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Decision Theory

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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 wealth a)Rabin (2000) paradox b)Buying/selling price gap (WTA/WTP disparity)

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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 wealth a)Rabin (2000) paradox b)Buying/selling price gap (WTA/WTP disparity)

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A Thought Experiment Would you like to have A)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

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

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

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

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Time Inconsistent Preferences: Time Choosing and Eating Simultaneously 70% choose chocolate

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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.

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Extremely thirsty subjects McClure, 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.

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Theoretical Framework Classical functional form: exponential functions. D(t) = t D(t) = 1, U t = u t + u t+1 u t+2 u t+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.

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Constant rate of decline -D'(t)/D(t) = rate of decline of a discount function

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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? 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

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Rapid rate of decline in short run Slow rate of decline in long run

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An Alternative Functional Form Quasi-hyperbolic discounting (Phelps and Pollak 1968, Laibson 1997) D(t) = 1, U t = u t + u t+1 u t+2 u t+3 U t = u t + u t+1 u t+2 u t+3 uniformly discounts all future periods. exponentially discounts all future periods.

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Building intuition To build intuition, assume that = ½ and = 1. Discounted utility function becomes U t = u t + ½ u t+1 u t+2 u t+3 Discounted utility from the perspective of time t+1. U t+1 = u t+1 + ½ u t+2 u t+3 Discount function reflects dynamic inconsistency: preferences held at date t do not agree with preferences held at date t+1.

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Application to massages = ½ and = 1 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 NPV in current minutes 15 minutes now 10 minutes now 7.5 minutes now 10 minutes now

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Application to massages = ½ and = 1 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 NPV in current minutes 15 minutes now 10 minutes now 7.5 minutes now 10 minutes now

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Exercise Assume that = ½ and = 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.

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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).

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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, …

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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-1Period tPeriod t+1 Pay cost θObserve current value of c Do task or delay again

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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: 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

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In equilibrium, the sophisticate needs to be exactly indifferent between acting now and waiting. Solving for c *, we find: So expected delay is:

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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

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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

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

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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

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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.

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Choi, Laibson, Madrian, Metrick (2002) Self-reports about undersaving. Survey Mailed to 590 employees (random sample) Matched to administrative data on actual savings behavior

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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

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osada_says_don_t_eat_the_marshmallow_yet.html osada_says_don_t_eat_the_marshmallow_yet.html Experiment in Stanford 32

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