**DecisionTheory**

**Plan for today (ambitious)** • Time inconsistency problem • Riskinessmeasures and gamblingwealth • Riskinessmeasures – the idea and description • Aumann, Serrano (2008) – economicindex of riskiness • Foster, Hart (2009) – operationalmeasure of riskiness • Buying and sellingprice for a lottery and theconnection to riskinessmeasures • Lewandowski (2010) • Twoproblemsresolved by gamblingwealth • Rabin (2000) paradox • Buying/sellingprice gap (WTA/WTP disparity)

**Let’s start…** • Time inconsistency problem • Riskinessmeasures and gamblingwealth • Riskinessmeasures – the idea and description • Aumann, Serrano (2008) – economicindex of riskiness • Foster, Hart (2009) – operationalmeasure of riskiness • Buying and sellingprice for a lottery and theconnection to riskinessmeasures • Lewandowski (2010) • Twoproblemsresolved by gamblingwealth • Rabin (2000) paradox • Buying/sellingprice gap (WTA/WTP disparity)

**A Thought Experiment** Would you like to have • 15 minute massage now or B) 20 minute massage in an hour Would you like to have C) 15 minute massage in a week or D) 20 minute massage in a week and an hour

**Read and van Leeuwen (1998)** Choosing Today Eating Next Week Time If you were deciding today, would you choose fruit or chocolate for next week?

**Patient choices for the future:** Choosing Today Eating Next Week Time Today, subjects typically choose fruit for next week. 74% choose fruit

**Impatient choices for today:** Choosing and Eating Simultaneously Time If you were deciding today, would you choose fruit or chocolate for today?

**Time Inconsistent Preferences:** Choosing and Eating Simultaneously Time 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 nowor 2x juice in 5 minutes 60% of subjects choose first option. • Choosing between juice in 20 minutesor 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.

**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*tutils 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

**Slow rate of decline ** in long run Rapid rate of decline in short 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. • 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** NPV in current minutes 15 minutes now 10 minutes now 7.5 minutes now 10 minutes now A 15 minutes now B 20 minutes in 1 hour C 15 minutes in 1 week D 20 minutes in 1 week plus 1 hour

**Application to massagesb = ½ and d = 1** NPV in current minutes 15 minutes now 10 minutes now 7.5 minutes now 10 minutes now A 15 minutes now B 20 minutes in 1 hour C 15 minutes in 1 week D 20 minutes in 1 week plus 1 hour

**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** • Period begins (assume task not yet done) • Pay cost θ (since task not yet done) • Observe current value of opportunity cost c (drawn from uniform) • Do task this period or choose to delay again. • It task is done, game ends. • If task remains undone, next period starts. Pay cost θ Observe current value of c Do task or delay again Period t-1 Period t Period t+1

**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: Likelihood of doing it in t+1 Likelihood of not doing it in t+1 Cost you’ll pay for certain in t+1, since job not yet done Expected cost conditional on drawing a low enough c* so that you do 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:

**How does introducing β<1 change the expected delay time?** If β=2/3, then the delay time is scaled up by a factor of

**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

**In equilibrium, the naif needs to be exactly indifferent** between acting now and waiting. • To solve for V, recall that:

**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

**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.

**Choi, Laibson, Madrian, Metrick (2002)Self-reports about** undersaving. Survey Mailed 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

**Experimentin Stanford** http://www.ted.com/index.php/talks/joachim_de_posada_says_don_t_eat_the_marshmallow_yet.html