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

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

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

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

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

Time Inconsistent Preferences: Time Choosing and Eating Simultaneously 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 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.

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.

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? 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

Rapid rate of 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,      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.

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.

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

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

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.

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-1Period tPeriod t+1 Pay cost θObserve 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: 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:

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

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