆ  t  
t
Tп‚ў
e
t
п‚ў
q
1 пЂ­ qt
Can rewrite derivatives as
(3)
T
пЂ­ rпЃґ
пЂ­ rT
пЂ­ rt
пѓ©
Л†
q  t   e d  qˆ  t  V  1  qˆ  t   T  t   e V  e V 
пѓЄпѓ« t
пѓєпѓ»
and
(4)
п‚ў
T
пЂ­ rпЃґ
пЂ­ rT п‚ў
пЂ­ rt
пѓ©
qˆ  t   e d  qˆ  t  V   1  qˆ  t   T   t   e V   e V  
пѓЄпѓ« t
пѓєпѓ»
• At t  t  , sums of first two terms in bracketed expressions are equal
• Because V   V , (3) bigger than (4)
пЂЄ
• Hence, (1) bigger than (2) for t  t ; (2) bigger than (1) for t  t 
20
Idea:
• passage of time has 2 marginal effects on expected
payoff from prospect
(i) brings nearer time at which payoff likely to be
realized
(ii) reduces probability of early realization
• effect (i) is proportional to expected payoff - - so
same for P and P п‚ў at t пЂЄ
• effect (ii) (negative) is proportional to V for P and
to V п‚ў for P п‚ў; because V п‚ў пЂѕ V , effect m ore negative
fo r V п‚ў th an fo r V
• so single-crossing property holds: P’s expected
payoff increases faster than that of P п‚ў whenever
expected payoffs equal.
21
2 ways to interpret preference reversal:
(i) once-and-for-all choice of P п‚ў o ver P
at time 0, and then an unexpected
opportunity to choose again at time
пЂЄ
t пЂѕ t (in which case the choice will
be P)
(ii) DM can switch at any time, in which
case will choose P п‚ў at t пЂЅ 0 and switch
пЂЄ
to P at t пЂЅ t .
22
• Proposition 1 pertains to positive payoffs,
but tax story entails negative payoffs
Proposition 1*: Maintain hypotheses of
Proposition 1 except assume V п‚ў пЂј V пЂј 0 .
пЂЄ
п‚ў
Then if DM indifferent between P and P at t ,
prefers P to P п‚ў for t пЂј t пЂЄ and P п‚ў to P for t пЂѕ t пЂЄ
23
Propositions assumed
(i) all early realization times equally likely
(ii) probability density of early realization same for both
prospects
(iii) no probability of late arrival
–
Can relax these assumptions
–
However, cannot dispense with all assumptions
about uncertainty
24
Example
– Consider  V , T 
an d пЂЁ V п‚ў, T п‚ў пЂ© , 0 пЂј V пЂј V п‚ў an d T пЂј T п‚ў
– In interval  0, t  , P  has zero probability of early
realization but P has positive probability
– Then at t , P  ’s chances for early realization are
the same as before, but P’s chances are dimmer
– So P  is more attractive (by comparison with P)
than before, contradicting hyperbolic discounting
25
Suppose
P пЂЅ пЂЁV , T
пЂ©
an d P п‚ў пЂЅ пЂЁ V п‚ў, T п‚ў пЂ©
q пЂЁ t пЂ© d en sity fo r realizatio n o f P at t пЂЁ t пЂј T o r t пЂѕ T
пЂ©
q п‚ў пЂЁ t пЂ© d en sity fo r realizatio n o f P п‚ў at t пЂЁ t пЂј T п‚ў o r t пЂѕ T п‚ў пЂ©
For t пЂј пЃґ пЂј T , let
qˆ   , t  
q пЂЁпЃґ
1пЂ­
пѓІ
t
0
пЂ©
q пЂЁ x пЂ© dx
Then qˆ  , t  is density of P being realized at 
conditional on t being reached with no previous
realization
26
Similarly
qˆ   , t  
q п‚ў пЂЁпЃґ
1пЂ­
пѓІ
t
0
пЂ©
q п‚ў пЂЁ x пЂ© dx
for t пЂј пЃґ пЂј T
is conditional density for P п‚ў
Assume
(5)
qˆ  t , t   qˆ   t , t  for all t  T
and
(6)
V qˆ ( t , t )  V qˆ   t , t  , for all t  T
Condition (6) is MLRP condition
Condition (7) rules out above example
27
Proposition 2: Under (5) and (6), if there
пЂЄ
exists t such that DM is indifferent
between P an d P п‚ў. Then DM prefers
пЂЄ
пЂЁ
пЂ©
пЂЄ
P п‚ў to P at t пЂј t and prefers P to P п‚ў at t пѓЋ t , T .
