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Decision Theory
Plan for today (ambitious)1. Time inconsistency problem 2. Riskiness measures and gambling wealth
Riskiness measures – the idea and description• Aumann, Serrano (2008) – economic index of riskiness• Foster, Hart (2009) – operational measure of riskiness
Buying and selling price for a lottery and the connection to riskiness measures• Lewandowski (2010)
Two problems resolved by gambling wealtha) Rabin (2000) paradoxb) Buying/selling price gap (WTA/WTP disparity)
Let’s start…1. Time inconsistency problem 2. Riskiness measures and gambling wealth
Riskiness measures – the idea and description• Aumann, Serrano (2008) – economic index of riskiness• Foster, Hart (2009) – operational measure of riskiness
Buying and selling price for a lottery and the connection to riskiness measures• Lewandowski (2010)
Two problems resolved by gambling wealtha) Rabin (2000) paradoxb) Buying/selling price gap (WTA/WTP disparity)
A Thought Experiment
Would you like to haveA) 15 minute massage now
orB) 20 minute massage in an hour
Would you like to haveC) 15 minute massage in a week
orD) 20 minute massage in a week and an hour
Read and van Leeuwen (1998)
TimeChoosing Today Eating Next Week
If you were deciding today,would you choosefruit or chocolatefor next week?
Patient choices for the future:
TimeChoosing Today Eating Next Week
Today, subjectstypically choosefruit for next week.
74%choosefruit
Impatient choices for today:
Time
Choosing and EatingSimultaneously
If you were deciding today,would you choosefruit or chocolatefor today?
Time Inconsistent Preferences:
Time
Choosing and EatingSimultaneously
70%choose chocolate
Read, Loewenstein & Kalyanaraman (1999)
Choose among 24 movie videos• Some are “low brow”: Four Weddings and a Funeral• Some are “high brow”: Schindler’s List
• Picking for tonight: 66% of subjects choose low brow.• Picking for next Wednesday: 37% choose low brow.• Picking for second Wednesday: 29% choose low brow.
Tonight I want to have fun… next week I want things that are good for me.
Extremely thirsty subjectsMcClure, Ericson, Laibson, Loewenstein and Cohen (2007)
• Choosing between, juice now or 2x juice in 5 minutes
60% of subjects choose first option. • Choosing between
juice in 20 minutes or 2x juice in 25 minutes 30% of subjects choose first option.
• We estimate that the 5-minute discount rate is 50% and the “long-run” discount rate is 0%.
• Ramsey (1930s), Strotz (1950s), & Herrnstein (1960s) were the first to understand that discount rates are higher in the short run than in the long run.
Theoretical Framework
• Classical functional form: exponential functions. D(t) = dt
D(t) = 1, , d d2, d3, ...Ut = ut + d ut+1 + d2 ut+2 + d3 ut+3 + ...
• But exponential function does not show instant gratification effect.
• Discount function declines at a constant rate.• Discount function does not decline more quickly in the
short-run than in the long-run.
Exponential Discount Function
0
1
1 11 21 31 41 51
Week (time = t)
Dis
cou
nte
d v
alu
e o
f d
elay
ed r
ewar
d
Exponential Hyperbolic
Constant rate of decline
-D'(t)/D(t) = rate of decline of a discount function
An exponential discounting paradox.
Suppose people discount at least 1% between today and tomorrow.
Suppose their discount functions were exponential. Then 100 utils in t years are worth 100*e(-0.01)*365*t utils today.
• What is 100 today worth today? 100.00• What is 100 in a year worth today? 2.55• What is 100 in two years worth today? 0.07• What is 100 in three years worth today? 0.00
Discount Functions
0
1
1 11 21 31 41 51
Week
Exponential Hyperbolic
Rapid rateof decline in short run
Slow rate of decline in long run
An Alternative Functional Form
Quasi-hyperbolic discounting(Phelps and Pollak 1968, Laibson 1997)
D(t) = 1, , bd bd2, bd3, ...Ut = ut + bdut+1 + bd2ut+2 + bd3ut+3 + ...
Ut = ut + [b dut+1 + d2ut+2 + d3ut+3 + ...]
b uniformly discounts all future periods.d exponentially discounts all future periods.
Building intuition
• To build intuition, assume that b = ½ and d = 1.• Discounted utility function becomes
Ut = ut + ½ [ut+1 + ut+2 + ut+3 + ...]
• Discounted utility from the perspective of time t+1. Ut+1 = ut+1 + ½ [ut+2 + ut+3 + ...]
• Discount function reflects dynamic inconsistency: preferences held at date t do not agree with preferences held at date t+1.
Application to massagesb = ½ and d = 1
A 15 minutes nowB 20 minutes in 1 hour
C 15 minutes in 1 weekD 20 minutes in 1 week plus 1 hour
NPV in current minutes
15 minutes now10 minutes now
7.5 minutes now10 minutes now
Application to massagesb = ½ and d = 1
A 15 minutes nowB 20 minutes in 1 hour
C 15 minutes in 1 weekD 20 minutes in 1 week plus 1 hour
NPV in current minutes
15 minutes now10 minutes now
7.5 minutes now10 minutes now
Exercise
• Assume that b = ½ and d = 1.• Suppose exercise (current effort 6) generates delayed benefits
(health improvement 8). • Will you exercise?
• Exercise Today: -6 + ½ [8] = -2• Exercise Tomorrow: 0 + ½ [-6 + 8] = +1
• Agent would like to relax today and exercise tomorrow.• Agent won’t follow through without commitment.
Beliefs about the future?
• Sophisticates: know that their plans to be patient tomorrow won’t pan out (Strotz, 1957).– “I won’t quit smoking next week, though I would like to
do so.”• Naifs: mistakenly believe that their plans to be patient will
be perfectly carried out (Strotz, 1957). Think that β=1 in the future.– “I will quit smoking next week, though I’ve failed to do so
every week for five years.”• Partial naifs: mistakenly believe that β=β* in the future
where β < β* < 1 (O’Donoghue and Rabin, 2001).
Example: A model of procrastination (sophisticated)Carroll et al (2009)
• Agent needs to do a task (once).– For example, switch to a lower cost cell phone.
• Until task is done, agent losses θ units per period.• Doing task costs c units of effort now.– Think of c as opportunity cost of time
• Each period c is drawn from a uniform distribution on [0,1].• Agent has quasi-hyperbolic discount function with β < 1 and δ = 1.• So weighting function is: 1, β, β, β, …• Agent has sophisticated (rational) forecast of her own future
behavior. She knows that next period, she will again have the weighting function 1, β, β, β, …
Timing of game
1. Period begins (assume task not yet done)2. Pay cost θ (since task not yet done)3. Observe current value of opportunity cost c (drawn from
uniform)4. Do task this period or choose to delay again.5. It task is done, game ends.6. If task remains undone, next period starts.
Period t-1 Period t Period t+1
Pay cost θ Observe current value of c
Do task or delay again
Sophisticated procrastination
• There are many equilibria of this game.• Let’s study the equilibrium in which sophisticates act whenever
c < c*. We need to solve for c*. This is sometimes called the action threshold.
• Let V represent the expected undiscounted cost if the agent decides not to do the task at the end of the current period t:
*
21 **
cc cV V
Cost you’ll pay for certain in t+1, since job not yet done
Likelihood of doing it in t+1
Expected cost conditional on drawing a low enough c* so that you do it in t+1
Likelihood of not doing it in t+1
Expected cost starting in t+2 if project was not done in t+1
• In equilibrium, the sophisticate needs to be exactly indifferent between acting now and waiting.
• Solving for c*, we find:
• So expected delay is:
* [ ( *)( * /2) (1 *) ]c V c c c V
*1 1
2
c
2
2
delay 1 * 2 1 * * 3 1 * *
1 * 1 *1*
1 1 * 1 1 * 1 1 *
1 11 1 1 2
*1 1 * 1 1 * *
E c c c c c
c cc
c c c
cc c c
How does introducing β<1 change the expected delay time?
1 11 12
delay given 1 221
1 1delay given =1 1 11 21 2
E
E
If β=2/3, then the delay time is scaled up by a factor of 2
Example: A model of procrastination: naifs
• Same assumptions as before, but…• Agent has naive forecasts of her own future behavior.• She thinks that future selves will act as if β = 1.• So she (falsely) thinks that future selves will pick an action
threshold of
* 21 1
2
c
• In equilibrium, the naif needs to be exactly indifferent between acting now and waiting.
• To solve for V, recall that:
**
[ ( *)( * /2) (1 *) ]
2 2 / 2 1 2
2 1 2
c V
c c c V
V
V
2 1 2
2
1 ***
2c
c
V
VcV
• Substituting in for V:
• So the naif uses an action threshold (today) of
• But anticipates that in the future, she will use a higher threshold of
** 2 1 2 2
2
c
** 2c
* 2c
• So her (naïve) forecast of delay is:
• And her actual delay will be:
• Her actual delay time exceeds her predicted delay time by the factor of 1/β.
1 1delay
* 2Forecast
c
1 1 1delay
** 2 2E
c
Choi, Laibson, Madrian, Metrick (2002)Self-reports about undersaving.
SurveyMailed to 590 employees (random sample)Matched to administrative data on actual savings behavior
Typical breakdown among 100 employees
Out of every 100 surveyed employees
68 self-report saving too little 24 plan to
raise savings rate in next 2 months
3 actually follow through
32
• http://www.ted.com/index.php/talks/joachim_de_posada_says_don_t_eat_the_marshmallow_yet.html
Experiment in Stanford