28
Example
Suppose
q пЂЁt пЂ© пЂЅ
and
qп‚ў пЂЁt пЂ© пЂЅ
пЃЎt
T
пЃЎt
Tп‚ў
Then (5) and (6) hold for пЃЎ sufficiently small
29
– But still one important ingredient to hyperbolic
discounting not yet accounted for
– Propositions show that preference reversals
consistent with hyperbolic discounting follow
from uncertainty about realization times
– However, choices are dynamically consistent
– The DM at t  0 would not choose to tie her
hands to prevent herself from switching at
tп‚іt
пЂЄ
30
• Yet Strotz notes people learn to save for Christmas
by putting money in illiquid low-interest savings
accounts: “Christmas accounts”
• In experiments, pigeons learn to commit
themselves not to switch from “patient” to
“impatient” choices
• must show how this arises naturally in a
evolutionary model
31
Can explain dynamic inconsistency and self-commitment as
follows:
– Suppose that, over evolutionary time, species faced
prospects of type discussed: ( V, T, q )
– Suppose, in any given choice problem between
P = (V,T, q ) and P п‚ў пЂЅ пЂЁ V п‚ў, T п‚ў, q п‚ў пЂ© , individual DM from
species can observe (V, T ) and пЂЁ V п‚ў, T п‚ў пЂ© , b u t n o t q an d q п‚ў.
– Then best DM can do is to choose between P an d P 
taking expectations over q and q п‚ў
– Evolution should endow her with urges and inclinations
that induce right choice.
– If q and q  satisfy (5) and (6), then may get preference
reversal from P п‚ў to P
32
– Important that uncertainty unobservable,
otherwise evolutionarily desirable to make
discount rates depend on uncertainty from
problem to problem
– Now suppose DM in a situation where
uncertainty differs significantly from
“typical.” Specifically, suppose q  q   0.
– Given inherited preferences, she will switch
from P п‚ў to P - - but this is dynamically
inconsistent
33
• But (thanks to evolution) many species can learn
from experience
• Suppose, DM faces choice between (V,T, 0 )
and пЂЁ V п‚ў, T п‚ў, 0 пЂ© recurrently
– each round, DM chooses a behavior probabilistically
– successful behaviors more likely to be chosen provided
DM does not have urge to behave otherwise
34
How does this model apply to experiments with
pigeons?
• large literature starting with Rachlin (1972) and
Ainslee (1974)
• stylized protocol:
– At t1 , pigeon makes choice (by pecking key) between
(V, T, 0) and пЂЁ V п‚ў, T п‚ў, 0 пЂ©
– later at t 2 , pigeon can switch choices
– Third key available at t1 - - if pecked, disables
switching option at t 2
35
Stylized findings:
(i) pigeon makes same t1 choice in most rounds
(ii) if chose P at t1 , unlikely to choose P п‚ў at t 2 ;
more likely to switch if chose P п‚ў at t1
(iii) unlikely to peck disabling key in early rounds
(iv) if peck disabling key in later rounds, more likely
to have chosen P п‚ў at t1
(v) if switch from P п‚ў to P in early rounds, more
likely to peck disabling key in later rounds
36
• finding (i)
– stable preferences
– consistent with bird maximizing something
(e.g., discounted calories)
• finding (ii)
– switch from P  to P is consistent with model, but
from P to P п‚ў is not
•
finding (iii)
– takes experience to discover that uncertainty is
atypical
• findings (iv) and (v)
– birds that switch from P  to P are the ones that gain
from disabling switch option
37
• Know something about pigeons, starlings, and
humans
– all have preference reversals consistent with hyperbolic
discounting
• Perhaps other species don’t exhibit preference
reversals
– if so, can try to correlate reversals with nature of
uncertainty they face
38
39
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